On some stochastic functional-integral equation

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1991)

A. A

(Gdansk) and A. E. R

(Voronezh) On some stochastic functional-integral equation

Abstract. We consider the existence and uniqueness problem for the stochastic equation x(t) = f ( t , l x , x), t e [0, T] , x(0) = 0, where

m

(Ix)(t) = j a(s, x )d s+ J b(s, x)dw(s),

о

w is a Wiener process, a(t), (j(t) < t and f ( t , y , ■), a(t, •), b(t, •) are functions of Volterra type. Our conditions on / , a, b, are weaker than the Lipschitz condition. Some examples are given and regularity of solutions is considered.

We consider the stochastic functional-integral equation (1) x{t) = f{ t, lx, x), fe[0 , T], x(0) = 0, where

aw /?(t)

{Ix)(t) = J a(s, x)ds-b j b{s, x)dw(s),

о о

w is a Wiener process and the functions f, a, b depend on all previous history of the process x on (0, t]. We assume some conditions on /, a, b, which are weaker than the Lipschitz condition in the second argument. Furthermore, we assume that for every и the operator (Fx)(t) — x(t)—f ( t , u, x) is invertible in a space CaT of continuous functions on [0, Г] which are zero for t = 0 and have bounded growth at zero. It is proved (see Section 2) that if, in addition, the functions /, a, b have at most linear growth, the equation (1) has a unique solution x on [0, T] and Ex belongs to Cj-. It is proved in Section 3 that the trajectories of x belong to C \ almost surely (a.s.). The method of successive approximations is used for the proof of the uniqueness and existence theorem (for similar ideas see [4], [5]). Section 3 contains some statements and examples connected with this method. Section 4 is devoted to the proof of the existence of a regular solution of (1) under some assumptions weaker than the assumption of linear growth of f, a, b.

1. Notations. Let (Û, P) bé a complete probability space, (f ^ 0) an increasing right-continuous family of sub-tr-algebras of the c-algebra ^ , Rd the

2 — Comment. Math. 30.2

d-dimensional Euclidean space with norm |-|, CT the space of continuous functions x : [0, T]->R d with norm ||x||r = sups<r |x(s)|, &t the c-algebra of cylindrical sets of the space CT,

& = cr{Fx (s, t]: s < t, F e ^ s},

^ — o{F x (s, Q xG : s < t , F e ^ s, G e& s}, J* = < t {F x (s, t]x G jX G2: s < t, F e ^ s, Glt G2e &s}.

The measurability of a matrix-d or vector d-valued function means the measurability of all its components.

Let w(t) = (w1^), ..., wp(t)) be a p-dimensional Wiener process. Set

||y|| = (tryy*)1/2 if y is a matrix. Let BT be the space of measurable random processes £(£, со) with a.s. continuous paths such that <^(t, со) is measurable with respect to ^ for each £ g [0, Г], £(0, со) = 0; set

||^||BT:= (E sup|^(s, co)|2)1/2,

where E is the mathematical expectation. Define ^ C j.= { x e C T: ||x||r>e: = sup|x(£)/£a| < oo},

t^T

B*T:= { £ e B T: ||£||B«: = sup t~a(E ||£(-, co)||2)1/2 < oo}, 0 <t^T

for some a ^ 0.

For detailed definitions and facts from the theory of random processes and stochastic differential equations see, for example, [2], [3].

2. Main results. We need the following assumptions:

(H0) The vector-valued function f = (/*), i = 1, ..., d, is defined on Q x [0, T] x CT x CT, the vector-valued function a = (a1) and the matrix-valued function b = (bu), i = 1, ..., d, j = 1, ..., p, are defined on Q x [0, T] x Cr , / g # , a, b e # ; f ( t , x, y )e C T a.s. for every x, y e C T a.s.

(Hj) t (£), a (t), f(t) ^ t for t e [ 0, T] and there exists a constant a ^ 0 such that

|/( t, U, lOI2 < H 1(r2*44N ,+IN ?,,,),

|a(t,x)|2 =S H 2( r 2l x | | 2 + (2“ ), К>(г,х)|2 « Я 2( Г 2(|||х||,2 + (2и),

for £ g [0, T], x, u, v e C j, where the constants y, f , el5 e2 fulfil the following conditions:

у < m in{l, a + i}, > - i , e ^ a - l ,

^ < a + i, e2 > - h e2 ^ a - 2 -

(H2) There exist continuous functions at , ot2: [0, T] x R + -►R+ : =

= [0, oo), h: [0, T]-»>[0, T] such that h(t) ^ t, aft, •), i = 1, 2, are convex and nondecreasing, a f - , u) is nondecreasing, a f t , 0) = a 2(t, 0) = 0, £e[0, T], and

I f i t , m 15 v ) - f ( t , u2, v)\2 < t ^ a f t , Г 2в||и1- м 2||?), Ia(t, u f — a(t, u2)I2 ^ tyia2(t, r 2e||M1-M2||Jt)), IHt, u f - b ( t , u2)I2 ^ tPia2(t, t~2a\\ul — u2\\l(t)), for £e[0, T], Mj, m 2, v g C j , where y1 ^ max{0, 2a —1}, ^ 2a.

(H3) There exists a continuous function к: [0, T]-*R + such that I fit, u, v f - f i t , u, v2)\ ^ k(t) K - i > 2||t(r)

for £e[0, T], u, vlt

2

C

and there exists L ^ 0 such that Jc:= sup /с (£)(r (£)/£)“ eL(t(0-r) < 1.

fe(0,T]

(H4) For each z0eC([0, Г], R+) the sequence {zn} defined by the formula y(t)

Zn + i (0 = Cot f t , K J a2(s, zn(h(s)))ds), re [0 , Г], о

where

y(t):= max max{a(s), /?(s)},

se[0,t]

is uniformly convergent to 0 as и -> oo, for

c

(1 - Щ ~ 2, К

2 (T 1 + yi~2a + 4T Pl~2a).

R em ark 1. The condition 1c < 1 is equivalent to lim sup к (£)( t (£)/£)“ < 1, sup k(t) < 1

t->0+ f*0

x(t) = t

(see [1]).

T

1. I f Assumptions (H0) — (H4) are satisfied, then in B j there exists exactly one solution of equation (1).

P ro o f. Define

IIIlf,a,L •= sups~ae~Ls|x(s)|, x e C f, (O.f]

||^||Bf>L:= sups-a e-Ls(.E||£(', со)\\2у / 2, Ç e B f (0,f]

The above norms are equivalent to || • ||t>a and |j * j|B« respectively. Take an arbitrary x 0e B j and define

*n + i(£) = f ( t , Ix n, xn + i), n = 0 , 1 , . . . , £ e [0 , Т].

Then

t~ae~Lt\xn + l { t) - x 0{t)\ < t~ae~Lt\f(t, Ixn, x n+1) - f { t , Ix n, x0)|

+ t~ae~Lt\f(t, Ix„, x 0)\ + t~ae~u \x0(t)\

< ( t (ty tf eL(m '■ 0 к (0 1| xn + ! - x 0 1| T(t) e ~ Lt(t)т (t) ~ + r ae~Lt\f( ty Ix n, x0)l +ll*o II t.«.L

< £\\Xn +

+ t~ae~Lt\f{t, lx n, x0)|

+ ll^oll*,*,!.»

so, using the inequality

s t

E sup |J (p(r, co)dw(r)\2 < 4 §E\q>(r, co)\2dr,

O ^ s ^ t О 0

we obtain for ô: = min{2 —2y, 1 — 2)9} > 0

(2) E\\xn + l —x 0\\la>L < 2(1 —£)_1( sup s~2ae~2LsE \ f (s, Ix n, x 0)\2

se(0,t) \

+ II^ o II^. l )

a(r)

^ ^ ( l d - sup s~2<xe~2Ls(s2x + E sup I j a{q, xn)dq\

se( 0,t) 1*6(0 ,s) 0

m

+ E sup I J b(q, xn)dw{q)|2))

re(0,s) 0

y(s)

>* ^ ^4 1 (l -h sup s _2ae_2is(y(s) j E\a{r, x„)|2dr

se( 0,t) 0

y(s)

+ 4 J E\b{r, x j 2dr))

0

s

^ A 2{\ + sup s -2ae~2Ls(s§(r~27E\\xn\\?+ r2ei)dr

se(0 ,f) 0

+ ]{r~2pE\\xn\\? + r2E2)dr))

0

^ 4 2( l + sup s~2ct(s$(r~2y + 2aE\\xn\\la'L + r2ei)dr

se(0,r) 0

+ |(r -2 ^ 2 .£ ||xJ 2^

+ r 2e2j

0

< Л 3(1+ (г£||х„||,2.„,1)

for some constants A x, A 2, A 3 > 0. Define g„(t) = Е\\хп\\1а,ь.

Then from the above we obtain

0n +

i(t)

A 3(l + tôgn(t)).

We prove that there exists a constant H 0 ^ 0 such that (3) 9 „ { t) ^ H 0, n = 0 , 1 , 2 , . . . , te [0 , Г].

Take tj > 0 such that l : = A 3tt < 1 and put h0 = (A3 + g0)(l— 1) 1, where g0 = ma.x[0jl]g0(t). It is obvious that g0{ t ) ^ h 0, te [0 , t j , and, by induction,

9n(t)<h> n = 0 , 1 , . . . , £e[0, r j .

From (2) we obtain for some constants A4, A 5 ^ 0 and £e[0, T]

0B + i(O < Л ( 1 + SUP s~2(s j r - 2y+2agn(r)dr+ ] r ' 2p + 2!Xgn(r)dr))

s e [ f j , f ] 11 11

ti

Now (3) follows easily by induction for H 0 = GeGt, where G = A 5 + supg0{t) + h0.

[ 0 . Г ]

We put z0 = H 0 and we prove that

(4) E\\xn + m- x m\\la,L < zm(t), n, m ^ 0, te [0 , T],

where {z„} is defined in (H4). (4) is fulfilled for m = O.If we assume that it holds for some m, then

\xn+m+l( t ) - x m+1( t ) \ r ae~u

= If( t , Ix m+„, xm+n + 1) - f ( t , Ixm, x m + l) \ r ae~Lt

^ ^"ll-^n + m + l Х щ + 1 II z(t),a,L

+ 1

f{t, Ix m+n,

+ 1) - f ( t , l x m, xm + 1)\t~,xe~Lt, so

E\\xn + m + 1- x m + 1\\ti0l'L ^ { l- J c ) ~ 2E sup a^s, \\Ixm+n- I x J 2s~2a) e ' 2Ls

s e ( 0 ,0

< C ax(t, E sup \\Ixn + m- I x m\\2s~2ae~2Ls)

s e ( 0 ,t )

y (s)

< C ax(t, sup 2[>’(s) j E\a(r, x m+n)-a (r, x j |2dr

s e ( 0 ,t) 0

y(s)

+ 4 ] E\b(r, x m+n)-b { r, xm)\2dr]s~2ae~2Ls) о

y(s)

^ C a f i t , sup 2[s J ryitx2{r, r~2aE\\xm+n- x m\\l(r))dr

se (0 ,t) 0

y(s)

+ 4 ] r * a 2(r, r~2aE\\xm+n- x mU(r))dr]s~2ae~2Ls)

0

y(s)

^ C<xt (t, К sup j a2(r, r~2ete~2LsE \\xm+n — xm\\t;ir))dr)

se(0 ,f) 0

' y(t )

^ C Cti(t, К J" $ 2 (Г 5 + n Хт\\цг),а,ь)^т)

0

y ( t )

^ C a ^ t, K J a2(r, zm(h(r)))dr = zm + 1(t), о

and (4) follows by induction. This means that { x j is a Cauchy sequence in Вт and so it has a limit x * e B j . A s usual we can prove that x* is a solution of (1). Uniqueness is based on Assumptions (H2)-(H4). The proof is complete.

R e m a rk 2. The case a = y = e1 =/? = e2 = y1 = — 0 is possible.

We obtain x * e B T in this case.

In some cases we can replace Assumption (H3) by the following:

(H5) fit, u, v) = fi( t, u)+ f2(t, v) and the operator ф defined by (фх)(t): = x ( t ) - f 2{t, x), t e l 0, T],

maps C j into itself, and has an inverse operator ф~1 which fulfils the Volterra condition and the Lipschitz condition

\\Ф~1х - ф ~ 1у\\г,а ^ ЯоЦх-ylk *, telO , T], . for some constant R0 ^ 0.

T

2. I f Assumptions (H0)-(H 2), (H4), (H5) are satisfied for C = R 0, then the assertion of Theorem 1 holds.

P ro o f. We have

II

II

^

II ^ФХп +1 ^Фх

II

= R q || x „ + 1 f 2{ x„ + 1 ) X q + / 2('» xo)lk«

< R0 ll/i(*» bc„)||t>a + R0 ||x0- / 2(-, X0)||,,«

and

I l + n + 1 ЭСщ + 1 II t,a ^ * 0 II Ф^т + n + 1 Ф%т + 1 IL,a

K

И^т + и+ 1 У г ( ’ -^m + n + l ) %m + 1 -t- / 2 ( *» -^m+l)llf,a

*0ll/l( ’ ^т + и) fli'»

so we complete this proof in the same way as the proof of Theorem 1 for L = 0.

3. Remarks. The lemma given below can be used for verifying Assump­

tion (H4).

L

. Suppose that Assumption (H3) holds and the problem

ÿ(t) = K a 2(t, Coc^hit), y(y(h(t])))), y(0) = 0, has a unique zero solution. Then Assumption (H4) is satisfied.

P ro o f. Since a l5 a2 are continuous and oc^t, 0) = 0, there exists L ^ 0 such that

Coe^T, (K/L) sup a2(s, M)) ^ M

(0.T)

for M := sup[0tT]z0(t). Define

*o(t) = MeLt,

y(t)

zn+l(t) = Cot^t, К f a2(s, z„(h(s)))ds), te[0, Г], n = 0, 1, 2, ...

о

The sequence {z„} is defined by the same formula as {zn}, but for a different z0.

It is obvious that z0 ^ z0 and we obtain

y(t) t

z^t) = Coc1(t, K J a2(s, MeLm )ds) ^ C a^t, К j e Lsa2(s, M)ds)

о о

t

^ C a ^ t, K J eLsds sup a2(r, M )) ^ Ca±(t, (K/L)eLt sup a2(r, M))

0 ( 0 . Г ) (0 ,T)

< eLtCa1(t, (K/L) sup a2(r, M)) ^ eLtM = z0(t).

(0 .Г )

Now, it is easy to prove by induction that

(6) zn +

zn ^ zn, n 0, 1, 2 ,. . . Put

yn(t) = K j a 2(s, zn(h(s)))ds, te [0 , T], и = 0 , 1 , 2 , . . .

о

Then we obtain

ÿn + i(t) = K<x2(t, zn + 1(h(t)))

( mt)) _ w

= K a2\t, Ccc^hit), K J a2(s, zn(h(s)))dsjj о

= K<x2(t, СаДВД, yn(y(h(t])))).

Using (6) we can easily prove by induction that the sequence {y„} is nonincreasing. Since

o < yn(t) < Уо(0> 0 ^ y'n(t) ^ K a2(t, C a1(/j(t), yo(# (0 ))))

for t e [О, Г], we deduce that there exists a continuous function у: [0, T] ->R+

such that {y„} is uniformly convergent to у on [0, Т]. It is easy to prove that у is a solution of the problem for the assumptions of the lemma, so у = 0. From (5) we obtain

zn+ i(t) = Cotjft, y„(y(0))>

so {zn} is uniformly convergent to zero. Since < zn, the same is true for {z„}.

This completes the proof.

Now we consider the following

E

. We will prove that if T ^ 1, y(h(t)) ^ t7 and a f t , и ) = A^ulnu, a2(t, и) = Л2ил or a f t , и) = Л1м“, a2(f, и) = A 2u\nu for ae(0, 1], у ^ 1/a and sufficiently small u, then Assumption (H4) is satisfied. It follows from the Lemma that it is sufficient to prove the following

P

1. I f T ^ 1 and a continuous, nonnegative function у fulfils the inequality

y(t) 5$ A $ f(y{s7))ds, t e l 0, Г], о

where

f ( x )

= _(ae)"1 ^{— x“lnx} for x e [0 , e v], for x > e~y, ae(0, 1], у = 1/a, A ^ 0, then у = 0.

P ro o f. First, we prove that y(t) — 0 for t e l 0, tf\ and sufficiently small tj > 0. Let re(0 , be fixed. It is obvious that there exists M ^ 0 such that

y t r K M t 1- 2', te [0 , Т].

Suppose that

y(t) < Btô+r, Btyf +r) ^ e ~ y,

for t e l 0, 1 1 ] and some ô, В ^ 0 . Then we obtain for te [0 , tf]

y{t) ^ - A jB as ^ +r)“ln(B s^+rV s

0

t

= - A B a \ sôsry(ô + r)\n(BllMÔ+r))s)ds

0

t

^ - ABay(ô + r)$sôB~r/MÔ+r))(B1/MÔ+r))sy\n{BllMÔ+r))s)ds о

t

^ ABay(ô + r)§sôdsB~r/MÔ+r)) sup ( — ur\nu)

0 u e ( 0 , l )

^ ABmy(ô+r)) у tô +1 (re) ~1 ^ A y (re) ~11\ Bmy(ô+r»tô+1~r.

ô + r

Take îj > 0 such that Ау^геУ1^ ^ Then

y(t) ^ ±BÔIMÔ+r))t{ô + l ~2r)+r, t e [ 0, t j . Now, we see by induction that if

<5„+1 = d„ + l - 2 r , <50 = 1 — 2r, B„+1 = B0 = M and

(7) n = 0, 1 ,...,JV 0,

then

(8) y ( t K S / * ' , n = О, 1 , JV0 + 1, t e [ 0 , t j .

We prove that (7) is satisfied for some t t > 0 and each N 0 ÿ 0. We have 0„ = ( n + l) ( l- 2 r ) and <V(y(<5„ + r)) < 1.

If we assume that Bn > 1 for every n ^ 0, then Bn + l ^ ^Bn and Bn ^ 2~nB0 -►(), n-+oo. This contradiction proves that there exists iV\ such that BNl ^ 1. It is easy to prove by induction that Bn ^ \ for n > N 1. This means that the sequence {Bn} is bounded and there exists t1 > 0 such that (7) is fulfilled for each N 0. It follows that (8) is satisfied for all N 0 and

y ( t ) ^ B ntin + 1){1- 2r)+r-+ 0, и — > oo, so y(t) = 0 for te [0 , tj].

Assume that у ф 0. Then there exists t2 e (0, T) such that y(t) = 0 for t e [0, t2] and у ф0 on (t2, t3) for each f3 > t2. In the case у > 1 we have ty2 < t2 and ty < t2 for t e l t 2, t2 + e] and some e > 0. It follows that

t

0 ^ y(t) ^ A f f(0)ds = 0, t e [ t 2, t2 + e],

о

which is a contradiction. In the case a = y = 1 we obtain z(t) < A j /(z(s))ds

о

for z(t) = y(t —12) and we can prove in the same way as in the first part of the proof that z(t) = 0 for t£ [0 , e] and some e > 0. We come to a contradiction again. This means that y = 0 on [0, Т]. This completes the proof.

Now we shall prove the following result on the trajectories of the solution x* of equation (1).

P

2. I f s2 > a — \ and the assumptions of Theorem 1 or 2 hold, then x*eC*r a.s.

P ro o f. It is sufficient to prove that f0 b(s, x)dw(s)eCr a.s. for x e B j. Let

x e B j and || x |||« ^ K. Then for every 0 < A < В we obtain P( sup t a|J b(s, x)dw(s)| > e )

A*ZtiZB

о

в

^ P( sup A ~a\§b(s, x)dw(s)| > e)

A ^ t ^ B

о

в

^ A ~ 2a£~2M J H 2(s~2p ИХд || + s2e2)ds

0

в

^ A ~ 2as~2 J H 2(s2e2 + s2(a~p)s~2aM\\x\\2)ds

о в

< A ~ 2a e ~ 2 H 2 f (s262 + Ks2(a " P))ds

0

/ о 2 е 2 + 1 l£ T )2 (c t — P )+ I N

" А ~2^ н { ^ х + г ( ^ ш ) « DA- 2^ BP' + ^

for p« = 2e9 +1 > 0, p9 = 2(a — B) + 1 > 0 and some constant D > 0. Setting A = 2 ~k~1d, В = 2~kd we have

t

P( sup t _a|J b(s, x)dw(s)| > e)

2 ~ k ~ 1d ^ t ^ 2 ~ kd 0

^ D22(k + 1)ad~2a£~2{2~kpidpi+ 2~kP2dP2)

< D1£~2(2k(2a~pl) + 2k(2a~P2)) ^ D2£~22kp

for some constants Dlt D2, where p — max(2a—pl5 2a—p2) < 0. Then we have

t

Se := P( sup |t- “J b(s, x)dw(s)| > e) 0

^ X D2E~22kp = D2 e ~2{ 1 - 2 p) - 1 = D 3 b ~2.

к = 0 Put

t

Qt = {o>: sup I t _aJb(s, x)dw(s)| = oo}.

O< t ^ d О

Suppose that P ^ ) > 0. If we take e > (2D3P(Q1)~1)112 then Р(Ох) ^ Se <

< ^P(f21), so P(Oa) = 0. This contradiction proves Proposition 2.

4. Regularity of solutions. Suppose that (H6) For every r > 0

Ia(t, x)\2, Ib(t, x)\2 < Pr(t), t e l 0, T], ||x||r ^ r, where fir(t) is an 3Ft-measurable function and j j Pr(s)ds < oo a.e.

(H7) There exist random functions N t: [0, T] xR d->Rd, i = 1, 2, 3, and a number r > 0 such that the functions N ft,- ) , i= 1 ,2 ,3 , iV3(-,x) are nondecreasing, N 3 is bounded on [0, Г] x [0, r] and

\a{t, x)| ^ N t {t, ||x||t), Ib(t, x)| ^ N 2{t, ||x||f),

\f(t, x, 0)| ^ N 3(t, ||x||() for te (0, T], ||x||f ^ r.

(H8) There exists e > 0 such that for every C1 > 0 , Tlf T2e(0, T], Tt < T2, each continuous, nondecreasing, nonnegative solution of the inequality

y(0

(9) y(t) ^ CA[ f N t (s, CN3(s, y(s)))ds о

7(0

+ [ | N 2{s, CN3(s, y(S))fds]42+‘ + l ] , t e [ T T 2), 0

extends to [Tl5 T2].

T heorem 3. Let Assumptions (H0), (H2), (H4), (H6)-(H8) and (H3) for L = a = 0 hold. Then there exists exactly one ^-measurable solution of equation (1) defined on [0, Т].

P ro o f. Define

an{t, x) =

f n(t, u, x) =

a{t, x), ||x||t ^ n

a(t> (и/1М1»> l|x||t > n

f( t , u, x), llMllt ^ n

f{t, {n/\\u\\t)u, x), \\u\\t > n

and define bn in the same way as an. It follows from Theorem 1 that equation (1) has exactly one solution хпе В т for a = an, b = bn, f — /„. Put

e„{co) = inf{te [0, T]: |x„(t)| > n}, coeQ.

It is obvious that x n(t) = x m(t), for t e [0, e„(œ)], n, and x„ is a solution of (1) on [0, £„((«)]. If we put

x*(t) = xn(t), te [0 , e„(co)], n = 1, 2,

then x* is well defined and is a solution of (1). To complete the proof we need to show that

(10) P(e(co)>T) = l t

where e(co) = lim„_>00e„(aj).

Suppose that (10) is not satisfied, i.e. P(e(co) < T) > 0. Define =

= {co: e(cc>) < T}. It is clear that

lim sup |x*(t)| = +oo for coe (21.

We will arrive at a contradiction with this condition. Put p(co) = sup{t > 0: |x*(t)| = r} for

Note that for te[p {со), e(a>)) we have

y(t) y(t)

j b(s, x*)dw{s) — w*( I |b(s, x*)|2ds),

о о

where w* is a Wiener process (see [3]). Therefore P(limsupw*(u)(nlnlnM)_1/2 = 1) = 1

U~*ao

(see [2]). It follows that there exist random values Я ^co), H 2{co) bounded a.s.

such that

w*(u) ^ Я 1(ш)п1/2+е + Я 2(ш), where s occurs in Assumption (H8).

It is easy to obtain the estimate

y(«) y(q)

||x*||f < CsupiV3(#, I j a(s, x*)ds| + | j b(s, x*)dw(s)|)

q*Zt

0 0

for C = (1 — lc)~2 (cf. the proof of Theorem 1). Now we have

7(0 7(9)

l|x*||t < CN3(t, J N ^ s , ||x*||s)ds+ sup|w*( J |fe(s, x*)|2ds)|)

0 q^t 0

7(f) 7(f)

<CiV3(t, J N ^ s , ||х*||5) ^ + Я 1(ш)( J \b(s, x*)\2ds)ll2+E + H 2(co))

о 0

7(f) 7(f)

< CN3(t, J N ,(s, ||х* У & + Н » ( J JV2(s, ||x*U 2<fs)1/2+, + H 2(o>))

0 0

We define

y(t) y(t)

y(t)= J N ^ s , ||x*||s)ds + H 1(co)(f N 2{ s , \\x*\\s)2ds)1/2+E + H2(œ).

о 0

Then

( 11 ) \x*\\t ^ CN3(t, y(t)),

y(t)

y ( t )

< I ^{N t (s,}

3(s, y(s)))ds v(t)

+ « ! < » ) ( J ^{N 2( s , CJV3 (}

, j-(s)))2) I/2+*<is + f f 2(<B).

0

It follows from Assumption (H8) that y extends to [p(œ»), е(ш)] for and condition (11) implies the boundedness of x* for coeQ. This contradiction shows that condition (10) holds true, and completes the proof.

R em ark 3. Below, we consider some sufficient conditions for assumption (H8) to be satisfied. We assume that e > 0 is sufficiently small and

<x(t), P(t) ^ t, t e l 0, Г].

a) Suppose that N 3(t, u) = A l + B 1u. Then (9) takes the form

y(t) y(t)

y ( t ) ^ H l ( J N 1( s ,A 1+ B 1y(s))ds+( S N 2(s, A l + B 1y(s))2dsYl2+E-fl).

о 0

Put p1 = 2(1+ 2e)-1 , qx = (1— рГ1)-1 = 2(1—2e)-1 . Then from the above we obtain

y(t) y(t)

y(ty 1 < н з(( f JVj(s, A 1+ B 1y(s))dsY1+ J ^{N 2( s} , A 1 + B1y(s))2d s + 1)

о 0

y(t) y(t)

Н г(у(1У‘">' J ^N,{s, J N 2(s, A 1 + Bl y ( s f d s + l)

0 0

y(t) ' y(f)

^ H4( f N ^ s , A 1 + B1y(s))Plds+ j N 2( s , A x-\-B1y(s))2d s + 1).

о 0

If

lV1(t, u) = C1(t)u\nllplu\nllPi\nu..A nllpi\n..Anu, N 2(t, и) = C2(t)upll2\nll2u\nll2\nu..A nll2\n..Anu,

T T

j C^sY'ds, J C2{s)2ds < oo, a.s.,

о о

then Assumption (H8) is fulfilled.

b) If a(t), /?(t) < t for t > 0 then Assumption (H8) holds true. Also, in this case we do not need Assumption (H7) to be satisfied.

c) Suppose that |b(t, x)| ^ b0. Then (9) takes the form

y(f)

y{t) « Я 3( J JV,(s, C N 3(s, y(s)))ds + 1).

0 If

N ^ t, и) = C1(t)uhi1/2Mln1/2ln u ...ln 1/2ln ...ln u , N 3{t, и) = C3(t)uln1/2Mln1/2ln u ...ln 1/2ln...lnM, and

T

J C1(s)C3(s)1+ôds < + oo о

for some <5 > 0, then Assumption (H8) is satisfied for any a, /1.

d) Suppose that for sufficiently large u,

N ±(t, u) = C1(t)uhi1/(2pi)uln1/(2pi)lnM...ln1/(2pi)ln ...ln u , N 2(t, u) = C2(t)uPl/2ln1/2uln1/2lnu.. An1/2ln .. Anu, N 3(t, u) = C3(t)wln1/2uln1/2ln u ...ln 1/2ln.. Anu.

In the same way as in case a) we deduce from (9)

?(0

ypi(t) < Я 4( J C1(s)pl/2C3(s)pl+£y(s)lny(s)lnlny(s)...rfs о

+ J C2(s)2C3(s)Pi +ey(5)lny(s)ln lny(s)... dsj.

0

We see that Assumption (H8) is satisfied if

T T T

J C1(s)Pll2dsi f C1{s)Pil2C3{s)pl+eds, \ C 2{s)2C2{s)p' +Eds< со

o o о

References

[1] A. A u g u s t y n o w ic z and M. K w a p is z , On the existence o f continuous solutions o f operator equations in Banach spaces, Casopis Pëst. Mat. I l l (1986), 261-219A

[2] R. S. L ip t s e r and A. N. S h ir y a e v , Statistics o f Random Processes, Springer, Berlin 1977.

[3] H. P. M c K e a n , Stochastic Integrals, Academic Press, 1969.

[4] A. E. R о d к i n a, On existence and uniqueness o f solution o f stochastic differential equation with heredity, Stochastics 12 (1984), 187-200.

[5] J. Tur'o, Existence and uniqueness o f random solutions o f nonlinear stochastic functional integral equations, Acta Sci. Math. (Szeged) 44 (1982), 321-328.

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