On the existence, uniqueness and parametric dependence on the coefficients of the solution processes in McShane's stochastic integral equations

Publicacions Matemàtiques, Vol 38 (1994), 11-24 . ON THE EXISTENCE, UNIQUENESS AN D PARAMETRIC DEPENDENCE O N THE COEFFICIENTS OF THE SOLUTIO N PROCESSES IN McSHANE'S STOCHASTI C INTEGRAL EQUATION S ADRIAN CONSTANTI N A bs tract In this paper we use the Schauder fixed point theorem and methods of integral inequalities in order to prove a result on the existence, uniqueness and parametric dependence on the coefficient s of the solution processes in McShane stochastic integral equation s 1 . Introductio n In this paper we consider the problem of the existence and uniquenes s of the solution processes in the stochastic integral syste m r  t (1) xZ ( t) = a Z( t) + E J g j ( S , x ( S )) dz j j=1 0 r  t h ; k (s, x(s) } dz i (s)dz k ( s ) , o Ç t Ç a, z = 1, . . . , ~z , 0 where the stochastic integrals involving dz 3 are interpreted as McShan e stochastic integrals . Some previous results were given by McShane [13] in the special cas e in which a z is not depending on time and the processes z i are sample continuous and by Elworthy [8] which does not require these conditions but considers stronger hypotheses on g t i and h ; k . Recently, Angul o Ibañez and Gutiérrez Jaimez [1] proved (under weaker requirements tha n in [8] and [13]) an existence and uniqueness theorem for (1) assuming 12  A . CoNSTANTI N that the functions g i i and h i i k satisfy a Lipschitz condition and the pro - cesses zá satisfy a K-condition, using an adequate version of the Fomin - Kolmogorov fixed point theorem . They considered in [2] the problem o f the convergence of the solution processes in McShane's stochastic integral equation systems with coefficients depending on a parameter unde r similar assumptions to [1] . Under the hypothesis of a weaker condition than the Lipschitz condition on and h t i ' k we prove an existence and uniqueness result for th e solution process of equation (1) applying Shauder's fixed point theore m [15] and methods of integrodifferential inequalities (see [3], [4], [5], [7] , [9]) . Under the same hypothesis we prove also the continuity with respect to the initial condition and we consider the problem of parametri c dependence of the solution processes on the coefficients, generalizing th e results of [2] . 2 . Preiiminarie s Let (S - 2, F, P) be a complete probability space and let {F, o t < a } be a family of complete u-subalgebras of F such that if o Ç s ç t Ç a then F s Ç Ft . Let L 2 be the space of all random variables y : SZ --> IR with finit e L 2 -norm 11 . II and let L2 be the space of all random variables x : S2 —> If ~ n with finite norm ii • jj ,~ , n II x IIn =  iIl2,  x = (Xi, . . . ,xn) E L . i = We say that the real valued second order stochastic process z on [o, a ] satisfies a K-condition if z is adapted to the F t ( i .e . z(t) is F t -measurabl e for every t E [o, a] } and 1E[(z(t) — z(s)) P /F s li  Kit — s i a .s . whenever o Ç s ç t < a, p = 1,2,4 . An example of such a proces s is a Wiener process with respect to the F t , o Ç t ç a . Let us define also for each second arder process x on [o, a] the nor m = sup {iix(t)ii} . tE[O,a l For the basic elements of the McShane stochastic calculus theory w e refer to McShane [13] and Elworthy [8] . Let us remind only that if f : ON MCSHANE TYPE STOCHASTIC INTEGRAL EQUATIONS  1 3 [o, a] —> L 2 is a measurable process adapted to the F t and if t -~ I is Lebesgue integrable on [o, a], then ([S], [9]), if z 1 and z 2 satisfy a K -condition, the McShane integral s J 0 Q f(s) dz i (s), J Q f(s) dz l ( s ) dz 2 ( s ) 0 exist, and the following estimates are tru e f a f ( s ) dz i (s ) a 1f (s) dzi(s) dz 2( s ) C j J a IIi 2 cu y ' J where C = (2 + 8Ka) 1 " 2 . We will use in Chis paper the following resul t Lemma (Bihari [3]) . Let v, h be positive, continuous functions o n 0 < t < a, and let q be a nonnegative constant ; further let w be a positiv e nondecreasing function on [0, oo) . Then the inequalit y t v(t) < q +  h(s)w(v(s)) ds, 0 < t < a , 0 implies the inequality L t v( t) < G -1 (G+  h(s) 1 \  / where G(u) = f~s} u > o, and a ' zs defined so that G(q) + f h(s) d s lies within the domain of definition of G -1 for o Ç t < a ' . For proofs of the lemma, see [3] and [5] . 3 . Existence and uniqueness of solution s Let C [O, a] denote the space of all processes x : [o, a] —> L2 which ar e continuous and adapted to the F t . A solution to the equation (1) o n [O,a] is a process x E C[0, a]which satsifies (1) on [O,a] . Let us assume tha t (H 1 ) the noise processes z j , j = 1, . . . , r, satisfy on [o, a] a K -condition ; ds, otÇa ' , 14 A . CONSTANTI N (H2) if f is any one of the functions h i . n, : [ o, a] x L2 —> L 2 , z = 1, . . . , n ; j, k = 1, . . . , r, then f (s, x) is continuous in x on L~ fo r every s E [0, Q] , and far any x E C[0, a] , the process t —> , f (t, x (t ) } is measurable and . F t -adapted with t --~ II f (t, x (t)} 1j 2 bounded o n [O,a] ; (H3) there exists a continuous, nondecreasing function w : R+ ~ R + with w(0) = o, w(t) > o for t ~ o and 1 ds  t d s hm  = oc, lim  = o 0 t—o t w(s)  t—}oo ~ w(s ) such tha t ll f ( t , x ) — f( t , y) I1 2 < w (ll x — II),  t E [O,a],  x , y E L z ; (H4) the initial condition a belongs to C[0, a] . Theorem 1 . Let us suppose that the hypotheses (H 1 )-(H 4 ) are satis - fied . Then there exists an unique solution of the equation (1) on [0, a] . Proof : Let us first prove the existence of a solution on [o, a] . W e define the operator T : C[0, a] —> C[0, a] b y r ¡ t Tx(t) = a(t) + E J g i (s, x(s)) dz ; (s ) j=1 0 r ~- E ft IZ i k( s, x(s)) dz j (s) dz k (s), 0  t < a . Let M = 2n sup lñ + 4rtC 2 (r+r 2 ) 2 Ka wher e tE[O,a]ll Ils ; (t~ o) II~,  sup  ,0)1 1 2 t E[o]  tE[O,a l i=1, . . .,n ; 3=1, . . .,r  i=1, . . .,n ; 7,k=1, . . ., r K = max  su p t E [o, ~, a n ; and let us consider ¡ d s G : (0, oo) —+ R, G(u) =  w(s ) (the hypotheses guarantee that G is a bijection from (0,00) to R) . ON MCSHANE TYPE STOCHASTIC INTEGRAL EQUATIONS  1 5 We also consider the functio n m(t) = G--1 (G(M) + 4nC 2 ( r + r 2 ) 2 t), o ç t ça . This function has the property that ¡ t m(t) = M + 4nC 2 (r + r 2 ) 2 J w(m(s)) ds, 0 < t < a , n o being the solution of the differential equatio n m ' (t) = 4nC 2 (r + r2)2w(m(t)), 0 < t < a , with initial condition m(0) = M . We consider the se t B = {x E C[0, a] : Ilx(t)~~n < m(t), 0 < t < a} . This is a closed, bounded andconvex subset of the Banach space C[0, a ] and we will show thatT(B) c B . Let x E B . We have then that t  1 1/ 2 fo t G C { J 9j (S , x(s)) Ii 2 dS 1 ~  J r ft  1/ 2 (h i t (s, x(M — g i i(s, 0)II + lig ;(s, 0)11) 2 d s l J <C (J t ( 2 II9 ;( s , x ( s )) — 9 ;( 5 , 0 )11 Z + Z I9~( S ,0 )11 2 ) ds 1 l o ( ¡t  2Kt }  C { 2 J w(llx(s)ii~) ds -E2Kt }  0 < t a , l  o  J and similarl y t h~k(s,x(s)) dz i (s) dz k (s ) f e  1 / 2  C{2  w(11x(s)iIn) ds + 2Kt }  0 < t < a . o deduce tha t IIT=(t)II„s  sup I eElo,ml ( ¡t  1 / 2  + 2n(r -}- r 2 )c  w(11x(s) iIñ) ds + Kt }  0 t < a , lo  J 1/2 16  A . CONSTANTI N thus (since x E B ) !i Tx ( t ) PIñ  t  1 / 2 a sup  1 w(m(s)) ds + Ka } tE[O,a]  f < 2n E Í p lla(t) + 4nC 2 (r + r2)2  w(m(s)) ds + 4nC 2 (r + r 2 ) 2 K a t ¡ t = M + 4nC 2 (r ~- r 2 ) 2 J w(m(s)) ds = m(t), 0 < t a . n 0 Hence T(B) c B . In a similar way we prove that if x E B, the n I ITx(t) — Tx(s) 11 n < 4nC 2 (r + r2)2 ft w(m(u)) du + 21la(t) — +4nC 2 (r+r 2 ) 2 (t—s)K, 0<s<t<a , from where we conclude that the set T(B) is equicontinuous . On the other hand we have for x, y E B tha t r  l 11 Tx ( t ) —Ty2( t)C li(( )) - 9~ sy ( s )) 11 2ds } j= l {f t 1 r  l { f t Ilh k (s,x(s)) — h(s,y(s)) 2 ds }  0  t < a . 7,k 1 From (H 3 ) and the continuity of g i j (s, x ) and h3 k (s, x ) in x we deduce b y the Lebesgue convergence theorem that T is continuous . An applicatio n o f Schauder' s fixe d poin t theore m enable s us to deduce that T has a fixe d point in B, thus equation (1) has a solution on [0, a] . Let us now prove the uniqueness of solutions far equation (1) . Suppose that there exist two different solutions x, y E C[o, a1] of equation (1) on some interval [o, a l ] with 0 < a l Ç a . Then there exist point s 0 C t < a1 with ii x (t) - -- y(t)II n> 0 . Let b be the lower bound of thes e t . We have then that l— y(b)l1n = = o, but l— y(t)ii n > 0 fo r b < t Ç b+ ,c3 with a certain number ,@ ~ 0 . ON MCSHANE TYPE STOCHASTIC INTEGRAL EQUATIONS  1 7 . We have that r  t  1 1/ 2  ll x2 ( t ) — yZ ( t ) Ii  E c ~~ Il9(5, x ( s )) — 9~( 5, ~J( S ))11 2 ds 5 .9=1  0 t  1/ 2 + É  C ¡ J Ij,k=1  l 0  ¡ ¡t  l i / 2 (Cr+Gr)J w(llx(s) — y(s) lIñ) ds i  lo  J thu s 11 x(t) — y(t)~~ñ Ç n(Cr + Cr 2 ) 2 f t w~~ i x(s) — y(s)Ii ñ ) ds, 0 G t a l . 0 Sílice w(0) = 0 and il x(t) — y(t) li n = 0 on [0, b] we obtain tha t f li x(t) — y(t) l n(Cr+Cr 2 ) 2 w(llx(s) — y(s)~~) ds, b t < b+~ , thus (denoting v(t) = li x(t) — y(t) ii ñ, b < t < b+ O) for every e > 0 w e have ¡ t v(t)  + n(Cr -}- Cr 2 ) 2 J w(v(s)) ds, b < t G b + ~ . b Let V E (t) = e + n(Cr + Cr 2 ) z fb w(v(s)) ds, b < t < b + /~ . We hav e then tha t VÉ (t) = n(Cr + Cr 2 ) Z w(v(t)) < n(Cr + Cr2)2w(V,(t)), b < t b+ ) 3 , thus (since V E (t) > 0 on [ b , b +~] ) V'(t)  < n(Cr + Cr 2 ) 2 , b < t < b + ,3 • w(V E (t) ) An integration yield s G(V E (t)) — G(V E (b)) < n(Cr + Cr 2 ) 2 (t — b) , thu s G(V E (t)) < G(V E (b)) + n(Cr + Cr 2 ) 2 (t — b ) < G(e) + n(Cr + Cr 2 ) 2 , b < t b + ~3 . 0 CtÇ al f bctcb+í3, 18  A . CONSTANTI N Since 0 < v(t) < V E (t) for b < t < b + /3, we deduce tha t G(v(t)) < G(e) + n(Cr + Cr 2 ) 2P , b c t < b + for every E > O . This leads to a contradiction since v(t) > 0 far t E (b, b + O] thu s G(v(t)) is a real number for b < t < b + but limG(E) = ---oo . E—> o Thus equation (1) has an unique solution on [0, a] . This completes the proof of Theorem 1 . ■ Remark 1 . Theorem 1 is not only an existence and uniqueness result , it provides also a bound for the solution x E C[0, al : lix(t)III < m(t), O < t  a . Remark 2 . If the hypothesis (H 3 ) is satisfied with w(t) _ L t on IR + (L > 0 being a real constant) we obtain the existence and uniquenes s theorem of Angulo Ibañez and Gutiérrez Jaimez [1] and as a special cas e (when a is not depending on time t and the processes z 3 are sample continuous) of this we obtain the existence and uniqueness result of McShan e [13] . Our requirements are weaker in some aspects with respect to thos e in [8] (Elworthy requires in [8] Lipschitz conditions on the function s and h~ k) . 4 . Continuity with respect to the  conditio n W e proved in the preceding section that if the conditions (H 1 )-(H 3 ) are satisfied, the stochastic integral equation s t T f j = (2 .k) x(t) = a k (t) + g i( s i x ( s )) dz7 (s ) r  t + E fh ik (sx(s))dz i (s)dz k (s) , 0<t<CL , have an uniquesolution x k E C[0, a] if a k E C [O, al, k ~ 1 . Theorem 2 . If the hypotheses (H 1 )-(H 3 ) are satisfied andzf al, Ct2 E C[0, a], then H X 1  x 2lll 2 n  C -' –1 ( G ( K 1IIIQ 1 — a 2111 2 n) + K 2) ON MCSHANE TYPE STOCHASTIC INTEGRAL EQUATIONS  1 9 where K l = 1 + r + r 2 and K 2 = nK l (C Z r + C 2 r 2 )a . Proof .• In a similar way to the proof of Theorem 1 we can stat e I —x z( t )ii 2 r 1/ 2 (t)—(t)11 +E c  (s, xi(s))—9~(s, x z( s ))11 2ds } j=1 u t T  1/ 2 + c ut Il h jk( s , x l( s )) — ~ jk ( s , x 2( S )) II 21, ~ 8 1 i,k=1  2 r ( 1 + r + r2 ) 1f w (11 x i( s ) —x 2( s )llñ) d s ~=1  0 T  f t + CZ  w (11 x l( s ) — x2(s)Il2n) ds  0 < t < a , thus (taking supremes on [0, a] to the initial condition term ) li x i( t ) —x 2( t )Ii 2 n ~ (1+r+r 2 )IIIc i —a zlli n ¡ t + n(1 + r -Ir 2 )(C 2 r -}- CZr2) J w (11 x 1( s ) — x 2( s ) Iiñ) ds, 0 < t < a . 0 By Bihari's inequality we obtain tha t 11 x i( t ) — x z( t )IIn ~ G —1 (G((1 + r + r2 )Ill a i — a 2 111 2 n + n(1 + r + r 2 ) (C 2 r + C Z r 2 )t), 0 t < a , so that illx l x 2 ill 2 n < G —1 (G(K 1  a 2 111 2 n ) + K 2 ) . E Theorem 3 . If the hypotheses (H 1 )-(H 3 ) are satisfied, cx, a n E C [o, a ] and lim Ick — alf i n = o, we have tha t k—>oa ~~~ IIIXk–XWn= 0 where x E C[0, a] is the solution of equation (1) and x k is the solutio n of (2 .k)with initial condition a k E C[0, a], k > 1 . Proof .• By Theorem 2 we have tha t 111 x k – x llll 5 G-1 ( G ( K illi a k – a lil 2 ) + K 2)