Non-Autonomous Kato Classes and Feynman-Kac Propagators

Transition Functions and Markov

Processes

5

In Lemma 1.1, the set K' can be chosen as follows. Let UQ be a relatively compact open subset of E for which K C UQ C UQ C U, where the symbol UQ stands for the closure of the set UQ. Then we can take K' — C/0. In addition, if the space E is second countable, then the following function can be chosen as the function (p in Urysohn's Lemma: 0, the set {x £ E : \f{x)\ > e} is a compact subset of E. Equipped with the supremum norm, Co is a Banach space. Since E is second countable, there exists a countable collection C of open relatively compact subsets of E such that every open subset O of E is a countable union of open sets from C. Let C = {Uj : j > 1} be an enumeration of C and choose a,j G Uj, j > 1. Let ifj G Co, j > 1 be a sequence of continuous functions on E such that 0 < ifij < 1 and 1. Then the algebra generated by the functions tpj, j > 1, separates points. By the Stone-Weierstrass theorem, this algebra is uniformly dense in the Banach space (Co, ||•!!„,). It follows that a sequence xn, n G N, of elements of E converges to x G E if and only if lim ifj (xn) = tpj (x) n—>oo

for all j > 1. It is not hard to see that the function p : E x E —> [0,1] denned by oo

p (x, y) = J2 2~j 1 ^ 0 ) - Vi(v)\.

(x,y)£Ex

E,

j=i

is a metric on the space E. This metric generates the topology of E. Hence, the space E is metrizable. It is also clear that E is a cr-compact space. 1.1.4

Stochastic

Processes

Let (Q, f, P) be a probability space, and let (E, £) be a state space. Suppose that Xs : Cl H-> E, S G /, is a family of random variables where I is an index set. Then the family Xs, s G J, is called a stochastic process. Throughout the book, we will use a bounded interval [a, b] or the halfline [0, oo) as the index set J, unless specified otherwise. The stochastic

6

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

process Xs, s £ I, can be identified with the mapping X : (s, w) H-> XS(LJ), (s, u>) £ I x £1. In addition, the process Xs can be considered as a random variable X : 0 -* E1 denned by X(w) = (X a (w)) s 6 / . For every u £ fi, the function s — i > X s (w), s £ /, is called a sample path. The path space E1 is equipped with the product cr-algebra ®/£. This u-algebra is generated by finite-dimensional cylinders of the form Ylsei E*, where Es £ £, s £ I, and Es ^= E for only a finite number of elements s £ I. Let Xs, s £ I, be a stochastic process, and let J = ( s i , . . . , sn) be a finite subset of / . The finite-dimensional distribution of the process Xs corresponding to the set J is the distribution of the random variable Xj : 0, —• EJ given by Xj(u) = (X3l(u),.. .,XSn(u)). It is defined as follows: »J(B)

= F[XJ€B],

B£BEJ.

Here BEJ denotes the Borel er-algebra of the space EJ. The distribution of the process Xs, s £ I, that is, the distribution of the random variable X is determined by the family of all finite-dimensional distributions. Each member of the family {(EJ,BEJ,(IJ)

:JCI,

J finite}

is a probability space, and the family {/jy : J finite} is a projective (or consistent) system of probability measures. This means that for finite subsets J and K oi I with J C K, the equality HK[(PK)-X{B)\=HJ{B)

holds for all B £ BEJ. given by

Here the projection mapping p^j : EK —> EJ is

PJ ( K W ) =* (<"«)sej -

(U').eK

G

E

*

(see Section 5.2). Let X] and X^, s £ I, be stochastic processes on the probability spaces (fii, Ti, Pi) and {Q.2, -^2, P2), respectively, and suppose that both processes have the same state space (E,£). The processes X^ and Xf are called stochastically equivalent if their finite-dimensional distributions coincide, i.e., if the equality Pi [X*j £B]=P2

[Xj £ B]

Transition Functions and Markov

Processes

7

holds for all finite subsets J of 7 and all B £ BEJ- Note that in the definition of the stochastic equivalence, it is not necessary to assume that the processes X^ and X% are defined on the same probability space. If the processes X] and X% are denned on the same probability space (fi, J7, P), then it is said that the process X% is a modification of the process X\ if for every s £ 7, P [X\ / Xf\ = 0. The processes X} and X% are called indistinguishable if P [X} ^ X% ] = 0 for all s £ I. It is not hard to see that if XJ and X^1 are continuous stochastic processes and X% is a modification of XI, then X} and X^ are indistinguishable (see [Revuz and Yor (1991)], p. 18).

1.1.5

Filtrations

Let (fljJ7) be a measurable space. A filtration Qt, t £ [0,T], is a family of sub-cr-algebras of T such that if 0 < t\ < £2 < T, then Qtl C Gt2- A two-parameter filtration QJ, 0 < r < £ < T, is a family of sub-cr-algebras of T that is increasing in t and decreasing in T. A stochastic process X s , s G [0,T], with state space {E,£) is called adapted to the nitration Qt, t G [0,T], provided that for every s G [0,T], X , is Ts/£-measurable. The following filtration is associated with the process Xs: Ft=a{Xs:Q<s
0
Recall that a (Xs : 0 < s < t) is the smallest cr-algebra containing all sets of the form f]s€J X~* (Bs), where J is any finite subset of [0, t], and Bs G £, s G J. Instead of Borel subsets of E, one can take, e.g., open subsets, or any other 7r-system which generates the cr-algebra £ (see Subsection 5.1 for the definition of a 7r-system). The cr-algebra a (Xs : 0 < s < t) is the smallest cr-algebra such that all the state variables X3 with 0 < s < t are measurable. The process Xs, s £ [0, T], generates the following two-parameter filtration: Tl =a(Xs:r

<s
0
The filtration T[ is sometimes called the internal history of the process Xs, s£[0,T].

Let (ft, T, P) be a probability space, and let Gt be a nitration. It will be often implicitly assumed that for every t £ [0,T], the cr-algebra Qt is complete with respect to the measure P. This means that if A £ J7, B C A, and F(A) — 0, then B £ Qt. In other words, all subsets of P-negligible sets belong to every cr-algebra Qt with t £ [0, T). If the filtration Qt is not

8

Non-Autonomous Kato Classes and Feynman-Kac Propagators

complete, then we can always augment it by the family M = {B C n : B C A, AeJ7,

F(A) = 0} .

More precisely, the augmentaion means that one passes from the cr-algebra Gt to the cr-algebra Qt = a {Gt,N). We refer the reader to Section 1.7 for more information on completions of u-algebras. 1.2

Markov Property

In this section we introduce Markov processes and formulate several equivalent conditions for the validity of the Markov property for a general stochastic process. Let (fi, T, P) be a probability space, and let Xt with t G [0, T] be a stochastic process on Q with state space E. Recall that the state space E is a locally compact Hausdorff topological space satisfying the second axiom of countability. Definition 1.1

A stochastic process Xt is called a Markov process if E[f(Xt)\F3]=E[f(Xt)\a(Xs)}

(1.1)

P-almost surely for all 0 < s < t < T and all bounded Borel functions / on E. The next lemma is well-known. It provides several equivalent descriptions of the Markov property. For the definition of two-parameter filtrations see Subsection 1.1.5. Lemma 1.2 Let Xa be a stochastic process on (fi,.F, P) with state space E. Then the following are equivalent: (1) Condition (1.1) holds. (2) For all t € [0, T], all finite sets { n , . . . , rn} with 0 < r\ < r 2 < • • • < r„ < t, and all bounded Borel functions f on E, the equality E [f(Xt)

\a(Xri,...,Xrn)]=E

[f(Xt)

| a(Xrn)]

holds P-a.s. (3) For all s G [0,T] and all bounded real-valued ^-measurable variables F, the equality E [F | Ts] - E [F | a(Xs)] holds F-a.s.

random

Transition Functions and Markov Processes

9

(4) For all s £ [0, T], and all bounded real-valued random variables G and F such that G is Fs-measurable and F is ^-measurable, the equality E[GF]=E[GE[F\a(Xs)]] holds. (5) For all s e [0,T], A € Ts, and B € T^, the equality P [A n B | a(Xs)} =F[A\

a(Xs)} P [B \ a(Xs)}

holds F-a.s. We refer the reader to [Blumenthal and Getoor (1968)] for the proof of Lemma 1.2. A similar lemma concerning the reciprocal Markov property will be established in Section 1.10 (see Lemma 1.20). Condition (5) in Lemma 1.2 states that for a Markov process the future and the past are conditionally independent, given the present. The aalgebra !FS is often interpreted as information from the past before time s, while the cr-algebra Fj, contains the future information. The time s is considered as the present time. In a sense, a Markov process forgets its past history. R e m a r k 1.1 If Xt, 0 < t < T, is a Markov process with respect to the probability measure P, then the time reversed process X1 = Xr-t, t E [0,T], is also a Markov process with respect to the same measure P. Indeed, it is not hard to see that condition (5) in Lemma 1.2 is invariant with respect to time-reversal. Therefore, Lemma 1.2 implies that the timereversed process Xt possesses the Markov property. The next assertion follows from Lemma 1.2: Lemma 1.3 Let Xs be a stochastic process on (O,^ 7 , P) with state space E. Then the following are equivalent: (1) For all 0 < s < t < T, and all bounded Borel functions f on E, the equality E[f(Xs)\^]=E[f(X3)\cr(Xt)] holds P-o.s. (2) For all t with 0 < t < T, all finite sets {u\,..., un} with t < u± < ti 2 < • • • < un < T, and all bounded Borel functions f on E, the equality E [/(X t ) | a(XUl,...,XUn)]

= E [f(Xt)

| a(XUl)}

10

Non-Autonomous Koto Classes and Feynman-Kac Propagators

holds P-a.s. (3) For all s with s 6 [0,T] and all bounded real-valued Ts-measurable random variables F, the equality E[F\J%\

=

E[F\a(Xs)]

holds P-a.s. (4) For all s with s € [0, T], the equality E[FG\ =

E[FE[G\
holds for all bounded real-valued random variables G and F such that G is Fs-measurable and F is J-^-measurable. (5) For all s £ [0, T], A e Ts, and B <E T?, the equality P [A n B | a(Xs)] =P[A\

a(Xs)] P [B \ a(Xs)}

holds F-a.s. 1.3

Transition Functions and Backward Transition Functions

This section is devoted to non-homogeneous transition functions. We will first introduce a forward transition probability function, or simply a transition probability function. Definition 1.2 A transition probability function P{r, x; s, A), where 0 < r < s < T, x 6 E, and A G £, is a nonnegative function for which the following conditions hold: (1) For fixed r, E. (2) For fixed r, (3) P(r, x; s, E) (4) P(r, x; s, A)

s, and A, P is a nonnegative Borel measurable function on s, and x, P is a Borel measure on £. — 1 for all r, s, and x. = JE P(r, x; u, dy)P(u, y; s, A) for all r < u < s, and A.

A function P satisfying Condition (3) in Definition 1.2 is called normal, or conservative. Condition (4) is the Chapman-Kolmogorov equation for transition functions. In applications, a transition function P describes the time evolution of a random system. The number P(r, x; s, A) can be interpreted as the probability of the following event: The random system located at x € E at time r hits the target A C E at time s.

Transition Functions and Markov

Processes

11

The next definition concerns backward transition probability functions. Definition 1.3 A backward transition probability function P(T, A; t, y), where 0
For fixed r , A, and t, P is a Borel function on E. For fixed r , t, and y, P is a Borel measure on £. The normality condition P(T, E; t, y) — 1 holds for all T, t, and y. The Chapman-Kolmogorov equation, that is, P(T, A; t,y)= f P{T, A; A,x)P(X,dx; t,y), JE

holds for all r < A < t, A G £, and y £ E. There is a simple relation between forward and backward transition probability functions in the case of a finite time-interval [0,T]. Here we need the time reversal operation 11-> T — t, t € [0, T]. It is easy to see that P is a backward transition probability function if and only if P{T,x;t,A) = P(T-t,A;t-T,x)

(1.2)

is a transition probability function. If a function P satisfies conditions (1), (2), and (4) in Definition 1.2, but does not satisfy the normality condition, then it is called a transition function. Similarly, a function P satisfying conditions (1), (2), and (4) in Definition 1.3 is called a backward transition function. If the condition P(T,x;t,E)
(1.3)

holds instead of condition (3) in Definition 1.2, then P is called a transition subprobability function. Similarly, if P(T,x;t,E)
(1.4)

then P is called a backward transition subprobability function. If P is such that (1.3) holds, then one can define a new state space EA = E U {A} where A is an extra point. If E is a compact space, then A is attached to E as an isolated point. If E is not compact, then the topology of E is that of a one-point compactification of E. The Borel cr-algebra of EA will

12

N'on-Autonomous

be denoted by £A.

PA(r,x;t,A)

Koto Classes and Feynman-Kac

Propagators

Put

= <

1,

ifx = A a n d A e A .

0,

if x = A and A £ .4

P(r,x;t,A),

if x € £ and A £ A

l-P(T,x;t,£),

if x € £ and A = {A}.

(1.5)

L e m m a 1.4 Lei P be a transition function satisfying (1.3), and let PA be defined by (1.5). Then PA is a transition probability function; moreover, the functions P A and P coincide on E.

Proof. It is clear that only conditions (3) and (4) in Definition 1.2 need to be checked for the function P A . For x £ E, (1.5) implies

P A (T, X; t, EA) = P(T, X; t, E) + P A (r, x; t, A) = P ( T , x ; t , E ) + l - P(T,x;t,E)

= \.

If x = A, then (1.5) gives P A (T, A ; i , £ A ) = 1. Therefore, the function P A is normal. Our next goal is to prove that the function P A satisfies the ChapmanKolmogorov equation. For x ^ A and A £ £, this fact follows from the Chapman-Kolmogorov equation for P . Let x € EA, T < r < t, A e £A, AG A, and put A = A\A. Then

PA(T,x;r,dz)PA(r,z;t,A)

J

= I PA(r,x;r,dz)PA(r,z;t,A)

+ f

JE

+ [ J{A}

+ /

PA(T,x;r,dz)PA(r,z;t,{A})

JE A

A

P (T,x;r,dz)P (r,z;t,A) V

PA(T,x;r,dz)PA(r,z;t,{A}).

/

(1.6)

Transition Functions and Markov

Processes

13

If x £ E, then (1.6), (1.5), and the Chapman-Kolmogorov equation for P give PA(T,x;r,dz)PA(r,z;t,A)

J =

P{T, X;

t,A)+

[ P{r,

X;

r, dz){\ - P(r, z; t, E)) + l -

P{T, X;

r, E)

JE

= P{T, x;t,A) + l - P(T, X; t, E) =

PA(T,

x-1, A) + P A ( r , x; t, {A}) =

PA{T,

X;

t, A).

(1.7)

It is easy to see that if x £ E and A £ A, then Lemma 1.4 follows from (1.7). A similar reasoning can be used in the case where x = A and A £ A, and in the case where x = A and A ^ A. • A nonnegative Borel measure m on (E, £) will be fixed throughout the book. The measure m is called the reference measure. We will write dx instead of m(dx) and assume that 0 < m(A) < oo for any compact subset A of E with nonempty interior. It is said that a transition function P possesses a density p, provided that there exists a nonnegative function p(r, x; s, y) such that for all 0 < r < s < T, the function (x, y) —> p (r, x; s, y) is £ <8> ^-measurable, and the condition P(r,x;s,A)=

/

p(r,x;s,y)dy

holds for all A € £. The Chapman-Kolmogorov equation for transition densities is P(T, X; t,y)=

/ p(r, X; r, z)p(r, z; t, y)dz JE

for all 0 < r < r < t and m x m almost all (x, y) £ E x E.

1.4

Markov Processes Associated with Transition Functions

Let E be a locally compact space such as in Section 1.1, and let P be a transition probability function. Our first goal is to construct a filtered measurable space (Q, J-, TJ), a family of probability measures PT,x, x £ E, T € [0,T], on the space (Q,^), and a Markov process Xt, t £ [0, T], such

14

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

that T[ = cr(Xs:T<s
0 < T
(1.8)

and Pr,x(X t G A) = P{T, x;t,A),

0
Ae £.

(1.9)

We begin with the construction of what is called the standard realization of such a process. Let f2 = J5' 0,T ' be the path space consisting of all functions mapping [0, T] into E. The space Q, is equipped with the cylinder cr-algebra T, which is the smallest a-algebra containing all sets of the form {UJ G Q :u(ti) G ^ i , . . . , ^ ^ ) G Ak}, 0 < ti <•••< tk < T and Ai e £ for all 1 < i < k. Such sets are called finite-dimensional cylinders. The process Xt is defined on the space Q. by Xt(u>) = u>(t). The cr-algebra T[ is defined by formula (1.8). The symbol PT,x, where 0 < T < T and x S E, stands for the probability measure on T^ such that P r , x (XT e A0, Xtl e Alt • • • , Xtk € Ak) = PT,X (w e fl: w(r) £ A0,iv(ti)

€AU---,

= XA0{x) /

P(ti,xr,t2,dx2)

P(T,X;tudxx)

J Ax

/

/

u(tk) G Ak) •••

JA2

P(tfc_i,Xfc_i;tfe,dxfc)

(1.10)

JAk

for all r < t\ < t2 < • • • < tk < T and Ai G £, 1 < i < k. Such a measure exists by Kolmogorov's extension theorem (see Appendix A). The process Xt is a Markov process with respect to the family of measures P r , x , that is, Er,x [f(Xt)\J^]

= Ea,x.f{Xt)

PT,x-a.s.

(1.11)

for all T < s < t, and all bounded Borel functions / on E. Recall that the symbol E T:X in (1.11) stands for the expectation with respect to the measure P T)X , and the left-hand side of (1.11) is the conditional expectation of f(Xt) with respect to the tr-algebra TTS. Taking the conditional expectation with respect to the cr-algebra o (Xs) on both sides of equality (1.11) and using the a (X s )-measurability of the expression on the right-hand side of (1.11), we see that condition (1.1) holds for all measures P r , x and all t, s with T < s < t < T. In a sense, the Markov property in (1.11) means that the past history of the process Xt does not affect predictions concerning the future of Xt. It is not hard to see that condition (1.9) holds for the process Xt. The expression on the left-hand side of (1.9) is called the marginal

Transition Functions and Markov

Processes

15

distribution of Xt, while the expression on the right-hand side of (1.10) is called the finite-dimensional distribution of Xt. It is not hard to see that for a Markov process, the finite-dimensional distributions can be recovered from the marginal ones. Next, we discuss general non-homogeneous stochastic processes (Xt,Gt^-r,x) defined on a general sample space (£1, Q) equipped with a two-parameter filtration Gl,0
PTliE-a.S.

(1.12)

for all r < s < t, and all bounded Borel functions / on E. In the definition of the standard realization of the process Xt, the distribution of the state variable XT with respect to the measure P TiX is Sx, the Dirac measure at x. Let / i b e a probability measure on (E,£). Then, replacing (1.10) with P r , M (u; efl : w(r) e A ) , ^ ) eAu--= j

dfi(x) /

J AQ

P(T,X;ti,dxi)

JAI

I P(tk-i,xk-i;tk,dxk),

/

,u(tk-i) P(ti,xi;t2,dx2)

£ Afc_i,a>(*fc) € Ak) •••

J A2

(1-13)

JAk

we get a stochastic process with the initial distribution at t — T equal to fi. Any two Markov processes associated with the same transition function P are called stochastically equivalent. Such processes may be defined on different sample spaces. Next, we will consider backward transition probability functions. Let P be such a function, and let t € (0, T] and y G E. Define a family of finitedimensional distributions on the path space Q = E^°'T^ equipped with the

16

Non-Autonomous Koto Classes and Feynman-Kac Propagators

cylinder a-algebra T by the following formula: ^

y

(u G ft : u){ti) G Alt • • • ,w(**-i) G i4fc_i,w(tfc) G Ak,w(t) P ( t i , da;i; £2, x2) • • • P(tk-i,dxk-i;

/

G A fc+1 )

tk, xk) (1.14)

P(tk,dxk;t,y)xAk+1{y),

where 0 < * i < t 2 < - - < t f e < t < T ' and At £ £, 1 < i < k+ 1. Here we use Kolmogorov's extension theorem to establish the existence of the measure P t , ! / . For all t G [0,T] and y £ E, the measure P*'y is defined on the <7-algebra J-^. Consider the standard realization Xs (u>) = u (s) on the path space (ft,.?-"). Then the process Xs has P as its backward transition function, that is, P{T,A;t,y)

=

Ft>*[XreA]

for all 0 < T < t < T and A G £. Moreover, the backward Markov property holds for Xs. This means that £*•» [/ (XT) | Ff] = E'-" [/ (X T ) | a (X s )] = E s - X »/ (XT) P ^ - a . s . for all bounded Borel functions / and all 0 < r < s < t < T. The process Xs has the terminal distribution at s = t equal to the Dirac measure Sy. In the case of a prescribed terminal distribution v at s = t, the finite-dimensional distributions satisfy P'-" (LOG CI: u(ti) GAi,---, /

P(ti,dxi\t2,

oj(tk-i) x2)---

GAk-i, w(tk) G Ak,w{t) G P(tk-i,

Ak+l)

dxk-i; tk, xk)

J' A i X -- xx A A jt ++ i

P(tk,dxk;t,y)du(y).

(1.15)

Since the relation described in (1.2) is a one-to-one correspondence between forward and backward transition functions, a backward transition probability function P generates a backward Markov process X1, 0 < t < T, with respect to the family of cr-algebras J-\ = a (Xr : t > r > T) = J-TZl and the family of measures P*'1. This means that Et,x

/(x T ) T\

= E*'X

[/(^)

Xs

E's.X'

/ (*T)

T)t,X

a.s.

for all 0 < T < s < t < T and all bounded Borel functions / on E. It is easy to see that the backward Markov property for the time reversed

Transition Functions and Markov Processes

17

process Xs — XT-S follows from the Markov property for the process Xt. Therefore, if Xt is a Markov process with transition function P, then the process Xr-t is a backward Markov process with backward transition function P. In general, any property of Markov processes can be reformulated for backward Markov processes. We simply let time run backward from T to 0. For instance, if P is a backward transition function satisfying the subnormality condition, then we can use the construction in Section 1.3 to extend P to a backward transition probability function on the space EA.

1.5

Space-Time Processes

A transition function P is called time-homogeneous if P(T,X;t,

A) = P((T + h)AT,x;

{t + h)AT, A)

(1.16)

for all 0 < T < t < T, h > 0, x e E, and A E £. The values of a time-homogeneous transition function depend only on the time span t — r between the initial and final moments. We will write P(t—r, x, A) instead of P(T, X; t, A) in the case of a time-homogeneous transition function P. The minimum (T + h) A T appears in formula (1.16) because the parameters T and t vary in a bounded interval. Definition 1.5 Let (CtjJ7) be a measurable space. A family of measurable mappings fls : Cl —> Cl, s > 0, such that i W . = 0t+., #o = I

(1-17)

for all t > 0 and s > 0, is called a family of time shift operators. The symbol / i n (1.17) stands for the identity mapping on CI. If a stochastic process Xt with state space (E, £) is given on a probability space (CI, P), then it is natural to expect the process Xt to be related to the time shift operators 6S as follows: Xt o # s = X(t+s)/\T

(1-18)

for all t £ [0,T] and s € [0, T]. Equality (1.18) is often used in the theory of time-homogeneous stochastic processes. For some sample spaces, it is clear how to define the family of time shift operators {"ds}. For instance, if Cl = E$ and the process Xt is given by Xt(uj) = w(t), then we can define the time shift operators by -ds(u>)(t) = u((t + s) A T).

18

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Let P be a transition probability function, not necessarily timehomogeneous, and let (Xt,J-[,FT>x) be a Markov process on (Q,F) associated with P. Our next goal is to define the space-time process Xt = (t, Xt), t G [0, T]. It is clear that the state space of this process must be the space [0, T] x E. However, it is not clear what is an optimal choice of the sample space for the space-time process and what is the transition function of the process Xt. It is natural to expect that the sample space for the process Xt has to be the space [0, T] x CI. We will show below that this is the case under certain restrictions. Let B be a Borel set in [0, T] x E. For every s G [0,T], the symbol (B)s will stand for the slice of the set B at level s, defined by (B)s = {x G E : {s,x) G B}. Consider the following function P (t, (r, x), B) = P (r, x- (r +1) A T, (B)( T + t ) A T ) , where (T,X) G [0,T] x E, t G [0,T], and B G be rewritten in the following form: P (t, (r, x), dsdy) = P{T, X; S,

B[ 0 ,T]X.E-

(1.19)

Equality (1.19) can

dy)d5{T+t)AT(s),

where 6 is the Dirac measure. If the transition probability function P has a density p, then formula (1.19) becomes P (t, (r, x), B)= f

P{T,

X; (T

+ t) A T, y ) ^

-' ( B ) ( T + I ) A T

= 11

P(i~,x;s,y)dyd6(T+t)AT(s).

(1.20)

The time variable in the definition of P is £ and the space variables are {T,X) G [0,T] x £ and B G £[0,r]x,ETheorem 1.1 The function P given by (1.19) is a time-homogeneous transition probability function. Proof. It follows from (1.19) that the function P is normal. Next, we will show that the function P satisfies the Chapman-Kolmogorov equation, which in the case of time-homogeneous transition functions has the following form: P ( ( t i + t 2 ) A T , ( T , a ; ) , B ) = [[

P(t1,(T,x),dsdy)P(t2,{s,y),B)

Jj[0,T]xE

(1.21)

Transition Functions and Markov Processes

19

for all (T,X) € [0,T] x E, B E B[0>T]xE, 0 < h < T, and 0 < t2 < T. By using the Chapman-Kolmogorov equation for P and equality (1.19) twice, we see that P ((ti + t2) A T, (r, x),B) = P (r, i ; (r + h +12) A T, ( B ) ( T + t l + t a ) A T ) = / P(T,x;(r

+

ti)AT,dy)

JE

P =

((T

/ /

+ ti) A T, y; „.

(T

+ ti +1 2 ) A T, (B) ( _p(T>x''s'dy)p(s'y''(s+t2)hT,(B)is+t2)AT)ds{T+tl)AT(s)

[0,T]xE

J he[o,r)xB

P (ti, (r,x),dsdy)P(t2,

(s,y),B).

This establishes (1.21). Therefore, the function P defined by (1.19) is a transition probability function. • Note that even if the transition function P has a density p, the measure B — i > P(t,(T,x),B) on B[otx]xE is singular with respect to the measure dtdm. The function P will play the role of the transition function of the space-time process Xt. There are several possible choices of the sample space fl for the spacetime process Xt- For instance, one can choose the full path space, that is, the space n = ([0,T\xE)[o'Ti, to be the sample space of the space-time process Xt. space-time process is defined by

(1.22) In this case, the

Xt() G O, and the time shift operators on the space CI are given by ds(
(1.23)

Since P is a transition probability function, the Kolmogorov extension theorem implies the existence of a family of measures P( r , x ) indexed by

20

Non-Autonomous Kato Classes and Feynman-Kac Propagators

(T, X) e [0, T]x E such that P(T,X)

Xti

G Ai,

• • • , Xtk

G Ak

= P(T,x) [fa (*i), w («i)) G A 1 ; • • • , (^ (tk), u (t fe )) G Ak] • r,x

^(T+tOAT G ( ^ ^ ( ( r + t ^ A T ) ' • • • '

^(r+tfc)AT G (^fc)v((r+tfc)AT)J

(L24)

for all 0 < ti < i 2 < • • • < tk < T and A{ G S[ 0 ,T]X£I 1 < i < k. This construction results in a time-homogeneous Markov process ( Xt,^ r t T ,P( T x ) j on the space Q,. The process Xt is our first version of the space-time process. However, it may happen so that the first component t H-» tp(t) of an element (
n,x (K1 (n*)) = i-

(i-25)

In formula (1.25), the symbol P * x stands for the outer measure on the space £,!°'T1 generated by the measure PT,X> {^T} denotes the family of time shift operators on 2?[°'T1 given by i?T(w)(t) = u>((r + t) A T), and I?~ 1 (J4) denotes the inverse image of the set A under the mapping i9T. Let fi* be the subspace of the space Q. consisting of all pairs (^)c,w) £ Q, such that 0, and UJ G Q*. This means that we choose linear functions t >—> (c + t) AT as the first components of the elements of fl* and the paths u> from SI* as their second components. Note that we can identify the space fi* with the space [0, T] x fi*, using the mapping j : £1* —> [0,T] x Q,* where j (y c ,w) = (C,LJ) with c G [0,T] and wGfi*. L e m m a 1.5

T/ie following equality holds for all (T,X) G [0,T] X £ : P^ TIX)

(n*) = i.

(i.26)

Transition

Functions and Markov

Processes

21

Proof. Given 0
{xtleAu---,xtkeAk}

= {(
.

(1.27)

For every T G [0,T], (1.27) gives d-1(C)={(ip,u):w((T

+

h)AT)e(A1)v{{T+ti)AT),

• • • , w ((r + tfc) A T) G ( A 0 „ ( ( T + t j k ) A r ) } .

(1.28)

Let 7To : £,[°,T1 H-> fi be the following mapping: 7To(w) = (<£o,w) where w G -Bl°'Tl and
= [w : w ((r + *0 A T) G M 0 ( T + t l ) A T , •••,W((T + tfe)AT)G(^0(.+tfc)Ar}-

(1-29)

Suppose that the set O* is covered by a countable family {Cj} of cylinders such as in (1.27). Then the family of cylinders WQ1 \$~x(Ci)\

>

defined in (1.29) covers the set 7r0-

1

(^- 1 (n*))=^.

Using (1.24) and (1.25), we get OO

OO

£P ( r ,z) (d) = £ P r , * ( V ($7\Ci))) > 1.

(1.30)

1=1

Now it is clear that equality (1.26) follows from (1.30). This completes the proof of Lemma 1.5.

•

It follows from equality (1.26) that one can restrict the probability space structure from (fl, J ^ P ^ x ) ) t o the set fi*. The resulting probability space will be denoted by f fi*,^j.,P(T]X) J. Summarizing what has already been accomplished, we see that the space-time process Xt can be defined on the probability space f fi*,^-j.,P(T>:t:) J as follows: Xt(
22

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Here w e ft*, c > 0, and <^c is the function on [0,T] given by <£c(£) = (c + t) A T. Using the identification j of the spaces ft* and [0, T] x ft*, we see that the space-time process Xt can be defined on the space [0, T) x ft* by

X,(c)w) = ((c+t)AT ) w(t)). The state space of this process is the space ([0,T] x E, Kfcrjxs)The space-time process Xt can also be defined for a general Markov process {Xt,J-t,WT,x) o n t n e sample space ft provided that there exists a family {fis} of time shift operators on ft satisfying condition (1.18). In this case, the space-time process Xt is defined on the sample space ft = {fc}o
(vc(t),Xt(uj)).

The family of time shift operators on the space ft is given by ds {) (t) = (V(C+ S )AT(*). tfsM(*)) •

It is not difficult to check that Xt o dT = X( T + t ) A T . As before, we can identify the space ft with the space [0, T] x ft, by using the mapping j : ft —> [0,T] x ft defined by j(
Xt(c,w) = ((c + i ) A T , X t H ) , and the time shift operators # s can be written as follows: 0,(c,w) = ((c + s ) A T , 0 , ( w ) ) . The state space of the space-time process Xt is [0, T] x E. Our next goal is to define the measure P(T,x) o n the er-algebra Tl. For the cylinders C defined in (1.27), we can use formula (1.24). For a general set A € F^, we extend this formula to P(r,x)(^)=Pr,x(7ro1(^1(A))),

where n0 : ft —> ft is defined by TTQ{UJ) = {(j>o,ui). Note that the time shift operator $T is ^ / L A / ^ /.Fj?-measurable. Moreover, since the equality Xt o n0 = (t, Xt) holds, the mapping TT0 is TIJT%measurable. It also follows that for any bounded .^-measurable random

Transition Functions

and Markov

Processes

23

variable F, E{TtX)[F] = Er,x

F O fiT O

7TQ

The space-time process Xt is a Markov process. The Markov property of the process Xt can be formulated as follows: E.(T,X) / [X{t+s)AT)

I ?% = E( S) x.) /

[Xtj

for all t G [0, T], s G [0, T], x G E, and all bounded Borel functions / on the space [0, T] x E. An equivalent formulation of the Markov property of the process Xt is E(T,*) [/ ( X t ) O #8 \F.

s,X.) f

\Xt)

Remark 1.2 The space-time process Xt associated with the given Markov process Xt is not simply the process t \-> (t,Xt). For instance, if the space-time process Xt is defined on the sample space [0, T] x 0, then Xt(c,ij)

= ((c +

t)AT,Xt(Lj))

where t G [0, T] and (c, u) G [0, T] x O. For c = 0, we get Xt (0, w) = (t,Jf t (w)). Our next goal is to discuss space-time processes associated with backward transition probability functions. Let P(T, A; t, x) be such a function, and put P(T, (i, x), B) = P ((t - T) V 0, (B) (t _ T) vo; t, x).

(1.31)

Here r G [0, T] plays the role of the time-variable, whereas (t, x) G [0, T]xE and 2? G B[O,T]XB a r e the space variables. The equality in (1.31) can be rewritten as follows: P(T, (t, x), dsdy) = P (s, dy; t, x) eW(t_T)Vo(s).

(1.32)

Theorem 1.2 Let P(T, A; t, x) be a backward transition probability function. Then the function P defined by (1-31) is a time-homogeneous transition probability function. Proof. It follows from equality (1.31) that P is normal. Next let T\ G [0, T], r 2 G [0, T], {t, x) G [0, T] xE, and B G B [0 ,T]XJ5- Then, using formulas

24

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

(1.31), (1-32), and the Chapman-Kolmogorov equation for the function P, we obtain P((r1+T2)AT,(t,x),B) = P((t-

(TI + r 2 ) A T) V 0, ( B ) ( t _ ( T l + T 2 ) A T ) v 0 ; t, x)

= [ P((t-

( n + r 2 ) A T) V 0, ( £ ) ( t _ ( r i + T . 2 ) A T ) v o ; (t - n ) V 0,2/)

P((t-Ti)V0,d»;t,x) = / / J

P((s-T2)V0,(B)(s-T2)vo;s,y)P(s,dy;t,x)dd{t_Tl)s,Q(s)

J[0,T]xE

-a

P{TU(t,x),dsdy)P(T2,{s,y),B).

(1.33)

l[0,T]xE

In (1.33), we used the equality

((* - n) v o - n) v o = (t - (n + r2) A T) V O. It is clear that (1.33) implies the Chapman-Kolmogorov equation for P. This completes the proof of Theorem 1.2. • Theorem 1.2 allows us to define the space-time process XT associated with the backward transition probability function P. Arguing as in the case of transition probability functions, we first choose the space Q defined by (1.22) as the sample space of the space-time process XT(ip,uj) = ([x T 1

GA1,---,XTkGAk\

= p".*) [(
(
= P*'x (^ ( t _ r i )vo G (^i) v ((t-Ti)vo) - ' ' ' >^(t-rfc)v0 G (-^fc)v,((t-Tit)vo)J • (1.34) In (1.34), P t , x is the family of measures denned by (1.14). The Markov process (XT, FT, p(*'x) J on the sample space fi is our first version of the spacetime process associated with the backward transition probability function P. The following family of time shift operators on Cl can be used in this

Transition Functions and Markov

Processes

25

case: ?.(^W) = M(T-*)VO)IW((T-J)VO)),

S>0.

(1.35)

The operators -d3 are backward shifts by s with respect to the time variable T. The time shift operators ds are connected with the space-time process XT as follows: Xrotfs = X(T_s)v0

(1.36)

for all s > 0 and r G [0, T], It is also possible to define space-time processes on smaller sample spaces as it has already been done in the case of transition probability functions. For instance, let P be a backward transition probability function, and let (XT, F[, P*'x J be a backward Markov process defined on the sample space Q and with P as its transition function. Assume that there exists a family of time shift operators i?5 on the sample space Cl such that XT o d„ = X ( r _ a ) v 0 for all s > 0 and r G [0,T]. Then the space-time process

(1.37) (xT,TT,P{t'xA

can be defined on the sample space [0, T] x fl by Xr{c,w)=

((T-C)V0,XT(C))

where (c, w) G [0, T]xCl and r G [0, T]. The time shift operators ^ s on the sample space [0, T] x f2 of the space-time process are defined by ?a(c,w)= ( ( T - C ) V 0,0,(2)) where s > 0 and (c, w) G [0, T] x Cl. It is clear that condition (1.36) holds for XT and i? s . 1.6

Classes of Stochastic Processes

In this section we introduce and discuss various classes of stochastic processes. Let (Xt, FJ, PT,X) be a stochastic process on (fi, T) with state space (E, £). A sample path of the process (Xt, FJ, PT,X) corresponding t o w e f i is the function s\-^> Xs (LJ) defined on the interval [0, T]. For a given u G O, the sample path s >—> XS(UJ) is often called the realization of LJ, or a realization of the process Xt.

26

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Definition 1.6 (1) A process (Xt,?J,FTtX) is called right-continuous if its sample paths are right-continuous functions on the interval [0,T). (2) A process (Xt, J-J, PT,x) is called left-continuous if its sample paths are left-continuous functions on the interval (0,T]. (3) It is said that a process (Xt, !F[,FT,x) is right-continuous and has left limits if its sample paths are right-continuous functions on the interval [0, T) and have left limits on the interval (0, T]. (4) It is said that a process (Xt,!Fl,'PTtX) is continuous if its sample paths are continuous functions on the interval [0,T]. If the process (Xt, FJ,fr,x) is right-continuous and has left limits, then the following notation will be used: \imX3(u>) = Xa-(u).

(1.38)

Since the process Xt is right-continuous, we also have \imXs(uj) = Xs(w). sit

(1.39)

Definition 1.7 A process (Xt, J^,FTiX) is called stochastically continuous if for all x G E, T G [0, T], and e > 0, , lim

FT,x(p(X!l,Xt)>e)=0.

Recall that the symbol p in Definition 1.7 stands for the distance on E x E. For e > 0 and y G E, put Gc(y) = {xeE:

p(x, y) > e} .

(1.40)

Then an equivalent condition for the stochastic continuity of the process Xt is as follows: for all x G E, r G [0, T], and e > 0, •

, Km

t — sl0;r<s
/ p ( s , y ; i , G e ( y ) ) P ( r , : r ; M 2 / ) = 0.

(1.41)

Jg

Definition 1.8 It is said that a process (Xt, T\', P T ,i) is strongly stochastically continuous provided that for all e > 0, *

. J ^ T

sup

Pr,x(p(Xa,Xt)>e)=0.

Transition Functions

and Markov

Processes

27

An equivalent condition for the strong stochastic continuity of the process Xt is as follows: for all e > 0, lim

sup

f P(s,y;t,Ge(y))P(r,x;s,dy)=0.

(1.42)

Definition 1.9 Let (fi,.F, P) be a probability space equipped with a filtration Tt,ti
The following assertions hold:

(1) Let Zt, 0 < t < T, be a right-continuous ft-martingale

with respect to

28

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

the measure P, and suppose that sup E\Zt\ < oo. t€[0,T)

Then the limit ZT = lim t |T Zt exists and is finite P-a.s. (2) Let Zt, 0 < t < T, be a right-continuous Tt-martingale with respect to the measure P, and suppose that the family of random variables Zt, t e [0,T), is uniformly integrable. Then ZT E L 1 , and Zt tends to XT in L1. Moreover, for all t £ [0, T],

Zt=E

[ZT I Ft]

F-a.s. The next assertion concerns martingales associated with transition probability functions. Lemma 1.6 Let P be a transition probability function, and let (Xt, Tl,PT,Z) be a Markov process with P as its transition function. Then for allO P (s,Xs; t, A) is an TTS -martingale on the interval [r, t) with respect to the family of measures P r , x . Moreover, if P has a density p, then for all 0 < T < t < T, x S E, and y G E, the process s — i > p(s,Xs;t,y) is an TTS -martingale on the interval [r, t) with respect to the family of measures P TjX . Proof. Using the Markov property and the Chapman-Kolmogorov equation, we see that for all r < r < s < t, E r , x [P (s, Xs; t, A)\T;]=

Er,Xr [P (s, Xs; t, A)}

= / P(r, Xr; s, dz)P(s, z; t, A) = P{r, Xr; t, A) JB

P riX -a.s. This proves the first part of Lemma 1.6. The proof of the second part is similar. This completes the proof of Lemma 1.6. D

1.7

Completions of cr-Algebras

This section is devoted to completions of cr-algebras with respect to families of measures.

Transition Functions

Definition 1.11

and Markov

Processes

29

A measure space (X, J7, fj.) is called complete if B&T,

n(B) = 0, A c 5 = > A e JF.

Let (X, T, n) be a measure space, and define a family of sets by M = {A : there exists B £ J7 such that A C B and fx(B) - 0} . Definition 1.12 The completion J7^ of the cr-algebra J7 with respect to the measure /i is the cr-algebra given by

A (a). If there exist Ai and A 2 such as in (b), then we set A = A\ and B = A2\Ai. It is clear that A £ T, B £ J7, A\A C A2\Ai = B, and H(B) = 0. (b)=>(c). If there exist Ai and A2 such as in (b), then we put A' = A\ and B = A2\AX. It follows that A' £ J7, B £ J7, AAA' C B, and fi (B) = 0. (c)=*>(b). If there exist A' £ J7 and B £ J7 such as in (3), then we set Ax = A'\B and A2 = 4 ' U B. It follows that Ai e T and A 2 £ J". Moreover, A\ C A. Indeed, if a; £ A\, then a; £ A' and x £ B. Suppose that x £ A. Then x £ A'\A C AAA' c B, which is a contradiction.

30

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Hence x £ A\, and the inclusion A\ C A holds. Next, we will prove that A C A2. Let x £ A and x £ A2. Then x £ A' and x £ B. Therefore, x £ A\A' C AAA' C B, which is a contradiction. It follows that A C A2. We also have A2\Ai C B, and hence y, {A2\A\) = 0 . • Remark 1.3 It is not difficult to prove that the family A consisting of all sets A satisfying any of the equivalent conditions in Lemma 1.7, is a cr-algebra. Using condition (a) in Lemma 1.7, we can show that A = T^. Indeed, since the inclusions T C A and H C A hold, we have T^ C A. Conversely, if A £ A, then condition (a) in Lemma 1.7 holds for the set A, and hence A £ F*1. Definition 1.13 Let (X, J7) be a measurable space, and let V be a family of measures on T. The completion Tv of the cr-algebra T with respect to the family V is defined as follows: nev A cr-algebra T satisfying T = J-v is called V-complete. Let (X, T, fi) be a measure space. Then the measure /x can be extended to a measure [IQ on T^ by setting fio(A) = fi (Ai), where A £ J711 and A\ is a set such as in part (b) of Lemma 1.7. It is not difficult to see that the number Ho(A) does not depend on the choice of the set A\ in (b). We will often use the same symbol \i for the extension \i§ of \i. The measure space (XJJ^J/J.) is called the completion of the measure space {X,!F,^) with respect to the measure /u. If V is a family of measures on T, then every measure fj. £ V can be extended to the cr-algebra !FV as above, and we will use the same symbol V for the family {/io : H £ V} consisting of the extensions of measures ^L £ V. It is not hard to see that the cr-algebra Tv is ^-complete. Next, we will discuss completions of nitrations generated by Markov processes. Let P be a transition probability function, and let Xt be a Markov process on (fi, J7) associated with P. The process Xt generates the following families of cr-algebras: J=rt =a(Xs:T<s
and

^

=

f]

FJ

(1.43)

s:t<s
where 0 < r < t < T. Both families FJ and J-J+ are increasing in t and decreasing in r. For every T £ [0,T], consider the family of measures on

Transition Functions and Markov Processes

31

the cr-algebra Fj, given by VT = {Pa,x : 0 < s < T, x G E}.

(1.44)

Suppose that {Gl)-> 0 < r < i < T , i s a family of cr-algebras such that it is increasing in t, decreasing in r, and satisfies the condition G[ C \Ttf*

(1.45)

for all 0 < T < t < T. In (1.45), VT is the family of measures defined by (1.44). For the sake of simplicity, we will denote the completion [GI]VT of the a-algebra G\ with respect to the family of measures VT by the symbol G\. Since the families of a-algebras {FJ} and {J~[+} satisfy condition (1.45), the families {£[} and {^7+} a r e well-defined. L e m m a 1.8

The following assertions hold:

(a) A set A belongs to the a-algebra Tl if and only if for all (s, x) E [0, r] x E there exist sets ASiX G F[, A's<x G [f%f'-x, and A's\x G [Jr^f"-X such that A U A'ttX = Aa>x U A'lx

and

F.,x (A's,x) = Fs<x «

x

) = 0.

(b) A set A belongs to the a-algebra J-^+ if and only if for all (s, x) G [0, r] x E there exist sets Aa%x G F[+, A!ax G [T^f"-X, and A!'sx G [JFf]P",x such that AUA'S<X = AStX U A's\x

and

P s , x ( i ^ x ) = Fs<x ( <

x

) = 0.

Proof. Let (X, T, y) be a measure space, and let G be a cr-algebra satisfying G C T^. Let A be a set for which there exists A' G G such that AAA' G T» and y(AAA') = 0. Then there exist A" G G, Ax G .P 4 , rx and A2 € ^ ' such that /J, (Ax) = 0, y, (A2) = 0, and AU Ax = A" U A2. Indeed, it is sufficient to take A" = A', Ai = A'\A, and A2 = A\A'. Then Ai C AAA', A2 c AAA', and hence Ai G P" and ^42 G ^ with /j,(A1) + n(A2) = 0. Now let A be a set such that there exist sets A", A\, and A2 for which the conditions formulated above hold. Set A' = A". Since AAA' = AAA" C Ail) A2, we have AAA' € T*1 and /i (AAA') = 0. Next, arguing as above, we see that Lemma 1.8 holds. •

32

Non-Autonomous Koto Classes and Feynman-Kac Propagators

Lemma 1.9 Let P be a transition probability function, and let Xt be a corresponding Markov process on (Q, IF). Then for allO < r
n c n+

(i-46)

and u:t
Proof. The inclusion in (1.46) is straightforward. Next, we will prove the equality in (1.47). Let A G JJ+, and let As,x G C\U:tf',x, and A'lx G [F£]r'-X be the sets from part (b) of Lemma 1.8. Then it follows from part (a) of Lemma 1.8 that A G C\u:t
fl

fl

(1.48)

u:t
holds. Let A G f]U:t oo, and put 4 S , X = limsup^ U n , a i X , A'StX = limsupyi; n]S]X , and A"tX = l i m s u p A ^ ^ . n—*oo

n—*oo

n—»oo

Since

for all n > 1, we see that -^ U A s , x

=

A J , X U -^s,x-

Therefore, At,t G J^ + , ^ ) X G [ ^ ] p - , < x G [ ^ ] p - « , and P s , x (A'tiX) = P s>x (A"iX) = 0. It follows from part (b) of Lemma 1.8 that A G FJ+. Moreover,

fl

^c-^+-

(!-49)

u:t
Now we see that (1.47) follows from (1.48) and (1.49). This completes the proof of Lemma 1.9.

•

Transition Functions

1.8

and Markov

Processes

33

Path Properties of Stochastic Processes: Separability and Progressive Measurability

It will be established in this section and in Section 1.9 that under certain restrictions on a transition probability function P, there exists a Markov process associated with P and with prescribed measurability or continuity properties of its sample paths. Let P be a transition probability function, and let (Xt, J-[,PT,x) be a Markov process on (f2,.F) with P as its transition function. If Xt is a modification of Xt, then the process Xt is not necessarily adapted to the filtration T\. However, if we replace the family J-J by its completion 3FJ, then it is not hard to prove that the process (X t ,.F t T ,P r]X ) is a Markov process, and any modification Y% of Xt is an ^"-adapted process. We will always assume that filtrations are complete when dealing with modifications of stochastic processes (see Subsection 1.1.5). Let J be a countable dense subset of the interval [a, b], and let g be an 2J-valued function defined on [a, b]. The function g is called minimally continuous with respect to the set J if for every t € [a, b] there exists a sequence tk & J such that tk —» t and g{tk) —> g(t) as k —» oo. Similarly, the function g is called minimally right-continuous with respect to J if for every t G [a, b) there exists a sequence tk € J such that tk I t and 9(tk) -> g{t) as k -> oo. The next definitions concern the separability and measurability properties of Markov processes. Definition 1.14 A Markov process Xt is called separable if there exists a countable dense set J C [0, T] such that the sample paths 11—> Xt(to) are minimally continuous with respect to J PTiX-almost surely for all (r, x) £ [0, T] x E. Similarly, a Markov process Xt is called separable from the right if there exists a countable dense subset J C [0, T) such that the sample paths t H+ X ( (LJ) are minimally right-continuous with respect to J PTtXalmost surely for all (r, x) G [0, T] x E. Definition 1.15 (a) A stochastic process Xt on a measurable space (Q, T) with state space (E,£) is called a measurable process if the function (s,u) H-> Xs(w) is BmtT] ® .F/£-measurable, where B[O,T] stands for the Borel a-algebra of the interval [0,T]. (b) Let Xt be a stochastic process on a measurable space (CI, J-) with state space (E, £), and let Tt be a filtration such that Xt is ^-adapted. The

34

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

process Xt is called .^-progressively measurable if for every t £ [0,T], the restriction of the function (S,OJ) >—• XS(OJ) to the set [Q,t] x CI is B[ott] ® .F t /£-measurable. Definition 1.16 Let P be a transition probability function, and let [Xt, Tl, P r ,x) be a corresponding Markov process with state space (E,£). The process Xt is called ^"/"-progressively measurable if for every r and t with 0 < r < t < T, the restriction of the function (s, w) — i > Xs (w) to the set [r,t] x f i i s B[r,t] ® J7"/5-measurable. The next result states that measurable processes always have progressively measurable modifications. Theorem 1.4 Let Xt be a measurable stochastic process on a probability space (fi, J-, P) and with (E, £) as its state space. Let Tt be a filtration such that the process Xt is Ft-adapted. Then there exists an J-t-progressively measurable modification Yt of the process Xt. Proof. Any measurable process can be identified with a i?[o,r] ® F /£measurable function (t,uj) i-> Xt(u>). Denote by ME the space of classes of equivalence of Tj£-measurable functions from the space Q into the space E, equipped with the metric d(f, g) = inf (e + P [w : p(f(u), g{u>)) > e]). Then the convergence in the metric topology of the space ME is equivalent to the convergence in probability. Moreover, if / € ME and / „ G ME are such that oo

£d(/,/„)
(1-50)

n=l

then / n (a;) —> f(u>) P-almost surely on Q. Any #[o,r] ® -F/f-measurable function / generates a function / : [0, T] —> . M E defined as follows: for t 6 [0, T], /(£) is the class of equivalence in ME containing the function U) H->

f(t,Lj).

A simple function is a function from the space [0, T] x Q into the space E assuming only finitely many values, each on a S[o,r] ® ^-measurable set. A simple product-space function is a simple function such that each value is assumed on a set that can be represented as a finite disjoint union of direct products of sets from S[O,T] a n d T. An elementary measurable process Yt is a stochastic process on the space (fi, J7, P) such that there exist a partition

Transition Functions and Markov Processes

35

{Afc : k > 1} of the interval [0, T] into Borel measurable sets and a sequence {fk • k > 1} of T/E -measurable mappings of the space ft into the space E such that Yt = fk for all t e AkDenote by DE the class consisting of all $[O,T] ® ^"/5-measurable functions / , for which the function J: [0,T] -> ME is B[0,T]/BJ^ -measurable and has a separable range. Here Bj^ denotes the Borel cr-algebra of the space ME- An equivalent definition of the class DE is as follows. A S[O,T] ® .F/f-measurable function / belongs to the class DE if the function / can be approximated by a sequence of elementary measurable processes in the sense of pointwise convergence on ft uniformly in t G [0,T]. Lemma 1.10 The class DE coincides with the class of all B[O,T] ®T IEmeasurable functions. Proof. Let us first prove the lemma in the case where E = R. It is not difficult to see that the class £>R is closed under pointwise convergence of sequences of functions and contains all simple product-space functions. By the monotone class theorem for functions, Lemma 1.10 holds for E = R. Our next goal is to prove Lemma 1.10 for any finite subset of R. Let Ro = {ci,c 2 , • • • ,Cn} be a finite subset of R equipped with the metric inherited from the space R. Next, using Lemma 1.10 for E = M, we see that if / : [0,T] x ft — i > Ro is a JB[O,T] ® F/Buo-measurable function and i s N , then the range < f(t) : 0 < t < T > of the function / can be covered by a countable disjoint family A\, k G N, of Borel subsets of the space MR so that f~l (A\) G B[otT], and the diameter of any set A\ is less than \. Therefore, there exists a sequence / , of elementary measurable processes such that fi{t) G MR0 for all t G [0,T], and moreover sup d(f(t,-)-fi(t,-))
(1.51) l

for all i € N. It follows from the second definition of the class DR 0 that / G -DR 0 . This establishes Lemma 1.10 for E = Ro. Next, we will pass from a finite subset Ro — {ci, C2, • • • , c„} of the space R to a finite subset EQ = {x\, X2, • • • , xn} of the given state space E. Let g : [0,T] x ft — i > EQ be a B[O,T] <8> J 7 //BB 0 -measurable function. Then for every 1 < j < n, we have g(t, w) = Xj on a set Bj e S[O,T] ® J7- The sets -Bj may be empty. It is also true that the nonempty sets Bj are disjoint and cover [0,T] x ft. Let us consider a #[O,T] ®T/B^o-measurable function defined by f(t, u>) = Cj on the set Bj with 1 < j < n. By the previous part

36

Non-Autonomous Kato Classes and Feynman-Kac Propagators

of the proof, there exists a sequence fa of elementary measurable processes such that fi(t) £ MRQ and inequality (1.51) holds. Replacing Cj by Xj in the function /,, we get a function g^. Taking into account (1.51) and the fact that p(xm,Xfc) < c cm — Ck\ for 1 < m < k < n, where c > 0 is a finite constant, we see that lim sup d(ff(i,-)>ffi(t>0) = 0. °te[o,T]

,_>0

(1.52)

Since gi is an elementary process and (1.52) holds, we have g £ DE0- It follows that any simple function s : [0, T] x Q —-> E belongs to the class DENow let / be a S[O,T] ® T/E -measurable function. Then the function / can be approximated pointwise by simple functions, and since the class DE is closed under pointwise convergence, we have / £ DEThis completes the proof of Lemma 1.10. • Let us continue the proof of Theorem 1.4. By Lemma 1.10, the class DE coincides with the class of all B\O,T\ ® .F/f-measurable functions. Approximating 6[o,T]/Sjq-measurable functions with separable range by simple functions, we see that for any measurable process Xt, there exists a sequence Y" of elementary measurable processes such that

<*PW)<2^T

(L53)

for all t £ [0,T] and n > 1. Recall that to every elementary process Ytn there corresponds a partition {A% : k > 1} of the interval [0, T] into Borel measurable sets and a sequence {/£ : k > 1} of T/E-measurable mappings of the space Q, into the space E such that Y™ = f% for all t £ A%. Fix n > 1 and 5 £ (0, T). Our next goal is to modify the process Y™ as follows. Put si = inf {t : t £ A%}. If sn £ An, then the new process Z? is denned for t £ Al by Ztn = X s n. If s£ £ A£, then we fix f£ e A% such that *£ - sfe < *> a n d P u t %t = -XtJJ f ° r a u * G -4fe- It i s c l e a r that the new processes J?" are elementary measurable processes. Since Xs is an adapted process, it is easy to see that for every t £ [0, T — 5], the restriction of the function (s, w) i-» Z?(w) to the space [0, t] x £1 is B[o,t] ®Tt+s/£-measurable. Moreover, inequality (1.53) implies d{XuZ?)<±;

(1.54)

for all t £ [0,T] and n > 1. By (1.50), Zt"(w) -> X;(w) as n -> oo for all £ G [0, T] almost surely on (1 Fix XQ £ E, and put I t (a;) = lim Z"(w)

Transition Functions and Markov Processes

37

if the limit exists, and Yt(u)) = xo otherwise. Then the process Yt is a modification of the process Xt. We will next show that the process Yt is progressively measurable. It is clear from the definition of the process Yt that it suffices to prove that every process Z " is progressively measurable. Since the process Xt is adapted, we see that for every u £ [0,T — S], the restriction of the function (s,u>) t-> Z™(u>) to the space [0,u] x 0, is #[o,u] <8> Fu+s/S-measurable. Now let t € [0,T]. Then using the previous assertion with um = t — ^ , m > mo, we see that the restriction of the function (s, w) — i > Z™{w) to the space [0,t) x Cl is S[o,t) <8> Tt/^-measurable. Since the process Xt is adapted, we see that the process Z " is progressively measurable for all n > 1. This completes the proof of Theorem 1.4. • Every sample path of a measurable process is a Bp^}/^ measurable function. The next assertion provides examples of progressively measurable stochastic processes. Theorem 1.5 measurable.

Every left- or right-continuous process is progressively

Proof. Let X be a right-continuous process. Fix r and t with 0 < T < t
7rfc = { r = s (

fc

)<4 fc )<...< s f

= f}

such that the mesh of the partition 7i> tends to zero as A; —> oo. For every k > 1, define a simple process Xk on [r,t] as follows: X% = X oo, if s € [SJ ', Sj + '), and X$ = Xt. It is clear that the function (s, a;) — i > X*(w) defined on [T, t] x Q, is B\Ttt] <8> F[ /£-measurable. It follows from the rightcontinuity of the process Xt that lim

X*{LJ)=XS(W)

fc—+00

for all s € [r,t] and w e fi. Hence, the function (s,w) i—»-X"s(w) defined on [T, i] x fi is #[T,t] ® .F t T /£-measurable. This means that the process Xt is progressively measurable. The proof of Theorem 1.5 is thus completed for right-continuous processes. The proof for left-continuous processes is similar. • The next result concerns separable processes and stochastic equivalence. Theorem 1.6 Let P be a transition probability function satisfying condition (1.41)- Then there exists a separable process (Xt,^,PT,x) on (Q,J-)

38

Non-Autonomous Koto Classes and Feynman-Kac Propagators

with state space (E, £) such that the transition function of Xt with P. Proof.

coincides

The following lemma will be used in the proof of Theorem 1.6:

Lemma 1.11 Let P be a transition probability function, and let Xt be a Markov process associated with P. Then for every (r, x) G [0, T] x E there exists a separable P r , x -modification XSI r < s
The next assertion is an important part of the proof of Lemma

Lemma 1.12 Let B G £, r G [0,T], and x e E. Then, there T exists a finite or countable sequence tk G [ ,T] depending on B for which ¥TiX[N{t,B)] = 0, t G [T,T], X G E, where N(t,B) = {Xtk G B : k>\,XtiB}. Proof. We will use the method of mathematical induction in the proof of Lemma 1.12. Let t\ be any number in [r, T]. If the numbers t\, £2, • • •, tk have already been chosen, then we put 7fe

= sup P r , x [Xtj € B : 1 < j < k, Xt i B] . t€[r,T]

If 7fc = 0, then we are done. If jk > 0, then there exists tk+i £ [r, T] such that sup P T;X [XtjeB:l<j<

k, Xtk+1

tB]>^.

te[r,T)

z

Put Nk(B) - {Xtj £B:l<j
Xtk+1 $ B) .

It is clear that the sets Nk(B) are disjoint. Therefore,

k

k

and hence, -jk —» 0 as k —> 00. Since P r ? x [N(t, B)] < 7^ for any k > 1, we have P r , x [N(t,B)] = 0 for all t G [T,T\. This completes the proof of Lemma 1.12. •

Transition Functions and Markov Processes

39

Now let Bi be a sequence of Borel sets in E. It follows from Lemma 1.12 that for every i > 1 there exists a sequence t\, k > 1, such that PT,x[N(t,Bi)]=0

(1.55)

for all t G [r, T\. By enumerating the set {t\ : i > 1, k > l } , we see that the sequence tk can be chosen independently of i. It is not hard to prove that the sets N(t,Bi) constructed for this sequence are subsets of the similar sets in (1.55). This means that (1.55) holds for the new sequence. Next we will show that more is true. Lemma 1.13 Let r G [0,T], x G E, and let Bt be a sequence of Borel sets in E. Then there exists a sequence tk G [T,T], k > 1, and for every t G [r, T] there exists a set N(t) G Tj- such that Pr,x [N(t)] = 0,

(1.56)

N(t,B)cN(t)

(1.57)

and

for all t G [r, T] and for all sets B which can be represented as a countable intersection of elements of the family {Bi}. Proof.

Put N(t) = ( J N(t, Bt) (here we use the sequence tk constructed

after equality (1.55)). Let B = f|. Biy N(t, B)c\J

{Xtk zBfk>\,Xti

Then Btj}

j

c\J{Xtk 3

G B^, k>\,Xti

Bij}=\jN(t,Bij)

c N(t).

(1.58)

3

Now it is clear that (1.56) follows from (1.55), while (1.57) follows from (1.58). • Let us return to the proof of Lemma 1.11. Denote by Cj the family of all open balls of rational radii centered at the points of a countable dense subset of E, and put Bi = E\d. It is clear that the family of sets which are representable as countable intersections of the sets from the family {Bi} contains the family of all closed subsets of E. Applying Lemma 1.13, we see that there exists a sequence tk G [r, T], and for every t G [r, T] there exists a set N(t) such that Pr,x[7V(i)] = 0 and N(t, C) C N(t) for all closed subsets C of E and all t G [r, T\.

40

Non-Autonomous Kato Classes and Feynman-Kac Propagators

Let Yt be a separable process on ft, and define a new process Xt as follows: Xt(w) = Xt{w) if t G {tk} or u £ N(t), and Xt(u>) = Yt(w) otherwise. Our next goal is to show that

Xt=Xt

=1

(1-59)

for all t G [T, T] and also to prove that Xt is a separable process. Indeed, if t G {U}, then lxt = Xt\ = ft. If t € {**}, then lxt ^ Xt\ C JV(i). This gives (1.59). Now we are ready to prove the separability of the process Xt- If t = ti for some i > 1, then Xti = Xti, and hence for all UJ G ft, the corresponding sample path is minimally continuous with respect to {U}. lit $. {tk} and ui G -/V(i), then X t coincides with a {ij}-separable process. If t £ {t,} and w $. N(t), then Xt(u>) = X t (w), and we proceed as follows. Suppose that X t (w) cannot be approximated by a subsequence of the sequence Xti{nv). Then there exists a ball C centered at Xt{u) such that Xti(w) £ C for all i > 1. Moreover, for every i > 1, we have Xtt{u>) G B and Xt{uj) £ B where B = E\C. Hence, w G N(t,B) C iV(i), which is a contradiction. Therefore, Xt (u>) can be approximated by a subsequence of the sequence Xti(uj), and this implies the separability of the process Xt. This completes the proof of Lemma 1.11. • Note that we have not yet employed the stochastic continuity condition in the proof of Theorem 1.6. This condition is needed to guarantee that any countable dense subset of [r, T] can be used as a separability set. Lemma 1.14 Suppose that P is a transition probability function satisfying the stochastic continuity condition (1-7), and let Xt be a corresponding Markov process. Fix (T,X) G [0,T] X E. Then any countable dense subset of [T,T] can be used as a separability set in Lemma 1.11. Proof. By Lemma 1.11, there exists a separable process Xt on [T,T] associated with a separability set {tk}- Let {SJ} be any countable dense subset of [r,T}. Next, we will show that for PTiX-almost all u> G ft and all k > 1, the element Xtk{w) of E belongs to the set A{UJ) consisting of all limit points of the set < XSi (w) >. By Fatou's Lemma and the stochastic

Transition Functions and Markov Processes

41

continuity of the process Xt, we see that for every k > 1, liminfp(xtt,X,4) >0] < lim P x

\iminfp(xtk,XSi)

< lim liminfP TiX n—»-oo i—>oo

> >

= 0.

(1.60)

Condition (1.60) means that for P r x -almost every w € Q, and every k > 1, the element Xtk(uj) of £ belongs to the set A(LJ). Therefore, the process Xt is separable with respect to the set {s;}. This completes the proof of Lemma 1.14. D The next result (Lemma 1.15) will allow us to get rid of the dependence of the process Xs in Lemma 1.11 on the variables r and x. The conditions in Lemma 1.15 are as follows. A family of stochastic processes X\T,X parameterized by (r, x) 6 [0, T] x E is given, and it is known that the sample paths of these processes possess a certain property. Our goal is to construct a single non-homogeneous process Xt from the processes X\T so that the sample paths of Xt possess the same property. Lemma 1.15 Let P be a transition probability function and suppose that for every pair (r, x) G [0, T]x E, a stochastic process is given on (fi,!F). Suppose also that X,

(r,x)

eB

P{j,x;t,B)

(1.61)

for all t with r < t < T and all B £ £. Let F be a class of E-valued functions defined on [0,T], and assume that the sample paths of all the processes X t ' T ' x) belong to F. Then there exists a Markov process Xt such that its sample paths belong to the class F and P is its transition function. Proof. Consider a new sample space (l = [ 0 , T ] x £ x ( l . This space will be equipped with the cr-algebra T=

{ACQ

: AT>X G T for all (T,X) G [0,T] X £?} .

Here AT^X = iu> : (r, x, ui) € A >. Define a stochastic process on il by Xt(r,x,u)

X\T

Ft = * (X.

T

{UJ) and consider the family of ^--algebras given by

<

s
•~fl

fined as follows: P.

Ar,

for every A £ T. It is not hard to

42

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

see from the definition of the process Xt that its sample paths belong to F. It remains to prove that Xt is associated with P. Let B £ £ and r < t. Then

'u = (u,y,w) eft : Xt(T'x) (w) e B

Xt£B

w : X t (r,x) (w) e f l ] = PT)X [w : X t (w) G B].

(1.62)

It follows from (1.61) and (1.62) that P is the transition function of the process Xt. This completes the proof of Lemma 1.15. • Finally, we are ready to finish the proof of Theorem 1.6. Let P be a given transition probability function satisfying the stochastic continuity condition, and let Yt be a Markov process on (fl, J7) with transition function P. By Lemma 1.11, for every (r, x) there exists a separable process i y , T < t < T, that is stochastically equivalent to the process Yt, r < t < T. Put

*

\x,

ifO
Then it is clear that the process Xj:T'x' is separable for every r and x. Since the stochastic continuity condition holds, Lemma 1.14 and the definition of the process X\T'X' imply that any countable dense subset of [0, T] serves as a separability set for the processes X^T'x'. Moreover, condition (1.61) holds. Let usfixa separability set {£&}, and define a class of functions F as follows. The class F contains all functions on [0, T] with values in E which are minimally continuous with respect to {tfc}. Now it is clear that we can apply Lemma 1.15, and get a separable process Xt with transition function P. This completes the proof of Theorem 1.6. • The next theorem concerns the existence of a progressively measurable Markov process associated with a given transition probability function P. T h e o r e m 1.7 Let P be a transition probability function satisfying the strong stochastic continuity condition: ,J.ini ^ t~slO;s
SU

P

( T ) a ; ) € [o i 4 ] x JS J

P{s,y;t,Gt(y))P{T,x;s,dy) = 0 E

Transition Functions and Markov Processes

43

for all £ > 0. Then there exists a progressively measurable process (Xt,Tl,Pr,x) on the space {£l,F) with P as its transition function. Proof. The strong continuity condition was discussed in Section 1.6 (see (1.42) and Definition 1.8). Let Xt be a strongly stochastically continuous Markov process with P as its transition function and assume that FJ is the filtration a (Xs : T < s
T]xE}.

Consider a sequence of partitions 0 = tft < t" < • • • < tm —T such that max (i? - *"_,) - > 0 a s n oo, and define a sequence of stochastic processes by if *!?_! < s < t™ with 1 < j < m„

X"

if s = T.

x7

It is clear that every process Xn is ^"/-progressively measurable. It follows from the definition of Xn and Definition 1.8 that lim

sup

sup

"->oo(T,x)e[0,T]xBs:0<s
Prx

lp(X?,Xs)>e}=0

(1.63)

for all e > 0. Next, using (1.63) and reasoning as in the standard proof of the fact that the convergence in measure of a sequence of functions implies the existence of an almost everywhere convergent subsequence, we see that there exists a sequence in f oo such that for every s € [0,T], X%sn converges to Xs P T)X -a.s. for all (T,X) £ [0,T] x E. Indeed, from (1.63) we see that there exists a sequence in | oo such that P(xi",xs)>

1 - 2"

for all s G [0, T], T G [0, T], and x e E. Therefore,

nul'W.*-)^} j>ln>j

for all s G [0, T], the sequence X\n Without loss Let A be the set

^

'

r G [0, T], and x G E. It follows that for every s G [0, T], converges P r , x -a.s. to Xs for every r G [0, T] and x G E. of generality, we can assume that in = n for all n > 1. of all (s, w) G [r, T] x f2 such that linin-joo X™(ui) exists.

44

Non-Autonomous Koto Classes and Feynman-Kac Propagators

It is clear that the set A is (fi[o,rj ® ^-measurable. Moreover, for every se[0,T], PT,X{W:(S,W)G.4} = 1

(1.64)

for all (T, X) € [0, T) x £\ Indeed, if s G [0, T] and (r, x) G [0, T] x E, then the set {ui : (s, u>) G .4} contains an ^"-measurable set AST such that ¥r<x(As(T,x)) = 1. This follows from the fact that if s G [0,T], then X£ converges to Xs P T)X -a.s. for every (T, X) G [0, T] x E. Define a new process Xs on Q by X,(w) = lim X™(UJ), if (s, u>) G ^4, and

Xs(bj) = xo for all (S,UJ) $ A, where xo is a fixed point in E. Since for all (r, x) G [0, T] x E, the process X* is an P^-modification of the process Xt, it is clear that PT)X Xt e B = P (T, X; t, B) for all 0 < T < t < T and all B G £. Moreover, since the processes Xn are .^-progressively measurable and (1-64) holds, the process Xt is ^-progressively measurable. This completes the proof of Theorem 1.7. • 1.9

P a t h Properties of Stochastic Processes: Continuity and Continuity

One-Sided

In this section we continue our exploration of the properties of paths of Markov processes. Specifically, we will study the processes with continuous sample paths and the processes for which the sample paths have only jump discontinuities. Theorem 1.8 Let P be a transition probability function, and let Xt be a Markov process with transition function P. Suppose that for all (T, X) G [0, T) x E the following condition holds: lim

PT,x-ess sup P (s, Xs; t, Ge (xs) ) = 0

t-si0)T<s
\

(1.65)

\ J J

for all e > 0, where Ge(y) is defined by (1-40)- The essential supremum in (1.65) is taken with respect to the measure PT,X- Then there exists a Markov process (X t ,^7,P T , x ) on (Q,,^) such that Xt is right-continuous, has left limits, and the transition function of Xt coincides with P. Corollary 1.1 Let P be a transition probability function satisfying the following condition: lim

supP(*,3/;*,G £ (j/))=0

(1.66)

Transition Functions

and Markov

Processes

45

for all e > 0, where Ge(y) is defined by (HO). Then there exists a Markov process (Xt,J-[,FT,x) on (Q.jJ7) such that Xt is right-continuous, has left limits, and the transition function of Xt coincides with P. Remark 1.5 It can be shown that condition (1.66) implies condition (1.41). Indeed, if a transition probability function P satisfies condition (1.66), then we have / P (s, y; t, Ge(y)) P(T, X; S, dy) < sup P (s, y; t, Ge(y)), JE

y&E

and hence condition (1.41) holds. Proof. The structure of the proof of Theorem 1.8 is similar to that of Theorem 1.6. We start with the following lemma. L e m m a 1.16 Let P be a transition probability function, and let Xt be a Markov process with transition function P. Suppose that condition (1.65) holds for a fixed pair (T,X) 6 [0,T) x E. Then there exists a ¥T>Xmodification Xt of the process Xt, r < t < T, which is right-continuous and has left limits. The process Xt depends on T and x. Proof. Without loss of generality, we may assume that Xt is a separable process on (fi,.F) (see Lemma 1.11). Fix r and x. The following random variables will be used in the proof: tp(e,r,8,t) = P TlX [p(Xs,Xt)

> e | J7]

(1.67)

where r < r < s 0. Denote a(e,8)=

sup

P r>x -ess sup
(s,t):0
(1.68)

u>

where 0 < 5 < T — r and e > 0. Using the properties of conditional expectations, we get
P T , x a.s.

(1.69)

Indeed, tp(e, r, s, t) = E T j I [PTjX [p(Xs,Xt)

>e\j^]\FJ]<

a(e, t - s).

Moreover, using the Markov property, we obtain a(e,6)<

sup (s,t):0
P r , x -esssup P {s, X3;t,Gc(Xs)). '

ui

(1.70)

46

Non-Autonomous Kato Classes and Feynman-Kac Propagators

Indeed, since f o r O < T < r < s < £ < T w e have T^ C J-J, it follows that a(e, S) <

sup

ess sup PStxs [p(Xs, Xt) > e]

(s,t):0
=

sup

u

P TiX -esssup P{s, Xs;t,

(s,t):0
Ge(Xs)).

u

By condition (1.65) for (r, x), we see that lima(e,<5) = 0

(1.71)

610

for every e > 0. In order to continue the proof of Lemma 1.16, we will need the following definition. Definition 1.17 Let / b e a subset of [r, T] and / be a function defined on J and taking values in E. Let e > 0 and k > 1. It is said that the function / has at least k e-oscillations on the set / provided that there exists a finite subset J = {si < S2 < • • • < Sk < Sfc+i} of the set / such that p(f(si),f(si+i)) > e for all 1 < i < k. Denote by Zk,e the class of all functions having at least k e-oscillations on /. We will say that the function / has a finite number of e-oscillations on / provided that there exists k £ N such that / ^ Zk>cLemma 1.17 For every subinterval I = [a,b] of the interval [T,T], the class of functions on I having a finite number of e-oscillations for every e > 0 coincides with the class of functions on I with no discontinuities of the second kind. Proof. Then

Suppose that the left limit of / does not exist at a point t € (a, b]. C=

inf

sup p(f(s)J(r))>0.

(1.72)

s:a<s
Now let e be such that 0 < e < (. Then it follows from (1.72) that / has an infinite number of e-oscillations on I. The proof of this fact in the case of the right-hand limit is similar. Conversely, let us assume that there exists e > 0 such that / has an infinite number of e-oscillations on I. Then there exists a sequence h = {ti < t\ < • • • < £fc+i}i k > 2, of finite subsets of [a, b] such that P (/ (*j) ' / (*i+i)) — e f° r a ^ 1 — J — ^- This implies the existence of a sequence (i£-i» i £ , 4 + i ) s u c h t h a t 4 + i - i i L - i -^ 0 as A; ^ oo. By passing to a subsequence, we may assume that t\ has a one-sided limit t. With

Transition Functions and Markov

Processes

47

no loss of generality, we may assume that tkk | t. This is a contradiction, since tkik T t, tkk_, -> t, and p ( / (%_-,) , f (tkk)) > e for all k > 2. This completes the proof of Lemma 1.17. D Let us continue the proof of Lemma 1.16. For every e > 0, k > 1, and any Borel subset H of the interval [r, T], define the following events: Tfc(e, H) = {w : the function t •-» Xt(uj) has at least k e-oscillations on H} (1.73) and roo(e,H)=f)Tk(e,H).

(1.74)

fc>i

Our goal is to prove that Pr,x[roo(€,[r,T])] = 0

(1.75)

for every e > 0. Then • T,X

|Jr0O(e,[T,ri) =o,

and hence, by Lemma 1.17, FTiX {u> : the function t >—> Xt (w) has no discontinuities of the second kind on [T, 2*1} = 1.

(1.76)

Let I = {ti < ti < • • • < tn} be a finite subset of the interval [T, T], Put $fc(e,/)=PT,x[rfc(e,J)|J71]

(1.77)

(3k{e,I) = ess sup $fc(e, J)(w).

(1.78)

and

L e m m a 1.18

For every k > 1, the following estimate holds: /3fc(e,/)<(2a(|,tn-i1))fc.

(1.79)

Proof. For any i with 1 < i < n, denote by A\ (e, /) the event consisting of all u G Ti(e,/) such that p(Xtj,Xtm) > e for some j and m with i < j < m < n. In addition, let us denote by Blk_l(e,I) the event consisting of all u G rfe_i(e,/) such that w has at least k — 1 e-oscillations on the set (
48

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

than k — 1. Then the events B\_l (e,I) are disjoint and ^-measurable, n

[J Bl-i

(e> -0 = r fe-i (e> -0> a n d moreover,

i=l n

rfe(e,/)cU[£i-iM)n4M)]. i=l

For every 1 < i < n, put /* = {tm : i <m
It follows that

n

$k(e,/)<52ETlI

-"r,x

XA^e./jXa^^e,/) | JTi | Ft!

n

K

XA\(e,I) i=l n

< X > - i ( e , J«)PTlX [i?UM) | JTJ i=i

= max

^-i^JOE^frfc-iCc,/)!^]

i:l
=

'

max Afc_i(e,/i)$i(e,/).

(1.80)

i:l
Now we see that in order to prove (1.79), it suffices to show that /31(e,J)<2a(|,sm-s1),

(1.81)

for any set J = { s i , . . . , sm} with T < s\ < • • • < sm < T. Define the following events: Ki(e,J)

= [p(XSl,X3j)

< | for all 1 < j < i - 1, and p(XSl,XSi)

> |}

for all 2 < i < m, and

Li(e,J)={p(X.t,X.m)>l} for all 1 < i < m. It is clear that the events Ki(e, J) are disjoint and Tl -measurable. Moreover, m

r x (6, J) C L^e, J) U ( J (tf<(e, J ) n L4(e, J ) ) . t=2

(1.82)

Transition Functions

and Markov

Processes

49

Indeed, using the triangle inequality, we obtain m i=2

Let UJ e Ti(e,J). Then CJ S Ki(e,J) for some i with 2 < i < m. Here i depends on CJ. If w ^ £•;(£, J), then it follows from

P(xai(w),xam(w))>p(x.M,xM)-p(x.M,xM)

>\

that w e L ^ e , J ) . This gives (1.82). It is not hard to see from (1.82) that *i(e, J) < Pr,x [Li(c, J) | ^ J J + X;P T ,x [Ki(e, J ) n ^ ( e , J ) | J ^ J i=2 m

< P r , x [Li(C, J) | J7J + £ E r , x [Er,x [x^(e,J)XLi(£,J) I ^ J I K] < FT>X [Liit, J) | J7J + £ > , , * [Xif((e,J)Er,« [XLi(e,J) I JT4] I - ^ J • i=2

(1.83) Therefore, taking the essential supremum in (1.83), we get A(e. •/) <

a

Q> s m - s i ) + a ( | , s m - s i ) ^ P r ,

x

[^(e, J) | J ^ J

j=2

< 2a K , s m - s i J . Now it is clear that (1.81) holds. Moreover, (1.81) and (1.80) imply (1.79). This completes the proof of Lemma 1.18. • Next, we will prove conditions (1.75) and (1.76). Fix e > 0. Since (1.71) holds, there exists 5 > 0 such that oc{\,5)<\.

(1.84)

Given e > 0, fix a number S for which (1.84) holds. Let [a, b] be any subinterval of [r, T] such that b — a < S. Then (1.79) gives lim sup {0k(e, I) • I finite and / C [a, b]} = 0. fe—»oo

50

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Let us subdivide the interval [T,T] into a finite family {Aj}, 1 < j < m, of intervals of length at most S. Recall that we assumed the separability of the process Xt. Moreover, any countable dense subset J = {st : £ > 1} of [T, T] may serve as a separability set for Xt on [r, T]. Fix such a set J and put Jn = {st • 1 < (• < n}, Jnj = J„ n A j , and anj = min{s : s G Jn,j}Then for every k > 1, we have

Pr,x [roc (e, Jn,j)\

< Pr,x [Tfc (e, J n > i )] = E r , x

Pr

1 and j with 1 < j < m, Pr,x Poo (e, Jnj)] = 0. It is easy to see that Too (e, J n Aj) = U n >iroo (e, J n j J ) . Therefore, P TiX [Too (e, J D Aj)} = 0 for all j with 1 < j < m. Moreover,

roo(e,J)= |J roofoJnAj), l<j'<m

implies PT)X [Too (e, J)] = 0. Since J is a separability set for Xt on the interval [T,T], we see that (1.75) and (1.76) hold. Denote by Q the event consisting of all w G Q for which t — i > Xt(u) has no discontinuities of the second kind on [r, T], and define a stochastic process by Xt(u>) = Xt(uj) if LJ £ fi, and by Xt(u) = x if u) £ Q\Q. It is clear that the process Xt is separable, and that its sample paths have no discontinuities of the second kind. Moreover, Xt is stochastically equivalent to Xt on [r, T]. It follows from the properties of the process Xt that the limit Xt(w) = lirrin-Kx, Xt+x (w) exists for all t G [r, T) and UJ e Q. The process X t is right-continuous on [0,T) and has left limits on a fixed t G [r, T), we have

(T,T}.

{x t ^x t }= |J i P (x t ,x t )>-i. m=1 ^

'

Moreover, for

Transition Functions

and Markov

Processes

51

It follows that • T,X

Xtj=XA<

lim

FT,x\p(Xt,Xt)> m

lim P T .

(xt,

lim

Xt+x

\ < lim

lim P TiX P(xt,xt+1)>-

m—>oo n—*oo

Now using the stochastic continuity of the process Xt, we get = 0. Therefore, the process Xt, [T,T], is a modification T,X Xt ^ Xt 1

of the process Xt, [T,T], and it follows that the process Xt has P as its transition function. This completes the proof of Lemma 1.16. • Finally, using Lemma 1.16 and Lemma 1.15 for the class F consisting of all right-continuous functions on [0, T] with left limits, we see that Theorem 1.8 holds. • The next result concerns the continuity of the sample paths. Theorem 1.9 Let P be a transition probability function, and let Xt be a Markov process with transition function P. Suppose that for all (r, x) S [0, T) x E the following condition holds: lim

PT,x-esssupP(s,Xs;t,Ge(Xs))

t-siO-,T<s
=0

\

(1.85)

)

for all e > 0. Then there exists a continuous process (Xt, J-^,FT}X) with P as its transition function. The next corollary follows from Theorem 1.9. Corollary 1.2 Let P be a transition probability function satisfying the following condition: ,. lim sup P(s,y;t,Gt(y)) t-s|0;0<s
= 0n

for all e > 0. Then there exists a continuous process (Xt,!Fl,FTtx) as its transition function. Proof.

(1.86) with P

The following lemma will be used in the proof of Theorem 1.9:

52

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

Lemma 1.19 Let P be a transition probability function, and let Xt be a Markov process with transition function P. Suppose that for a given pair (T,x)e[0,T)xE, lim

Vrx-esssupP(s,Xs;t,GJXs))

t-slO;r<s
t —S

'

V

=0

(1.87)

/

for all e > 0. Then there exists a continuous PT)X -modification Xt of the process Xt. The process Xt depends on r and x. Proof. By Lemma 1.16, there exists a PTiX -modification Yt of the process Xt which is right-continuous and has left limits. Let TTi = {r = t« < t\ < • • • < 4 _ x < ^ = T } ,

i>1

be a sequence of partitions of the interval [r, T] such that 7TJ-|_I is a refinement of 7Tj for a l i i > 1. Let us also assume that 5i = m a x { 4 + i - t \ :0

tends to 0 as i tends to cc. Note that condition (1.87) still holds if we replace Xt by Yt. Put z(e,<S) =

sup

PT,X- ess sup P

(s,Ys;t,Gc(Ys))

(s,t):0
and ?(e, (5) =

sup

Pr,ar ess sup

P( 5 ,n;i,G e (F s )) f-s

(s,t):0
for all e > 0 and 5 > 0. Then, using (1.67)-(1.70), we obtain Tli — 1

EMK^'^+J^ fc=0 n;-l

= £ ET,X [PT>I [/> (Yti,Yti+i) > e | ^ ] ] < £ a (e,4 +1 - 4) fc=o

< E - (e> «i+i - 4) = E Ci+i - 4) fe=0

k=o

i,fc+1 t%

fcj

— fl

rii-1

< E (4+1 - 4) % ffc+1 - 4) < (T - T)Z(€, 5i) < Tz(e, 5t fc=0

Transition Functions

and Markov

Processes

53

fcE^K^'^J^^0-

(L88)

By (1.85), z (e, Si) —> 0 as i —• oo. Therefore Tli-l

fe=0

For every z > 1 and e > 0, put Bi(e)

= {, e n: f e max n / (y t ,M,r t . + i H) >

and oo

Cj(e) = \jBi(e). i=j

It is easy to see, using (1.88) and passing to a subsequence of the sequence 7Tj, that without loss of generality we may assume that lira P T , x [C>(e)]=0. j—>oo

The sequence of events Cj(e) is decreasing. Hence, for C(e) = O

Cj(e),

we have PT,X [C(e)j = 0. The complement D(e) of the event C(e) can be described as follows: u £ D(e) if and only if there exists j > 1 depending on u) and such that

for all i > j . Recall that the process Yt is right-continuous and has left limits. For all t € (r,T], put Yt-(cj) — limSft Ys(ui). Our next goal is to show that if u £ D(e), then p(Yt-{u),Yt{Lj))<2e

(1.90)

for all t E (r, T}. Indeed, let w G £>(e), and let j be the integer corresponding to UJ in the definition of D(e). Assume that p(Ys_(uJ),Ys(u))>2e for some s € (r, T]. It is not hard to see that this inequality contradicts the inequality in (1.89). Hence, (1.90) is satisfied. Moreover, if w € H D(e), then (1.90) holds for any e > 0. This establishes the continuity condition for all a; £ Q D(e) and all t G {T,T\. Since the process Xt is right-continuous £>0

54

Nov.-Autonomous

Kato Classes and Feynman-Kac

Propagators

at t = r and P T]X [ne>oD(e)j = 1, there exists a continuous modification Xt of the process Yt. This completes the proof of Lemma 1.19. • It is not hard to see that Lemma 1.15 with the class F consisting of all continuous functions on [0,T] implies Theorem 1.9. • Remark 1.6 Note that if r and x in Theorem 1.9 are fixed, and if Xt is a Markov process with transition function P, then there exists a continuous PTjX-modification Xt of the process Xt. Remark 1.7 Let P be a transition probability function, and let Xt be a Markov process associated with P. Fix r with 0 < T < T and x e E, and let A be a closed subinterval of the interval [r, T\. Suppose that lim

FTX-esss\ipP(s,Xs;t,Ge(Xs))

t—s|0;s,teA

'

=0

\

/

for all e > 0. Then, arguing as in the proof of Lemma 1.16, we see that there exists a PT>x-modification Xt of the process Xt, which is right-continuous and has left limits on the interval A. Similarly, if P is a transition probability function such that lim

P r x- ess sup P (s, Xs; t, GJXS))

t-«i.0;s,t6A t — S

\

=0 I

for all e > 0, then there exists a PTiX-modification Xt of the process Xt, which is continuous on the interval A. Next, we will formulate two results concerning the path properties of supermartingales (Theorems 1.10 and 1.11 below). The proofs of these results will be omitted, and we refer the reader to Section 2.9 in [Yeh (1995)] for more information. Theorem 1.10 Let Z% be a supermartingale on a filtered probability space (fi, J-, Tu P) where 0 < t < T. Denote by Q the set of all rational numbers in [0, T], Then there exists a set A £ T such that P(A) = 0, and the limits lim Zr(w)

and

lim

ZJw)

exist for all UJ £ fi\A. The set A in the formulation of Theorem 1.10 can be chosen independently o f t e [0,T].

Transition Functions and Markov Processes

55

Theorem 1.11 Let Zt be a super-martingale on a filtered probability space (f2,.F, .F t ,P), and assume that Tt is an augmented right-continuous filtration. Then the function t i—• E [Zt] is right-continuous if and only if there exists a modification Zt of the process Zt, which is right-continuous and has left limits. It is clear that a martingale is also a supermartingale, and for a martingale Zt, the function t >—> E [Zt] is equal to a constant. Hence, Theorem 1.11 implies that every martingale on a filtered probability space satisfying the conditions in Theorem 1.11 has a modification that is right-continuous and has left limits. 1.10

Reciprocal Transition Functions and Reciprocal Processes

Let Xt be a Markov process with transition function P. Then the time reversed process Xt = Xr-t is a backward Markov process with backward transition function P given by P(T,A;t,y)

= P(T -t,y;T

-T,A)

(see Section 1.4). If the motion of a random system described by the process {Xs : T < s starts at y £ E at time t, one says that the process XS is pinned at y at time t. Our goal is to study Markov processes possessing both features described above, that is, we would like to understand whether it is possible to simultaneously pin the process Xs at x € E at time r and at y £ E at time t. This problem goes back to Schrodinger (see [Schrodinger (1931)]) and Bernstein (see [Bernstein (1932)]). More references can be found in Section 1.13. Let (fl, J7, P) be a probability space, and let Xt be a stochastic process on Cl with state space (E, £). For all r < u < v < t, put J^V

F? = o-(Xs

:T<S
or v

<s
The family of c-algebras {J^ V J7?} will play an important role in the present section. Definition 1.18 The process Xs, s £ [r, t], is called a reciprocal Markov process on the interval [r, t] provided that for all u, s, and v with r < u <

56

Non-Autonomous Kato Classes and Feynman-Kac Propagators

s < v < t and all bounded Borel functions / on E, the equality E [/ (X3) \FZvr?]=E[f(X.)\
(Xu, Xv)]

(1.91)

holds P-a.s. Condition (1.91) is called the two-sided Markov property of the process Xs. The o--algebra a(Xu,Xv) in (1.91) is interpreted as the present information about the random system described by the process Xs, the cr-algebra J-£ V J^ contains the information about the system before the moment u in the past and the moment v in the future, and finally the cralgebra !F% contains the information about the system after the moment u in the past and the moment v in the future. The following lemma gives several equivalent conditions for the given stochastic process Xs to be a reciprocal Markov process on the interval

M]Lemma 1.20 Let Xs be a stochastic process on (Q, !F, P) with state space (E,£). Then the following are equivalent: (1) Condition (1.91) holds. (2) For all u and v with r < u < v
| a(Xri,...,Xrn,Xu,Xv)]

= E [f(Xs)

\ a(Xu, Xv)]

holds for any bounded Borel function f on E. (3) For all u and v with r
=V[A\ a(Xu, Xv)] P [B \ and B € F%.

a(Xu,Xv)]

Transition Functions and Markov

Processes

57

Remark 1.8 Condition 5 in Lemma 1.20 states that the future and the past are conditionally independent if the present is known. Here the present, the past, and the future are interpreted in the sense of reciprocal processes. Condition 5 was used in [Jamison (1970); Jamison (1974)] as the definition of a reciprocal process. Proof.

(1) ==• (2).

This implication follows from the inclusions cr(Xu, Xv) C a (Xri,...,

XTn, Xu, Xv) C T^ V T%.

(2) = • (3). Assume that condition (2) in Lemma 1.20 holds. In order to prove the implication (2) = » (3), we will first show that the equality E[F\a(Xri,...,Xrn,Xu,Xv)]

=E[F\


(1.92)

holds if the random variable F has a special form. Let m

F=

Y[fj(Xaj), i=i

where fj, 1 < j < m, is a bounded Borel function on E and u < s\ < S2 < • • • < sm < v. We will prove equality (1.92) by induction with respect to m. Since we assumed that condition (2) holds, (1.92) is valid for m = 1. Next, suppose that (1.92) is true for m G N and for all finite sets {SJ : 1 < j' < rn} with u < s\ < S2 < • • • < sm < v. Let {s\,..., s m + i } be a subset of [u, v] with m + 1 elements. Without loss of generality, we can assume that u < sj < S2 < • • • < s m +i < v. Put x — a (Xri,..., Xrn, Xu, Xv) and 7i — a {Xri,...,

XTn,XSl,...,

XSm,Xu,

Xv).

Then the tower property of conditional expectations gives m+l

E

n h (*.,) i ^ =

= E

E

n/j(^)E[/„,+i(xara+1)|w]|^ J'=l

(1.93)

58

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

It follows from condition (2) in Lemma 1.20 that E [fm+i (XSm+1)

\H]=E

[fm+i (XSm+1)

\a(X3m,Xv)]

.

(1.94)

Therefore, (1.93) gives m+l

E

n fj(x») i ^

= E l[fi(Xaj)E

[fm+1 {XSm+1)

\a(XSm,Xv)]

| f

(1.95)

Our next goal is to prove that there exists a bounded Borel function g on E x E such that E [fm+i (XSm+1)

\a(X3rn,Xv)]=g(XSrn,Xv)

(1.96)

P-almost surely. Actually, we will prove a more general known assertion: If F is a bounded a (XSm, Xv)-measurable random variable on O, then there exists a bounded Borel function g on E x E such that F =

g(XSm,Xv)

(1.97)

P-almost surely. Denote by H the class of all real functions F on Q for which there exists a bounded Borel function g on E x E such that equality (1.97) holds. The class H is a vector space containing the constant functions. Moreover, the product of any two functions from 7i belongs to H, and any function of the form XA (XSm)xB (Xv) with A G £ and B G £ belongs to TC. Next, let hn € 71, n 6 N, be an increasing uniformly bounded sequence of positive functions. Then

K = gn{xSm,xv), neN, where gn is a bounded Borel function on E x E. It is easy to see that without loss of generality we can assume that the sequence gn is increasing as n —> co (otherwise, we can replace the sequence gn by the sequence max gi). Then, Ki
sup K = ( sup£„ ) (XSm, n

\ n

J

Xv),

Transition Functions and Markov

Processes

59

and thus sup n hn £ H. It follows from the monotone class theorem for functions (Theorem 5.2) that any bounded a (XSm, X„)-measurable random variable on fi belongs to H. Therefore, equality (1.96) holds. It is clear that (1.95) and (1.96) imply that m+l

E

= E 3= 1

n / J W S (*.».*») i-F

(1.98)

3=1

P-almost surely. Applying the induction hypothesis to the right-hand side of (1.98) and using (1.94) and the tower property of conditional expectations, we get m+l

E

ruwi^

E X\f3(X3.)g{XSm,Xv)\a(Xu,Xv)

3= 1

E

JJfjiX^E

3= 1

[fm+1 (XSm+1)

| a{XSm,Xv)]

|

a(Xu,Xv)

3= 1

:E

Y[fj(XSj)E

[fm+1 (XSm+1)

| 7i] |

a(Xu,Xv)

3= 1

m+l

E E

n f*x'i) i n

a \XU, Xv)

3= 1

m+l

= E

(1.99)

J\f3{XSj)\a{Xu,Xv) 3= 1

By the monotone class theorem and (1.99), E{F\

T]

=E[F\a(Xu,Xv)}

(1.100)

for any bounded ^-measurable random variable F. Next, applying the monotone class theorem again and using the definition of conditional expectations, we obtain E[F\j^\/J^}=E[F\a(Xu,Xv)}. Therefore, condition (3) in Lemma 1.20 holds. (3) =* (4). This implication follows from the definition of conditional expectations, the

60

Non-Autonomous Kato Classes and Feynman-Kac Propagators

formula poo

G=

roo

X{G->x}d\, (1.101) Jo Jo and Tonelli's theorem. (4) = • (5). Suppose that condition (4) in Lemma 1.20 holds, and let A G T^ V T%, B G F%, and D G o(Xu,Xv). Then, using condition (4) with G = XA and F = XBXD, we obtain E

X{G+>x}d\-

[XCXAXB]

= E

[XAE [X£>XB

I o'v-Xui-Xu)]]

= E [XDXAE [XB J ff(X„, X„)j] .

(1.102)

Therefore, E

[XAXB

I

= E [XAE [XB I < T ( X U > X „ ) ] | a(X u ,X„)] = P [A | a(Xu,Xv)} P [B | ff^.X,,)] , (1.103)


and hence condition (5) in Lemma 1.20 holds. (5) = > (1). Suppose that condition (5) in Lemma 1.20 holds. Then, comparing (1.102) and (1.103), we see that (1.102) holds. Let D = Q and G = f(Xs). It follows from (2.110) that E\XAf{X3)}

= E [XAE [f(Xs)

| a{Xu,Xv)}}

.

Next, using the definition of conditional expectations, we get E [f(Xs)

| Tl V F t \ = E [/(*,) |

a(Xu,Xv)]

for all s with u < s < v. Therefore, condition (1) in Lemma 1.20 holds. This completes the proof of Lemma 1.20.

•

It has already been established in Section 1.2 that the Markov property is invariant under time-reversal (see Lemma 1.2 and Lemma 1.3). In fact, more is true. Lemma 1.21 Let (£],.F, P) be a probability space, and let Xt with 0 < t < T be a Markov process with respect to the measure P. Then Xt is a reciprocal process with respect to P. Proof. We will prove that equality (1.91) holds. Suppose that 0 < r < u<s
Transition Functions

and Markov

Processes

61

J^-measurable random variable. Let / be a bounded Borel function on E. It follows from the properties of conditional expectations that E[Yf(Xs)Z\a(Xu,Xv)] = E [E [Yf(Xs)Z = E[f{X,)E

| a (XU,XS,XV)}

\ a (Xu, Xv)]

[YZ \a(Xu,Xs,Xv)]

\a(Xu,Xv)]

.

(1.104)

Next, using the properties of conditional expectations and the Markov property of the processes Xr and XT-T (see Lemma 1.2 and Lemma 1.3), we see that E[YZ\a(Xu,Xs,Xv)] = E [YE [Z | a (Y, Xu,X„

Xv)] \ a

= E [YE [Z | a {Xv)] | a {XU,XS, = E [Z | a (Xv)} E[Y\a

Xvj\

(XU,X3,XV)]

=

E[Z\a(Xv)]E[Y\a(Xu)]

=

E[Z\cr(Xv)]E[Y\a(Xu,Xv)]

= E[YE[Z\a(Xv)}

(XU,XS,Xv)]

\a(Xu,Xv)]

= E [YE [Z | a (Y, Xu, Xv)} | a (Xu, Xv)}

= E [E [YZ J a(Y,XU,XV)}

|

a{Xu,Xv)}

= E[YZ\a{Xu,Xv)}.

(1.105)

Therefore, (1.105) and (1.104) give E [Yf(Xs)Z

| a(XU,XV)]

= E [YE [Z \ a(Xu,Xv)} = E[Y\a

\

{Xu, Xv)} E[Z\a

a(Xu,Xv)} (Xu, Xv)] . (1.106)

Now it is clear that (1.106) implies (1.91), and hence Xr is a reciprocal Markov process. This completes the proof of Lemma 1.21. • The remaining part of the present section is devoted to reciprocal transition probability functions. Such a function describes the evolution of a random system as follows. The value of a reciprocal transition probability function is the probability of finding the system inside a set B € £ at an intermediate moment of time knowing the location of the system at the initial and the final moments of time.

62

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

Definition 1.19 A function Q(T, X; S, B; t, y) where 0 < r < s < t Q (r, x; s, -B; £, y) is £ ® £-measurable for all B £ £ and 0 < r < s < ( < T. (iii) For all (C, D) £ £ x £, (x,y) £ E x E, and 0 < T < u < v < t < T, the following equality holds: /

Q(T,x;v,dz;t,y)Q(T,x;u,C;v,z)

JD

= / Q (T,x;u,dw;t,y)Q Jc

(u,w;v,D;t,y).

(1.107)

The equation in (iii) is an analogue of the Chapman-Kolmogorov equation for transition probability functions. The following equalities follow from (1.107): Q(T,x;v,D;t,y)=

/ Q(r,x;u,dw;t,y)Q(u,w;v,D;t,y)

(1.108)

JE

for all 0 < r < u < v < T and D £ £; and Q(T,x;u,C;t,y)=

/ Q (r,x;u,C; v,z)Q (r,x; v,dz\ t,y)

(1.109)

JE

for all 0 < T < u < v < T and C £ £. The equality in (1.107) can be interpreted as follows. Suppose we know that a random system with the transition law Q is at x £ E at the initial moment r and at y £ E at the final moment t. Let u and v be such that r < u < v < t, and let C £ £ and D £ £. Then the probability of the system hitting the target C at the moment u and then the target D at the moment v in forward motion is equal to the probability of hitting the target D at the moment v and then the target C at the moment u in backward motion. L e m m a 1.22 Let Q(T,x;s,A;t,y) be a reciprocal transition probability function. Then for all (t, y) £ [0, T] x E, the function Pt,v{T, x; s, A) = Q(T, X; S, A; t, y)

Transition Functions and Markov Processes

63

is a transition probability function, while for all (r, x) e [0, T] x E, the function PT,x{s,A;t,y)=Q(T,x;s,A;t,y) is a backward transition probability function. Proof. we get

Using condition (iii) in Definition 1.19 with C = E and D = A,

Pt,v(T, x; s, A) = Q(T, X; s, A; t, y) " /

Q (r, x; s, dz; t, y) Q (r, x; u, E; s, z) Q

•

(T, X; U,

dw; t, y) Q (u, w; s, A; t, y)

/

- /

Pt,y (r, x; u, dw) PttV (u, w; s, A).

(1.110)

On the other hand, using condition (iii) in Definition 1.19 with C = A and D — E, we obtain PT,x(s, A-1, y) = Q(T, X; s, A; t, y) = / JA •

/

Q(T,x;s,dw;t,y)Q(s,w;v,E;t,y) Q

(T, X;

v, dz; t, y) Q (r, x; s, A, v, z)

= I PTiX{v,dz;t,y))PTyX{s,A;v,z).

(1-1H)

JE

Now it is clear that Lemma 1.22 follows from (1.110) and (1.111).

•

Let Q be a reciprocal transition probability function. Next, we will construct a stochastic process Xt associated with Q. Fix (T, X) € [0, T] x E and (t, y) 6 [0, T] x E with 0 < r < t < T, put Q, = E^°'T\ and consider the process Xs{ui) = w(s) for all ui £ Q, and 0 < s < T. Let T be the cr-algebra generated by finite-dimensional cylinders. Then there exists a probability

64

Non-Autonomous Kato Classes and Feynman-Kac Propagators

measure P(T,t),(z,j/) °n i^,^7)

sucn

P(r,t),(x,v) [XT G A0, XSl

= XA0(x)XAn+1(y)

/

that

£ Ai,...,

XSn

G An, Xt

G

An+i]

Q{T,x;S!,dzi;t,y)

JA1


(1.112)

where Ai G 5 for all 0 < i < n + 1 and T < s\ < • • • < sn < t. The existence of the measure P(r,t),(x,y) follows from the Kolmogorov extension theorem, since the expression on the right-hand side of (1.112) defines a projective system of measures. This fact is a consequence of the equalities in (1.108) and (1.109). More precisely, suppose that for some k with 1 < k < n we have Ak = E. Then (1.109) implies the equality

Q (sk-i, Zk-i; " /

sk+1,C;t,y)

Q(sk-i,Zk-i;sk,dzk;t,y)Q(sk,Zk;sk+i,C;t,y),

C G £.

In other words, the number of integrations on the right-hand side of (1.112) can be reduced by one. If k = n, then applying (1.108) we arrive at the same conclusion. Now it is not hard to see that (1.112) defines a projective system of measures. A special case of (1.112) is

p

(T,t),(*,») [ * « € A\ = Q(T> x< s> A\ *. v)

(L113)

for all 0 < T < s < t < T, x € E, y G E, and A G £. Formula (1.113) corresponds to the case n — 1. The initial distribution of the process X3, T < s < t, is the Dirac measure 5X, while the final distribution of X8 is 5y. Since the present state of the reciprocal process X3, T < s < t, can be interpolated from the past and future information, the joint distribution of XT and Xt plays an important role. Next, we will construct a reciprocal process Xt with a prescribed joint distribution /i of XT and Xt. Let ^ be a probability measure on £ £. Then there exists a probability measure P(T,t),M on

Transition Functions and Markov Processes

65

(Q, F) for which P(r,t),/i [XT 6 AQ, XS1 S A i , . . . , XSn € An, Xt € A n + 1 ] = // dfi(x,y) J JA0xAn+i JAi

Q{T,x;sudz1;t,y) d^jt.y)

•Ma

(1.114)

-M„

where Aj £ £ for all 0 < i < n + 1 and T < Si < • • • < sn < t. The existence of the measure P(T,t),/n follows from the Kolmogorov extension theorem. Note that our previous notation P(T,t),(x,y) is nothing else but P(T,t),*rx<5B- We will call the measure p, the initial-final distribution of the process Xs on the interval [r, t]. T h e o r e m 1.12 Let Q be a reciprocal transition probability function, and let the measures P(r,i),(x,y) be defined by (1.112). Then the following conditions hold for the process Xs: (1) For allO

xeE,yeE,

Ae £, and B € £,

(r,t),(x,y) [XT € A, Xt € B) = 5X X Sy(A X B).

(2)ForallO
E(u,vUxM

[f(Xs)},

(1.116)

and E :r,t),(»,v) [/(*.) I K v J7] = j f(y)Q («, *«;», <*y;«, *„) • (i-H7) (4) For allO
66

N'on-Autonomous Koto Classes and Feynman-Kac Propagators /loWP( Ti t),(z,y)-a.S..-

E(T,0,<*,„) [F I K V f%\ = E(r,t),(x,„) [^ | a (Xu, XV)]

(1.118)

and E(r,t),(x,„) [^ | ^

VT*t] = E{u>vUXM

[F].

(1.119)

Proof. Condition (1) in Theorem 1.12 is a corollary of (1.112). Our next goal is to prove that condition (2) holds. Fix T < u < s < t, and let AQ,AI,..., An+i be an arbitrary finite family of Borel subsets of E. By the monotone class theorem and the definition of the conditional expectation, condition (2) follows from the following assertion: The equality P(r,t),(x,j/) [XT € A), Xri G Ai,...,

Xrk_1 £ Ak-i, Xrk G Ak,

Xrk+i € Ak+1, • • • , -X'rn £ ^ n , Xt G A n + i J = E(r,t),(a;,i/) [Q (W. Xu] S, Ak\V,Xv),XT Xrk~i

£ Ak-i, Xrk+1

holds for all finite subsets {r\,..., that

£ A0,Xri

€ A\, . . . ,

G Afc+i,. . •, XTn G v4 n , Xt G A n + i J (1.120)

r n } of the set (r, t) \ {(u, s) U (s, v)} such

r < r i < • • • < rfc_i = u < r f e = s < w = r f c + i < • • • < rn < t. It is not hard to see that the equality in (1.120) is a consequence of (1.112). We will next prove condition (3) in Theorem 1.12. Note that (1.101) with / instead of G and Tonelli's theorem imply equality (1.117). The fact that the expressions on the right-hand side of (1.116) and (1.117) are equal follows from (1.112). This shows that the function (x,y) >-> E(T)t))(X)j,) [/ (Xs)} is Borel measurable. Therefore, the right-hand side of (1.116), that is, the expression TE>(u,v),(xu,xv) [f {Xs)], is measurable with respect to the aalgebra a (XU,XV). Using the definition of the conditional expectation and formula (1.112), we see that the expressions on the right-hand side of (1.115) and (1.116) are equal. For any s with u < s < v, put F — f (Xs). Then the equalities in part (4) of Theorem 1.12 can be obtained from part (3) of this theorem. Equality (1.118) follows from part (3) of Lemma 1.20. The fact that the expressions on the right-hand side of (1.118) and (1.119)

Transition Functions

and Markov

Processes

67

coincide is a consequence of (1.112) and the monotone class theorem. More precisely, (1.112) implies that E

(r,t),(*,v) [F I o-(XU,XV)]

= KMl(Xu,Xv)

[F]

(1.121)

for F — XA-, where A = {XT G AQ, Xn G Ai,...,

Xrk^

G A fc -i, Xrk G Afc,

-^i-fc+i G j4fc+i, • • •, -Xrn S A n , Xt G A n + i } . Here each Aj, 0 < j < n + 1, is a Borel subset of E and r < r\ < • • • < T-fc_i = u
AeS,

and B

e£,

P(T,t),M [*r €A,Xt£B]=

n(A x B).

(2) For all 0 < r < u < s < v
P (r>t)>/1 -a.s.

(3) The two-sided Markov property with respect to the measure P(r,t),M holds for the process Xs. This means that for allO
I K V J ? ] = E(T,t),M [f{X.)

| o- (Xu, Xv)]

P(T,t),M-o.a.

The next part of the present section is devoted to the relations between Markov processes and reciprocal Markov processes. Let Q be a reciprocal transition probability function. It is said that Q possesses a density q if there exists a function q (T, X; s, Z; t, y) such that for all r, s, and t with 0 q (r, x; s, z; t, y) is a Borel function on E x E, and the condition Q{T,x;s,A;t,y)=

/ q(r,x;s, JA

z;t,y)dm(z)

(1.122)

68

Non-Autonomous Kato Classes and Feynman-Kac Propagators

holds. In (1.122), m is the reference measure (see Section 1.3). The following construction goes back to Schrodinger (see [Schrodinger (1931)]). Let P be a transition function possessing everywhere positive density p. Here we do not assume that P is normal. Consider the so-called derived density 9(T l;S 2;i y)

'

'

'

=

p(T,»;t,y)

'

(1 123)

'

where 0 < T < s
*-('. *;*.») =[2n(t-r)]^eXP{Jir7j}-

(L124)

The reciprocal transition probability function Q with derived density q given by (1.123) is associated with a reciprocal Markov process Xs satisfying the conditions in Theorem 1.12. The density q is nothing else but the conditional density of the process Xs subject to the conditions XT = x and Xt = y. Indeed, for any open neighborhood U of the point y € E, we have P

\X ^A\X

GlJ]-

Vr,AXs€A,Xt€U]

pr,x [x.eA\xteir\_

T

IAXUP( >

X

S

Z

Prx [Xt e u}— S

> ' I)P( '

Z 1

^'

z

2)dzidz2

Jup{r,x;t,z2)dz2 By shrinking U to y, we get the following formula: P

rx- <= A I y - „1 -

IAP(T'x's'z^P(s,zi;t,y)dz1

Let jtx be a Borel probability measure on E x E, and let Xs be the reciprocal Markov process on [r, t] with derived density q and such that /i is its initial-final distribution; that is, l*(A xB)=

P (Tit)iM [XT G A, X t E B]

(1.125)

for all Borel subsets A and B of E. We will denote by fiT and /xt the marginal distributions of /x. They are defined as follows: fiT(A) = /i(A x E)

Transition Functions and Markov Processes

69

and Ht{B) = ^{E x B). For all u and v with r < u < v < t, the symbol HUtV will stand for the joint distribution of the random variables Xu and Xv with respect to the measure P(T)t),M. Jamison posed and solved the following problem in [Jamison (1974)]: Determine under what restrictions on the initial-final distribution fi a reciprocal Markov process Xs on the interval [T, t] associated with the derived density q is a Markov process with respect to the measure P(T,t),/xT h e o r e m 1.14 Let p(r,x;s,y) be a transition density, and let q(T,x;s,z;t,y) be the derived density given by (1.123). Fix r and t with 0 < T < t < T, and let Xs be a reciprocal process on [T, t] with transition density q and the initial-final distribution \x given by (1.125). Then the following are equivalent: (1) Xs is a Markov process with respect to the measure P(T)t),M(2) There exist a-finite Borel measures uT and vt on E such that KG) -

If

P(T, x-1, y)dvT{x)dvt(y)

(1.126)

for all Borel measurable subsets G of E x E. Proof. (2) =$• (1). Suppose that there exist measures vr and ut for which condition (1.126) holds. Let T < si < S2 < • • • < sn < t and Ai G £ for all 1 < i < n. Given n > 1, put E x. E x Ai x ••• x An-i

= J4^_!.

Then, using equalities (1.123) and (1.126) and making cancellations, we get P(T,t),„ [XSi e Au 1 < * < n] = /

q{T,x;si,zi;t,y)...q(sn-i,zn-i;sn,zn;t,y) q (sn-i, Zn-i; sn, zn; t, y) d[i(x, y)dzi...

L

dzn-idzn

p (r, x; si, z i ) . . .p (s„_i, Zn-i; sn, zn)

A„-i

p(s

xA„

\t,y)dvT{x)dvt{y)dzi.

..dzn-idzn.

(1.127)

Denote by / the following function: f(zn-i)

=

J A n X E p ( s n - i , 2n-i! sn, zn)p(sn, zn\ t, JEP(sn-l,Zn-i;

t, y)dvt{y)

y)dzndvt(y)

70

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

The function / depends on s n _i, sn, and An. It follows from (1.127) that P(r,t),,i [X,i € At, 1 < i < n) = /

dvT(x)dvt(y)p(T,x;si,Zi)

. . .p (sn-2,

L

Zn-2', Sn-i,

Zn-i) p (sn-i,

Zn-i;t,y)

f(zn-i)dZi

.

..dZn-i

dp,(x,y)q(T,x;si,z1;t,y)

.. .q (sn-2, zn-2; sn-i, zn-i\t,y) = E (Tit)iM [/ {Xtn_x)

f(zi)dzi

...dzn-i

,XSieAi,l
(1.128)

Therefore, (1.128) gives P(r.o,/x [Xsn G An | Tln_^

= /(XSn_J

= P ( T i t ) i / 1 [XSn G A , | a ( X S n _ J ] .

(1.129)

The cases where s\ = r and s n < i; or si > r and s n = t; or si = r and sn = t are similar. Now it is not hard to see that the Markov property holds for Xs. (1) = » (2). Let us suppose that the reciprocal process Xs is simultaneously a Markov process on [r, t] with respect to the measure P(T,t),M. For all s with r < s
l(s, A, B, C) = p(Ti0iM [XT eA,xse

B,

xteC\.

Then the following equalities hold: I(s,A,B,C)=

/

dp.(x,y) /

JAxC

= 1 JAXC

q(T,x;s,z;t,y)dz

JB dli(Xiy)

f JB

P(r,*;s,z)p(s,z,t,y)dz_

P\T,x;t,y)

On the other hand, since Xs is a Markov process, there exists the Markov transition P*(u, Xu;s, A). It follows that / dfj,T(x) / Pf{T,x;s,dz)Pf(s,z;t,C). (1.131) JA JB Let u and s be such that T < u < s < t. Then, since the process Xs is simultaneously reciprocal and Markov, and the Borel cr-algebra of E is I(s,A,B,C)=

Transition Functions and Markov

Processes

71

countably generated, we have pf (u,x;s,B)

=

pf (u,x;t,dy)

q(u,x;s,z;t,y)dz

(1.132)

for /i u -almost all x £ E and all B £ £. Let us denote by AQ the set of all x £ E such that (1.132) holds for all B &£. Then /x„ (S\A 0 ) = 0. Our next goal is to prove that for /i u -almost all x £ E, the measure B \r-> P? (u,x;s,B) and the reference measure m are mutually absolutely continuous. Indeed, let m(B) = 0. Then (1.132) gives Pf(u, x; s, B) = 0 for ^ - a l m o s t all x G E. On the other hand, if x £ AQ and P? (u, x; s, B) = 0, then (1.132) implies Pf (u, x; t, dy) q(u, x; s, z\ t, y) = 0.

/ dz

Since p is a strictly positive function and P* (u,x;t,E) = 1, we get m(B) = 0. This completes the proof of the mutual absolute continuity of the measures B H-> pf (u, x; s, B) and m. It follows from the reasoning above that Pf(u,x;s,B)=

pf(u, x; s, z)dz

(1.133)

JB

for all x G A0 and B G £. In (1.133), pf(u,x;s,z)=

/ Pf(u,x;t,dy)q{u,x;s,z;t,y). JE We will next prove that the measure D >-> P? (u, z; t, D) is absolutely continuous with respect to the measure fit for /i u -almost all z G E. Indeed, we have Ht{D) = f dfj,u(x)Pf (u, x; t, D)

(1.134)

for all D G £. Suppose that C G £ is such that /ut(C) = 0. Then it follows from (1.134) with D = C that Pf (u,x;t,C)=0

for /x„-almost all x £ E.

(1.135)

Using the Markov property, we see that for every D £ £, Pf (u,x;t,D)=

Pf (u,x;s,dy)Pf

(s,y;t,D)

(1.136)

/uu-almost everywhere on E. Let us denote by A\ the set of those x £ E for which (1.136) holds for all D £ £. It follows from the separability of £

72

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

that /u„ (E\Ai) = 0. Fix xi £ A0r\Ai such that (1.135) holds for xi. Note that xi depends on the set C. Then (1.136) and (1.135) with x = x\ show that Pf(s,y;t,C)

=0

(1.137)

for Pf (u,xi; s, -)-almost all y G E. Since xx £ AQ, equality (1.137) holds for m-almost all y G E. Now we see that for all XQ G AQ n A\, equality (1.137) holds for P ' (u, x0; s, -)-almost all y & E. Then (1.136) with D = C gives Pf(u, xo; t, C) = 0. We have thus completed the proof of the fact that the measure D H-» P ^ (U, Z; £, Z?) is absolutely continuous with respect to the measure fit for /i u -almost all z G E. Indeed, the equality Pf(u, xo; t, C) = 0 holds for all C with Ht(C) — 0 for all XQ belonging to the set AQC\A\. This set is independent of C and such that fiu (E\ {AQ n A{)) = 0. It follows that there exists a function P such that Pf («, z; t,D)=

[ Pf (u, z; t, y) d(it(y)

(1.138)

JD

for all £> G £ and for /i u -almost all z £ E. Now it is not hard to show that for every u G [r, £) there exists a function (z,y) >—> P* (u,z;t,y) such that it is £ (g> immeasurable and coincides with the function (z, y) i-> P (u, z; i, y) /xu x jut-almost everywhere. Indeed, using (1.134) and (1.138), we get / dnu(z) JE

/ P (u,z;t,y)dnt(y)=

Ht(E) = l,

JE

and hence, there exists a probability measure v on £ ® £ such that iv(Ax.D)= / d/xu(z) / P"'

(u,z;t,y)dfj.t(y).

JD

JA

This measure is defined as follows. For any set G G £ <8> £, v{G)=

P~f

/ d// u (z) / JE

(u,z;t,y)dfit(y)

JGZ

where Gz = {y G E : (z,y) G G}. It is easy to see that the measure v is absolutely continuous with respect to the measure fiu x fit. Let us denote An

~

the Radon-Nikodym derivative —. r by PA Then for all u with d (/xu x lit) T < u < t, the function (z, y) H-> P-f (U, Z; i, y) is £ <8> ^-measurable, and moreover, / dnu{z) \ P JA

JD

(u,z;t,y)dni (y)

Transition Functions and Markov

73

Processes

= f dfiu(z) f Pf(u,z;t,y)dtH(y) JA

(1.139)

JD

for all Borel sets A and D in E. It follows from (1.139) and from the separability of the cr-algebra £ that for every u with r
= Pf

(u,z;t,y)

for fj,u x //t-almost all (z, y) G E x E. Let us fix s with r < s < t. Then, using (1.130), (1.131), and (1.133), we get [

f

MXiy)

JAXC

P(r,x;s,z)p(s,z;t,y)dz

JB

P(T,x;t,y)

I pf(T,x;s,z)dzPf(s,z;t,C).

= I d(ir(x) JA

(1.140)

JB

Now it would be natural to use equality (1.138) with u = s in (1.140). However, we only know that equality (1.138) holds for /^-almost all z £ E. The following equality justifies the use of (1.138) with u = s in (1.140): fi3(B) =

dnT(x)pf

(T,X;S,B)

— / dz / d^T{x)pf

(T,X;S,Z)

.

Therefore, f

Mx,y)

JAXC

[

P(r^s/z)p(s,z;t,y)dz

JB

P{T,x]t,y) f

= / dfj,T(x) \ p (r,x;s,z)dz JA

JB

/ P (s,z;t,y)dfj,t(y).

(1-141)

JC

It is not hard to see that (1.141) implies , , . p(r, X; S, z)p(s, z; t, y) , , , . , . . f, .—f. ., dn(x, y) —7 dz = diiT{x)dpLt{y)pI(r, x; s, z)P (s, z; t, y)dz. P(T,x;t,y) Hence, there exists ZQ E E such that the following representation holds for

J i \ i 4. \Pf(T>x'>s>zo) . / \ „ p (s,zo;t,y) dfi(x,y) = p{T,x;t,y)—± r-dfiT(x) x — -—Ldfit{y). p{T,x;s,z0) p{s,z0;t,y) This shows that the implication (1) =>• (2) holds. The proof of Theorem 1.14 is thus completed.

•

74

Non-Autonomous Koto Classes and Feynman-Kac Propagators

R e m a r k 1.9 Assume that the conditions in Theorem 1.14 hold. Then (1.129) implies 1Rr P I T-rl - SBP(s>x>'>t>y)dvt(v)El>.tUx.,y)[F] Hr,t),n [? \
»

, . (1.142)

where the random variable F is measurable with respect to the cr-algebra Tl- Indeed, the function / appearing in the proof of Theorem 1.14 can be represented as follows: fAnXEp(sn-i,zn-i;t,y)q(sn-1,zn_1;sn,zn;t,y)dzndi>t(y) J {z-n-i) = '—~ fEP(Sn-l,

Zn-1] t,

y)dvt(y)

s

JEP {sn-i, Zn-i; t, y) dvt{y)Q ( s n - i , Zn-ii n, An; t,y) dz, IEP(

;t,y)dvt{y)

(1.143)

It is not hard to see that (1.129) and (1.143) imply • (r,t),A»

•X-Sr, G Ann \ J•> s„ a.

fEp(sn-i,XSn_1;

t, y) dvt{y)Q (sn-i,XSn_1; fEp(sn-i,

sn, An; t, y) dzn

X8n_1; t, y)dut{y)

IEP{s^-^XSn_1]t,y)dvt{yyS'(sn_1,t),(xSri_1,y)

[XSn <S An] .

(1.144)

JE P(«n-i, ^ „ _ ! ; t, y)dut(y) Therefore, (1.142) follows from (1.144) and from the Markov property of the process Xs. A formula, similar to formula (1.142), holds in the case of a general probability measure \i on £ x £. Lemma 1.23 Suppose that the conditions in Theorem 1.14 hold. Then the following are true: (1) For all T < r < s < t and all J-* -measurable random variables F,

%,*),, [F I JT] /

JEXE

P{T,X;r,Xr)p(s,Xa;t,y)E{s,tUxs,y)

[

JEXE

[F]

p(r,x;r,Xr)P(s,Xs;t,y)^^

d x

^ 'f>

P\J-,x\t,y)

P{r,x;t, y)

(1.145)

Transition Functions and Markov

Processes

(2) For all T < r < s
75

random variables F,

E(r,t),M [F | F.] [

p(T,x;r,Xr)p(s,Xt;t,y)E(T
JExE

f{x'f

[F] d

p(T,x;r,Xr)p(s,Xs;t,y)

^ f

p[T,x;t,y)

Note that the measurability properties of the random variable F in parts (1) and (2) of Lemma 1.23 are different. Proof. Part (1). Let r < s\ < S2 < • • • < Sfc < Sfc+i • • • < sn < t, and let Ai G £ for all 1 < i < n. Then, using (1.123) and (1.126) and making cancellations, we get P(T.t).M [X"i

£Ai,l
= /

q{T,x;si,Zi;t,y)...q(sn-i,Zn-i;sn,zn;t,y) q (s„_i, z n _i; sn, zn; t, y) dfj,(x, y)dzx...

dzn^xdzn

d^(x,y) p (T, X; SI, zi).. .p (s„_i, Zn-i; sn, zn) p(r,x;t,y)

I

ExExAiX-xAn

V ( 5 n , zn \t,y)dzi...dzn-\dzn.

(1-147)

For the sake of shortness, we will use the following notation: B = E x E x Ak+i

diiix

v]

x • • • x An and du(x, y) = , .. p (T, X; t, y)

Let us define the function / by f{zi,Zk) I p {T,X; s\, zi)p (sk, zk; sk+i, zk+i) • • •p(sn,zn;t,y)dvdzk+i _

• --dzn

JJB

/

p(T,x;si,zi)p(sk,zk;t,y)di/

JExE

This function depends on si, Si with k < i < n and Ai with k + 1 < i < n. It is easy to see from (1.147) that W(T,t),v, [X3i G Ai, 1 < i < n] = /

p(r,x;si,zi)...

JExExAi_x---xAk

p (sfc, zk; t, y) f(zi, zk)dv(x, y)dzx

...dzk

76

Non-Autonomous

J

Koto Classes and Feynman-Kac

Propagators

ExExAiX--xAk

...q(sfc-i,2fc-i;sk,zk;t,y)

f(zi,zk)dfi(x,y)dzi

...dzk

= E(T,t),M [/ (XS1 ,XSk),X3ieAul
(1.148)

It follows from (1.148) that P ( r , t ) i / I [X3k+1 G Ak+1,...,

X3n G An | J £ ] = / (X S1 , XSk).

(1.149)

Moreover, we have / p (r, x; si, ^i)p (sfe, Zjt; s fe+ i, z fe+ i) • • -p (sn, zn; t,y) du(x, y)dzk+i

---dzn

JB f

„, „. _ . x P(T,X;S!,ZI)

IEXE JE

dfj,(x,y)

p(r,x;t,y)

/ P (sfc, zk;t, y) q {sk,zk; sk+i, zk+i; t, y) J' A, Ak+iX---xAn • q(sn-i,zn-i;sn,zn;t,y)dzk+i • • -dzn

L

ExE

p(r,x; SI, zi)p{sk,

zk;t,y)

dfi(x, y) P{T,x;t,y)

[XSk+lGAk+l,...,X3neAn].

(1.150)

Now the definition of the function / , (1.149), and (1.150) in the case where si = r and sn = s give (1.145) for all random variables of the form n

F= n xAAxSi). i=k+l

By the monotone class theorem, equality (1.145) holds for all bounded ^"/-measurable random variables F. It is not hard to see that this implies equality (1.145) for all ^/-measurable random variables F. Equality (1.146) can be obtained from equality (1.145) by using time-reversal and noting that the reciprocality is preserved under time-reversal. This completes the proof of Lemma 1.23. • Let VQ and v? be c-finite Borel measures on E, and let p be a strictly positive transition density (not necessarily normal). A pair (fo, v-r) is called an entrance-exit law provided that f J ExE

p (0, z; T, y) du0(x)duT{y)

= 1.

(1.151)

Transition Functions and Markov Processes

77

It follows from Jamison's theorem (Theorem 1.14) that there exists a one-toone correspondence between entrance-exit laws and initial-final conditions fi on £ x £, for which the process X3 associated with the derived reciprocal density q is Markovian with respect to the measure P(O,T),M- This one-toone correspondence is given by K

[

VT) <=>H(AXB)=

p(0, x; T, y)du0{x)duT{y)

(1.152)

JAxB

for all A € £ and B e£. Given an entrance-exit law (^o, VT), we define the functions h and h by h{s,x)=

I

p(s,x;T,y)dvT(y)

JE

for all s with 0 < s < T and x € E; and f>-(s,y) = /

p(0,x;s,y)dvo(x)

JE

for all s with 0 < s < T and y £ E. These functions are strictly positive, but it is not excluded that h(s, x) or h(s, x) may be infinite. However, for all s with 0 < s < T, the function x H-> h (s, x) is finite m-almost everywhere. This follows from the equalities / dv0(z) / dxp(0, z; s,x)h(s,x) JE

= /

JE

p(0, z;T,y)dv0{z)dvT{y)

= 1-

JExE

Here we used the strict positivity of p, the Chapman-Kolmogorov equation, and the definition of the entrance-exit law. If s = 0, then we can only claim that h(0,x) is finite i^o-alrnost everywhere. Similarly, for all 0 < s < T, the function x — i > h(s,x) is finite m-almost everywhere, and the function I H / I (T, x) is finite j/r-almost everywhere. Next, we will introduce the transition functions P 1 and Pi. Unlike the transition functions in Section 1.3, the functions P 1 and Pi are defined almost everywhere. More exactly, we put P 1 (r, x; t, A) = —^— H[T,X)

/ p (r, x; t, y) h {t, y) dy

(1.153)

jA

for 0 < r < t < T, x £ E, and A € £. Moreover, we set P 1 (r, x; T, A) = — ^ -

/ p (r, i ; T, y) ^ T ( y )

fi (T> a:) 7v4

(1.154)

78

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

for 0 < r < T and x E E. For every T and t with 0 < T < t < T and every A E £, the function x H-> P 1 (r, x; t, A) is defined m-almost everywhere with the exceptional set independent of t and A. On the other hand, for all 0 < T < T, the function x H-+ P 1 (T, X; T, A) is defined i/0-almost everywhere with the exceptional set independent of A. Put

P1(T,A;t,y)=

/ /i(r,a;)i)(r,a;;t,2/)da;^JA n(t,y)

(1.155)

for 0 < r < t < T and i / e £ , and

P1(0,A;t,y)

= [ p(0,x;t,y)dvo(x)T-!—7A h(t,y)

(1.156)

for 0 < t < T and i / e £ . Then for every r and £ with 0 < T < t < T and every A E £, the function y >—• P\ (T, A; i, y) is defined m-almost everywhere with the exceptional set independent of r and A. On the other hand, for all 0 < t < T, the function y — i » Pi (0, A; t, y) is defined i/^-almost everywhere with the exceptional set independent of A. It is easy to check that the function P 1 is a transition probability function and the function P\ is a backward transition probability function, if we exclude the exceptional sets described above. Theorem 1.15 Let p be a strictly positive transition density, and let (VQ, VT) be an entrance-exit law with respect to p. Let \L be the probability measure defined in (1.152), and denote by q the derived reciprocal transition probability density associated with p. Let P 1 and P\ be given by (1.153)(1.156). Then the process Xt(ui) = u{t), u> € Cl, 0 < t < T, where Q. = E\°'T>, has the following three realizations: a reciprocal process with the entrance-exit law (i and the reciprocal transition density q; a Markov process with the initial condition h (0, x) duo (x) and the transition function P 1 ; and a backward Markov process with the final condition h (T, x) dvT (x) and the backward transition function P\. Proof. Let 0 < s\ < s2 < • • • < sn < T and Ai E £ with 0 < i < n + 1. Then it is not difficult to see that the finite-dimensional distributions of all

Transition Functions and Markov Processes

79

the processes described in the formulation of Theorem 1.15 are given by

/

dfj,(x,y) /

JA0xAn+i

J

q{0,x;s1,z1;T,y)q(s1:zi;s2,z2;T,y) AiX--xAn

•••q (s n _i, zn-i; sn, zn; T, y) dz\... = /

dzn

du0(x)p(0,x;si,z1)p(si,z1;s2,z2)

JA0x---xAn+i

• • -p(sn, z„; T, y) dzx...

dzndvT(y).

This completes the proof of Theorem 1.15.

•

Theorem 1.15 is taken from [Nagasawa (2000)] (see Theorem 3.3.1 in [Nagasawa (2000)]). A triplicate nature of the stochastic process Xt is clearly seen from Theorem 1.15. The three representations of the process Xt, that is, the reciprocal, the Markov, and the backward Markov are called the Schrodinger, the forward Kolmogorov, and the backward Kolmogorov representation of the process Xt, respectively. Many of the ideas discussed in this section go back to Schrodinger, Bernstein, and Kolmogorov. These ideas found applications in quantum mechanics (see the references in Section 1.13).

1.11

P a t h Properties of Reciprocal Processes

Let p be a strictly positive transition density, and denote by q the corresponding derived transition probability density given by (1.123). Fix (r, x) € [0, T] x E and (t, y) e [0, T] x E, and suppose that the measure H = 5X x 5y is the initial-final distribution in formula (1.114). The resulting measure will be denoted by f(T,t),(x,y)- This measure satisfies

P(r,t),(T,y) [XT e AQ, XSl e A\,..., = XA0(x)XAn+1(y)

/

XSn e An, Xt s

Q(T, X\ si,dzx ;t,y)

J Ax Sn — 1) Zn — 1 i &n i

dzn;t,y) JA2

J An

An+X\

80

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

(see (1.112)). Using the definition of the derived density q and making cancellations, we obtain J

(r,t),(x,i/) [XT £ AQ, XSl £ Ai,..., P(r,x;t,y)

XSn £ An, Xt £ An+{\

JAlx-xAn

• • -p{sn,zn;t,y)dzi

•••dzn.

(1.157)

If the transition density p satisfies the normality condition, then an equivalent form of the previous equality is W(T,t),(x,y) [XT £ AQ, XSl £ Ai,..., =

XSn £ An, Xt £ An+i]

XA

° i g ) X A ; + 1 ! V ) Er,x [p (aw, X.n; t, y), XSl £ Ax,..., P\T, x, i, y)

XSn £ An].

Moreover, using the monotone class theorem, we get P r„T P ™'<*»> [ A ] ~

ET,x[XAP(s,Xs;t,y)} P(r,x;t,y)

(1 158)

'

for all A £ ?J with T < s < t. If, in addition, a strictly positive function p(r, x;t,y) is simultaneously a forward and a backward transition probability density, then

*™<*-v)

[B]

-

(1 159)

p(r,x;t,y)

'

for all B £ Tl with r < s
°

n
and UT =

**>•


p{0,x0;l,yo) Then it is easy to see that (vo, VT) is an entrance-exit law. Hence Theorem 1.15 can be applied. Since h (s, x)=p (s, x; T, yo) and Ji (s, y) =

p (0, x0; T, y0)'

we see that the process Xs(w) = w(s), w £ Q, 0 < s < T, in Theorem 1.15 has the following three properties. It is a reciprocal process with M = $x0 x $vo a s * n e entrance-exit law and q as the reciprocal density; a

Transition Functions and Markov Processes

81

Markov process with the initial condition 6Xo and the transition function P

T,VO

s i v e n °y P

T,yo (T,x;t,A)=

„.T,,\

p(T,x;t,y)p(t,y;T,y0)dy

= f q(T,x;t,y;T,y0)dy

(1.161)

JA

for 0 < T < t < T, x G £ , and 4 G £, and by P^yo(r,a;;r,J4)=x^(2/o)

(1.162)

for 0 < r < T, x G £ , and A G £\ and, finally, a backward Markov process with the final condition Syo and the backward transition function Px 'x° given by P°,X0 (T,i4;t,y) = — —- / p{0,x0;t,y) JA

p(0,xo;T,x)p(T,x;t,y)dx

L

q(0,xo;T,x;t,y)dx

(1.163)

for 0 < T < t < T and y G E, and by ??>Xo(0,A;t,y)

= XA(xo),

(1.164)

for 0 < t < T, y G E, and A G £. As before, we denote by P(o,r),(s0,i/o) t n e measure on Q = E^°'T^ associated with the entrance-exit law (1.160) and the reciprocal transition density q; by PQ'%° the measure on fi corresponding to the transition function PTIVO given by (1.161) and (1.162) and the initial condition 5Xo; and by P 0 '"° the measure on fi associated with the backward transition function P®>x° defined by (1.163) and (1.164) and with the final condition 5Vo. If p is a transition density, then we denote by Po,x0 the measure on Q. associated with the density p and the initial distribution SXo. If p is also a backward transition probability density, then we denote by P r ' y o the measure on Q associated with the backward transition density p and the final distribution 5yo. By Theorem 1.15, the measures P(o,T),(xo,2/o)> ^o,'xa' anc ^ ^o,'xo° c ° i n c i d e on the (7-algebra Tj.. It is not hard to prove that if p is a transition probability density, then the measure P 0 'x° is absolutely continuous with respect to the measure Po,x0 o n every cr-algebra 3-^ with 0 < r < T. Similarly, if, in addition, p is a backward transition probability density, then the measure PQ'"" is

82

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

absolutely continuous with respect to the measure FT'yo on every c-algebra T^ with 0 < r < T. Suppose that under certain restrictions on the transition density p the following conditions hold: (a) There exists a continuous P(O,T),(X0,I/O)" modification of the forward Kolmogorov representation of the process Xt on the half-open interval [0,T); (b) l i m t j r X ( = Vo ]P(o,T),(:ro,!/o)~amiost ev~ erywhere. Then the same restrictions guarantee the existence of a continuous modification of the reciprocal process Xt on the closed interval [0, T]. Similarly, suppose that under certain restrictions on p, the following conditions hold: (a) There exists a P( 0) T),(xo,!/o)" mo ^ mcat i on °f t n e forward Kolmogorov representation of the process Xt, which is right-continuous and has left limits on the half-open interval [0, T); (b) l i m t j r X t = j/o F>(o,T),(x0,2/o)"ahTlost everywhere. Then the same restrictions imply the existence of a P(o,T),(x 0 ,!/o)" mo< ^ mcat i on °f the reciprocal process Xs that is right-continuous and has left limits on the closed interval [0, T]. We will use these ideas in the proof of the following assertion. T h e o r e m 1.16 Let p(r,x;t,y) be a strictly positive function that is simultaneously a forward and a backward transition probability density, and let XQ € E and yo £ E be given. Denote by q the derived reciprocal transition probability density associated with p, and consider the Schrddinger representation of the process X t (w) = w(i), u> £ fi, 0 < t < T, on the space £1 — J5[°>T] with respect to entrance-exit law (1.160) and the reciprocal transition density q. Suppose that for all e > 0, lim / *i°

lim

p(0,x0;t,y)dy

= 0,

(1.165)

JGt(x0)

sup /

p(r,x;t,z)dz

— 0,

t-TlO;0
lim

sup /

(1.166) V

p(T,z;t,y)dz

= 0,

;

(1.167)

t-rl0;0
and lim/

p(t,y;T,yo)dy

= 0.

(1.168)

Then there exists a W,(Q>T),(x0,ya)-rn°dification Xt of the process Xt that is right-continuous and has left limits on [0,T].

Transition Functions

and Markov

Processes

83

Proof. Conditions (1.165) and (1.166) in the formulation of Theorem 1.16 guarantee the existence of a Po^xo'^odification Yt of the process Xt that is right-continuous and has left limits on the half-open interval [0, T) (see Remark 1.7). Since the measure P 0 'x° is absolutely continuous with respect to the measure P 0]Xo on every cr-algebra J® where 0 < r < T, there exists a P0'^"-modification Yt of the process Xt which is right-continuous and has left limits on the half-open interval [0,T). Since P(0,T),(*o,yo)=<xo>

the process Yt is a P(o,T),(xo,2/o)"m°dification of the process Xt on [0,T). On the other hand, since p is a backward transition probability density, conditions (1.167) and (1.168) imply the existence of a P T ' yo -modification Zt of the process Xt that is left-continuous and has right limits on the half-open interval (0,T] (see Remark 1.7). Using the absolute continuity of the measure Pj£'"° with respect to the measure FT'yo on every cr-algebra Fj, with 0 < r < T and the equality P(0,T),(x o ,j/o) = P 0 , x o ° '

we see that there exists a P(o,T),(To,yo)"mo(lification Zt of the process Xt on the interval (0, T] that is left-continuous and has right limits on the interval (0, T}. Using the fact that the processes Yt and Zt are P(o,T),(x0,3/o)~ modifications of the process Xt, we see that there exists a P(o,T),(x0,?/o)" modification Xt of the process Xt having right and left limits on the closed interval [0,T]. Finally, by redefining the process Xt as we did at the end of the proof of Lemma 1.16, we get a process Xt satisfying the conditions in Theorem 1.16. • The next assertion concerns continuous modifications of reciprocal processes. Its proof is similar to that of Theorem 1.16, and we leave it as an exercise for the reader. Theorem 1.17 Let p (r, x; t, y) be a strictly positive function which is simultaneously a forward and a backward transition probability density, and let xo e E and yo £ E. Denote by q the derived reciprocal transition probability density associated with p, and consider the Schrodinger representation of the process -Xt(w) = w{t), OJ € £1, 0 < t < T, on the space Q, = £ , I 0,r l with respect to the entrance-exit law (1.160) and the reciprocal transition

84

Non-Autonomous Kato Classes and Feynman-Kac Propagators

density q. Suppose that the following conditions hold for every e > 0:

lim- /

p(0,x0;t,z)dz

= 0,

(1.169)

lim

sup/

p(T,y;t,z)dz

= 0,

(1.170)

lim

sup /

p(T,z;t,y)dz

= 0,

(1-171)

t-TlO;0
and !lm^7

/

p{t,z;T,yo)dz = 0.

(1.172)

TTien t/iere exists a continuous P(o,r),(x0iVo)-modification Xt of the process In [Jamison (1975)], a different condition is used to ensure the validity of the equality limXt = yo- In the next theorem, Jamison's condition is employed instead of the conditions in (1.167), (1.168), (1.171), and (1.172): T h e o r e m 1.18 Let p (T, X; t, y) be a strictly positive transition probability density, and let XQ E E and yo 6 E. Denote by q the derived reciprocal transition probability density associated with p, and consider the Schrodinger representation of the process Xt(ui) = u>(t), u> G Q., 0
sup

^TyeGc(yo)

p(t,y;T,yo)

=0

for all e > 0. Then there exists a F^o,T),(x0,y0)'mo^fica^on Xt that is right-continuous and has left limits on [0,T].

(1.173) Xt of the process

T h e o r e m 1.19 Let p (r, x; t, y) be a strictly positive transition probability density, and let XQ G E and yo £ E. Denote by q the derived reciprocal transition probability density associated with p, and consider the Schrodinger representation of the process Xt(w) — w(t), w £ Q,, 0 < t < T, on the space £1 = £[°. r ] with respect to the entrance-exit law (1.160) and the reciprocal transition density q. Suppose that conditions (1.169), (1.170), and (1.173)

Transition Functions and Markov Processes

hold. Then there exists a continuous ¥^otT),(x0,vo)'mo^fica^on process Xt-

85

%t of the

Remark 1.10 It is not assumed in Theorems 1.18 and 1.19 that p is a backward transition probability density. Proof. Arguing as in the beginning of the proof of Theorem 1.16, we see that there exists a P ^ T ) , ^ , ^ - m o d i f i c a t i o n Y of the process Xt that is right-continuous and has left limits on the interval [0,T). We will next show that Jamison's condition (1.173) implies the equality lim.t-\T Xt = Vo P(o)T)i(Xo,!/0)-almost everywhere. The following lemma has an independent interest. A special case of this lemma will be used in the proof of Theorem 1.18. Lemma 1.24 —, '

T y

For every T and t with 0 < r < t < T, the process

' ' y w T < s < t, is a P(o i r),(a;o,yo)" mar ^ n 3 a ' e with respect to

the filtration TTS V a (Xt), r < s
We will first show that E (0,T),(xo,!/o)

p(T,XT;t,Xt)

[p(s,Xs;t,Xt)\

1.

"

(L174)

Indeed, using the normality condition for p and the Chapman-Kolmogorov equation twice, we get E,(0,T),(xo,2/o)

'p(T,XT;t,Xty p(s,X3;t,Xt)_

q (0, x 0 ; r, zx\ T, y0) q (r, zx;s, z2; T, y0) q (s, z2; t, z3\ T, y0) JE*

p(r,zi-t,z3)

p(s,zr,t,z3)

dz\dz2dz3

p (T, Z\ ; t, z3) p (0, x 0 ; T,z{)p (r, zx;s, z2) p (t, z3; T, y0) ,n r N dzldz2dz3 JB p (0, x0; T, 2/0) 1 ,n rr s dzidz3p{T,zi;t,z3)p{Q,Xo;T,zi)p(t,z3;T,y0) p(0,xo;T,yo) JBxE 3

=

=

—

7F, r—V / P(*> Z 3;T,2/0)^3 / p(0,xo;l,yo) JB JE

~H\ ^—V / p(t,z3;T,yo)p(0,x0;t,z3)dz3 p{0,xo;T,yo) JB This gives (1.174).

p(T,zi;t,z3)p(0,x0;T,zi)dzi = 1.

86

N'on-Autonomous

Kato Classes and Feynman-Kac

We will next prove that for all r < Si < s2 E,(0,T),(x0,yo)

p(r,XT;t,Xt) p(s2,Xa2;t,Xt)

Propagators


Kvv(Xt)

p(T,XT;t,Xt) p(si,XSl;t,Xt)

(1.175)

P(o,T),(xo,2/o)"a-s- Indeed, using the normality condition for p, the measurability of the random variable p (r, XT; t, Xt) with respect to the
(0,r),(x 0 ,2tt)

p(T,XT;t,Xt) p(s2,XS2;t,Xt)

^VoiXt)

= P (T, X T ; t, Xt) E(0,T),(xo,y0) -j———— P(s2,XS2;t,Xt) 1 p(s2,X32;t,Xt) 1 = P (T, XT\t, Xt) E{sut),(xn,xt) p(s2,XS2;t,Xt)_ q{s1,XSl;s2,z;t,Xt) = p(T,XT;t,Xt) / — dz JE P{S2,z;t, Xt) p(T,XT;t,Xt) ' ' v\ / P(si,XSl;s2,z)dz p(si,Xs i\l-> A t ; JE _ p(r,XT;t,Xt) p(sx,XSl;t,Xt)' = P (T> XT; t, Xt)

E(O,T),(X0,!/O)

\r3iVo{Xt)

.£?,

VJ4

This establishes (1.175). It follows from (1.174) and (1.175) that Lemma 1.24 holds.

(1.176)

D

We are now ready to finish the proof of Theorem 1.18. Using Lemma 1.24 with T = 0 and t — T, we see that the process — ——- is a p(s,Xs;T,y0) nonnegative P ^ r ^ ^ ^ - m a r t i n g a l e . By Theorem 1.3 and equality (1.174), this martingale converges to a finite limit as s f T P(o,r),(x0,s/o)"a'most: surely. It follows that p(s, XS;T, yo) converges to a strictly positive limit IP(o,r),(:ro,i/o)~amlost s u r ely as s T T. By condition (1.173), we see that this can only happen if Xs converges to yo P(o,r),(xo,vo)"a^mos* s u r e i v a s s | T. Combining the previous assertion with the fact that there exists a P(o,r),(xo,i/o)~mocufication Ya of the process Xs that is right-continuous and has left limits on the interval [0, T), we see that there exists a P(o,T),(x0,3/o)~ modification X3 of the process Xs that is right-continuous and has left limits on the closed interval [0,T].

87

Transition Functions and Markov Processes

This completes the proof of Theorem 1.18. The proof of Theorem 1.19 is similar. • If the transition density p is not normal, then the measures P r > x o and W'Vo employed in the proof of Theorem 1.16 do not exist. However, by changing the assumptions in Theorem 1.16 and taking into account the remarks before the formulation of Theorem 1.16, we see that the following assertion holds. Theorem 1.20 Let p(r,x;t,y) be a strictly positive transition density, and let xo € E and yo € E. Denote by q the derived reciprocal transition probability density associated with p, and consider the Schrodinger representation of the process Xt(u) = w(£), LJ £ CI, 0 < t < T, on the space ft = E^°'T' with respect to the entrance-exit law (1.160) and the reciprocal transition density q. Suppose that for all e > 0, lim/

lim t-Tio-fl
P(0,xo;t,y)p(t,y;T,yo)dy

suP-7 xeE

^

p ( r , x ; T , y0)

7/

= 0,

p(T,x;t,z)p(t,z;T,y0)dz

(1.177)

= Q,

JGe(x)

(1.178) lim sup-— —-r / p(0,xo;T,z)p(T,z;t,y)dz t-T|o ; o
= 0, (1.179)

and lim/ ^TJGe(yo)

p(0,xo;t,z)p{t,z;T,yo)dy

= 0.

(1.180)

Then there exists a P(0,T),(x0,y0)~modification Xt of the process Xt that is right-continuous and has left limits on [0,T]. A similar theorem holds for continuous modifications. It is based on Theorem 1.17. We leave this case as an exercise for the reader. In the next definition, we introduce pinned measures. Definition 1.20 Let (T,X) e [0,T] x E, (t,y) £ [0,T] x E, and let p be a strictly positive transition density on E. Denote by Xs the process T v XS{OJ) = w(s) on the space O = E^°' \ Then the measure p^ x on (Q, !FJ)

88

Non-Autonomous Koto Classes and Feynman-Kac Propagators

satisfying /#» (XT €A0,X3leAi,...,XSn

G An,Xt

= p(r,a;;i,y)P( T ,t),(x,y) [XT eA0,XSl = XA0{x)XAn+1(y)

/ J

G An+1)

eAi,...,X3n

eAn,Xt

€ An+i]

P(T,X;SI,ZI)P(SI,Z1-,S2,Z2) AI*....XA„

•••p(sn,zn;t,y)dzi...dzn

(1.181)

where r < si < S2 < • • • < sn < t and Ai G £ for 1 < i < n, is called the pinned measure associated with p, (T,X), and (£, y). It follows from (1.157) and (1.181) that M r " = P ( r . a1! *> 2/) P (r,t),(x,y)-

Remark 1.11 If the density JD is normal, then in addition to (1.181) the following formula holds:

/ 4 * (XT eAo,xaieAi,...,xSn G A„,xt e > W i ) s = XA0 0*0x,*„+1 (y)Er,x b ( n, -X"Sn;«, y), XSl e Ax,...,

x3n e A n ] . (1.182)

The pinned measure ^yx is defined on the measurable space ( E [ ° ' T 1 , !FJ). However, if Xs is a Markov process on a smaller path space, e.g., on the space of continuous paths or on the space of left- or rightcontinuous paths, then certain difficulties may arise at the endpoints r and t. To avoid these difficulties, the measure fi^x is usually restricted to the CT-algebra F[_ = a (Xs : T < s < t) in the case of right-continuous processes Xs having left limits, while in the case of left-continuous processes having right limits, the measure /4'x is restricted to the cr-algebra F{+ =a{Xs:r

<s
Lemma 1.25 Let 0
=

ETtX[Fp(u,X»]t,y)].

Transition Functions

and Markov

Processes

89

The proof of Lemma 1.25 is not difficult, and we leave it as an exercise for the reader.

1.12

Examples of Transition Functions and Markov Processes

In this section we discuss several well-known examples of stochastic processes (see Section 1.13 where more references can be found). We also introduce pinned processes or "bridges".

1.12.1

Brownian

motion

and Brownian

bridge

The state space E of Brownian motion is d-dimensional Euclidean space E.d equipped with the Borel cr-algebra B^d. The d-dimensional Lebesgue measure plays the role of the reference measure m on B.d. Recall that the d-dimensional Gaussian transition probability density Pd is defined by Pd(r, x; t, y) = ^

_* ^

exp { ~ ^ y

}

(see Section 1.11). Since pd depends only on the differences s = t — T and z = x — y, we will use the notation gd (s, z) = pd (T, X; t, y). The function Pd is a transition probability density associated with a time- and spacehomogeneous transition probability function. The density pd is normal, since ixi 2

r

(27r)d/2

/

e

2

dx = 1.

The fact that gd satisfies the Chapman-Kolmogorov equation can be obtained from the following well-known formula for the Fourier transform of the Gaussian density:

*<«•«- (dps

e K dx

L ~^ "

IKJi 2

For any (T, X) € [0, oo) x R d , there exists a measure P TjX on the path space

90

Non-Autonomous Koto Classes and Feynman-Kac Propagators

'°° with the finite-dimensional distributions given by

PTiX[XT eA0,XtleAi,...,xtn = XA0(X)

e An\

/

9d(ti,xi-x)gd(t2-ti,x2-xi)---

JA!X---xA„

9d (tn -tn-i,xn-

xn-i) dxi--- dxn,

(1.183)

where r < ii < t2 < • • • < tn, x e R d , and At e SRd for 0 < i < n. The Gaussian transition function P is defined by P{T,x;t,A)=

/ JA

gd(t-T,x-y)dy.

The function P satisfies condition (1.86) in Corollary 1.2. Indeed,

P(s,y;t,Ge(y))

1

/

__J__

f Js-yf,

^-^x:|x-y|>e(2^-S))^eXP\

*-*

=

2(t-S)/to

* ^ /:w>« (27r(t - *))-/» exp \~2(rr7)}dz

*ih ( ^ 1 M >^>- exp {"^} "u(1.184) and it follows from (1.184) that r hm

sup

4-s-^O+^jjd

P(s,y;t,Ge{y)) —^ = 0 t - S

for all e > 0. By Corollary 1.2, there exists a continuous stochastic process (Bt,T{ ,WTtX) with state space Rd such that its finite-dimensional distributions are given by (1.183). The space Q of all continuous paths from [0, oo) into Rd can be taken as the sample space in this case. The process Bt is called Brownian motion in M.d. It follows from (1.183) that PTtX [BT &Ao,BtleAlt...,Btn = P0,o [-Bo eAo-x,Btl-r

e An) eAi-x,...,BU-T

£

An-x\.

The measure P 0iX is denoted by P x . Brownian motion starting at x — 0 at moment T = 0 is called a standard Brownian motion, and the measure

Transition Functions and Markov

Processes

91

space (C (R+; Rd) , C, P 0 ,o), where C is the cylinder cx-algebra of C (R+; R d ), is called the Wiener space. We denote the components of Brownian motion Bt by B\, 1 < i < d. For a given pair (r, x) £ [0, oo) x Rd, Brownian motion has the following properties: (1) Every component B\ of Bt is a one-dimensional Brownian motion with transition density gi. The components of Bt are PTjX-independent. (2) P T > X [BT = x} = l.

(3) For T < ti < t2 < • • • < tn, the increments Btl, Bt2 — Btl, •••, Btn — Bt„_i of Brownian motion are P TiX -independent. (4) The increments of Bt are stationary; that is, for all r < s < t and h>Q,Bt—B3 has the same P^-distribution as Bt+h — Bs+h(5) Brownian motion is a Gaussian process with mean ET<xBt = x, r < t, x G R d , and the correlation matrix given by E r , a (Bj - x') (fl£ - x^') = (s -

T)

A (t -

T)

,

where r < s, r < i, 1 < i, j < d, and x = (a; 1 ,.. .,x d ) G Md (see [Revuz and Yor (1991)] for the properties of Gaussian processes). If Bt is a d-dimensional Brownian motion starting at x at time r, then —Bt is a d-dimensional Brownian motion starting at —x at time r. For a standard Brownian motion Bt, the finite-dimensional distributions of the processes Bt and tBi/t, t > 0, coincide. They also coincide with the finitedimensional distributions of the process Bt+h — Bh for every h > 0. Moreover, for any a > 0, the finite-dimensional distributions of the process Bt are the same as those of the process a~1^Bat (scale-invariance). If Q is an orthogonal dxd matrix, then the process QBt is a d-dimensional standard Brownian motion (orthogonal invariance). If Brownian motion Bt starts at x at time r, then Bt + y is a Brownian motion starting at x + y at time r (translation-invariance). A d-dimensional Brownian motion Bt is a P TlX -martingale with respect to the filtration F[. There are several other interesting martingales related to Brownian motion, for instance, if Bt is Brownian motion starting at x at time r, then the process |-B t | 2 —dt,t> T, is a P TjX -martingale with respect to the filtration Tl', r < t. If d = 1, then for every a > 0, the process exp {aBt - \aH) is a P T);c -martingale with respect to the filtration J~1 •, T
92

Non-Autonomous Koto Classes and Feynman-Kac Propagators

Next, we will discuss the reciprocal process and the pinned measure associated with Brownian motion. Fix T > 0. Since the Gaussian density is strictly positive, we can define the derived transition probability density qd using formula (1.123). An important link between the reciprocal process with transition density qd and the original Brownian motion is provided by the following formula: qd(T,x;s,z;t,y) (t — s)x + (s — r)y 2 z—

1 d/2

(S-T)(t-S) 2TT

exp <

t-T

\

>. (1.185)

,(s-r)(t-s) t-T

t-T

This formula can be established as follows: qd (r, x; s, z; t, y) 9d(s-T,zx) gd(t- s,zgd(t-T,xy) 2TT(S - r)(t -

y) \x-y\2 \x - z\2 2(t - T) ~ 2(s -T)~

exp

s)

\z-y\2' 2(t - s) _

(t — s)x + (s — r)y 2 ^ z—

1 2TT

(S - T)(t - S)

d/2

exp <

t-T Cs-T)(t-s) t-T

t-T

where 0
x) gd(t-

= 9d(t-T,x-y)

/

s,z-

y) dz

qd(T,x;s,z;t,y)dz

JRd

=

9d(t-T,x-y).

Let (r, x) e [0, T] x Rd and (t, y) G [0, T] x Rd. Since the density gd satisfies the conditions in Theorem 1.17, there exists a continuous reciprocal stochastic process B,^vl, T < s < t, such that its reciprocal transition probability density is equal to the derived density qd obtained from the

Transition Functions and Markov

Processes

93

density gd as in formula (1.123). The process B]fcy\ 3 is called the Brownian bridge between (r, x) and (t,y), or Brownian motion pinned at x at time T and at y at time t. Lemma 1.26

The finite-dimensional distributions of the process

*.=*{$,..

r<s
with respect to the measure W(T,t),(x,y) coincide with the finite-dimensional distributions of the following processes: (a) The process Xl = V t ^ B ^ ^

+ ( l - ^ - \ x +

S

^ y ,

r<s
(1.186)

r<s
(1.187)

with respect to the measure F(o,i),(o,o)(b) The process x

l

= ^-B(.-r)(t-r) t—T t-s

+ ( 1 - ^ ^ J x + ^^y, y t—TJ t—T

with respect to the measure Po,o(c) The process X ^ V t ^ ^ B ^ - ^ B . y ^ l - ^ y + ^ y ,

r<s
with respect to the measure Po,oProof. The finite-dimensional distributions of the process Xs, r < s < t, are given by F

(r,t),(x,v) [*** € i4j : 1 < i < n] .

n—1

= /

qd{T,x;si,zi;t,y)Y[qd(si,Zi\Si+i,zi+i;t,y)dzi...dzn

JA1x--xAn

i = 1

(1.189) where r < si < s 2 < • • • < sn < t and Ai £ Bua for 1 < i < n. Put t — Si \ Wi =

t —T ( Zi t — Si V

t —T t — Si t —T

t —T x -

J

Si — T \ t —T y = Zi - x t —T I t — Si

Si — T t — Si

y,

94

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

and (Si-T)(t-T) t - Si

for 1 < i < n. Then the finite-dimensional distributions of the process X% are given by n Bj.j-THt-r) t-si

'0,0

P 0 ,o

G

n—1

9d (Si, wi) TT gd($i+i -5i,wi+1 r*

i ^ /•

TT -.— n

r

/ -i d

FT :

i=l

-wi)dw1...dwn

n—i

gdih^^Y]

9d(si+i-s,

u>i+i -Wi)dzi..

.dzn

»

/

dzi...dzn

' (si — r)(t — r ) £ — zr ffd I : .T i t — s\ t — si n-l

: l < i < n

G Ci : 1 < i < n

P

/ n

T

I A t — Si Si — T . Ai x y t — Si \ t —T t — T

Bj.i-rHt-T)

t ~

x

- -.

51—7" V

t — s\

(t-r)2(si+i-Si)

(t-r)zi+i

(t - s i + i ) ( i - Si)

t - s.»+i

(t-r)zi t - Si

(t - r)(s i + i - s»)y ( i - s i + 1 ) ( i - Sj)

(1.190) where r < s\ < S2 < • • • < sn < t and Ai G B^d for 1 < i < n. Put pi =

t — T t — Si

where 1 < i < n. Then, it is not hard to see that

formula (1.185) implies the equalities qd(r,x;si,zi;t,y)

= pdgd (pi(si

-r),pizi

- x - ~—y)

(1.191)

and qd(si,Zi;si+i,Zi+i;t, Pi+l9d

y)

I Pi+lPi (Si+l - Si) , pi+iZi+i

- PiZi -

pi+lp,

(Si+l -

Si)

t-T

(1.192)

Transition Functions and Markov Processes

95

for 1 < i < n—1. It follows from formulae (1.189)—(1.192) that the processes Xs and X% are stochastically equivalent. By the previous results, the process -Bj^o) x is stochastically equivalent to the process (1 - X)B * . Hence, the process B^

^^

is stochasti-

t —s

cally equivalent to the process

Bs=j_. Now the scaling properties of t — T *-' Brownian motion imply that the process yjt — T B 1 ' 0 S^J_ is stochastically equivalent to the process

(0.0), t_T

t —s

BU-TW-T) t — T t-s

. It follows that the process X]

is stochastically equivalent to the process X%. Finally, we will prove the stochastic equivalence of the processes X2S and X^. In the proof, we will use the fact that the process (1 — A) B \ is stochastically equivalent to the process B\ — XBi where 0 < A < 1. Indeed, it follows from the properties of Brownian motion that the processes B\ — XBx, \Bi — XBx, \Bi _!, and (1 — X)B x are all stochastically equivalent. S — 7"

Moreover, the process B ^ B\ is stochastically equivalent to the t—r t —T t—s process Ba=j_. This follows from the property of Brownian motion t — T *-« formulated above. Here we take A = s — r . Multiplying by y/t — T and t —T using the scaling property of Brownian motion, we see that the processes Xg and X% are stochastically equivalent. This completes the proof of Lemma 1.26. D The stochastic equivalence of the processes Xs and X% in Lemma 1.26 provides an alternative proof of the existence of continuous versions of Brownian bridges.

1.12.2

Cauchy process

and Cauchy

bridge

The transition density p^ of a d-dimensional Cauchy process is defined by pcd(T,x;t,y)

= cd{t-T,x-y),

0
xGMd,

y&Rd,

where

<*,(«, *) = r ( g ± l )

'

(s,,)e[o,oo)x^.

96

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

It is known that the following formula holds for the Fourier transform of cd(t,£) = e-m,

t>0,££Md.

(1.193)

It is not hard to see using (1.193) that the density pd is normal and satisfies the Chapman-Kolmogorov equation. A Markov process associated with the transition density pd is called a Cauchy process. Since the Cauchy density satisfies the conditions in Corollary 1.1, there exists a realization Xt of a Cauchy process that is right-continuous and has left limits. Cauchy processes have the following scaling property. For every a > 0, the process a~1Xat is P^o-stochastically equivalent to the process XtThe next lemma provides an example of a martingale related to a Cauchy process. Lemma 1.27 Let Xt be a Cauchy process. Then for every £ £ Rd, x G B.d, and T > 0, the process t >—> exp {i£ • Xt + £|£|} is a complex P T)X martingale on the interval [r, oo). Proof. By the Markov property and formula (1.193), we see that for 0 < s < t, ET,X [exp {i£ • Xt + i|£|} | Fs] = e'leiE.,*. [e***] i e*l«l1 / e*vCd (tfe-s,X y)dy = e *l€l e «-*. e -(t-.)iei ^cd(t-s,X ss-y)dy

= exp{i£-Xa + 8\t\}. This completes the proof of Lemma 1.27.

•

Our next goal is to define a pinned Cauchy process. Since the density pd is strictly positive, formula (1.123) can be used for a Cauchy process. According to this formula, the derived transition probability density qd is given by

,S(r,*;S,.;«,9) = r ( i ± I ) < i ^ i > ( i - T ) 2 + |x-y|2

7T ((* - T)2 + \Z - X\*) {(t - S)2 + \y - *P) _ (1.194)

Transition Functions and Markov Processes

97

where 0 < r < s < t < T and x, y, z e Rd. Fix (r,x) e [0,T] x Rd and (t, y) € [0, T] x Rd with T < £. Since the Cauchy density a satisfies the conditions in Theorem 1.16, there exists a reciprocal process -X/*'"\ , T < s
1.12.3

Forward bridges

Kolmogorov

representation

of

Brownian

Brownian motion and Cauchy process are homogeneous Markov processes. Next we give an example of a non-homogeneous transition density and nonhomogeneous Markov process. Such examples can be obtained from the forward Kolmogorow representation of Brownian bridges. Similar examples can be constructed using the forward Kolmogorow representation of Cauchy bridges and other pinned processes. Let XQ 6 Rd and yo 6 K d . In Subsection 1.12.1, we defined the Brownian bridge B^'^3, s £ [0,T], between (0,x 0 ) and {T,y0). In this case, the reciprocal transition density is given by the formula qd (r, x; s, z; t, y) (t - s)x + (s-

1 2 T ('-;W-«))

d/2

exp <

r)y 2 1

t-T

,(s-r)(t-5) t-T

>, (1.195)

where 0
T,y0{T^x'^^A)

= I JA

Qd{T,x;t,y;T,y0)dy

where 0 < T < t < T and qd is defined in (1.195).

98

1.13

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Notes and Comments

(a) The following list is a sample of books devoted to Markov processes [Dynkin (1960); Dynkin (1965); Dynkin (1973); Dynkin (1982); Dynkin (2000); Dynkin and Yushkevich (1969); Bhattacharya and Waymire (1990); Blumenthal and Getoor (1968); Chung (1982); Chung and Zhao (1995); Doob (2001); Ethier and Kurtz (1986); Gihman and Skorohod (1974); Gihman and Skorohod (1975); Gihman and Skorohod (1979); Jacob (2001); Jacob (2002); Jacob (2005); Meyer (1966); Revuz and Yor (1991); Sharpe (1988); Stroock (2005)]. Our presentation of the path properties of Markov processes in Sections 1.8 and 1.9 is similar to that in [Gihman and Skorohod (1975)]. (b) Kolmogorov's papers [Kolmogorov (1931); Kolmogorov (1933)] are important early contributions to the theory of non-homogeneous Markov processes. (c) Progressively measurable stochastic processes were first introduced and studied in [Chung and Doob (1965)]. Theorem 1.4 concerning the existence of progressively measurable modifications of measurable processes was established in [Chung and Doob (1965)] (see also [Meyer (1966); Doob (2001)]). (d) Reciprocal processes were studied in [Jamison (1970); Jamison (1974); Jamison (1975)]. The concept of a reciprocal process and a reciprocal transition function goes back to Schrodinger (see [Schrodinger (1931)]) and Bernstein (see [Bernstein (1932)]). Reciprocal processes are used in stochastic quantum mechanics to model some aspects of the behavior of quantum mechanical systems. The readers who would like to learn more about reciprocal processes and their use in quantum mechanics may consult the following books: [Nelson (1967); Nelson (1988); Aebi (1996); Nagasawa (1993); Nagasawa (2000); Chung and Zambrini (2003)], and the following articles: [Cruzeiro and Zambrini (1994); Cruzeiro, Wu, and Zambrini (2000); Privault and Zambrini (2004); Privault and Zambrini (2005); Roelly and Thieullen (2002); Roelly and Thieullen (2005); Thieullen (1993); Thieullen (1998); Thieullen and Zambrini (1997); Thieullen (2002); Truman and Davies (1988); van Casteren (2000)]. An important paper on time reversal of Markov processes is [Chung and Walsh (1969)]. For more information on time symmetries of Markov processes see [Chung and Walsh (2005)]. (e) Brownian motion is probably the most popular example of a Markov process. For more information on Brownian motion see [Chung (1982);

Transition Functions and Markov Processes

99

Chung and Zhao (1995); Chung and Walsh (2005); Durrett (1984); Johnson and Lapidus (2000); Kahane (1997); Kahane J.-P. (1998); Karatzas and Shreve (1991); Revuz and Yor (1991)]. (f) Cauchy processes belong to the class of Levy processes. This means that they are right-continuous, have left limits, and their increments are independent and stationary. More precisely, a Cauchy process is a symmetric stable process of index 1 (see [Bertoin (1996); Sato (2000); Barndorffet al (2001); Schoutens (2003); Applebaum (2004)] for more information on Levy processes).

This page is intentionally left blank

Chapter 2

Propagators: General Theory

2.1

Propagators and Backward Propagators on Banach Spaces

Propagators are two-parameter families of bounded linear operators on a Banach space satisfying the flow condition (forward propagators) or the backward flow condition (backward propagators). Let B be a Banach space, and denote by L (B, B) the space of all bounded linear operators on B. The symbol / will stand for the identity operator on B. Definition 2.1

A two-parameter family {W (t, T) e L (B, B) : 0 < r < t < T}

is called a propagator on B provided that the following conditions hold: (1) W(t, T) = W(t, X)W(X, T) for 0 < r < A < t < T. (2) W{T,

T)

= I for 0 <

T

< T.

Conditions (1) and (2) in Definition 2.1 are called the flow conditions. Definition 2.2

A two-parameter family of operators {Q(T, t) € L(B, B):0
is called a backward propagator on B provided that the following conditions hold: (1) Q(r, t) = Q(T, X)Q(X, t) for 0 < T < X < t < T. (2) Q(t, t) = / for 0 < t < T. Conditions (1) and (2) in Definition 2.2 are called the backward flow conditions. 101

102

N'on-Autonomous

Kato Classes and Feynman-Kac

Propagators

There are simple relations between propagators and backward propagators. If T > 0 is given, and Q is a backward propagator on a Banach space B, then the family of operators defined by W(t,T) = Q(T-t,T-T),

0
(2.1)

is a propagator. Moreover, if Q is a backward propagator on a Banach space B, and a family of operators is defined by W (t, r ) = Q* (r, t) where Q* (T, t) is the adjoint of Q (r, t), then W is a propagator on the space B*. Here B* stands for the dual space of B. Similarly, if W is a propagator on B, and Q (r, t) = W* (t, T), then Q is a backward propagator on B*. A propagator W is called strongly continuous if for every x £ B, the B-valued function (£, r) —• W (t, r) x, 0 < r < t < T, is continuous. A propagator W is called uniformly bounded if \\W(t,T)\\B^B<M

for allO


If r and t are such that 0 < r < t < oo, and for every compact subset K of the set {(r, t) : 0 < r < t < oo}, the estimate \\W(t,r)\\B^B<MK holds for all (£,r) e If, then W is called a locally uniformly bounded propagator. A propagator W is called separately strongly continuous if for every fixed t and x € B, the function r —• W(t,r)x is continuous on [0,t], and for every fixed r and x £ B, the function £ —> M/(£, r)x is continuous on [r, T] (if T = oo, then we consider the interval [t, oo) instead of the interval [t, T}). Similar definitions apply in the case of backward propagators. The next theorem states that the joint continuity and the separate continuity are equivalent if forward or backward propagators are locally uniformly bounded. T h e o r e m 2.1 Let W be a propagator on a Banach space B. following are equivalent for W:

Then the

(i) The strong continuity, (ii) The strong separate continuity and the uniform local boundedness. The same assertion holds for a backward propagator Q on B. Proof. We will prove Theorem 2.1 for a backward propagator Q. The case of propagators is similar. By the uniform boundedness principle, condition (i) implies condition (ii). Next, let Q be a strongly separately continuous and locally uniformly

Propagators:

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Theory

103

bounded backward propagator. Let 0 < r < t < T, and suppose t' and r' are close to t and r, respectively. We will first assume that t > r. Then for T' close to r-, we have t > T'. Using the local uniform boundedness condition and assuming that t' > t, we see that for every x £ B,

I=\\Q(r',t')x-Q(T,t)x\\B < \\Q(T\ t')x - Q(T', t)x\\B + 113(7-', t)x - Q(T, t)x\\B < \\Q(T', t)(Q(t, 0 * -x)\\B + \\Q(r', t)x - Q(T, t)x\\B < M \\Q{t, t')x - x\\B + \\Q(T', t)x - Q(T, t)x\\B . It follows from the separate continuity condition that lim

7 = 0.

(2.2)

t'—>t,T'—>T

If t' < t, then I < \\Q(T', t')x - Q(T', t)x\\B + \\Q(T', t)x - Q(T, t)x\\B < \\Q(r', t')(x - Q(t', t)x)\\B + \\Q(T', t)x - Q(T, t)x\\B < M \\Q(t', t)x - x\\B + \\Q(T', t)x - Q(T, t)x\\B , and we see that formula (2.2) also holds for t' < t. Finally, let r = t < T' < t'. Then the separate continuity condition implies that for every e > 0 there exists A > 0 such that A > r and ||Q(r,A)ar-ar|| B < e .

(2.3)

If follows from the local uniform boundedness condition and from (2.3) that I=\\Q(T,,t')z-z\\B < \\Q(T', t')x - Q(T', \)X\\B

+ \\Q(T', X)x - Q(r, \)x\\B

+ \\Q(T, X)X -

< | | Q ( T ' , t')(x - Q(t', \)x\\B

+ \\Q(T', X)X - Q(T, X)X\\B

+e

< M \\Q(t', X)x - x\\B + < M \\Q{t', X)x -

\\Q(T', X)X

Q(T, X)X\\B

+ \\Q(T',X)x-Q(T,X)x\\B < M \\Q(t', X)x + (M + l)c.

Q(T, X)X\\B

-

Q(T, X)X\\B

+ M \\Q(T,

X)X

x\\B

+e

- x\\B

+e + \\Q(T',

X)X

-

Q(T,

X)X\\B

(2.4)

In (2.4), M depends on t. Next we get from (2.4) and from the separate continuity condition that there exists S > 0 such that for r < T' < t' < T+S, we have I < (2M + 2)e. Therefore, (2.2) holds for r = t < T' < t'.

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This completes the proof of Theorem 2.1.

D

A simple example of a propagator is as follows. Let St be a semigroup of linear operators on a Banach space B, and consider the following twoparameter family of operators: St-T, 0
=

St-T.

Then U is a propagator on B, and Y is a backward propagator on B.

2.2

Free Propagators and Free Backward Propagators

Recall that we denoted by E a locally compact Hausdorff topological space satisfying the second axiom of countability. Let p : E x E —> [0, oo) be a metric generating the topology of E, and let £ be the cr-algebra of Borel subsets of E. The symbol m will stand for the reference measure. The measure m is a nonnegative Borel measure on (E, £), and we always assume that 0 < m(A) < oo for any compact subset A of E with nonempty interior. We will next define various function spaces on the space E. The symbol BC will stand for the space of all bounded continuous functions on E equipped with the norm ||/|| 0 0 = 8 u p | / ( x ) | ,

feBC

(2.5)

x€E

We denote by Co the subspace of the space BC consisting of all bounded continuous functions on E vanishing at infinity. More precisely, a function / belongs to the space Co if for every e > 0 there exists a compact set Kcj in E such that |/(ai)| < e for all x G E\Kej. It is not hard to prove that Co is a closed subspace of the space BC. The symbol BUC will stand for the space of all bounded uniformly continuous functions on E. More precisely, a function / belongs to the space BUC provided that for every e > 0 there exists 5 > 0 such that for all x € E and y £ E with p (x, y) < S, the inequality \f{x) — f(y)\ < e holds. The space BUC is a closed subspace of BC. For 1 < r < oo, we denote by U£ the space of all Borel functions on E such that 11/11, = {J

\f(x)\rdx}

^oo.

(2.6)

The symbol Iff will stand for the space of all bounded Borel functions on

Propagators: General Theory

105

E equipped with the norm sup | / ( x ) | . xeE We denote by Lr, 1 < r < oo, the Lebesgue space on E with respect to the measure TO. The norm on the space U is defined by (2.6). We denote by £m the cr-algebra obtained by completing the Borel c-algebra £ with respect to the reference measure m. As usual, it is assumed that the elements of Lebesgue spaces are classes of equivalence modO of £mmeasurable functions on E with respect to the measure m. For r = oo, the norm of a function / G L°° is given by ll/lloo = e s s s u p x e s |/(a:)| where esssup^gg |/(x)| = inf {a : \f(x)\ < a

for m-almost all i £ f i } .

In the remaining part of the present section, we discuss backward propagators generated by transition functions. Let P (r, x; t, A) be a transition subprobability function, and define a family of operators on Lf by Y{TMX)

f/s/(»)^.*;Mv),

ifo
\/(a0,

ifr = t a n d O < r < T

for all x e E and / 6 L^. It follows from the subnormality condition that Y is a family of contraction operators on Lf. Moreover, the ChapmanKolmogorov equation shows that Y is a backward propagator on Lf. The family Y defined by (2.7) will be called the free backward propagator associated with the transition function P. Suppose that the transition subprobability function P possesses a density p; that is, there exists a nonnegative function p(r, x; s, y) such that JA

for all A £ £. In this case, the free backward propagator Y is defined on the space L°° by

Y(T1t)m

= {s*ny)p{T'x't'v)dv> ^/(x),

* ° ^ < ' *

r

(2.8)

if r = t and 0 < r < T

for all x e E and / e L°°. The operator Y(T, t) maps the space L°° into the space Lf.

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If P is a backward transition subprobability function, then we define the free propagator associated with P by

[f{y),

\IT — t and 0 < r < T

for all x £ E and / £ Lf. If p is a backward subprobability transition density, then the free propagator has the following form:

[f(y),

IIT = t and 0 < r < T

for all x £ E and / £ L°°. Let us recall that in the case of a finite timeinterval [0,T], there exists a one-to-one correspondence P(T,x;t,A)

= P(T-t,A;T-T,x),

0 < T
between forward and backward transition functions. Therefore, if P is a given backward transition subprobability function, then the free propagator associated with P can be defined by U{t,T) = Y{T-t,T-r),

(2.11)

where Y is the backward free propagator associated with P.

2.3

Generators of Propagators and Kolmogorov's Forward and Backward Equations

In this section we continue the study of propagators on Banach spaces. Our goal is to show that propagators are related to non-homogeneous evolution equations. We will first discuss general propagators and then consider free propagators generated by transition functions. The non-homogeneous equations which are studied in the present section are called Kolmogorov's forward and backward equations. Let B be a Banach space and let ip be a B-valued function on the interval (0,T). If the function

symbol ——(r) will stand for its derivative from the right, that is, or dr

MO

h

Propagators:

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Theory

107

Similarly, if the function ip is differentiable from the left, then the symbol — — ( T ) will stand for its derivative from the left, or v ;

dr

hio

-h

Suppose that Q = Q{r, t), 0 < r < t < T is a backward propagator on a Banach space B. For every r with 0 < r < T, consider a linear operator on B defined as follows: A

i \

i.

Q(T

— h,r)x

—x

.„ „ .

A-(T)x

= hm— ~ . (2.12) Mo h The domain of this operator is the set D(A-(T)) of all x £ B for which the limit in (2.12) exists. The operators A_(r), 0 < r < T, are called the left generators of the backward propagator Q. For every t € (0,T], denote by D-(t) the set of all x e B such that the function T H-» Q(r,t) is differentiable from the left on (0,t), and by .F(i) the set of all x £ B for which l i m Q ( t - M ) a : = x.

(2-13)

The next result explains in what sense the left generators are related to the backward propagator Q. Theorem 2.2 Let Q be a backward propagator on B, and let t G (0,T]. Then for every x £ D-(t)C\F(t), the function u (r) = Q(r,t)x is a solution to the following final value problem on (0, i):

f

£M--MrWr).

pi4)

limu(T) = a;. Proof. Let a; € £>-(£) ("1 F(t). Then, using the properties of backward propagators, we obtain d~u, , ,. Q(T - h,t)x -Q(T,t)x —— (T) = lim '-j5r fcio -h -- lim ^ Mo

T

~ H' T^T'

^X ~ g ^ T ' -/i

^

_lim(Q(r-/i,r)-/)Q(r,^ MO = -A-{j)Q{r,t)x

-h =

-A-(T)U{T).

108

Non-Autonomous Koto Classes and Feynman-Kac Propagators

In addition, the equality limu(r) = x follows from the definition of the set Tit

F(t). This completes the proof of Theorem 2.2.

•

Our next goal is to introduce the family of right generators of the backward propagator Q. For every T with 0 < r < T, consider a linear operator on the space B given by A+(T)X v

=

Q (r, r + h) x — x h

lim-

hj.o

(2.15)

The domain D (A+(T)) of this operator is the set of points x e B for which the limit in (2.15) exists. The operators A+(T), 0 < T < T, are called the right generators of the backward propagator Q. For every t € (0,T], denote by D+(t) the set of all x G B such that the function T H-> Q(r,t) is differentiable from the right on (0, t). T h e o r e m 2.3 Let Q{r,t), 0 < T < t < T, be a strongly continuous backward propagator on B, and fix t with 0 < t < T. Then for every x e D+(t), the function U(T) = Q (T, t) x is a solution to the following final value problem on (0, t): ( d+U

-(r) = dr limzi(r) = x.

-A+(T)U{T),

(2.16)

T\t

Proof. Let x € D+ (t). Then, using the strong continuity of the backward propagator Q, the Banach-Steinhaus theorem, and the definition of the set D+(t), we obtain d+u ,. _,. —— =hmQ(T,T OT

MO

,^d+u + h)—— OT

d+u Q(r + h,t)x-Q(T,t)x dr h Q(T + h,t)x - Q{r,t)x + limQ(T,T + /i) Mo h Q(T + h,t)xQ(r,t)x — lim Q(T,T + h) Mo h Q(T, t)x - Q(T, T + h)Q(r, t)x lim Mo h Q(r,T + h)-I = —lim Q{r,t)x. Mo h

= lim Q(T,T + h) Mo

(2.17)

Propagators: General Theory

109

It follows from (2.15) and (2.17) that Q(T, t)x G D (A+(T)) and the equation in (2.16) is satisfied. In addition, the equality limu(r) = x follows from the T"T*

strong continuity of Q. This completes the proof of Theorem 2.3.

•

Now we turn our attention to propagators on B. The generators in this case are defined exactly as in the case of backward propagators. Suppose that W is a propagator on a Banach space B. For every t with 0 < t < T, consider a linear operator on the space B given by 7 /^

i.

W(t

+ h,t)x-x

.„.,„,

A+(t)x = lim — i -^ . (2.18) v Mo h ' The domain D (A+ (t) J of this operator is the set of points x £ B for which the limit in (2.18) exists. The operators A+(t), 0 < t < T, are called the right generators of the propagator W. For every T e [0,T), denote by D+(T) the set of all x £ B such that the function t H-> W(t,r)x is differentiable from the right on (r, T), and by F(T) the set of all x € B for which limt^(T + /i,T)a; = ar. (2.19) Mo Theorem 2.4 Let W be a propagator on B, and fix T with 0 < r < T. Then for every x £ D+(T) n F(T), the function u{i) = W(t,r)x is a solution to the following initial value problem on (r, T):

^

w

= i + ( «<),

(22o)

limu(t) = x. Ur

Proof. Let x £ D+(T) n F(T). Then, using the properties of propagators and the definition of the set D+(T), we obtain d+~,.s ,. W(t + h,t)x-W(t,T)x -K7u(t) = hm —i '— dt MO h ,. W(t + h,t)W(t,r)x-W(t,T)x = hm ; MO h _ {W{t + h,t)-T)W(t,T)x Mo h = A+(t)W(t,T)x = A+(t)u(t).

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Koto Classes and Feynman-Kac

Propagators

In addition, the equality limu(i) = x follows from the definition of the set t\.T

F(r). This completes the proof of Theorem 2.4.

•

Let W b e a propagator on B. For every t with 0 < t < T, consider a linear operator on the space B given by r ,N ,. W(t,t — h)x — x A_(t)a; = lim————• . w MO h The domain D (A-(t))

,„.,, (2.21)

of this operator is the set of points x G B for which

the limit in (2.21) exists. The operators A-(T), 0 < t < T, are called the left generators of the propagator W. For every T G [0, T), denote by D-(T) the set of all x G B such that the function t H-> W(t, r) is differentiable from the left on (r, T). Theorem 2.5 Let W be a strongly continuous propagator on B, and fix r with 0 < r < T. Then for every x G D- (r), the function u (t) = W (t, T) X is a solution to the following initial value problem on (r, T): (

^ ( t )

=

A_(t)u(t),

dt

(2.22)

limw(£) = x. tJ.T

Proof. Let x G D-(t). Then, using the strong continuity of the propagator W, the Banach-Steinhaus theorem, and the definition of the set -D-(i), we obtain

HO

K

' [ dt

-h

hiO

-h

llmW(t,t-h)W^-h^X-W^X

= hiO

V

'

'

-h

W(t, T)X ~ W(t, t - h)W(t, T)X h.10

-h

J

Propagators:

General

Theory

111

It is not hard to see that (2.21) and (2.23) imply W(t,r)x G D (A-(t)Y Moreover, the equation in (2.22) is satisfied. In addition, the equality limu(t) = x follows from the strong continuity of W. This completes the proof of Theorem 2.5.

•

Our next goal is to discuss what happens if we differentiate Q and W with respect to "wrong" time variables. Let Q be a backward propagator on a Banach space B. For every r with 0 < r < T, put D*+{T)=

p|

D(A+(t)),

t:r
where D(A+(t))

is the domain of the operator A+(t) defined by (2.15).

Theorem 2.6 Let Q be a backward propagator on B, and fix r with 0 < r < T. Then for every x G D\(T) and t with r
=

Q(T,t)A+(t)x.

Proof. Let x G D*+{T). Then, using the properties of backward propagators and the definition of D+(T), we obtain d+Q(T,t)x dt

_

Q(T,t +

h)x-Q(T,t)x

hio

h Q(T,t)Q(t,t

hio

h)x-Q(T,t)x

Q(r,t)]imQ{t't+uh)x-X hio h

= =

+ h

Q{T,t)A+{t)x.

This completes the proof of Theorem 2.6.

•

For a propagator W an & Banach space B and t G (0, T], put

Dl(t)=

f|

D(A-(T)),

T:0
where D

(A-(T)J

is the domain of the operator A_(r) defined by (2.21).

Theorem 2.7 Let W be a propagator on B, and fix t with 0 < t < T. Then for every x G D*_{i) and r with 0 < T < t, d~W(t,r)x ^

=

,-r , , -W(t,T)A-(T)x.

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Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

Proof. Let x e D*_{t). Then, using the properties of propagators and the definition of D*_(t), we obtain d~W(t,T)x dr

_

W(t,T-h)x-W(t,r)x h

hio _

W(t, T)W(T,

r-h)x-

hio

W(t, T)X

h -W(t,r)hmW^T-k)X-X h

=

K

=

-W(t,T)A-(r)x.

' hio

This completes the proof of Theorem 2.7.

•

Next we will discuss Kolmogorov's forward and backward equations for transition probability functions. Given such a function P, let us consider the Banach space B = Lf and the free backward propagator Y defined by formula (2.7). Instead of the strong generators A_(r) (see formula (2.12)) and A+(T) (see formula (2.15)), we will consider generators in a weaker sense. The new generators have larger domains than those of the generators A_(r) and A+(T). We will use the topology a (Lf ,M) on the space Lf, where the symbol M stands for the space of all finite signed Borel measures on E. For every r € [0,T), denote by Dw ( A ^ ( r ) ) the set of all / e Lf for which the a (Lf, M)-limit

AM(T)f =

*(Lf,M)-limY{T-h'hT)f-f

exists. This means that there is a function A^(r)f

€ Lf

//"w^°a//(r"T)/"^

such that

<2-24>

for all v e M. The operators A^(T), 0 < r < T, are called the left a (Lf, M)-generators of the backward propagator Y. The next simple lemma provides an equivalent condition for the convergence of a sequence of functions in the topology a (Lf ,M). Lemma 2.1 A sequence of functions hk € Lf converges to a function h G Lf in the topology a(Lf,M) if and only if lim hk(x) exists for all k—HX>

x e E, and sup \hk(x)\ < oo. k,x

Propagators: General Theory

Proof. that

113

Suppose that hk —> ft in the topology a(Lf,M).

This means

lim / hkdu = / hdu ^<*>JE

(2.25)

JE

for all ;/ G M. Put 1/ = 5X. Then the sequence ftfc converges to ft pointwise on E. Moreover, the uniform boundedness of the sequence {hk} follows from the Banach-Steinhaus theorem. Now assume that the sequence hk converges pointwise on E to ft, and moreover, it is uniformly bounded. Then h £ Lf, and by the dominated convergence theorem, we see that hk —* h in the topology a (Lf,M). • For every r with 0 < r < T and every / £ Dw (A™ (r)), we have lim Y(r-h,r)f(x)-f(x) hio h

= AM{T)f{x)

(2i2g)

and sup |y^-^^)/(')-/WI (/i,x)6(0,r)x£ ft

< oo.

(2.27)

This follows from Lemma 2.1. By (2.26), (2.27), and the dominated convergence theorem, for all / £ Dw (A^(T)), the pointwise derivative from the left i (x) of the function r H-> K(T, t)f(x) coincides with its a (^|°, M)-derivative from the left. This means that for all u € M, lim / y(r-h,t)f(X)-Y(r,t)f(x) hioJE ft Y(T-h,t)f(x)-Y(r,t)f(x) I im Y{r-h,mx)-Y(r,mx)Mx) = / llim h IE JE L° ft Let t G (0,T], and denote by F™(t) the set of all functions / G Lf such that KmY(t-h,t)f(x)

= f(x)

for all x €. E. The uniform boundedness of the family of functions Y (t — h,t) f in the space Lf follows from the definition of the free backward propagator Y. By Lemma 2.1, Y (t — h,t)f —> / in the topology a (Lf,M) as ft | 0. For any t with 0 < t < T, denote by D™(t) the set of all functions / G Lf such that the function r i-» Y(T, t)f is cr (Lf,M)differentiable from the left on (0, £).

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Kato Classes and Feynman-Kac

Propagators

Theorem 2.8 LetO
?-1(T,X)

=

-AM(T)U(T)(X)

9T limu(r, x) = f(x)

(2.29)

Tit

for all x G E. Remark 2.1

In final value problem (2.29), the derivative ——(r, x) and or the equality limw(T, x) = f(x) are understood in the sense of pointwise convergence, or in the sense of convergence in the space a (I/g°, M). Proof. Let / e D™ (t) n F™{t). Then, using the properties of backward propagators and the definition of the set D^(t), we see that for every v e M, t u{r-h)-u{r)dv hioJEE -hn —

Hm

Y{r-h,t)f-Y(r,t)fdiy h (Y(T-h,T)-I)Y(T,t)f = - lim / dv nh IE

=

_]]mr HoJ E h-lUJE

-L

A?(T)Y(T,t)fdl>

= - f A"(T)u(r)dv.

(2.30)

JE

Now it is easy to see that Theorem 2.8 follows from (2.26), (2.28), (2.30), and the definition of the set F^{t). This completes the proof of Theorem 2.8. • The equation in (2.29) is called Kolmogorov's backward equation. Kolmogorov's equation is a linearization of the Chapman-Kolmogorov equation. One can find examples of transition probability functions by solving Kolmogorov's backward equation. This can be done, for instance, if the set D^(t) is large enough, and if final value problem (2.29) is uniquely solvable. More precisely, suppose that for every open set O C E, there exists a sequence /„ G D^(t) such that sup |/„(a;)| < oo and lim fn(x) — Xo(x) for n,x

n—>oo

all x £ E. By Lemma 2.1, the sequence / „ converges to xo in the topology a (X/£°,M). Therefore, the sequence Y (r,t) fn(x) converges as n —> oo for

Propagators:

General

Theory

115

all x G E and 0 < r < t. Here we use the dominated convergence theorem. Put

Y{T,t)Xo(x)=

lim

Y(r,t)fn(x).

Then it is not hard to see that Y (r, t) xo does not depend on the approximating sequence / „ . It is also clear that

P{T,x-t,0)

=

Y(T,t)Xo{x).

Fix r , t, and x such that 0 P (r, X; t, A) is a Borel measure. Define a family V of Borel sets by V = {B e £ : P ( r , x; t, 5 ) = ^ ( T , * ) X B ( Z ) } - Then X> is a d-class and contains all open subsets of E. By the monotone class theorem, T> = £. This allows us to recover the transition function P from Y. Next we will introduce the family of right a (Lf, M)-generators of the backward propagator Y. As in formula (2.15), for every r with 0 < r < T, we consider the set Dw (A+ (r)) of all functions / € Lf for which the limit A%(T)f =

*{Lf>,M)-1im nliJ

Y(r,T + h

h)f-f

(2.31)

exists. The operators A+ (r) are called the right a (Lf, M)-generators of the backward propagator Y. For any t with 0 < t < T, denote by D+(t) the set of all functions / S Lf such that the function T —> Y(r,t)f is c (L~, M)-differentiable from the right on (0, t). Theorem 2.9 Let Y be an o (Lf ,M)-continuous backward propagator on the space Lf, and fix t with 0 < t < T. Then for every f G D^(t), the function U(T) =Y (T, t) f is a solution to the following final value problem

on(0,t): ( d+u (T,X) dr

=

l i m u ( r , x) =

-A¥(T)U(T)(X),

(2.32) f(x)

for all x G E. R e m a r k 2.2

d+i

The derivative ——(r, x) and the equality or limu(T)(a;) = f(x)

116

Non-Autonomous Kato Classes and Feynman-Kac Propagators

in final value problem (2.32) are understood in the sense of convergence in the space (Lf, a (Lf, M)). Proof. Let / £ D^(t), and suppose that hn is a sequence of positive numbers such that hn j 0. Let v e M be a nonnegative Borel measure on E. Then, using the fact that Y is a a (L|?,M)-continuous backward propagator on the space Lf and reasoning as in the proof of (2.17), we get r g+u r g+u / ——dv= lim / Y (r, r + h„) —— dv JE dr n^ooJB dr = hm / Y

(T, T

+ hn)

+ lim / F ( T , T . / E.

—

+ /I„)

d^

y(T + /ln,*)/-K(T,t)/

dv. (2.33)

/in

Our next goal is to prove that lim / Y(T,r + hn)

d+u dr

Y(T +

hn,t)f-Y(T,t)f hn

di/ = 0. (2.34)

It follows from the a (Lf, M)-continuity of Y that for every n G N, the set function vn(B)=

i Y{r,T + hn)XB{x)du{x),

Be£,

JE

is a Borel measure. It is not hard to see that the set function \i defined by

L

n=\

is a finite Borel measure on E. Define a sequence of functions on E by 9n

d+u dr

Y(r +

hn,t)f-Y(r,t)f hn

Then,

d+u

[ Y(r,T + hn) dr JE

Y(r +

hn,t)f-Y(T,t)f h„

dv=

\ JE

gn{x)dvn{x). (2.35)

Since / e D+(t) and Lemma 2.1 holds, lim gn(x) = 0

(2.36)

Propagators:

General

Theory

117

for all x € E, and su

( 2 - 37 )

Plffnloo < °°-

n

It follows from (2.36) and Egorov's theorem that for every e > 0 there exists B £ £ such that H(E\B) < e and \gn{x)\ < e, x£B,

n e N.

(2.38)

Moreover, since every measure vn is absolutely continuous with respect to the measure fi, and lim vn(B) = v(B) for all B € £, the Vitali-Hahn-Saks n—>oo

theorem (see Section 5.5) implies that lim sxxpvJB) = 0 .

(2.39)

M(B)iO n

Now it is not hard to see that equality (2.34) follows from (2.35), (2.37), and (2.39). Next, using (2.34) and the properties of backward propagators, we see that

[p*,JE

dT

lim /r(x, T + M r ( T + " - ' ' / - n r ' t ) / ^ h

n->°°JE

= lim

n

/ Y(T,t)f-Y(T,T

+

hn)Y(T,t)f

dv

n->ooJE

~ ~."St/, YiT-rlK)-'Y(r,

W,

(2.40)

It follows from (2.40) that Y(r,t)f € Dw (A*£(r)) and that the equation in (2.32) holds in the space (Lf, a (Lf, M)). In addition, the equality lim / u{r)du = I fdv follows from the continuity of Y on the space T T* JE JE

{Lf,
= [ d(i{x) [ P(T,x;t,dy) JE

JB

= [ JE

P(T,x;t,B)dfi{x),

118

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

and the last expression is a Borel measure on E. Let 0 < t < T, and denote by Dw ( ^ ( i ) ) the set of all /x G M such that the limit m

= lhnW(t

m + v ;rA

'

h,t)KB)-m h

+

hio

exists for all B G £. Then, by the corollary to the Vitali-Hahn-Saks theorem formulated in Subsection 5.5 of the Appendix (see Corollary 5.2), A+(t)/x is a Borel measure on E for all ix G Dw (A+(t)\. We will denote by F£ (T) where 0 < r < T the set consisting of all /x G M such that \imW(T

+

h,T)fi(B)=fi{B)

for all B G £. For every r G [0,T), denote by D + ( T ) the set of measures /tx G M such that the function £ — i > iy(£, T)/X is setwise differentiable from the right on {T,T). The next result concerns Kolmogorov's forward equation. Theorem 2.10 Let Y be the free backward Lf -propagator corresponding to a transition probability function P, and let W be the propagator on the space M given by W(t,r) = Y(r,t)*, 0 < r < t < T. Suppose that 0 < r < T and /x G D+(T) f]F£(r). Then the M-valued function v defined for all t G (T,T) by the formula v{t){B) = W (t,r)/x(B), B G £, is a solution to the following initial value problem: ( d

~(t)(B) = A£+(t)u(t)(B), M Unu/(t)(B) = M(B)

(2.42)

tlr

for all B G £. Proof. We have already established that the family W is a propagator on the space M. If xx G x9^.(r), then using the properties of propagators and the definition of the set x9^(r), we get 9+V

- lim W{t

+ M)/i(B)

- lim W ^

+ hl

T)/i(jB)

(t)(B)

- l i m hio

(

^

+

^W^ /l f)

'

~

W

" ^(*'TM3) ^

"/)ff(f'T);j(g) h

r)M(B)

Propagators:

General

Theory

119

A£tW(t,T)(i(B)=A£tv(t)(B)

=

for all B € £. In addition, the equality limi/(t)(B)

= n(B), B e £, follows

from the definition of the set F£{T). This completes the proof of Theorem 2.10.

D

The equation in Theorem 2.10 is called Kolmogorov's forward equation, or the Fokker-Planck equation. If the Dirac measure Sx belongs to the set D+{r) for every x £ E, and if problem (2.42) is uniquely solvable for all /i € D+(T), then we can recover the transition function P from the formula P(r,x;t,A)

=

W(t,T)St(A).

Let Y be the free backward propagator on the space Lf associated with a transition probability function P. For every r with 0 < r < T, put

2?f*(r)=

f| ^ « W ) . t:r
where the operator A?f(t) is defined by (2.31). Then the following theorem holds: T h e o r e m 2.11 Let Y be the free backward propagator on the space Lf associated with a transition probability function P. Then for every r with 0
£

for all t e

Proof. Um

hlOjE

=Y(T,t)A™(t)f

(2.43)

{T,T).

Let / e f Y(r,t

+

D+'*(T).

h)f-Y{r,t)f h

Then for every /z <E M , _ ^

/ Y{r,t)(Y(t,t

hlOJE

+ h)f - f) h

(2.44) By Lemma 2.1, the cr(Lf, M)-convergence of a sequence hk & Lf to a function h £ Lf is equivalent to the pointwise convergence and the uniform boundedness of the sequence hk in the space Lf'. It is not hard to prove that

„(L?,M)_m"(MXr(M + W - / ) „ r ( r , t ) A , m

(2 . 45)

120

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Next, passing to the limit under the integral sign on the left-hand side of equality (2.44) and using (2.45), we get

J

Y{T,t + h)f-Y{T,t)f H

=

h

JE W

I

m

JE

for all (i G M. It is clear that (2.46) implies (2.43). This completes the proof of Theorem 2.11.

•

Finally, let Y be the free backward propagator on the space L|° associated with a transition probability function P, and let W(t, r ) = Y(T, t)*. Then W is a propagator on the space M (see the discussion before the formulation of Theorem 2.10). For every r with 0 < T < T, consider the operator

K(TMm . J j f c M M , n|0

B e £,

(2.4T)

fl

and denote by D (As_(r)\ the subspace of the space M consisting of all measures fJ. for which the limit in (2.47) exists for all B € £. By the corollary to the Vitali-Hahn-Saks theorem formulated in Subsection 5.5 of the Appendix (see Corollary 5.2), for // £ D (A^_(T)), yl£(r)/i is a Borel measure on E. Put D£J*(t)=

f)

D(A£_{T)).

(2.48)

T:0
Then the following theorem holds: Theorem 2.12 Let Y be the free backward Lf -propagator associated with a transition probability function P, and let W be the propagator on the space M given by W(t,r) = Y(r,t)* where 0 < r < t < T. Then for any t with 0
-^llli

Proof.

= -W{t,r)Ai(r),.

(2.49)

Let ( i £ M . Then we have W{t,r)n(B)=

I P(T,x;t,B)dn(x)

(2.50)

JE

for all B S £. It follows from (2.50) that if a sequence /u^ £ M converges to \x G M setwise, then the sequence W(t, r)fik converges to W(t, T)H setwise.

Propagators:

General

Theory

121

For/iG D£J*{t), we have W(t,

T - fe)/x - W(t, T)H

_

W(t,

T)(W(T,

T - ft)/i -

/x)

Now using the previous remark and passing to the limit as h [ 0 in (2.51), we get (2.49). This completes the proof of Theorem 2.12. •

2.4

Howland Semigroups

Propagators are two-parameter generalizations of semigroups. Conversely, if a propagator is given, then, under certain restrictions, one can define a semigroup by introducing an extra time variable. For instance, let Q be a backward propagator on a Banach space B, and let T > 0 be a positive number. Denote by Lg D ([0,T], B) the space of all bounded Bvalued strongly measurable functions on [0, T] equipped with the norm

11/11^= sup ||/(t)||B. te[o,T]

Definition 2.3 Let Q be a backward propagator on a Banach space B, and let T > 0. The Howland semigroup SQ(£) on the space Ug(\Q,T],B) is defined by SQW(T)

=

Q(T,

(r +1) A T)f((r

+ t)AT),

t £ [0,T],

(2.52)

where f e Lf

([0,T],B).

Lemma 2.2 Lf([0,T\,B).

The family of operators 5g(t) is a semigroup on the space

Remark 2.3 Note that the semigroup SQ(t) is defined on a finite interval [0, T]. If T = oo, then the Howland semigroup is defined by SQ(t)f{T)=Q(T,T Proof.

+ t)f(T + t),

0
0
We will first prove the following two assertions:

(1) If / € Lg'([0,ri,B),then SQ(t)(f) e Lf([0,T\,B). (2) For all s > 0 and t > 0, SQ({S +1) A T) = SQ(S) O SQ{t), and moreover SQ (0) = /, where / stands for the identity operator on the space Lf([0,T\,B).

122

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Let / € L g ( [ 0 , T ] , 5 ) . Then there exists a sequence of simple functions sn : [0,T] -> B such that for all u £ [0,T], \\f(u)-sn(u)\\B -> 0 as n —> oo. Fix £ € [0, T], and define a sequence of simple functions by Sn(r) = Q(T, (r +1) A r ) s „ ( ( r + t ) A T ) where T € [0,T]. Then, for all r E [0,T], l|5Q(t)/(r)-S-n(r)||B = \\Q(r, (r + t)A T)f((r <

\\Q(T, (T

+t)AT)-

Q(T, (r + t)A T)sn((r

+1) A T)\\B

+ t) A T ) | | B ^ B ||/((r + t) A T) - sn((r + t) A T ) | | B - 0

as n —» oo. Therefore, the B-valued function Sq{t)f is strongly measurable. Moreover, l|5gW/(r)||B<||Q(r,(r + i ) A T ) | | B ^ B | | / ( ( r + i ) A r ) | | B , and hence, sup \\SQ(t)f(T)\\B

< co.

T€[0,T]

It follows that 5 Q ( < ) / e Lf([0, T\,B). This establishes condition (1) above. We will next prove condition (2). It is clear that £Q(0)/(T)=Q(T,T)/(T)=/(T).

Moreover, using the properties of backward propagators, we see that for all s€ [0,T] a n d i e [0,T], SQ(s) O SQ(t)f(T)

= Q(r, (T + s) A T) (5 Q (t)/) ((T + s) A T)

= Q(r, (r + S) A T ) Q ( ( T + s) A T, ((r + s) A T + t) A T)

f(((r + s)AT

+

= Q(r, (r + s) A T)Q((T

= Q(T, (T +

8

t)AT) + s) AT,(T

+ t) A T)f((r

+ s + t) A T)f{{r

+ s +1) A T)

+ s + t ) A T ) = 5 Q ((s +1) A T)f(r).

This establishes condition (2). The proof of Lemma 2.2 is thus completed.

•

An element x of the space B can be identified with the constant function / x ( r ) = x, T € [0, T\. By taking this identification into account, we get the

Propagators:

General

Theory

123

following formula connecting the backward propagator Q with the Howland semigroup SQ: SQ{t)fx{T)

= Q(T,(T + t)AT)x

(2.53)

for all t G [0, T], r e [0, T], and x e B. Next, we will discuss Howland semigroups associated with propagators. Definition 2.4 Let W be a propagator on a Banach space B, and let T > 0. The Howland semigroup Sw on the L%>([0, T], 5 ) is defined for all r > 0 by Sw(r)f(t)

= W{t, (t-r)V

0)/((t - r) V 0)

(2.54)

where / G Lg>([0,T],£). Arguing as in the proof of Lemma 2.2, we can show that SW(T) is a semigroup on the space L^([0,T],B). The propagator W is related to the semigroup Sw as follows: Sw(T)fx{t) = W(t,(t-T)V0)x

(2.55)

for all t G [0, T], r G [0, T], and a; G B. Howland semigroups associated with free propagators and free backward propagators admit a probabilistic description. Let P be a transition probability function, and let Xt be a corresponding Markov process with state space (E,£). Recall that in Section 1.5 we discussed space-time processes associated with the process Xt (see Section 1.5 for the definition of the transition function P, the family of measures P(T)a;), and the sample space Cl of the space-time process Xt). Let Y be the free backward propagator on the space Lf associated with the transition function P. As we already know, the backward propagator Y can be expressed in probabilistic terms as follows: K(T, *)/(*) = E r , * / ( * t )

(2-56)

for all 0 < r < t < T, x G E, and / G Lf. In addition, for the Howland semigroup Sy(t) associated with the backward propagator Y, the following probabilistic characterization is valid: SY(t)F(T,x)

= ETiXF ((r + t ) A T , X ( T + t ) A T ) = E ( T , X ) F ( x t )

(2.57)

for all r G [0,T], t G [0,T], x G E, and F G L% ([0,T], L£°). This can be derived from (2.52) and (2.56).

124

Non-Autonomous Kato Classes and Feynman-Kac Propagators

Now let P be a backward transition probability function, and let XT be a corresponding backward Markov process. In Section 1.5, we constructed a time-homogeneous transition probability function P and the family of measures pC- 1 ). The function P was used as the transition function of the space-time process XT. Let U be the free propagator on the space Lf corresponding to the backward transition probability function P. Then the following formula is valid: U{t,T)f{x)=&xf(xT)

(2.58)

for all 0 < r < t < T, x 6 E, and / G Lf. For the Howland semigroup Sir (T) associated with the propagator U, we have Sv(T)F(t,

x) = E*'XF ((t - r) V 0, X ( t _ T ) v 0 ) = E{t'x)F

(xT)

(2.59)

for all T G [0,T], t G [0,T], i G E, and F G L ^ ([0,T],Lf). This follows from (2.54) and (2.58).

2.5

Feller-Dynkin Propagators and the Continuity Properties of Markov Processes

Let us recall that in Section 2.2 we defined the following spaces of continuous functions on the space E: the space BC of all bounded continuous functions on E; the space BUC of all bounded uniformly continuous functions on E, and the space Co of all functions from BC which vanish at infinity. Definition 2.5 A backward BC-propagator is called a backward Feller propagator. A backward Co-propagator is called a backward Feller-Dynkin propagator. If a backward L|°-propagator Q is such that Q(T,t)eL(L?,BC) for all 0 < r < t < T, then it is said that Q possesses the strong Feller property. If a backward Lf -propagator Q is such that Q(T,t)eL{Lf,BUC) for all 0 < r < t < T, then it is said that Q satisfies the strong BUCcondition.

Propagators:

General

Theory

125

Remark 2.4 If Q is a backward L°°-propagator, then one can replace the space Lf by the space L°° in the definition of the strong Feller property and the strong B[/C-condition. It is not hard to see how to define forward Feller and Feller-Dynkin propagators, and also propagators possessing the strong Feller property, or satisfying the strong i?£/C-condition. It this section, we begin the study of Markov processes associated with free backward Feller-Dynkin propagators. Let P be a transition probability function, and let Y be the corresponding free backward propagator. Our first goal is to prove that if Y is a strongly continuous backward FellerDynkin propagator, then the class of all Markov processes associated with P contains a process Xt such that its sample functions are right-continuous and have left limits. In the following sections, we will show that the process Xt possesses additional properties. Given a transition probability function P, we start with the standard realization Xt of a corresponding Markov process on the probability space (n,F[,FTtX} where ft = £ l 0 ' r l . It is denned by Xt(u) = u(t) where u> £ Ct. By Q, will be denoted the space of all E-valued functions defined on the interval [OjT], which are right-continuous and have left limits in E. Then we have Q C ft. Put Xt{w) = u(t), w £ ft, t £ [0,T], and let J J , 0 < T < t < T, be the er-algebra generated by Xs with T < s < t. For every r € [0, T) and x £ E, denote by P T|X the probability measure on TT determined by FT,X [Xtl £Bu...,Xtn£Bn)=

PT,X [ x t l 6 B i

Xtn £ £ „ ] .

(2.60)

Here r < h < • • • < tn < T and Bj £ £, 1 < j < n. Let us denote by !F^+, t £ [T,T), the cr-algebra defined by Tl+=

Q

T[.

(2.61)

t<s
Theorem 2.13 Let P be a transition probability function such that the corresponding free backward propagator Y is a strongly continuous backward Feller-Dynkin propagator. Then the processes {Xt^T'ljr,Pr,x) and ( Xt, Ft •> ^T,X ) are stochastically equivalent. Remark 2.5 It follows from Theorem 2.13 that the process (Xt, .FtT+, PT,X) is a Markov process with P as its transition function. Moreover, all the sample paths of Xt are right-continuous and have left limits.

126

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

It also follows from the proof of Theorem 2.13 that for every w G O , the set {Xt(u>) :t£ [0, T]} is relatively compact in E. Proof.

For every function f £ Co and every e with 0 < e < T, put Ne(f)(t,z)

= f

Y(t,s)f(z)ds

Jt+e

where 0 < t < T - e and z € E. Lemma 2.3 Fix r with 0 < r < T and x G E. Then for every nonnegative function f £ Co and every e with 0 < e < T — T, the process t — i > Nc(f) (t,Xt), t e [T,T — e], is a supermartingale with respect to the filtration Tl and the measure PT>X. Proof.

It is clear that the process t i-> Ne(f) (t,Xt\

filtration T[. implies that E.r,x

is adapted to the

Moreover, for r < t\ < ti < T — e, the Markov property

Ne(f) (t2,Xt2) | Ft] =E t i ] j f t i [Ne(f) (t2,Xt2)] .

(2.62)

Since Y is a backward propagator, E ti,Z

Ne(f)(t2,Xt2)}=

= /

[

Y{t2,s)f(xt2)

E-

Y (h, t2) Y (t2, s) f(z)ds=

f

Jt2+e

<[

Y (h,s)

ds f(z)ds

Jt2+e

Y(t1,s)f(z)ds

= Ne(f)(t1,z)

(2.63)

for all z € E. Finally, (2.63) with z = Xtl and (2.62) show that Lemma 2.3 holds. • Let us continue the proof of Theorem 2.13. It follows from Lemma 2.3 and Theorem 1.10 that for every f £ Co and T £ [0,T), there exists a set A r G T such that PT)X (AT) = 0, and for every integer n > 1, the process Z?(f)(uj) = nJ

n

Y(t,s)f(xt(u))ds,

T
has left and right limits, tonZr(/)(w) sjt.sGQ

and

Urn Z ? ( / ) ( w ) , sit,seQ

(2.64)

Propagators:

General

Theory

127

for all u G fi\Ar and £ G [T,T - i ] . The set A r does not depend on n and i. Moreover, AT can be chosen independently of / € Co- Indeed, let /& be an everywhere dense countable subset of Co- Then it is not hard to see that there exists an exceptional set A r G f with PT)X(A) = 0 such that the limits in (2.64) with / replaced by fk exist for all w G fi\AT, t e [T,T- ^ ] , and k > 1. It is also easy to prove that if a sequence gk G Co converges uniformly to a function g, then we have Y(t,s)gk(z)ds->n Y (t, s) g (z) ds •/ uniformly in z as k —> oo. It follows that the exceptional set A r can be chosen independently of / G Co, n > 1, and t with T
J

lim

sup

n->OO

zG£,0
Y(t,s)f(z)ds-f(z)

0

for every / G Co- Hence, there exists a set AT G T for which P T X(AT) = 0, and for every f £ Co the limits lim f(xs(u))

and

lim f (xJu))

(2.65)

exist for all ui G fi\A T and all t with T < t < T. The exceptional set AT does not depend on / G Co and t G [r, 71]. The following well-known fact concerning nonnegative supermartingales will be used in the proof of Theorem 2.13. L e m m a 2.4 Let (fi, F, Tt, P) be a filtered probability space, and let Zt be a nonnegative Tt-supermartingale. Denote by Q a countable dense subset of the interval [0, T]. Then Zt > 0,

inf Zs = 0 = 0 s
for all t G [0, T]. Proof. Fix t with 0 < t < T, and let Q5 = {sj < • • • < s ^ } increasing sequence of finite subsets of the set Q D [0, t] such that

Qn[0,t] = ( j Q i .

be an

128

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

For all j > 1 and n > 1, put Tj>n = m m

|4:

i
In addition, if LJ G

min

fi\

Z, < —

then we put Tj,„(w) = T. Next, using the properties of supermartingales and the Tsi -measurability of the event < r^n = s°k \, we get E Zu

min Zi < - = E Zt, TjyTl < T, ZTjn < — l
"k

n

= fc=i E E •^t, ''"j.n — S , Z k

= EE E

Zt I ^

Tjn

<

j T j . n — Sk, Z j < —

rrij

<Efc=iE

Z.j i Tin — Sk, Z j < — "k J k n

< - . (2.66) n

Passing to the limit in (2.66) as j —» oo and then as n —> oo, we get E Zu

inf Zs = 0

0.

(2.67)

s
Now it is clear that Lemma 2.4 follows from (2.67).

•

Let us return to the proof of Theorem 2.13. Fix a countable dense subset Q of the interval [0, T], and for every r G [0,T), consider the event TT consisting of all w G CI for which there exists r G Q n [r, T) such that the set Xs(o»), s G Q n [r, r], is not contained in any compact subset of E. Then the following lemma holds: Lemma 2.5 Let f be a strictly positive function from CQ. Then for every T with 0 < r < T, TT=

\J

i w :I

reQn[r,T) I

inf

/

s€Qn[r «n[r,r],/s

Y(r,u)f(xr{u))du>0,

^r

Y(s,u)f(xs(w))du = o\. v

'

(2.68)

Propagators: General Theory

Proof.

129

Our first goal is to prove that for every r G [r, T) and z £ E, /

Indeed, suppose that / /

Y(r,u)f(z)du>Q.

(2.69)

Y (r, u) f(z)du = 0. Then the equality

Y (r, u) f(z)du =

Jr

du Jr

f(y)P (r, z; u, dy), JE

and the strict positivity of the function / imply that for almost all u € [r, T] with respect to the Lebesgue measure, we have P (r, z; u, E) = 0. This contradicts the definition of transition probability functions. Therefore, inequality (2.69) holds. Next, fix a sequence sn € [T,T] such that lim sn = s where r < s < T. We will show below that for a sequence zn G E, the following two conditions are equivalent: (1) There exists a compact subset C C E such that zn £ C for all n > 1. (2) The inequality inf /

Y(sn, u)f(zn)du>0

(2.70)

holds. Indeed, if condition (1) does not hold, then there exists a subsequence znk of zn such that lim g(znk) = 0 for all g G Co- It follows from the strong fe—>oo

continuity of Y on Co and from the condition lim sn = s that f

Y(s,u)f(znk)du-

Js

f

Y(snk,u)f(znk)du^0

(2.71)

JSnk

as k —» oo. Next, the strong continuity of Y on Co gives lim f

Y(s,u)f(znk)du

= 0.

(2.72)

By (2.71) and (2.72), we see that inequality (2.70) does not hold. Hence, the implication (2) =*• (1) is valid. On the other hand, if condition (1) above holds, and condition (2) does not hold, then there exists a sequence nk such that lim znk = z where k—*oo

130

Non-Autonomous

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z € C; moreover, lim /

Y(snk,u)f(znk)du

= 0.

(2.73)

k

^°°Jsnk

Since Y is a strongly continuous backward propagator on Co, and (2.69) holds, we have lim /

Y(snk,u)f(znk)du=

Y (s,u) f{z)du > 0.

k—>oo /„

/.

This contradicts (2.73). Therefore, the implication (1) =>• (2) holds. The proof of Lemma 2.5 is thus completed.

•

It follows from Lemma 2.3 with e = 0, Lemma 2.4, and Lemma 2.5 that for all r G [0, T) and all cc G £ , the equality PT,X [TT] = 0 holds. Therefore, PT,X fi\rr = 1. Moreover, for all w G f2\r T and r G Q ("1 [r,T), there exists a compact set C C E such that Xj(w) G C for all s G Q n [T, r]. The set C depends on r and u>. Given r G [0,T), put

S r = n \ (AT u r T ), where A r is the complement of the event consisting of all u> G tl for which the limits in (2.65) exist for all f € Co and all t with T
lim XJw)

(2.74)

sj.t,«6Q

exist for all t with T
of the first limit in (2.74) is similar. Let us consider the subset Q of the set f2, consisting of all w G E^°'T^ such that the following conditions hold: the limits lim ui(s) and lim LJ(S) *lt,s€Q

sRsGQ

exist for all t G [0, T), and for every r G Q n [0, T) there exists a compact subset CT{u)) of E for which w(s) G C r (w) where s G Q n [0, r]. It is not

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131

difficult to show that for all r and x, (2.75)

n

where P * x denotes the outer measure generated by the measure P r , x - Indeed, for every r G [0,T) and x G E, the set fi contains the set £lT,x, consisting of those w G fir for which w(s) = x for 0 < s < T. Since fiT 1, we have P* fiT = 1. Now it is clear that (2.75) holds. Therefore, one can restrict the probability space structure from Q, to O. The resulting probability space is denoted by (Q, J7, PT)X J. Let us define a stochastic process on Q, by lim Xs, Xt = { »lt,st

H0
XT,

Then, it is not difficult to see that the process Xt is right-continuous and has left limits on the interval [0,T). It is also clear that the process Xt, t G [T,T), is .F t r + -adapted. Here

*T+

n %• t<.s
Lemma 2.6 The process lXt,^+,PT>x) sition function P.

is a Markov process with tran-

Proof. Let r < s < t < T, and choose sn G n (s, t) and tn G Q n (t, T) with sn I s and tn [ t as n —> oo. Let .4 G ^ J + . Then A G ^ J n for all n > 1, and since X u is a Markov process,

Er,x [/ (*t„) XA] = Er,« [ESn,^sn [/ ( X )

XA

^T,x[Y(sn,tn)f(^XSn)xA]

(2.77)

for every f £ Co- It follows from the strong continuity of Y on Co and from (2.77) that E. ;x

[f (Xt) XA] = Kx [Y (S, t) f ( £ ) X^

132

Non-Autonomous Kato Classes and Feynman-Kac Propagators

for all A G J~l+- Here we used the fact that ET]X is the restriction of ET>I to ft. Therefore, E.T,X

'f(xt)\^+]=Y(s,t)f(xs)

(2.78)

for all / G Co- In the case where t = T, (2.78) is also true. Indeed, we may assume tn =T for all n > t, and use the equality XT = XTFor all x G E and 0 < r < t < T, we have E T, , [/ (Xt)}

= ET)X [ s U m Q / ( * . ) ] =

lim

ETtX\f(xs)]

s

H m Q I r , x [/ ( x s )

lim

r(r,S)/(x)

sit,s€
Y(T,t)f(x)=ET,x[f(xt)'

(2.79)

Here we used the strong continuity of Y. In the case where t = T, (2.79) also holds since XT = XT- Next, using Urysohn's Lemma, the monotone class theorem, and approximating bounded Borel functions by simple functions, we see that (2.78) and (2.79) hold for all bounded Borel functions / on E. Consequently, the process Xt is an P riX -Markov process with respect to the filtration !F[+, r
and •

Lemma 2.7

The process ( Xt, Ft+i ^T,X ) is a modification of the process

(Xt,T[,FT,x);

that is, for every 0 < r < t < T and x G E, the equality

P r x Xt = Xt

= 1 holds.

Proof. Let / and g be functions from the space Co, and fix x G E, t G [0, £], and T with 0 < T < t. Let sn be a sequence in Q n (t,T) such that sn 11. Then the Markov property of the process Xu, 0
Er,x [/ (*.„) 9 (Xt) = Er>x Er,x [/ (Xsn) | fl] 9 (Xt) = ET|X K~xt [/ {*.„)] 9 (Xt)} = E r x \Y(ttsn)f(Xt))g(Xt)].

(2.80)

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General

Theory

133

Passing to the limit in (2.80) as n —> oo and taking into account that Y is strongly continuous on Co, we obtain Kx

[/ ( * t ) 9 ( * t ) ] = ^m%,x

= lim^^x

[f (*«„) g

(Xtj

[(Y (t, sn) f ( X t ) ) g {xt)\

= ET,X [/ (Xt) g ( X t ) ] = ET,X [/ (Xt) g ( * t ) ] •

(2.81)

Next, formula (2.81) with g = f and the stochastic equivalence of the processes Xt and Xt (see Lemma 2.6) give

f{xt)-f(xt) f (X t ) 2 - 2ET;X [/ (X t ) / ( x t ) = Er

x

f [Xtf

- 2ET,X [/ (xt) f (Xt)

/ ( X t ) 2 ] = 0 (2.82)

for all / G Co, 0 < r < f < T, and a; 6 £ . Let //c G Co with HAU^ < 1, k > 1, be a sequence such that the closure of its linear span coincides with Co- For all x, y G E, put oo

p(a;,y) = ^ 2 - f c | A ( a : ) - / f e ( j / ) | . Then p is a metric on E x E generating the same topology as the metric p. It follows from (2.82) that ET,x[p(xt,Xt)] = 0 for all 0 < r < t < T and x G E. This completes the proof of Lemma 2.7.

(2.83)

•

Now we are ready to finish the proof of Theorem 2.13. By equality (2.60) and Lemma 2.7, PT,X [Xtl eBlt...,XtnGBn]=

Pr,x [Xtl G £ i , . . . , X t „ G £ „ ]

(2.84)

for all T < ti < • • • < tn < T and Bj G £ with 1 < j < n. It is not hard to see that Theorem 2.13 follows from (2.84). •

134

2.6

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

Stopping Times and the Strong Markov Property

In this section we discuss the Markov property with respect to random times. In the next definition, an important family of random times is introduced. Definition 2.6 Let {Q.,Q,Qt,W), r < t < T, be a filtered probability space. A function S : Q —» [T,T] is called a ^-stopping time if for every t e [r, T], the event {S < i) belongs to the cr-algebra Qt- If T = oo, then it is assumed that S takes values in the extended real half-line R+. We will often write "a stopping time" instead of "a C/f-stopping time" if the filtration is fixed. It is not hard to see that 5 is a stopping time if and only if the process t H—> X{S
u> € ft.

Definition 2.7 Let (Q,£,£ ( ,P), T < t < T, be a filtered probability space, and let 5 be a St-stopping time. The cr-algebra Qs is defined as follows: An event A £ Q belongs to Qs if and only if A n {5 < t) £ Qt for every t G [r, T]. It is not hard to see that Qs is a cr-algebra. The cr-algebra Qs is usually interpreted as the information contained in the process Xt before or at time 5. If S\ and 52 are stopping times such that S\ < S2, then QSl^Qs2-

(2.85)

Indeed, if A G QSl, then A n {5 2 < t} = A n {Si < t} n {5 2 < t}, and since the events An {Si < t} and {S2 < t} belong to the cr-algebra Qu the event A n {S 2
Propagators: General Theory

Gg* and G$l

are

135

defined as follows:

gll=a(xs.:S'ESil) and GSsl=o-(s',Xs,:S'£Sssl). Put Gs = Gs and Gs = G%- It is not hard to see that for all stopping times Si and S2 with Si < S2 the following inclusion holds: £f* C Sf 1 . On the other hand, it is not clear whether Gs C Gs- However, if the process Xt is progressively measurable, then Gs C Gs C Gs-

(2.86)

The inclusions in (2.86) follow from the fact that S is C/s-measurable and from the following lemma: Lemma 2.8 Let (Xt,Gl), 0 < r T.\t is clear that the resulting process is progressively measurable on [0,00). Consider the stopped process XSM, t £ [T,T\. For every u e [T,T] and B £ £, we have {XSAt e B } n { S A f < 4 = {XSA(tA„) £B}f){SAt
(2.87)

Since the composition of measurable functions is measurable, the random variable XsA(tAu) 1S ^tA«/^- m e a s u r a t>le. It follows that the event {^SA(tAu) € -B} belongs to the cr-algebra GJAW a n d therefore it also belongs to the cr-algebra GZ,- In addition, the event {S At
136

Non-Autonomous Kato Classes and Feynman-Kac Propagators

Our next goal is to define and study cr-algebras related to the future behavior of a stochastic process Xt after a stopping time S, and also aalgebras containing the present information about Xt. It seems promising to choose the cr-algebras Qj. and GT to represent the future of the process after the stopping time S. However, this is not a good choice, if we want to use these cr-algebras in the formulation of the strong Markov property (see Definition 2.14 below), since the events belonging to the cr-algebras GT and GT m a v depend on the past history of the process before the stopping time S. It is easy to see that GT C GT- O*1 * n e other hand, it is not clear whether these cr-algebras coincide, or if the cr-algebras in (2.86) coincide. Note that the stopping time S is not necessarily measurable with respect to the cr-algebra a (Xs). As a result, we can choose the cr-algebra a (Xs) or the cr-algebra a (S, Xs) as a storage of information contained in the process Xt at the stopping time S. Next, we turn our attention to cr-algebras representing the future of a stochastic process Xt after the stopping time S. One possibility to define such a cr-algebra is to imitate the definition of the cr-algebra Gs representing the past before S. Let us first note that if a stochastic process Xt is progressively measurable, then Gs=

f^

{AeGT--

An{S
(2-88)

t:0
Indeed, equality (2.88) follows from the equality

{AeGT-

An{S
= {AeGT-

An{S
XSM) VGt}, 0 < t < T,

and from the fact that for progressively measurable processes, tr{SAt,XSM)cg°SMC$ (see Lemma 2.8). For a stopping time S, define the family of events GS by the following: AeGS <^>An{S
(2.89)

for all t with 0 < t < T. It is not hard to see that the family Gs in (2.89) is a cr-algebra. Note that an equality similar to the equality in (2.88) does not hold for this cr-algebra. Indeed, the event {S < t} does not necessarily belong to the a-algebra GT- The next equality shows that the cr-algebra GS

Propagators:

General

Theory

137

is not large enough to store the information about the process Xt between S and T: gs =
(2.90)

The equality in (2.90) can be obtained as follows. By (2.89) with t = T, we see that if A G Qs, then Aea(S,Xs,XT). Moreover, if A € a (S, Xs, XT), then A£a{S,Xs)VQT for all t e [0, T]. On the other hand, we have {S
An{St>S}£a(S,Xs)vgT.

(2.91)

Condition (2.91) is equivalent to the following condition: for all ti and t2 with 0 < tx < t2 < T, {S' >t2>t1>S}ea(S,Xs)V

9%.

(2.92)

Indeed, this equivalence can be easily established by taking into account the equality

{S' >t2>h>s}

= {h > s} n {s'

>t2>s}.

Example 2.1 Let S be a stopping time, and let a random variable S' be defined by S' = <j> (S, Xs) where <j> is a Borel function on [0,T]xE. Assume that S' is a stopping time such that S < S'. Then we have 5" £ N(S). This follows from the fact that the event {S < t < S'} belongs to the cr-algebra a (S, Xs)- As an example, we can take S' = (a + S) A T where a > 0. Another example is given by S' = S V p where 0 < p < T.

138

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Definition 2.10 A stopping time S is called a terminal stopping time if for every pair of fixed times (£1,^2) with r < ti < t?, < T, the event {ti < S < £2} belongs to the cr-algebra GilRemark 2.6 Not all stopping times are terminal. For instance, the stopping times (a + S) A T, where a > 0 and S is a given stopping time, are not necessarily terminal stopping times. E x a m p l e 2.2 Let S and 5" be stopping times with T < S < S' < T. Assume that S' is a terminal stopping time in the sense of Definition 2.10. Then S' G M(S). Indeed, {S
= {S
{S' >t}ea

(S, Xs) V QlT.

Lemma 2.9 Let Si, S2, and S3 be stopping times such that Si < S2 and Si < S3. If S2 G M (Si) and S3 G TV (Si), then S2 V S3 G M (Si) and S2AS3eM(Si). Proof.

Lemma 2.9 for the stopping time S2 V S3 follows from

{S 2 V S3 > t > Si} = {S 2 > t > Si} U {S3 > t > Si} G a (SuXSl)

V $,.

The proof of Lemma 2.9 for the stopping time S2 A S3 is equally simple. • The next definition will be important in our presentation of the strong Markov property (see Theorems 2.17 and 2.20 below). The cr-algebra Q^y in Definition 2.11 is a good candidate to represent the future information about the process Xt after the stopping time S. Definition 2.11 Let S be a stopping time. Then the a-algebra Q^'v is defined as follows: Q^y = a (S V p, XSvP • 0 < p
The c-algebra QT' is a special case of the : S'GM

and QT (S)),

(

: S'

>S}.

' are defined as follows:

G?{S)

= a (S\ Xs> : S' G M

(S)). (2.93)

Propagators:

General

Theory

139

It is clear that if S is a stopping time and M = {SV p : 0 < p
Let S be a stopping time. Then the cr-algebra Q^^

@pS)=a(S,,Xs>

is

: S' &N (5).).

For the
Let S be a stopping time. Then

gs c QST* c g*(s) c g$.

(2.94)

Proof. The first inclusion in (2.94) follows from equality (2.90). Since SV pe Af(S) for all p e [0, T], we get the second inclusion in (2.94). The third inclusion is obvious. This completes the proof of Lemma 2.10. • It is interesting to notice that for a right-continuous stochastic process Xt on (fi, F) and the two-parameter filtration F[ generated by the process Xt, the second and third cr-algebras in (2.94) coincide. Lemma 2.11 Let Xt be a right-continuous stochastic process on (Cl,F) with state space E, and let S be a F\-stopping time. Then Ql»=Q$W.

(2.95)

Proof. The inclusion Q^,'v C QT follows from Lemma 2.10. In order to prove the opposite inclusion, let us consider the family Tit, t € [0,T], of vector spaces of random variables on O defined as follows. For every t e [OjT1], the vector space Ht consists of all bounded random variables F on Q such that F is measurable with respect to the cr-algebra a (S, Xs) V a (Xp : t < p
140

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

Ht contains all random variables of the form n

fo{S,Xs)Y[fj{pj,XPj), 3= 1

where fj : [ 0 , T ] x £ - * R , 0 < j < n , are bounded Borel functions, and t < pi < • • • < pn < T. By the monotone class theorem, Ht contains all bounded random variables which are measurable with respect to the cr-algebra a (S, Xs) V a (XSVp • t t > S} belongs to the cr-algebra a(S,Xs)Va(Xp:

t
It is not hard to see that the event {S' > t > S} belongs to the cr-algebra a (S, Xs) V<7(SVp, XsvP

:t
Hence, the same is true for the event {S' > t} = {S' > t > S} U {S > t}. It follows that the stopping time S' is measurable with respect to the cralgebra cr (S V p, XSvp •• 0 < p < T). By Theorem 2.20 (this theorem will be obtained below), we see that the state variable Xs> is also measurable with respect to the cr-algebra a(SVp, XsvP -0 < p
gfs) c gp holds. This completes the proof of Lemma 2.11.

•

The next definition concerns the strong Markov property in the case of a family of stopping times. Definition 2.14 Let M be a family of stopping times that is closed with respect to the operations A and V. A stochastic process (Xt, 0, Qt, P) with state space E is called a strong Markov process with respect to the family M provided that the following conditions hold: (1) The process Xt is progressively measurable. (2) For every pair of stopping times Si, S2 € M with 52 > S\ and every bounded Borel function / on [0, T] x E, the equality E [f(S2,XS2)

I gSl]=E[f(S2,Xs2)

\a(SuXSl)]

(2.96)

Propagators:

General

Theory

141

holds P-almost surely. The progressive measurability of the process Xt is assumed in Definition 2.14, because this implies the measurability of X$ with respect to the aalgebra Gs (see Lemma 2.8). There are more versions of the strong Markov property with respect to the family M, namely E [f(XS2)\G3l]=E[f

(XS2) | o (XSl)]

P-a.s.

(2.97)

for all bounded Borel functions / on E, or E f(S2,XS2)

| QSl] =E[f{S2,XS2)

\o-(SuXSl)]

P-a.s.

(2.98)

for all bounded Borel functions / on [0, T] x E, or E [f(XSa)

\gSl]=E

[f(XS2)

\a(S1,XSl)}

P-a.s.

(2.99)

for all bounded Borel functions / on E. Recall that it is assumed in equalities (2.97), (2.98), and (2.99) that S 2 > Si. The next theorem provides several equivalent conditions for the validity of the strong Markov property with respect to the family M (see Definition 2.14). This theorem is related to Lemmas 1.2 and 1.20. Recall that for S G .M, we put M (S) = {S' G M : S' > S}. Theorem 2.14 Let M be a family of stopping times that is closed with respect to A and V, and let Xs be a progressively measurable stochastic process on (fi, T, P) with state space (E, £). Then the following are equivalent: (1) For all pairs Si e M. and S2 € M. with S2 > Si and all bounded Borel functions f on [0, T] x E, the following equality holds: E [f (S2, XS2) I QSl] = E [/ (5 2 , XS2) I a (Si,XSl)} (2) For any S € M and any bounded gT F, the equality

(

F-a.s.

(2.100)

-measurable random variable

E[F\gs]=E[F\a(S,Xs)] holds P-a.s. (3) For any S G M, any bounded gs-measurable random variable G, and any bounded real-valued 0T -measurable random variable F, the equality E[GF]=E[GE[F\a(S,Xs)}]

142

Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

holds. -M(S) (4) For any S £ M, A £ Gs, and B £ QT y ', the equality P [A n B I a(S, XS)] =F[A\

a(S, Xsj] P [B \ a(S, Xs)]

holds P-a.s. Proof. (1) = • (2). Suppose that condition (1) in Theorem 2.14 holds. The most important step in the proof of the implication (1) = > (2) is to show that the equality E[F\gs]=E[F\

(2.101)

a(Xs)]

holds P-almost surely provided that m

F=

Ylfj{Si,XSi)

(2.102)

3= 1

where for any 1 < j < m, fj is a bounded Borel function on [0, T] x E, and Sj is a stopping time from M. {S), satisfying S < S\ < ••• < Sm < T. We will obtain equality (2.101) using the method of mathematical induction. Since we assumed that condition (1) holds, equality (2.101) is satisfied for m = 1. Next, assume that (2.101) is true for a given integer m > 1 and for all finite sets {fy :l<j <m\ from M (5) with 5 < Si < S2 < • • • < Sm < T. Let Si < • • • < Sm+i from M. (S), and put

be an increasing family of stopping times

H = cr [QsiS\,X§i,...,Sm,^sm)

•

Then, using the tower property of conditional expectations, we get m+l

E m

JJfj (Sj,X§.) I fm+l [Sm+l,Xgm+iJ E

[[fj ( s , j . ^ 5 j E fm+i [sm+i,Xgm+ij

I Gs

| n

Gs

(2.103)

Propagators: General Theory

143

It follows from condition (1) in Theorem 2.14 and from the inclusions Xo C H C Qa that E :E

(2.104)

Therefore, (2.103) gives m+l

E

ii/»-(^^)i^ 3=1

= E 3= 1

(2.105) L e m m a 2.12 that

There exists a bounded Borel function g on [0, T]x E such

E fm+i (sm+i,Xsm+i)

| a (5m,X§mjJ = g

[Sm,Xgmj

Proof. Lemma 2.12 follows from the Radon-Nikodym theorem. Indeed, consider the following Borel measures: (J-i {B) = E fm+i

ISm+i, Xgm+1 J , {Sm, Xgm J e B

and ^(B)=¥[(sm,X-Sm)eB where B belongs to the Borel cr-algebra of [0,T] x E. It is easy to see that the measure [i\ is absolutely continuous with respect to the measure /i2- Denote the Radon-Nikodym derivative of the measure ^i with respect to the measure ^2 by g. Then the equality in the formulation of Lemma 2.12 holds. Note that H2 is the image measure of the measure P under the mapping ^Sm,XgmJ. This completes the proof of Lemma 2.12. •

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Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Let us return to the proof of Theorem 2.14. It follows from Lemma 2.12 and equality (2.105) that m+l

E I m

E I I U (Shx~s)) 9 (Sm,XSm)

| Gs

(2.106)

j=i

Applying the induction hypothesis to the right-hand side of (2.106) and using (2.104) and the tower property of conditional expectations, we get m + lL n-t-

E

n

I I fi (§> X~Si) 9 (Sm, X~Sm) I a (S, XS) 3=1

E I J fj [Sj,Xg. j E / m + i [Sm+i,Xgm+i =E

j | a {Sm,X§mJ

\a(S,Xs)

m

T{ f3 (sjtXSj)E[fm+1

( 5 m + l l X § m + i ) | n] |

a(S,Xs)

m+l

:E E

UfifaXsJlH

v(S,Xs)

3=1

m+l

Ylfi(sjtXh)\a{S,Xs)

(2.107)

3=1

This establishes condition (2) in Theorem 2.14 in the case where the random variable F is given by (2.102). Our next goal is to prove the equality in (2) in the case where F = X {(~SuXSi)eBl}x---xX{Csm,XsJeBmy

(2.108)

In (2.108), S i , . . . , Bm are Borel subsets of [0, T] x E and Si,..., Sm are stopping times from the class M (S). The sequence of stopping times Si, 1 < i < m, is not necessarily increasing. For 1 < k < m, define an increasing sequence of stopping times S'k G M (S) by S'k = m i n { 4 V • • • V Sjk : 1 < ji < • • • < jk < m} .

Propagators: General Theory

145

Let n be a permutation of the set { 1 , . . . , m}, and define the event A„ by

^ = {(%^a.) eBl

(^^O6*"}-

Then it is not hard to see that {(SUX~S1)

e B1,...,{srn,X~Sm)

6 Bm)

= (jAr-

Next, using the inclusion-exclusion principle, we get F =

X

{{~SuX§i)eBl}X---XX{(~sm,x~sJeBm}

= 2 ^ XA„ if

2_^

XA„nA„,

7r, 7r':7T^7r'

+

Y.

XA.rv^ru,,,, - • •' •

(2.109)

Since S^ is an increasing sequence of stopping times, (2.107) and (2.109) give E[F\gs]=E[F\a(S,Xs)], where F is defined by F = X

x

{{~SuXSi)eBl}

---xX{(sm,x§jEBm}-

It follows from /»00

G

=

/ Jo

/-DO

X{G+>A}dA- / Vo

X{G->x}dX

(2.110)

that condition (2) holds for all functions F such that m

F = l[fj(sj,X~Sj). j=i

Finally, using the monotone class theorem, we obtain condition (2) as it is formulated in Theorem 2.14. (2) = » (3). This implication follows from the properties of conditional expectations. (3) = > (4). Suppose that condition (3) in Theorem 2.14 holds, and let A £ Qs, B €

146

Non-Autonomous

§™{S\ and D e a(S,Xs). F = XBXD, w e obtain E

[XDXAXB]

Kato Classes and Feynman-Kac

Propagators

Then, applying condition (3) with G = XA and

=E

[XAE [XDXB

\ <* (S,Xs)]]

= E[XDXAE[XB\
(2.111)

Therefore, E [XAXB

I a (5, Xs)]

= E [ X A E [XB | * (S, Xs)]

= F[A\a(S,

Xs)] F[B\a

\ o (S,

Xs)]

(S, Xs)] .

(2.112)

This implies condition (4) in Theorem 2.14. (4) => (1). Suppose that condition (4) in Theorem 2.14 holds. Then it is not hard to show that (2.111) is valid. Let Si G M and S2 £ M. be stopping times such as in condition (1). Taking D = Q in (2.111) and using (2.110) with G = f (S2, Xs2), we obtain E

[XA!

(5 2 , XS2)] = E

[XAE

[/ (5 a , X 5 2 ) | a (Si, X S l )] ] .

It follows from the definition of conditional expectations that E [f(S2,XS2)

I GSl] =E[f(Si,XSt)

\a(SuXSl)]

.

This implies condition (1) in Theorem 2.14. The proof of Theorem 2.14 is thus completed.

•

The next theorem concerns the strong Markov property with respect to the cr-algebra Qs- Theorem 2.15 below contains an extra condition (condition (2)) in comparison with Theorem 2.14. The reason why we did not include this condition in the formulation of Theorem 2.14 is that we do not know whether Qs = QsTheorem 2.15 Let M be a family of stopping times that is closed with respect to A and V, and let Xs be a progressively measurable stochastic process on (fi, T, IP) with state space (E, S). Then the following are equivalent: (1) For all stopping times Si £ M and S2 £ M with Si < S2, E \f(XS2)

\gSl]=E

[f(XS2)

I a (XSl)]

F-a.s.

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(2) For all stopping times S G M and S £ M with S < S, any finite set of stopping times Si, 1 < i < n, such that Si < S for all i, and any bounded Borel function f on E, the equality E [f (XS) | a (XSl ,...,XSn,

XS)} = E [/ (Xs) |
holds P-a.s. (3) For any stopping time S € M. and any bounded QT random variable F, the equality E F\Gs

=

(

-measurable

E[F\a(Xs)]

holds P-a.s. (4) For any stopping time S G M., any bounded Qs -measurable random variable G, and any bounded QT ^ -measurable random variable F, the equality E[GF]=E[GE[F\a(Xs)}] holds. (5) For any stopping time S £ M., the equality P [A n B I a(Xs)]

=F[A\

holds P-a.s. for all A e Qs and B £

<J(XS)} P [B \ a(Xs)] g^(s).

Proof. We will only prove the implications (1) =>• (2) and (2) = » (3). The remaining implications can be obtained as in Theorem 2.14. (1) = • (2). Suppose that condition (1) in Theorem 2.15 holds, and let S, S, Si,..., S„ be such as in condition (2). It is not hard to see that condition (2) follows from condition (1) applied to the pair S and S and from the following inclusions:

a(Xs)Ca(XSl,...,Xsn,Xs)cgs. (2) = » (3). Suppose that condition (2) in Theorem 2.15 holds. Then, reasoning as in the proof of the implication (1) =>• (2) in Theorem 2.14, we see that the equality E [F \o-(XSl,.

..,XSn,Xs)]

=E[F\


(2.113)

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Non-Autonomous Koto Classes and Feynman-Kac Propagators

holds for any G? -measurable random variable F. Next, replacing the cr-algebra cr (Xst,..., Xs„, Xs) by the cr-algebra Gs in formula (2.113) and using the monotone class theorem, we get condition (3) in Theorem 2.15. This completes the proof of Theorem 2.15. • Remark 2.7 By using the inclusion-exclusion principle as we have already done in the proof of the implication (1) =»• (2) in Theorem 2.14, we can show that the following condition is equivalent to conditions (l)-(5) in Theorem 2.15: (2') For any pair of stopping times S € M. and S £ A4 with S < S, any increasing finite sequence of stopping times {Si : 1 < i < n} such that Si < S for 1 < i < n, and any bounded Borel function / on E, the equality E [/ (XS) \a(XSl,...,

XSn,Xs)]

= E [/ (X§) | a (Xs)]

holds P-a.s. An assertion, similar to Theorem 2.15, holds for the cr-algebras Gs and GT • The proof of this fact is similar to the proof of Theorems 2.14 and 2.15, and we leave it as an exercise for the reader. Theorem 2.16 Let Ad be a family of stopping times that is closed with respect to A and V, and let Xs be a progressively measurable stochastic process on (fi, !F, P) with state space (E, S). Then the following are equivalent: (1) For all pairs of stopping times S\ 6 M and S% € M with S2 > Si, the following equality holds: E f(S2,XS2)

J a S l ] = E [f(S2,XS2)

I a(SuXSl)}

F-a.s.

for all bounded Borel functions f on [0, T] x E. (2) For any pair of stopping times S G M and S € M with S < S, any finite set of stopping times {Si : 1 < i < n] such that Si < S for 1 < i < n, and all bounded Borel functions f on [0, T) x E, the equality E f [S,Xgj f(s,Xs)

I a(S\,Xs1,.

..,Sn,XSn,S,Xs)

\a(S,Xs)

holds F-a.s. (3) Let S 6 M.. Then for any bounded GT F, the equality E F\Gs]=E[F\a(S,Xs)]

-measurable random variable

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holds P-a.s. (4) For any S £ M, any bounded Gs-fneasurable random variable G, and any bounded real-valued QT -measurable random variable F, the equality E[GF}=E[GE[F\a{S,Xs)]] holds. (5) For any S e M, A^Gs, P [A n B |


and B e Q^S\ =P[A\

the equality

(7(5, XS)] P [B \ a(S, Xs)]

holds P-a.s. R e m a r k 2.8 Arguing as in the proof of the implication (1) ==> (2) in Theorem 2.14, we can show that the following condition is equivalent to conditions (l)-(5) in Theorem 2.16: (2') For all pairs of stopping times S e M and S £ M with S < S, all increasing finite sequences of stopping times {Si : 1 < i < n} such that Si < S for 1 < i < n, and all bounded Borel functions / on [0, T] x E, the equality E [/ (S,Xs)

\
= E [/ (s,X§)

|

a(S,Xsj

holds P-a.s. We conclude the present section by the definition of the strong Markov property of a stochastic process. It is based on condition (2) in Theorem 2.14 and on our choice of the future cr-algebra. Definition 2.15 A stochastic process {Xt,G,Gt,1?) with state space E is called a strong Markov process if for every stopping time S and every C w

bounded GT -measurable random variable F, the equality E[F\Gs)=E[F\a(S,Xs)}

holds P-almost surely.

150

2.7

Non-Autonomous Kato Classes and Feynman-Kac Propagators

Strong Markov Property with Respect to Families of Measures

This section is a continuation of Section 2.6. It is devoted to strong Markov processes with respect to families of stopping times and families of measures. Let (Xt,QJ,PT,x) be a non-homogeneous Markov process on {Q.,J-) with state space (E,£). Fix T € [0,T], and consider a pair (Si,S2) of Q\stopping times with T < S\ < S2 < T. One of the possible ways to define the strong Markov property with respect to the pair (Si, 52) and the family of measures {Ps,^ : r < s < T, y € E} is the following: ET,X [f (S2,XS2)

\GrSl]= E S l , x S l [/ (S2,XS2)}

P TlX -a.s.

(2.114)

for every bounded Borel function / on [r, T] x E. However, in order for the expressions in equality (2.114) to be well-defined, one has to impose certain restrictions on the process Xt and the stopping time S2. Let us first assume that Xt is an ^-progressively measurable process (see Definition 1.15). Then, since Si is a stopping time with T < S\ < T, the random variable X$ is QTS -measurable. Indeed, by the progressive measurability of the process Xt, the function (t, ui) 1—> Xt (OJ) is #[T'r]
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stochastic process implies its progressive measurability (see Theorem 1.5), and the measurability of 52 with respect to the cr-algebra ^ T J ' V implies the measurability of Xs2 with respect to the same cr-algebra (see Theorem 2.17 below). Now we are finally ready to define the strong Markov property with respect to a pair of stopping times. Definition 2.16 Let P be a transition probability function, and let (Xt,Gl,fT<x) be a corresponding adapted Markov process. Fix r e [0,T], and suppose that Si and 52 are S^-stopping times with r < Si < S2 < T. Then the process Xt is called a strong Markov process with respect to the pair of stopping times (5i,52) and the family of measures {PS]!/ : r < s < T, y £ E} provided that the following conditions hold: (1) The process Xt is right-continuous. (2) For every B £ £, the function (T,x,t) >—> P(r,x;t,B) S[o, T] - measurable. (3) The stopping time S2 is £ T 1,v -measurable.

is

#[O,T]

®£ ®

(4) The equality ETtX[f(S2,XS2)\gTSi]=Es1,xSl[f(S2,Xs,)}

P T , x -a.s.

(2.115)

holds for all bounded Borel functions / on [T, T]x E. Recall that our choice for the cr-algebra representing the future after the stopping time Si is the cr-algebra QTl' . Hence, condition (3) in Definition 2.16 can be interpreted as follows: the stopping time S2 resides in the future after the stopping time Si. It remains to show that under the restrictions described in conditions (l)-(3) in Definition 2.16 the function on the right-hand side of (2.115) makes sense. Moreover, we will establish that this function is measurable with respect to the u-algebra a (Si, Xgj). We will prove this fact for more general functions of the form E s ^ s [F], where F is a £/ T 1,v -measurable random variable. Note that by condition (3) in Definition 2.16, 52 is QT1,S/measurable. Moreover, using condition (1) in Definition 2.16 and Theorem 2.17 below, we see that Xs2 is 5 T 1,v -measurable. Therefore, f(S2,Xs2) is GT1,V-measurable for any bounded Borel function / on [T,T] X E. Lemma 2.13 Let P be a transition probability function, and let (Xt, GJ, FV,x) be a corresponding Markov process. Fix r € [0, T] and x € E, and suppose that S is a Q\ -stopping time with r < S < T and F is a bounded GT'W-measurable random variable. Suppose also that condition (2)

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N'on-Autonomous Koto Classes and Feynman-Kac Propagators

in Definition 2.16 holds. Then the expression Es,xs \F] *s well-defined, and moreover, there exists a bounded Borel function g on [0, T] x E such that Es,xs[F}

= g(S,Xs).

(2.116)

Remark 2.9 It follows from (2.116) that the random variable Ms,xs [F] is a (S, Xs)-measurable. Proof. Let "H be the vector space of all bounded QT' v -measurable random variables F satisfying the following conditions: (i) ^s,xs [F] exists PTiX-almost surely. (ii) There exists a bounded Borel function j o n [ 0 , T ] x £ such that the equality in (2.116) holds. It is not hard to see that H contains the constant functions and is closed under the convergence of uniformly bounded nonnegative nondecreasing sequences of its elements. Moreover, for any random variable F of the form F = / (5 V p,Xsvp), where / is a bounded Borel function on [T,T] X E and T < p < T, we have F € H. Indeed, the random variable F can be rewritten in the following form: ESM,X S ( I J ) (UO

[F] = Es(oi),x S M H [/ ( 5 M

= Y ( S M , S( W ) \fp)f

(S(w) Vp)

v

P>xs(u,)\/p(-))]

(XS{u){w))

= f f (5(w) V p, y) P (S{u), XS(UJ) ( w ); S(u) V p, dy) .

(2.117)

JE

Next, put 9(s,z)=

/ / (s V p, y) P (s, z; s V p, dy). JE

Then it follows from (2.117) and condition (2) in Definition 2.16 that F Let F be a random variable defined by

eH.

n+l

F=Y[fj(SVpj,XsvPj),

n>0,

3= 1

where T < p\ < p2 < • • • < pn < pn+i < T, and for every j with 1 < j < n + l, fj is a, bounded Borel function on [T,T] X E. Our next goal is to prove that any such random variable F belongs to H. We will use the method of mathematical induction in the proof. For n = 0, the assertion

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153

above follows from (2.117). By the Markov property for constant times, we have n+1

E S(u>),Xs(u)(u)

I]/;(%)V ft ,Is M v ft (.)) 3= 1

= E S M , x S M ( w ) Hfj(S(u)VPj,Xs{u)s,Pi(-)) j=i

E

=

S ( w ) V p „ , X s ( „ ) V p n ( t j ) [/n+1 {S(U}) V

Pn+l,XS{oj)Vpn+1(-))]

(2.118)

ES(CJ),X,S(u>) M

j=l

where gj — fj for 1 < j < n — 1 and 9n{s,y)

= fn(s,y)Es,y

[/„+i (sV/)B+l,XsVp„+1(-))] •

Next, using condition (2) in Definition 2.16, we see that the function gn is a bounded Borel function. Taking into account (2.118) and the fact that F S H for n = 0, we see that induction gives F € H for all n > 0. By the monotone class theorem for bounded functions, we conclude that the space H contains all bounded C/ r ' v -measurable random variables. This completes the proof of Lemma 2.13. • The next theorem has already been used in the beginning of the present section. In our opinion, Theorem 2.17 has an independent interest. Theorem 2.17 Let Xt, r < t < T, be a right-continuous stochastic process with state space E, and let (Si, £2) be a pair of QJ -stopping times with r < Si < S2 < T. Suppose that S2 is measurable with respect to the a-algebra QT1: . Then the state variable Xs2 is measurable with respect to the same a-algebra. Proof.

Define a family of random variables by S2,n = Si +

T-Si 2"

'2"(52-5i)"

T-Si

, n> 1.

We will need the following lemma in the proof of Theorem 2.17.

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Non-Autonomous

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L e m m a 2.14 For every n > 1, the random variable S2,n is a stopping time. Moreover, S2 < S ^ . n + l < S2,n

< 52 +

(2.119)

—.

Proof. It is not difficult to see that (2.119) follows from the definition of the function x 1—> \x\ (see Subsection 1.1.1). The fact that S2,n is a stopping time can be established as follows. For every r < t < T, we have 2"

{S2,n
f

u{

2"(52-5i)' T-S!

k \ n {52,„ < t)

fc=0 v-

S = {5i = S2 < t} U j j S.S1 + ^ - ^ (fc - 1) < S2 < Si + T -k
2™

fc=i *•

(2.120) It is also true that {S2 < t}\{Si

=

=S2
|J

= {Si

<S2
{Si
(2.121)

r€Qn[r,t]

Now it is not hard to see that {S\ = S2 < t} 6 Q\. For every k > 1, we have

{si + 7^(k-i)<S2<s1 Si + ^iJ1 = {ft + ^ ^

+

^^k
{k - 1) < S2 < t J n J S2 < S, + 1 ^k n {S2 < i)

n\sl + ^-¥^k
+ ^-^-k<S2
(2.122)

We will next prove that for every k > 0, the random variable Si H k is a stopping time. We have already established this fact for k — 0. The case where k = 2" is also simple. In the case where 1 < k < 2™, we have

{si + ^k
- fcT = {Sl<( *2" 2"-fc

This implies the assertion above.

A^

G &.

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155

Let us go back to the proof of Lemma 2.14. By taking into account the fact that the random variable S\ H

——k is a stopping time and

reasoning as in the proof of (2.121), we see that the event on the right-hand side of (2.122) belongs to the cr-algebra Qt. Now it is clear that Lemma 2.14 can be obtained from (2.120), (2.121), and (2.122). • It follows from the right-continuity of the process Xt and from (2.119) that in order to establish the £ T 1,v -measurability of the state variable Xs2, it suffices to prove that the state variables Xs2iTV, n > 1, are QTl,vmeasurable. Consider the following events:

{*.» = Sl + ^ A f c } = [S2,n = (l " £ ) * + YnT) (2"123) where 0 < k < 2". It is not hard to see that the events in (2.123) belong to the cr-algebra QT1' . On these events, the state variables Xs2,n coincide with X

S1 + ^p~k

= X

(l-2*k)S1

+ &T-

Consider the following approximating sequence for Si: T •

2m{Sl~T)

S\,m —T-\ Since the process Xt is right-continuous, it suffices to prove that the state variable Xi, k \a , * ^ is 0^ x ' v -measurable. ^1— j r r j J l , m + 27r J

J

Let 0 < £ < 2 m . Then it follows from £ \ £ 2m J 2

m

~~

that on the event

{^=(1-^)-+^},

(2.124)

the equality

holds. Here we take into account that p (k, £) > Si. Therefore, on the event in (2.124) we have X

(}-Jk)Si.m+&T

= xs1vP(k,e)-

(2.125)

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Non-Autonomous Kato Classes and Feynman-Kac Propagators

Now it is not hard to see that the QT1' -measurability of the state variable Xs2 follows from the ^| 1 , v -measurability of the events in (2.123) and (2.124) together with equality (2.125). This completes the proof of Theorem 2.17. • The next definition is motivated by condition (3) in Definition (2.16). Definition 2.17 Let Xt be a stochastic process, and let r £ [0,T]. The relation •< is defined on the set Sj. x <Sj. as follows: S x < S2 «=> Si < S2 and S2 is ££ 1,v -measurable.

(2.126)

Lemma 2.15 Suppose that Xt is a right-continuous process, and let r £ [0, T]. Then the relation •< defined in (2.126) is a partial ordering on the set o<ji x o^p. Proof. The reflexivity and antisymmetry of the relation X are clear. Next let Si, S2, S 3 £ Sj. x Sf with Si X S 2 d S 3 . Then, it is clear that Si < S3. Since S2 is GT1,y-measurable, S2 V a is also G? ' v -measurable for all a € [T, T]. It follows from the right-continuity of Xt that the function (S2 V a,Xs2va) is QT1,y-measurable (apply Theorem 2.17 to S2 V a and Si). Therefore, the cr-algebra QT2'V is contained in the cr-algebra QT1,V. Now the measurability of S3 with respect to the <j-algebra QT2'V implies that S3 is also G^1' -measurable. This completes the proof of Lemma 2.15. D Let P be a transition probability function, and let {Xt,Gt,^T,x) be a corresponding Markov process on Q. with state space E. Suppose that for every r £ [0, T], a family FT<X, x £ E, of probability measures is defined on the measurable space (fi, C/J.). Put

M°T=

[j

Sf.

re[0,T]

Here the symbol Sf stands for the family of all ^-stopping times where T
DSf.

Definition 2.18 A family M C Mj> is called an admissible family of stopping times provided that the following two conditions hold: (i) For every T £ [0,T], the family M(T) is closed with respect to the operations V and A.

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(ii) For every r G [0,T], the conditions Si G M(T), S\ < S2 imply the condition Si •< S2.

S2 G M(T),

and

The next definition is a generalization of Definition 2.16. Definition 2.19 Let P be a transition probability function, and let (Xt,Gt^T,x) be a corresponding Markov process. Suppose that M is an admissible family of stopping times. Then the process Xt is called a strong Markov process with respect to the family of stopping times M and the family of measures {PT)X : 0 < r < T, x G E} provided that the following conditions hold: (1) The process Xt is right-continuous. (2) For every B £ 8, the function (T,x,t) i-» P(T,x;t,B) S[o,T]-measurable. (3) The equality ET,X [f (S2,XS2)

I FSl] = E S l , X s i [f(S2,XS2)}

is £[O,T] ® £ ®

P T , x -a.s.

(2.127)

holds for all r G [0,T], a; G £;, 5i G M{r), S2 G A ^ ( T ) with Si < S2, and all bounded Borel functions / on [r, T] x £\ Lemma 2.16 Let Xt be a strong Markov process with respect to an admissible family of stopping times M. and the family of measures PTiX. Then formulae (2.96) and (2.98) hold for every measure P r , x and all QJ -stopping times Si€M (T) and S2 G M (r) with r < Si < S2 < T. Proof. Let Xt be a strong Markov process with respect to the family M and the family PT<X, and let Si and #2 be stopping times such as in the formulation of Lemma 2.16. Then the random variable F = f (S2,Xs2) is Qp,v^measurable (see Definition 2.18 and Theorem 2.17). Therefore, equality (2.127), Remark 2.9, and the properties of conditional expectations give E S l ,x S l [f(S2,Xs2)}=ETtX

[f(S2,XS2)

I GTSl]

= ET,X [ET,X [f(S2,XS2) =

I GTSl]

\a(Si,XSl))

ET>x[f(S2,XS2)\a(Si,XSl)].

This implies condition (2.96). Condition (2.98) follows from (2.96), since

o-iSuXs^cg^cg^. This completes the proof of Lemma 2.16.

•

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The next theorems provide equivalent conditions for the validity of the strong Markov property with respect to a family of stopping times M and the family of measures PT)X- Recall that if M. is a family of stopping times, then the^family M(S) is denned by M(S) = {5" £ M(T) : S' > S} and the symbol QT ( stands for the a-algebra given by gP{s)

=a{S',Xs,:S'

£M(S)).

Theorem 2.18 Let P be a transition probability function, and let (Xt, Ql, P r ,x) be an adapted Markov process with P as its transition function. Suppose that Ad is an admissible family of stopping times, and assume that conditions (1) and (2) in Definition 2.16 hold. Then for every r £ [0,T] and x £ E, the following are equivalent: (1) For all QJ -stopping times S\ £ M (T) and S2 £ M (T) with r < Si < S2 < T and all bounded Borel functions f on [r, T] x E, the equality

Er,x [f{S2,XS2) I SJJ =Es l l X s i [f{S2,XS2)\ holds FTiX-a.s. (2) For all stopping times S £ M{T) and all bounded real-valued QT ^ measurable random variables F, the equality ETtX [F I Ql] = Es,xs [F] holds FT:X-a.s. (3) For any stopping time S £ M (r), any bounded Gg-measurable random variable G, and any bounded QT -measurable random variable F, the equality E r>x [GF] = ETtX [GEStXs [F]} holds. (4) For any A£gTs,

B £ g™{S), and S £ M (r), the equality

P r , x [Af)B\

a{S, Xs)} = Fr>x [A | a(S, Xs)} Fs,Xs [B]

holds FTiX-a.s. Proof. We will only prove the implications (1) =£• (2) and (4) =4> (1). The remaining implications can be obtained as in Theorem 2.14. (1) = * (2).

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Suppose t h a t condition (1) in Theorem 2.18 holds. It will be first shown t h a t the equality E r , x [F | Ql] = Es,xs

[F]

(2.128)

holds for all random variables F given by

where for every 1 < j < m, fj is a bounded Borel function on [0, T] x E, and Sj £ M (S) is a stopping time such t h a t S < Si < • • • < Sm < T. We will use the m e t h o d of mathematical induction in the proof of equality (2.128). It follows from condition (1) in Theorem 2.18 t h a t (2.128) holds for m = 1. Next, let m > 1 and assume t h a t (2.128) is true for all 1 < k < m and for all finite subsets <Sj : 1 < j < k > of M (S) with S < Si < S2 < • • • < Sk < T. For any family of stopping times Si < • • • < Sm+i contained in the class M. (S), denote TC = a [Gs,Si,X^,...,Sm,Xg^j

with m + 1 elements

.

Then, using the tower property of conditional expectations, we get m+1 E

r,x

n

i=i

m = ET 3=1

/

..A

= ET,:

Y[fj {Sj,Xs.)ET,x

[/m+i (sm+i,Xgm+i)

| W| | Qs

Now it follows from condition (1) and from the inclusions

*(sm,X~Sm)cHcQlm

. (2.129)

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Non-Autonomous

Koto Classes and Feynman-Kac

Propagators

that

/m+l [Sm+l,Xgm+i) I ftj

ET,

[fm+1 (5 m + i,X § m + i jJ.

= E

(2.130)

Next, we obtain from (2.129) and (2.130) that m+l

n/i(^'x5,)i^

ET,

" 1

m

ET

J J / j ^5j,X g jEg m)Xgm |/ m + i ^S m+ i,Xg m+ jJ | ^5 i=i

(2.131) Since M. is an admissible family, the stopping time 5 m +i is QsTmm',v - measurmm,v able. Moreover, Theorem 2.17 implies that Xg is Q->S T ' - measurable. It follows from Lemma 2.13 that there exists a bounded Borel measurable function g on [T, T] X E such that

g\Sm,XSm)

- E § m X . m |/m+i

[Sm+1,X§m+ij^

(2.132)

Therefore, (2.131) and (2.132) give m 771 + - J -l1

E.T , X

n m

— Er,:r

Ylfi{Si,xh)g(sm,xSm)\rs

(2.133)

By equality (2.130), the tower property of conditional expectations, and the induction hypothesis applied to the right-hand side of (2.133), we have m+l

ET,

Ilfi{Si,x§j)\gi

_j = l

m ••Es,. XS j= l

m

Es,x s

J J / j ( S ^ X g J E ^ ^ [/m+1 (S' m+1 ,X §m+i J J'=I

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161

771

=

E

S,XS

1 1 /?' ( S i>- X 'sJ E S.*s fm+l \Sm+i,Xgm+i)

|W

j=i

n+1 = Es,x s

E

II £($.*§,) I w

S,XS

3=1

m+1

= Es, x s

iH(^*s,)

(2.134)

J' = l

It follows from (2.134) that equality (2.128) holds for any random variable F given by 771

F=

Hfj(sj,X~Sj)

where fj, 1 < j < m, is a finite sequence of bounded Borel functions on [T,T] x E, and Sj G M (S), 1 < j < m, is a finite sequence of stopping times satisfying r < S < Si < ••• < Sm < T. Our next goal is to prove the equality in condition (2) in Theorem 2.18 for a function F given by F = X

{ (§i.*Sl)eBl}

X

" 'X

X

«S-**.>M'

where B\,..., Bm are Borel subsets of [r,T] x £ , and Si,...,Sm are stopping times from the class M (S) satisfying Sj > S for all 1 < j < m. Define an increasing sequence of stopping times Sk by S'k = min lsjl

V • • • V Sjk : 1 < j i < • • • < j k < m j ,

1 < k < m.

It is clear that S'k £ M (S). For any permutation 7r of the set { 1 , . . . , TO}, put

^ = {(s;(1)I^(i))eBi,...>(s;(m)>^{m))€Bm}. Then we have

77

Next, using the inclusion-exclusion principle, we get F

= *{ ( § . . * * ) € * }

X

• • •X

X

{(Sm,XsJ^Bm}

162

Non-Autonomous

Kato Classes and Feynman-Kac

22

= Z2XA^TV

7r,

+

Propagators

XA„nA„,

7Tr:ir^nf

X)

XA^A„,nA„„ - • • • •

7T, 7 r ' , TT"-.TTy^TV1 ,-n^lt"

(2.135)

,7r' T ^ 7 T "

Since S£ is an increasing sequence of stopping times from the class M (S), equalities (2.134) and (2.135) imply that ET,I [F I Gs] = Es,xs [F]. By formula (2.110), the previous equality holds for all functions F which can be represented as follows: m

F=

l[fj(sj,X-S]).

Now using the monotone class theorem, we see that condition (2) in Theorem 2.18 holds. (4) = » (1). Suppose that condition (4) in Theorem 2.18 holds, and let A £ (?£ , B G §¥iSl), Si € M(T), and S 2 G M(T) with r < Si < S 2 < T. Then, using condition (4) in Theorem 2.18 with S = Si, we get FT,X [AnB\a(SuXSl))

= ET,X

[B]

[XAFSUXSI

\
P T):r -a.s. Taking the expectation ETyX in the previous equality, we see that P r , x \Br\A}= for all A e GTSl and B G G^(Sl)•

ET,X [Fsi,xSl [B] XA] It follows that

E r , x [/ (S 2 , X S a ) x^] = ET,X [ESuXsi

[/ (S 2 , XS2)]

Xi4 ]

,

and hence ET,X [f(S2,XS2)

| 0 J J =ESuXsi

lf(S2,XS2)}

This establishes condition (1) in Theorem 2.18. The proof of Theorem 2.18 is thus completed.

P r ,*-a.s.

D

The next theorem is similar to Theorems 2.14 and 2.18. Its proof is omitted.

Propagators:

General

Theory

163

Theorem 2.19 Let P be a transition probability function, and let (Xt,Gt^T,x) be an adapted Markov process with P as its transition function. Suppose that M. is an admissible family of stopping times, and assume that conditions (1) and (2) in Definition 2.16 hold. Then, for every T G [0, T] and x G E, the following are equivalent: (1) For all Qf -stopping times S\ G M. (T) and S2 G M. (T) with r < S\ < S2 < T, and all bounded Borel functions f on [r, T] x E, the equality I GTSl]

ET,x [f(S2,XS2)

=ESuxsJ(S2,Xs2)

holds P T)X -a.s. (2) For all stopping times S G M(T) and S G M (S), all finite families of stopping times {Si : 1 < i < n} such that r < Si < S for 1 < i < n, and all bounded Borel functions f on [r, T] x E, the equality o~ (Si, Xs1 ,...,Sn,

ET =

Xsn, S, Xs)

ESlxs[f(s,X§)~

holds P r>x -a.s. (3) For any stopping time S € M(T) random variable F, the equality T Er,x F\G S

and any bounded QT

-measurable

=Es,xs[F}

holds ¥r<x-a.s. (4) For any stopping time S £ A4 (T), any bounded Gs-measurable random variable G, and any bounded real-valued GT -measurable random variable F, the equality ET,X \GF] = ET,X [GEs,xs [F]} holds. (5) For A G Gs, B & GT PT,X [AHB\

> and any stopping time S G M (T), the equality a(S,Xs)]

= PT,X [A I a(S,Xs)]

Fs,xs [B]

holds P r>x -a.s. Finally, we are ready to define the strong Markov property of a stochastic process with respect to a family of measures.

164

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

Definition 2.20 Let P be a transition probability function, and let (Xt,Ql,PTtX) be an adapted Markov process associated with P. It is said that the process Xt is a strong Markov process with respect to the family of measures PT,X if for every r e [0, T], x e E, every ^-stopping time S with C w

T < S
(2.136)

for all T
[F]

(2.137)

where F is any bounded Gp -measurable random variable. Let Y be the free backward propagator associated with P. It is defined for all 0 < r < t < T and / e Co by the following formula: Y{r,t)f(x)=

[

f(y)P(r,x;t,dy).

JE

Recall that for a given stopping time S, the symbol £^' v stands for the cr-algebra a (S V p, Xsvp '• 0 < p < T) (see Definition 2.11). It is clear that for a stopping time S with r < S
0 < p
:

T
Theorem 2.20 Let P be a transition probability function, ( X t , ^ , P T i X ) be an adapted Markov process associated with P. that the following conditions hold:

and let Suppose

(1) The process Xt is right-continuous. (2) For every B € £, the function (T, X, t) i-> P (r, z; t, B) is B[o,r] ® £ ® #[o,r] -measurable.

Propagators:

General

Theory

165

(3) For any function f e Co and t € (0,T], the function (T,X) H-» Y (r, t) f(x) is continuous from the right in r on the interval [0, i) and continuous in x on E. Then Xt is a strong Markov process with respect to the family {P T)X }. Remark 2.10 ing condition:

Condition (3) in Theorem 2.20 is equivalent to the follow-

lim

EtiX[f(Xt)]=Et0,X0[f(Xt)]

siso,x—*xo

for all XQ S E, T < SQ < t, and f £ Co- It follows from conditions (1) and (3) in the formulation of Theorem 2.20 that lim ESiXs [/ (Xt)} = E S0 , X

[/ (Xt)],

P TiX -a.s.

(2.138)

for all x € E, f e C0, and 0 < r < s 0 < t < T. Proof. Let T € [0,T], x € E, and let S be a ^-stopping time with T < S
r,x

[F | gTs] = Es<Xs

[F]

(2.139)

holds PT)X-almost surely for any bounded QT' -measurable random variable F. We will derive equality (2.139) from the following assertion. For all r e [0,T], x e E, all bounded Borel functions / on [T,T] X E, and all T
ET,x[f(svp2,xSvP2)\gTSVPi] = E s v p 1 , x S V p i / ( 5 v / 9 2 , X S V p 2 ) Pr,x-a.s.

(2.140)

Indeed, if (2.140) holds, then the validity of (1) = > (2) in Theorem 2.18, where the family M is defined by .M = { S V p : r < p < T } , implies equality (2.139). It remains to prove equality (2.140). Let r < p\ < P2 < T, and put S = S V p\. Then S V p2 = S V p2- Consider the following approximations from above of the stopping time S: ~ Sm=T

T-T

+ ——

•(S-0 T-T

m> 1.

166

Non-Autonomous Kato Classes and Feynman-Kac Propagators

Next we will prove that for all m > n and all functions f £ Co ([r, T] x E),

f(snvp2,x~Snyp2)\gzm

Er, :

= E~sm,x-sj{SnVP2,X~Snyp2)

P T , x -a.s.

(2.141)

Let us first show that (2.141) implies (2.140). Assume that (2.141) holds, and let w G fi. Then the right-hand side of (2.141) evaluated at w is equal to E

§m(„),XSmluy{u)f

{Sn{")V

P2,X~Sn{ul)yp2{-))

= Y (Sm (w), Sn (to)) f (Sn(u>) V p2) (Xg n ( w ) V p 2 ) ,

(2.142)

where the function / (Sn (u) V p2 J is defined by / (Sn(w) V P2) (y) = / (Sn(w) V p2, y) , » G £ . By the right-continuity of the process Xt, condition (2) in Theorem 2.20, and equality (2.142), we see that lim Es , , „ B ) ( B ) / ( g , ( w ) v f t l X j , ( u ) V B ( . ) ) m-»oo ,5'"^a'''Asm(, • EW l M ( ^ ( ^ ) V f t - ^ ) v W 0 ) c

(2-143)

for P r , x -almost all w £ f l . Let A G 0 | . Then A £ QZ for all m > n, and hence (2.141) gives /(5„VP2,^Vp2),A = E.

Ee

Y_

/(5„Vp2,^snVp2

,A

(2.144)

Passing to the limit as m —> oo in (2.144) and using (2.143), we get E. -,x[f(snVp2,X~s^p2),A = ET

E

§[f(snVP2,X-SnVp2),A

(2.145)

for all / G CQ([T,T] X E). NOW taking into account the continuity of the function / and the right-continuity of the process Xt, we see that the dominated convergence theorem can be applied in (2.145). Therefore, ET,:

'f(svP2,X~SVp2),A

Propagators: General Theory

= ET

E

3.*s

f{sVp2,X~SVp2),A]\.

167

(2.146)

The random variable S is (/--measurable (use the definition of the cr-algebra Q-,). Moreover, the state variable Xs is also C/i-measurable. This follows from Lemma 2.8 and from the fact that the right-continuity of a stochastic process implies its progressive measurability. Hence, the random variable ^s,x- f \S v P2^svP2) *s ^ s - m e a s u r a b l e . Now, using the definition of conditional expectations and equality (2.146), we see that equality (2.140) holds for all f e Co ([r, T] x E). By Urysohn's Lemma and the monotone class theorem, equality (2.140) holds for all bounded Borel functions. This establishes the implication (2.141) ==>• (2.140). Next we will prove equality (2.141). The following lemma will be needed in the proof. Lemma 2.17 Let n be a given nonnegative integer. Then for every integer m with m > n and every integer k with 0 < k < 2m, there exists a unique integer (.^m such that

k < 2m-ne<£]m < 2m-n + jfe - 1. Proof. It is easy to see that Lemma 2.17 holds for m = n. Now suppose that m ^ n. Then the inequalities in the formulation of Lemma 2.17 are equivalent to the inequalities k2n-m

< e(n)^

< ^n-m

+

1

_ yi-m_

(2.147)

It is clear that the interval [fc2"~m, k2n~m + 1 - 2™~m] contains at most one integer. We will show that

4*2, = [k2n-m + 1 - 2"" m J = \2n~mk] .

(2.148)

Let I = \2n-mk\. Then 2n~mk <£< 2n~mk + 1, and hence k < 2m~n£ < k + 2m~n. It follows that k < 2m~nt
< £ < k2n-m

+ 1 - 2n~m

and i < [k2n~m + 1 — 2n~m\. Combining the previous equality with the facts that I = [2 n-m fc] and there is at most one integer between k2n~m and k2n~m + 1 - 2 n " m , we see that (2.147) holds. This completes the proof of Lemma 2.17. •

168

Non-Autonomous

Kato Classes and Feynman-Kac

Propagators

We are finally ready to finish the proof of Theorem 2.20. Note that it only remains to establish equality (2.141). For n > 1, m > n, and 0 < k < 2 m , choose v£m as in Lemma 2.17. Then for Sm defined by

(S-r)

T-T

Sm = r +

T-T

we have

{S™ = r + ^ f c } C {§ n = r + ^ C }

•

(2.149)

Indeed, the event on the left-hand side of (2.149) is the same as the event f 2™ ( S - r ) ) < k — 1 < —•£ — < k >, while the event on the right-hand side of T T

"

I

J

, .

2" f 5 - T)

'- < C ,)

(2.149) coincides with the event { 4?m - 1 < — £

fc,m

Now

we see that (2.149) follows from Lemma 2.17. Next put f(Jfc,m) = r + ^ - = ^ f c =

(I-1-)T+^-T

and «(«)

t (n, k,m) = T+

Z^ffrl km

2"

^'

-•

~ \

fc,m

«(«) I

2"~ /

.

k,m.rp

2"

'

Then, it follows from the properties of the number P£'m that t (k, m) < t(n,k,m). Therefore, (2.149) gives = t ( f c , m ) | C JS„ = t(n,

fc,m)|.

Let A G gz . Then, using (2.150), we get

2m

- £ E T , S [/ (§n V p 2 , X § n V p 2 ) , A n { s m = tffc.m)}]

(2.150)

Propagators: General Theory

] T E r , x f(t(n,k,m)Vp2,Xt fc=0 2m = ^2^r,x [Er,i / (t (n, fc, m) V p2, ' =0 k=0

169

),An{sm=t{k,m)}} Xt(nXm)yp2),

An{sm = t(k,m)}\gj{k:,m)

(2.151)

It follows from the definition of the cr-algebra Q\

that

An{sm = t(k,m)}egiik<m). Therefore, (2.151) gives

ET,x[f(snVP2,X~Snyp2),A =

^^T,X ' =o fe=0

| E T , I / (* (", fc, m)\J p2, X t ( n i f c i m )v P 2 ) | £tTt(fc,m) (

An{§ m = t(fc,m)}] .

(2.152)

Next, applying the Markov property for constant times and using (2.150) in (2.152), we obtain

, [f(snVp2,X~Snyp2),A

E.T x

= ET , £

YlEt(k>m)>XHk,m)

[f (l (n> fc> m )

V

^2, *t(n,fc,m)Vp2)]

.fc=0

X{Sm=t(fc,m)}> ^ •

(2.153)

Since the r a n d o m variable 2m t(fe,m),Xt(fcm) [/ (i ( l , k, m) V P2> -^t(n,fc,m)VP2)] X{s m= t(fc,m)} fc=0 is C/~ -measurable, inclusion (2.150) and equality (2.153) give E.

-4f{snvpi,x~SnVp2)\gzm_ ^Z^t(k,m),xt(k,m) k=0

[f (t (n, k, m) V p2, X t ( n > f c ) m ) V p 2 )] X { § m = t ( f e , m ) }

170

Non-Autonomous

= HE§m,X-Sm

Kato Classes and Feynman-Kac

[f (Sn V p 2 , X g n V p 2 ) ]

Propagators

X{sm=t(k,m)}

k=0

= E

f(snvP2,x~SnVp2)'

This implies equality (2.141). The proof of Theorem 2.20 is thus completed.

•

Remark 2.11 Fix r € [0, T] and x £ E, and denote by Q^'y the completion of the cr-algebra Q^y with respect to the measure P r , x . Arguing as in the proof of Theorem 2.20, we see that the ^y' v -measurability assumption for the random variable F in the formulation of the strong Markov property can be replaced by the QT1,V -measurability of F. This means that under the conditions in Theorem 2.20, the GT' -measurability of the random variable F implies the strong Markov property in the following form: ^r,x[F\Gs1]=^S1,XSl[F] P TiX -almost surely. The next assertion follows from Theorem 2.20. Corollary 2.1 Let P be a transition probability function, (Xt,Gl,¥TtX) be an adapted Markov process associated with P. that the following conditions hold:

and let Suppose

(1) The process Xt is right-continuous. (2) For every B e £, the function (T, X, t) t-> P (r, x; t, B) is S[ 0 ,T] ® £ ® S[o,r] -measurable. (3) For every f E Co and t 6 (0,T], the function (T,X) t-> Y (r,t) f{x) is continuous from the right in T on the interval [0, t) and continuous in xeE. Let M. be an admissible family of stopping times. Then the process Xt is a strong Markov process with respect to the families M and {PT,x}Next we will give examples of families of stopping times which can be used as the families M(T) in Corollary 2.1. More examples will be given in Section 2.10. Example 2.3 Let P be a transition probability function, and let {Xt,Gl,PT,x) be a corresponding Markov process. Assume that the process Xt is right-continuous, and fix r £ [0,T]. For a given (/^-stopping time S with T < S < T, consider the following family of stopping times:

Propagators:

General

Theory

171

M = M(T) = {(S + p) A T : p > 0}. It is not difficult to prove that the family M satisfies the following condition: for all S\ G M and 52 G M with r < Si < S2 < T, the stopping time S2 is measurable with respect to the ex-algebra a (Si) C GTl'y. In the next example, we will consider a more general case. E x a m p l e 2.4 In this example, the family M = M(T) consists of certain functions of a given stopping time. Let r G [0, T], and let S be a ^-stopping time with r < S < T. Suppose that the process Xt is right-continuous, and consider the family $ of all functions cp : [r, T] —• [r, T] satisfying the following conditions: (1) ip is continuous on [T,T]. (2) There exists a point u(tp) G [r, T] such that the function