J
essup d.(w))
W
1
Given 0 E r2(SZ,C(J,Rn); F0), J = [r,0], consider the stochastic delay
equation (with finite random delays): p
dx(t) =
q
h.(x(tr.))dt +
E
j=1
E i=1
g (x(td ))dz(t), i
t> 0
i
x0=0 The random delays r3, di, j = 1,...,p, i = 1,...,q, in (IV) may not all be essentially bounded away from zero; and so a stepbystep direct integration of (IV) is not in general possible. basic Existence Theorem (11.2.1).
Nevertheless we can still apply our First we need to check that (IV) does ti
indeed satisfy Conditions (E) of Chapter II. 92(Q,Rn), 9:c2(Sj,C(J,Rn)) ti
h(W)(w) =
p E
j=1
>
Define h:L2(S2,C(J,Rn)) 
C2(c,L(Rm,Rn)) by ti
h.[w(w)(r.(w))], 9(W)(w) = J
3
for W E L2(S2,C(J,Rn)), a.a. w E 0.
q E
9i[w(w)(di(w))]
i=1
To see that h, g are globally Lipschitz,
let L > 0 be a common Lipschitz constant for all the hj's and gi's.
Then,
if W11W2 E C2(S2,C(J,Rn)), we have
167
Ilh(W1)h(W2) 1Ir2(p.Rn)
< P
E
Ihj[w1(w)(rj(w))h.[P2(W)(r.(W))]I2 dP(W)
J
j=1
Q
p E
p L2
J
j=1
< p2 L2
0
1k)1
lip (w)(r.())  W (w)(r.(w)12 dP(W)

2
J
1
3
w2112 2
r (,C)
Similarly g is Lipschitz with Lipschitz constant qL.
It remains now to
verify the adaptability condition E(iii) of §(II.1).
To do this it is
11
sufficient to show that for each measurable, for t > 0.
It,
E L(Q,C;Ft), 2 h('P) and g(W) are Ft
2 n Let W E L (2,C) be Ftmeasurable, and P:J X C > R
the evaluation map (s,rl) Ea n(s), s E J,n E C.
Then p is continuous, and
since each rj is F0measurable, it follows that
W
>
h('P)(w) _
[P(r .(W),w(W))]
h
j=1
j
J
ti
is F measurable. ti
Therefore h(W) E r2(D,Rn;F ); similarly
t
g(W) E 92(SI,L(Rm,Rn);Ft).
Hence all the conditions of Theorem (11.2.1) are
satisfied and so a unique strong solution
ex
E C (c,C([r,a],Rn)) of the
stochastic DDE (IV) exists with initial process e.
The trajectory
{ext:a > t > 01 is defined for every a > 0, is (Ft)adapted and has con
Moreover each map
tinuous sample paths. Tt:c2(c,C;F0)
`
'
e
c2(c,C;Ft),
t > 0,
ext
is globally Lipschitz, by Theorem (11.3.1). Now suppose in (IV) that z is mdimensional Brownian motion w adapted to
(F t)0., where each Ft is suitably enlarged by adding all sets of measure zero, i.e. we get the stochastic DDE of It; type q
p
dx(t) =
Z
j=1
h.(x(tr .))dt + J
x0=nEC(J,Rn) 168
J
E
i=1
g (x(td.))dw(t),
t > 0
(V)
with deterministic initial condition n.
Note that the coefficients in (V)
ti
can also be viewed as F0 ® Borel C measurable maps H:S2 x
G:Q x C(J,Rn) > H(w,n) =
n
L(Rm,Rn) given by p E
ti
n
C(J,R ) > R
j=1
q
ti
h(n(r.(w)))
,
G(w,n) =
i
E i=1
1
1
So (V) becomes the stochastic RFDE with random
a.a. w E sl, all n E C(J,Rn).
coefficients:
ti
ti
0 < t< a
dx(t) =
x0=n Because of the randomness in the coefficients of (V)' we cannot apply the results of Chapter III to obtain a Markov property for the trajectory field of (V).
However, if the delays rj, di are all independent of the Brownian 0 < t < a}, we shall show that the trajectory field
aalgebra Fa E { n xt:0 < t < a,
n E C} of (V) is generated in a natural way by a random
family of continuous timehomogeneous Feller processes on C, each of which adapted to the Brownian filtration
u < t})0<. First,
(Ft
note that when the delays are all deterministic the coefficients in (V)' become nonrandom and Theorems (111.1.1), (111.2.1) imply that the trajectory field gives a continuous timehomogeneous Feller process on C(J,Rn) adapted to (Ftw)O
Secondly, when all the delays rj, di are simple functions, an
associated random family of continuous timehomogeneous Feller processes on C can easily be constructed (Lemma (3.2)) and one can then approximate the general equation (V) by a sequence of stochastic simpledelay equations (Lemma (3.3)).
For simplicity of exposition, and without loss of generality, we deal with the case of only two F0measurable delays
r0,d0:52 R which are independent
of Fa, viz: dx(t) = h(x(tr0))dt + g(x(td0))dw(t),
0 < t < a
x0 = n E C(J,Rn)
where h:Rn ' Rn, g:Rn Lemma (3.1):
L(Rm,Rn) are Lipschitz maps.
Suppose the delays r0, d0 E LOO(Q J o ;FD) are simple functions 169
with respect to the delay aalgebra F0 = a{r0,d0} c F0 i.e. there are dm,
c FD such that
< r, m' = 1,...,k and {c
0 < rm',
k
E rmX
r0 =
m'=1
e
m =1
St
k
E dm X is
d0 =
For each m' = 1,...,k, let nxmo:0
,
U
= S2.
Stm
m.=1
s C([r,a],Rn) be the unique solution of 0 < t < a
dxm'(t) = h(xm'(trm'))dt + g(xm'(tdm'))dw(t),
(VI(m'))
xm' = n Then x(t)
nxm'(t)X
E
m' =1
2
r < t < a, gives the unique solution of (VI)
SP
n
n
in t (S2,C([r,a],R )) starting at n E C(J,R ) and adapted to (Ft)0
Proof:
By the definition of x, x
E
for all t > 0.
in
m' =1
Stm
k
Then
x
E nx
=
0
m'=1
n.
f
Also since r0, d0 are simple functions,
0 h(x(tr)) =
E
h(nxm'(trm'))X
,, g(x(td0))=m Eg(nxm'(tdm'))X 1
,;
m' =I
and so by Theorem (I.8.3)(i) for the stochastic integral one has n(0) +
x(t) = {
E fo h(1xm,(urm,))x d m'=1
h(x(ur 0))du +
n(0) +
ln(t)
170
J
k E
0 m'=1
0
m
g(nxm'(udm'))dw(u)} 0
Jot
0 < t < a
,
X
S'
g(nxm (udm ))dw(u),0
t E J
n(t)
J
t
rt J
E x m'=1
t E J
n(t)
rn(o) + {
du +
h(x(ur0))du +
t E J
J0t
g(x(ud0))dw(u),
0 < t < a
nxm'
is (Ft)aadapted, then so is x.
As each
tion of (VI) in £2(Sj,C([r,a],Rn)).
For a.a. W' E S, let
nxw'
Hence x is the unique solu
o
E c2(Q,C([r,a],Rn)) be the unique solution
adapted to (Ft)o
dx(t) = h(x(tr0(w'))dt + g(x(td0(w')))dw(t)
0 < t < a (VI(w'))
x0
= n
with fixed delays r0(w'), d0(w') E [0,r].
The trajectory fields
{nxt:t E [0,a], n E C, of E 0 therefore correspond to a random family
{Xw
:w' E S2}
of continuous timehomogeneous Feller processes on C(J,Rn) adapted to (Ft)O
w' E sl,n E C, t E [0,a]}
Let
denote the associated random family of transition probabilities of the Xw''s. Po(nxt)1 for the distribution
Note that we retain the notation
of the trajectory field {nxt: t E [0,a], n E C} of (VI).
In (VI) suppose the delays r0, d0 are simple, independent of
Lemma (3.2):
Fa and satisfy the notation of Lemma (3.1).
t E [0,a],
Let
n E C, m = 1,...,k} be the family of transition probabilities associated with the trajectory fields {nxt': t E [0,a], n E C, m' = 1,...,k} for (VI(m')), m' = 1,...,k.
Then for any B E Borel C, n E C, t E [0,a], we have k E
P(w',O,n,t,B) =
,
pm (O,n,t,B) x
,(w')
a.a. W' E S2
m'=1 and
JQ p(w',O,n,t,B)dP(w') = P(0,n,t,B) = P{w:w E Sl, nxt(w) E B}.
Remark Note that for each B E Borel C, the map (w',n,t) +> p(w',O,n,t,B) is FD
0 Borel C 0 Borel [0,a]measurable.
!_roof:
By Lemma (3.1), the trajectory field of (VI) is given by k
nx (w) =
t
n E C.
The
E
m'=1 SZm'
nxm
t
,
(w)x
,(w)
a.a. W E sl, t E [O,a],
Stn
form a partition of S2 and are all independent of nxt'.
Let 171
w' E Q. dm
.
Then there is an m' such that w' E. 52 and so r(w')
Hence
nxw
= nxm
d(w') _
and
P(w',O,n,t,B) = P{w:nxtw"(w) E B} = P{w:w E R,nxtm'(w) E B)
=
for every B E Borel C.
pm (O,n,t,B) k E
Thus p(w',O,n,t,B) =
,
pm (O,n,t,B)X
m' =1
for all w' E 12.
,(w')
S2m
is Borel C ® Borel [0,a]measurable (Theorem
Since each
it is clear that p(.,O,.,.,B) is FD ® Borel C 0 Borel [O,a] measurable.
fQ
Also, by independence of xand nxtm' for t E [O,a], we get k E
p(w',O,n,t,B)dP(w') =
,
pm (O,n,t,B)P(cim )
m' =1
k
E
m '=I fQ k =
1
j
XB(nxtm'(w))dP(w)
`
X
Qd
(w)O(w)
XB(nxtm,(w))XOM,(w)dP(w) nxtm,
=
JXB( m'=l
(w)X
.(w))O(w)
XB(nxt(w))dP(w) = P{w:w E S2, nxt(w) E B} for every B E Borel C.
o
The following is a general approximation lemma for stochastic DDE's:
Suppose the coefficients h, g in (VI) are globally bounded and
Lemma (3.3):
Go
Lipschitz.
00
C'(SIjbou;
Let {rk}k=1, {dk}k=1 be increasing sequences in
such that rk ' r0 and dk
d0 as k '
in E2(SIJ ° ;FD).
FD
Let nxk and nxk'w
denote solutions of the stochastic DDE's: dxk(t) = h(x(trk))dt + g(x(tdk))dw(t),
0 < t < a (VIk)
xk = n 0
172
and
dxk,w'(t)
=
h(xk,w'(trk(w')))dt + g(xk'w8(tdk(w')))dw(t)
xk,w' = n
0
} (VIk(w' )
0
respectively, for k = 1,2,3,..., a.a. W E Q.
Then the map
0 3 w' > nxw E C2(a.C([r,a],Rn)) is in rOO(S2,r (S2.C([r.a1 .Rn));FD
lim J
Ilnxk(w)  nx(w)II2 dP(w)
k
= 0
and
lim
Ilnxk'w'(w)  Ixw'(w)II2 dP(w)}dP(w') = 0
{J
J
kc
w' ES2
wES2
Remark For each n E C(J,Rn), the random family {nxw':w' E Sl} has a unique version nX E 9 2(S2 x St, C([r,a],Rn); FD 0 Fa) such that for a.a. w' E S2,
nxw
a.s.
Suppose h and g are bounded by a common bound K > 0 and have a common
Proof:
Lipschitz constant L > 0.
Let rk ' r0, dk ' d0 as k
Consider
Fix n E C(J,Rn) and take any integer k > 0.
{h(nxk(urk))  h(nx(ur0))}du
Jt0 nxk(t)
{g(nxk(udk))g(nx(ud0))}dw(u), 0
0
nx(t) =
in r2(SI;FD).
ft
t E J
0
By Doob's inequality, we have, for any t E [O,a], IJv{h(nxk(urk))h(nx(urk))}du E
sup Inxk(v)nx(v)l2 < 4E sup 0
l2
0
v
sup
+ 4E O
I
{h(nx(urk))h(nx(ur0))}dul2
J 0
+ 4K1 Jt
Ell g(nxk(udk))g(nx(udk))II2 du
0
173
0t
+ 4K1
Ell g(nx(udk))  g(9x(ud0))II2 du, some K1 > 0,
I
denotes the operator norm sup {IA(e)l:e E Rm, lel < 1} for each
where IIAII
A E L(Rm,Rn).
Therefore, by the Lipschitz condition., one gets t
E
sup r
Inxk(v) nx(v)l2
< 4aL2 J
Elnxk(urk)nx(urk)I2du 0
Jt
+ 4aL2
Elnx(urk)  9x(ur0)12du 0
t + 4K1L2
J
OElnxk(udk)nx(udk)I2du + 4K1L2 J OElnx(udk)nx(ud0)I2du (1)
for t E [O,a]. Clearly,
Elnxk(urk)nx(urk)I2 < E
sup r
Inxk(v)  nx(v)I2
(2)
Elnxk(udk)nx(udk)I2 < E
sup r
Inxk(v)  nx(v)I2
(3)
and
for all u E [O,aJ.
We must estimate the second integrand Elnx(urk)  nx(ur0)l2 in (1) for sufficiently large k, uniformly in u E [0,a].
To do this, let E > 0 and note
that by uniform continuity of n on J there is a 0 < 6 < 1, 6 = 6(E,n) such that for s1,s2 E J, Is1s2I < 6 , we have In(si)n(s2)I2 < E. so by convergence of {rk}k=1 in that Elrkr0 I2 < c' = 62E for all
Let E' = 62E,
there exists k0 = k0(c,6) such k
> k0.
Suppose k > k0 and for any
u E [0,a] denote by
X(r0u), X(rku), XA, XBk the characteristic functions of the sets {w:w E St, r0(w) < u}, {w:w E Q, r0(w) > u.}, {w:w E Q, rk(w) < u}, {w:w E Q, rk(w) > u} in F
D
and
A = {(w,v):w E SI, V E [O,a], v + r0(w) > u}, Bk = {(w,v):w E 0, v E [0,a],
v + rk(w) < u} in FO 0 Borel [O,a], respectively. the stochastic integral, we may write
174
Using Theorem (1.8.3) for
r
rk
ulur r0g(nx(vd0))dw(v)
fur0
h(nx(vr0))dv + if r0 < u
rkh(nx(vr0))dv nx(urk)nx(ur0)
_
u
u0rk
+
+ n(0)n(ur0), if rk < u and r0 > U.
n(urk)  n(ur0) ,
if rk > u
a
J X(r0
+
(nx(vd0))dw(v)
10 X(r0
a
+ Jo X
('v)9 (nx(vd0))dw(v)
X
(rk u) Bk
/0
[n(urk)n(ur0)]
+ En(O)  n(ur0)]X(rk
X(rou)
Therefore, a
Elnx(urk)  nx(ur0)I2 < 6a
( EX(r 0
+ 6K1 Ja EX
(r
0
0
k
II9(nx(vd0 ))II2 dv k
+ 6a
a
II9(nx(vd0))II2dv
J
+ 6EIn(0)n(ur0)I2 X(rku) +
<
6EIn(urk)n(ur0)I2X(rk>u)
12K2aEX(r0u)(urk)
+ 6EIn(0)n(ur 0)I2X(rku) + 6EIn(urk)n(ur0)I2X(rk>u)
(4)
175
Now
Eln(O)  n(ur0)12X(rku)
1iur0i<6 In(O)n(ur
0)12X(rk < u)X(r0 >u) dP
fIuroI>6 In(O)n(ur 0)I2X(rk < u)X(r0 +
< c + 411n II 2 P{w:wE S2 ,
>u)
dP
(r0(w)  rk(w)) > 6)
C
< E + d IIn IIC EIrOrkl2 < E + 4IIn II2 E, because k > k0.
(5)
Similarly, for k > k0, we get
EIn(urk)n(ur0)I2X(rk
> u)
I<6In(urk)n(ur0)I2X(rk
I Ir r 0
+
0
>U)
dP
k k1>6
< E + s IIn III E IrOrkl2 < (1 + 4IIn I102)E Combining
dP
k1<6
In(urk)n(ur0)I2X(rk
f1rr
>u)
(6)
(4), (5) and (6), one gets <.12K2(a+K1)EIrOrk12
Elnx(urk)nx(ur0)I2
+ 12(1 + 4IInII2 )E
< 12K2(a+K1)62E + 12(1 + 4IInhIc )E
< 12[K2(a+K1) + 1 + 4IIn IIc ]E for all u E [O,a], k > k0.
(7)
Note that k0 is independent of u E [O,a].
A similar argument applied to the last integrand in (1) yields for every
E>0, a k 0 > 0 such that Elnx(udk) nx(ud0)I2 for all u E [O,a], k > k0. (7) and (8) to obtain
176
<
12[K2(a+K1) + 1 + 4IInhlc ]E
(8)
Now put together the inequalities (1), (2), (3),
t E
sup Inxk(v) nx(v)I2 r
<
(4aL2+4K1L2)
sup IN k(v)nx(v)I2 du
E
f
0 rtv
+ 12a(4aL2 + 4K1L2)[K2(a+K1)
for k > k0 and for all t E [O,a].
+ 1
+ 411n II2]c
(9)
By Gronwall's lemma, it follows that,
if k > k0, then E
Inxk(v) nx(v)I2 sup r
for all t E [0,a].
<
]ee4L2(a+K1)t
48aL2(a+K1)[K2(a+K1)+1+4IInIIC2
In particular, take t = a to get
xk(w) nx(w)II2dP()< 48aL2(a+K1)[K2(a+K1)+1+4IIn1I2]ce4L2(a+K1)a
J II
for all k > k0; i.e. {nxk}k1 converges to nx in 92(S2,C([r,a],Rn)).
To demonstrate the F
D_measurability
of the map SZ 3 w'  nxwe E
2(52,C([r,a],Rn)) assume for the moment that the delays rk, dk are simple
ones of the form rk =
rm1'k Xm',k,
E
m=l
dk =
Ek
dm',k
m=1 rm',k, dm' k
for some nk > 0, 0 <
< r,
52m',k E FD,
1 < m' < nk, k = 1,2,3,...
Denote by nxm',k E l2(S2,C([r,a],Rn)) the unique solution of the stochastic DDE
dxm" k(t) = h(xm'k(trm',k))dt + g(xm''k(tdm'k))dw(t), xm',k 0
= n
with fixed (deterministic) delays rm k = 1,2,3,...
.
0
,k, dm
k,
for each m' = 1,...,nk,
Clearly the approximations Ox 'w :w' E
2}
are then given
by nxk'wo()
E
=
1
nxm''k(w)Xek(w)
(10)
m for a.a. w, w' E 52, k = 1,2,3....
.
Since the simple delays can always be
chosen so that rk < rk+1' dk < dk+1 for all k > 1 and rk(w')  r0(w'),
dk(w')  d0(w') as k >  for a.a. w' E 52, it follows from what has already
177
been proved that
= lim nxk,w
nxw
in t (S2,C([r,a],Rn)) for a.a. w' E 0.
k,
But (10) implies that the map S2 3 w' , ; nx w E L 2 (c,C([r,a],Rn)) is FDmeasurable and 0 x S2 3 (w',w) ' 'Ix kw (w) E C([r,a],Rn) is FD ® Fameasurable for each k > 1. By the StrickerYor Lemma (Theorem (1.5.1)) the random family {nxw' :w' E Q j admits an FD 0 Fa measurable version nX:S2 x S2  C([r,a],Dr) ,
such that for a.a.
E S2,
w'
nxw
Moreover the map
a.s.
S2 3 w' + nxw' E L2(SZ,C([r,a],Rn)) is F D_measurable and is in fact essent
ially bounded because, for a.a. w' E S2, t E
lnxw (t)I2 < 3IInII2 + 3 E
sup
r
< 311n 11C
sup
h(nxw,(ur(w')))dul2
I
0
sup I1t g(nxw'(ud(w')))dw(u)I2 0[t