Dynamic Systems and Applications 16 (2007) BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS 1. INTRODUCTION. x t + t

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1 Dynamic Syem and Applicaion 16 (7) BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS MICHA L KISIELEWICZ Faculy of Mahemaic, Compuer Science and Economeric, Univeriy of Zielona Góra, Podgórna 5, Zielona Góra, Poland ABSTRACT. Exience of oluion o backward ochaic differenial incluion i conidered. The paper conain he baic noion dealing wih backward ochaic differenial incluion. AMS (MOS) Subjec Claificaion. 47H4, 49J53, 6H5. 1. INTRODUCTION Given meaurable e-valued mapping F : [, T R m Cl(R m ) and H : R m Cl(R m ) by a backward ochaic differenial incluion BSDI(F, H) we mean relaion (1.1) x E x T H(x T ) [ x + F (τ, x τ)dτ F ha have o be aified a.. for every T by a cádlág proce x = (x ) T defined on a complee filered probabiliy pace P IF = (Ω, F, P, IF) wih a filraion IF = (F ) T aifying he uual hypohei (ee [14). E[x + F (τ, x τ)dτ F denoe he e-valued condiional expecaion (ee [4, [5) of he e-valued mapping Ω ω x (ω) + F (τ, x τ(ω))dτ R m wih repec o he ub-σ-algebra F F. If P IF i given hen x, aifying condiion preened above, i aid o be a rong oluion o BSDI(F, H). In a general cae we can look for yem (P IF, x) aifying condiion (1). Such yem are aid o be weak oluion o BSDI(F, H). I i clear ha if x i a rong oluion o BSDI(F, H) on P IF, hen a pair (P IF, x) i i weak oluion. Backward ochaic differenial incluion can be reaed a ome generalizaion of backward ochaic differenial equaion of he form [ (1.) x = E h(x) + f(τ, x τ, z τ )dτ F a.. where he ripe (h, f, z) i called he daa e of uch equaion (ee [, [3, [7, [13). Uually, if we conider rong oluion o (1.) apar from (h, f, z), a probabiliy pace P = (Ω, F, P ) i alo given and a filraion IF i defined by a proce z by aking IF z = (F z ) T, where (F z ) T i he malle filraion aifying he uual Received Augu 3, $15. c Dynamic Publiher, Inc.

2 1 MICHA L KISIELEWICZ condiion and uch ha z i IF z -adaped. Proce z i called he driving proce. In pracical applicaion he driving proce z i aken a a d-dimenional Brownian moion or i i a rong oluion o a forward ochaic differenial equaion. he cae of weak oluion o (1.) apar from h and f a probabiliy meaure µ on he pace D(IR d ) of d-dimenional cádlág funcion on [, T i given and i weak oluion wih an iniial diribuion µ i defined a a yem (P IF, x, z) In aifying (1.) and uch ha P z 1 = µ and every IF z -maringale i alo IF - maringale. Le u oberve ha in paricular for a given weak oluion (P IF, x) o BSDI(F, H) wih H(x) = {h(x and F (, x) = {f(, x, z) : z Z} for (, x) [, T IR m, where f and h are given meaurable funcion and Z i a nonempy compac ube of he pace D(IR d ), here exi (ee [8, Th. II.3.1) a meaurable IF-adaped ochaic proce (z ) T wih value in Z uch ha [ (1.3) x = E h(x) + f(τ, x τ, z τ )dτ F For given probabiliy maure µ and µ T on IR m, we can look for a weak oluion (P IF, x) o BSDI(F, H) uch ha P x 1 = µ and P x 1 T = µ T. If F and H are uch a above hen here exi a meaurable and IF-adaped ochaic proce (z ) T uch ha (1.3) i aified and uch ha E[h(x) + f(τ, x τ, z τ )dτ = udµ IR m. If f(, x, z) = f(, x) + g(z), wih g C(D(IR d ), IR m ), hen g(v)dλ τ dτ = udµ h(u)dµ T E f(τ, x τ )dτ D(IR d ) IR m IR m where λ = P z 1 for [, T. In ome pecial cae weak oluion o BSDI(F, H) decribe a cla of recurive uiliie under uncerainy (ee [7). To verify ha uppoe (P IF, x) i a weak oluion o BSDI(F, H) wih H(x) = {h(x and F (, x) = {f(, x, c, z) : (c, z) C Z}, where h and f are meaurable funcion and C, Z are nonempy compac ube of C([, T, R + ) and D(R m ), repecively. Similarly a above we can elec a pair of meaurable IF-adaped ochaic procee (c ) T and (z ) T wih value a C and Z, repecively and uch ha [ (1.4) x = E h(x) + f(τ, x τ, c τ, z τ )dτ F a.. for T. In uch a cae (1.4) decribe ome cla of recurive uiliie under uncerainy, where (c (, )) T denoe for fixed [, T he fuure conumpion. Le u oberve ha in ome pecial cae a rong oluion x o BSDI(F, H) on a filered probabiliy pace P IF wih he conan filraion IF = (F) i a oluion o a backward random incluion x cof (, x ) wih a erminal condiion x T H(x T ) a.. for a.e. [, T. A uual cof (, x ) denoe he convex hull of he e F (, x ). Throughou he paper we aume ha P IF = (Ω, F P, IF) i a complee filered probabiliy pace wih a filraion IF = (F ) T aifying he uual hypohee. Given a..

3 BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS 13 P IF we denoe by ID(IF, R m ) he pace of all m-dimenional IF-adaped cádlág prcee on P IF and by S (IF, R m ) he e of all m-dimenional IF-emimaringale x uch ha x S = E[up [,T x <. We have S (IF, R m ) ID(IF, R m ). I can be proved (ee [14, Th. IV.1, Th. V..) ha (S (IF, R m ), S ) i a Banach pace. The preen paper i mainly devoed o properie of oluion e of weak coninuou oluion o BSDI(F, H). I i organized a follow. Secion conain ome properie of he e-valued condiional expecaion of e-valued inegral. In Secion 3 ome meaurable elecion heorem are given. Exience heorem o BSDI(F, H) are given in Secion 4 and Secion 5. Finally, in Secion 6 a weak compacne of he e X (F, H) of all coninuou weak oluion o BSDI(F, H) i proved.. CONDITIONAL EXPECTION OF SET-VALUED INTEGRALS Le (Ω, F, P ) be a probabiliy pace and le G be a ub-σ-algebra of F. Given an F-meaurable e-valued mapping Φ : Ω Cl(R m ) wih a nonempy e S(Φ) of all i F-meaurable and inegrable elecor here exi (ee [4) an unique (in he a.. ene) G-meaurable e-valued mapping E[Φ G aifying (.1) S(E[Φ G) = cl L {E[ϕ G : ϕ S(Φ where cl L denoe he cloure operaion in L(Ω, G, R m ). We call E[Φ G he mulivalued condiional expecaion of Φ relaive o G. Thi condiional expecaion ha properie imilar o hoe of he uual one. For example, we have A E[Φ GdP = A ΦdP for every A G, where inegral are underood in he Aumann ene (ee [5, Prop. 6.8). I can be proved (ee [5, Prop. 6.) ha for given meaurable and inegrably bounded e-valued mapping Φ, Ψ : [, T Ω Cl(R m ) one ha Eh(E[Φ G, E[Ψ G) Eh(Φ, Ψ), where h i he Haudorff meric on Cl(R m ). Le G : [, T Ω Cl(R m ) be meaurable and inegrably bounded, i.e. here i m L([, T Ω, R + ) uch ha G(, x) m(, ω), a.e., where R + = [, ) and G(, ω) = up{ g : g G(, ω. A uual we denoe by S(G) a e of all inegrable elecor for G. We have S(G) = {g L([, T Ω, R m ) : g(, ω) G(, ω) a.e.}. I i eay o verify (ee [8) ha S(G) i nonempy and decompoable, i.e. ha for every f, g S(G) and E β T F one ha 1 E f +1 E g S(G), where β T denoe he Borel σ-algebra of [, T and E i he complemen of E. In paricular, if G(, ω) are convex ube of R m for (, ω) [, T Ω, he e S(G) i a convex weakly compac ube of L([, T Ω, R m ). Then i i alo a cloed ube of hi pace. For he given above G we can define an Aumann inegral Φ(ω) = G(, ω)d depending on a parameer ω Ω. By Aumann heorem (ee [8, Th. II.3.) G(, ω)d i a nonempy, convex compac ube of R m for every ω Ω. Furhermore, G(, ω)d = for ω Ω. Hence and ([8,Th. II.3.1) we obain he following reul. co G(, ω)d

4 14 MICHA L KISIELEWICZ Propoiion.1. Le G : [, T Ω Cl(R m ) be meaurable and inegrably bounded. Then a e-valued mapping Φ : Ω Conv(R m ) defined by Φ(ω) = G(, ω)d for ω Ω i meaurable. Proof. By virue of ([8,Th. II.3.8) i i enough only o verify ha he funcion Ω ω (p, Φ(ω)) R i meaurable for every p R n, where (, A) denoe a uppor funcion of A Cl(R m ). By he meaurabiliy of G and i inegrably boundedne a funcion [, T Ω (, ω) (p, G(, ω)) R i meaurable for every p R m (ee [8, Remark II.3.5). By virue of ([8, Th. II.3.1) for every p R m one ha (p, Φ(ω)) = (p, G(, ω))d for ω Ω. Hence he meaurabiliy of he funcion Ω ω (p, Φ(ω)) R follow for every p R m. Therefore Φ i F-meaurable. Propoiion.. Le G : [, T Ω Cl(R m ) be meaurable and inegrably bounded and le Φ(ω) = G(, ω)d for ω Ω. Then S(Φ) i a nonempy convex weakly compac ube of L(Ω, F, R m ). Furhermore, ϕ S(Φ) if and only if here i g S(co G) uch ha ϕ(ω) = g(, ω)d for a.e. ω Ω. Proof. By Propoiion.1, Φ i F-meaurable. I i alo inegrably bounded, becaue Φ(ω) m(, ω)d for a.e. ω Ω. Therefore (ee [8, Th. III..3) S(Φ) i a nonempy convex weakly compac ube of L(Ω, F, R m ). For every g S(co G) a func- ion ϕ(ω) = g(, ω)d i a meaurable elecor for Φ, becaue of ([8, Th. II.3.) we have Φ(ω) = co G(, ω)d for ω Ω. I i alo inegrably bounded, becaue ϕ(ω) m(, ω)d for a.e. ω Ω. Then ϕ S(Φ) for every g S(co G). Aume now ϕ S(Φ). Then for every A F one ha E A ϕ E A Φ, where E A ϕ = ϕdp A and E A Φ = ΦdP. Le ε > be given and elec a meaurable pariion A (Aε n) Nε n=1 of Ω uch ha E A ε m(, )d < n ε/n+1. For every n = 1,..., N ε here i a gn ε S(G) uch ha E A ε n ϕ = E A ε n gε n(, )d. Le g ε = N ε n=1 1 A ε n gε n. By he decompoabiliy of S(G) one ha g ε S(G). We have g ε S(co G) becaue S(G) S(co G). Taking a equence (ε k ) k=1 of poiive number ε k > uch ha ε k a k we can elec g S(co G) and a ubequence, denoed again by (g ε k ) k=1, of (g ε k ) k=1 weakly converging o g in L([, T Ω, R n ), becaue S(co G) i a weakly compac ube of L([, T Ω, R n ). For every A F and k = 1,,... here i a ube {n 1,..., n p } of {1,..., N εk } uch ha A A ε k ni for i = 1,,..., p and A A r = for r {1,,..., N εk } \ {n 1,..., n p }. Therefore Nεk E Aϕ E A g ε k (, )d E A A ε k n ϕ E ε A A k n g ε k n (, )d = n=1 p E A A ε k ni ϕ E ε A A k i=1 p i=1 ni g ε k E ε A k ni m(, )d ε k n (, )d

5 BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS 15 for every k = 1,,.... On he oher hand for every A F we alo have E Aϕ E A g(, )d E T Aϕ E A g ε k (, )d + E A g ε k (, )d E A g(, )d ε k + E A g ε k (, )d E A g(, )d for k = 1,,.... Hence i follow ha E A ϕ = E A g(, )d for every A F, T becaue ε k and E A gε k T (, )d EA g(, )d a k. Therefore ϕ(ω) = g(, )d for a.e. ω Ω. Corollary.3. If G : [, T Ω Cl(R m ) i meaurable and inegrably bounded hen ( ) { } S G(, )d = g(, )d : g S(co G). Corollary.4. If G : [, T Ω Cl(R m ) i meaurable and inegrably bounded and G i a ub-σ-algebra of F hen ( [ ) S E G(, )d G = { [ } E g(, )d G : g S(co G). Proof. I i enough only o ee ha he e H = {E[ g(, )d G : g S(co G i a cloed ube of L(Ω, G, R m ). By he properie of condiional expecaion and he properie of he e S(co G) i follow ha H i a convex weakly compac ube of L(Ω, G, R m ). Therefore H i a cloed ube of L(Ω, G, R m ). 3. MEASURABLE SELECTION THEOREMS Le x = (x ) T be an IF-adaped m-dimenional cádlág proce on P IF. Given a meaurable and uniformly inegrably bounded mulivalued mapping F : [, T R m Cl(R m ) we denoe by F x a e-valued mapping defined on [, T Ω by eing (F x)(, ω) = F (, x (ω)) for (, ω) [, T Ω. I i clear ha F x i meaurable and IF-adaped, i.e., i i β T F-meaurable and uch ha for every fixed [, T a mapping Ω ω (F x)(, ω) R m i F -meaurable. In wha follow we hall denoe by S IF (F x) a e of all meaurable and IF-adaped elecor for F x. Le u oberve ha F x i meaurable and IF-adaped if and only if i i Σ IF -meaurable, where Σ IF = {A β T F : A F for T } and A denoe a ecion of a e A β T F a [, T. Therefore, immediaely from Kuraowki and Ryll- Nardzewki meaurable elecion heorem (ee [8, Th. II.3.1) i follow ha for he given above F and x he e S IF (F x) i nonempy. In he general cae we hall alo denoe by S IF (G) he e of all meaurable and IF-adaped elecor for a given

6 16 MICHA L KISIELEWICZ meaurable IF-adaped and inegrably bounded e-valued mapping G : [, T Ω Cl(R m ). Similarly a above we can verify ha S IF (co G) i a nonempy convex and weakly compac ube of L([, T Ω, Σ IF, R m ). We hall prove he following meaurable elecion heorem. Theorem 3.1. Le G : [, T Ω Cl(R m ) be a meaurable IF-adaped and inegrably bounded e-valued mapping. Aume x = (x ) T i an m-dimenional meaurable proce on P IF uch ha E x T <. Then [ (3.1) x E x + G(τ, )dτ F a.. for every T if and only if here i g S IF (co G) uch ha [ (3.) x = E x T + g(τ, )dτ F a.. for every T. Proof. Suppoe here i g S IF (co G) uch ha (3.) i aified. Then for every T one ha [ [ x = E x T + g(τ, )dτ F = E g(τ, )dτ F [ +E x T + g(τ, )dτ F and Therefore E[x F = E x = E [ x T + g(τ, )dτ F [ x + g(τ, )dτ F a.. for T. Hence by Corollary.4 i follow ha ( [ ) x S E x + G(τ, )dτ F for T. Therefore, (3.1) i aified a.. for T. Aume ha (3.1) i aified for every T a.. and le m L([, T Ω, R + ) be uch ha G(, ω) m(, ω) for a.e. (, ω) [, T Ω. For every T one ha E x E x T + E m(, )d <. By virue of Corollary.4 x i IFadaped. Le η > be arbirarily fixed and elec δ > uch ha δ < T and up T δ E +δ m(τ, )dτ < η/. For fixed [, T δ and τ + δ we have x E[x τ + τ G(, )d F a.. Therefore, for every A F we ge E A (x x τ ) τ τ E A G(, )d. Then E A (x x τ ) E A G(, ) d E +δ m(, )d < η/ for every T δ and A F. Therefore, up τ +δ E A (x x τ ) η/ for every A F and fixed T δ. Le τ =, τ 1 = δ,..., τ N 1 = (N 1)δ < T Nδ. a..

7 BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS 17 Immediaely from (3.1) and Corollary.4 i follow ha for every i = 1,,..., N 1 here i g η i S IF (co G) uch ha [ E x τ i 1 E x τi + τi τ i 1 g η i (, )d F τ i 1 =. Furhermore, here i g η N S IF(co G) uch ha [ E x τ N 1 E x T + g η N (, )d F τ N 1 =. τ N 1 Define g η = N 1 i=1 1 [τ i 1,τ i )g η i + 1 [τ N 1,T g η N. By he decompoabiliy of S IF(co G) we have g η S IF (co G). For fixed [, T here i p {1,,..., N 1} or p = N uch ha [τ p 1, τ p ) or [τ N 1, T. Le [τ p 1, τ p ) wih 1 p N 1. For every A F one ha ( [ ) E A x E x T + g η (, )df [ τp+1 E A (x x τp ) + E x τ p E x τp+1 + g η (, )dτ F τp τ p τp + E A (E[x τp+1 F τp x τp+1 ) + E g η (, )d + ( [ τp+1 + E A E g η (, )d F τp τ p [ + E x τ N 1 E x T + [ τp+1 E g η (, )dτ F τ p ) + + g η (, )dτ F τn 1 τ N 1 [ + E A (E[x τn 1 F τn 1 x τn 1 ) + E A (E g η (, )d F τn 1 τ N 1 [ ) E g η (, )d F τ N 1 E x τ i E N + i=p [ x τi+1 + τi+1 τ i +δ up E A (x x τ ) + E m(, )d+ τ +δ g η (, )d F τi [ +E x τ N 1 E x T + g η (, )dτ F τn 1 τ N 1 E A (E[x τi+1 F τi x τi+1 ) + N + i=p [ τi+1 ) E g η (, )d F ( τ i [ + E A E g η (, )d F τn 1 τ N 1 N i=p ( [ τi+1 E A E g η (, )d F τi τ i [ ) E g η (, )d F. τ N 1

8 18 MICHA L KISIELEWICZ Bu F F τi and N i=p Hence i follow for i = p, p + 1,..., N 1. Then for A F one ha ( E A E N i=p [ τi+1 τ i E A (E[x τi+1 F τi x τi+1 ) =, g η (, )d F τi E [ τi+1 τ i g η (, )d F ) = ( [ T [ ) E A E g η (, )d F τn 1 E g η (, )d F =. τ N 1 τ N 1 E A for fixed T and A F. ( [ ) x E x T + g η (, )d F η Le (η j ) j=1 be a equence of poiive number converging o zero. For every j = 1,,... we can elec g η j S IF (co G) uch ha (3.) i aified wih η = η j. By he weak compacne of S IF (co G) here are g S IF (co G) and a ubequence (g η k ) k=1 of g η j ) j=1 weakly converging o g in L([, T Ω, Σ IF, R). Then for every A F F one ha lim k E A g η k T (, )d = EA g(, )d. On he oher hand for every fixed [, T and A F we have ( [ ) E A x E x T + g(, )d F ( [ ) E A x E x T + g η k (, )d F + ( [ [ ) + E A E g η k (, )d F E + g(, )d F η k + E A g η k (, )d E A g(, )d for k = 1,,.... Therefore E A (x E [ ) x T + g(, )d F = for every A F and fixed T. Bu x and E[x T + g(, )d F are F -meaurable. Then x = E[x T + g(, )d F for T wih (P.1). Corollary 3.. Le G : [, T Ω Cl(R m ) be meaurable IF-adaped and quare inegrably bounded. If x = (x ) T i meurable, aifie (3.1) a.. for every T and E x T < hen x S (IF, R m ) and x = x + M + A, where M = E[x T + g τdτ F E[x T + g τdτ F and A = g τdτ for T wih g S IF (cog) uch ha x = E[x T + g τ dτ F a.. for T.

9 BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS 19 Proof. The reul follow immediaely from he repreenaion x = E[x T + g τ dτ F given in Theorem 3.1 (ee [3, Lemma 1.1). In wha follow we hall aume ha F : [, T R m Cl(R m ) and H : R m Cl(R m ) aify he following condiion (A): (i) F i meaurable and uniformly quare inegrably bounded by a funcion m L ([, T, R + ), (ii) H i meaurable and bounded by a number L >, (iii) F (, ) i Lipchiz coninuou, i.e. here i k L ([, T, R + ) uch ha h(f (, x 1 ), F (, x )) k() x 1 x for a.e. [, T and x 1, x R m, where h i he Haudorff meric on Cl(R m ), (iv) here i a random variable ξ L (Ω, F T, R m ) uch ha ξ H(ξ) a.. We hall prove now ha condiion (A) imply he exience of ome pecial equence of ucceive approximaion for BSDI(F,H). Theorem 3.3. Le F : [, T R m Cl(R m ) and H : R m Cl(R m ) aify condiion (A). There exi a equence (x n ) n=1 of S (IF, R m ) defined by x n = E[ξ + f n 1 τ dτ F a.. wih f n 1 S IF (F x n 1 ) for n = 1,,... and T uch ha x n E[x n + 4E( k(τ) up τ T x n x n 1 F (τ, xn 1 τ )dτ F a.. and E up u T x n+1 u x n u dτ) for n = 1,... and T. Proof. Le u oberve (ee [8, Th. II.3.13) ha for every m-dimenional meaurable and IF-adaped procee x and y on P IF and every f x S IF (F x) here i f y S IF (F y) uch ha f x (ω) f y (ω) = di(f x (ω), F (, y (ω)) h(f (, x (ω)), F (, y (ω)) k() x (ω) y (ω) for a.e. [, T and ω Ω. Furhermore, by properie of H here i ξ L (Ω, F T, R m ) uch ha ξ H(ξ) a.. Le (x ) T be an m-dimenional meaurable IF-adaped proce on P IF uch ha x T = ξ a.. and le f S IF (F x ). Define x 1 = E[ξ + f τ dτ F a.. for T. By Corollary 3. we have x 1 S (IF, R m ). Selec now f 1 S IF (F x 1 ) uch ha f 1 f = di(f 1, F (, x )) for a.e. T wih (P.1). Then f 1 f k() x 1 x a.. for a.e. T. Define x = E[ξ + f 1 τ dτ F a.. for T. We have x S (IF, R m ). Coninuing he above procedure we can define x n+1 f n S IF (F x n ) uch ha f n f n 1 = E[ξ + k() x n x n 1 f n τ dτ F a.. for T wih a.. for a.e. T and n =, 3,.... By Corollary 3. we alo have x n S (IF, R m ). Hence i follow [ x n+1 x n E fτ n fτ n 1 dτ F [ E k(τ) up x n x n 1 F τ T

10 13 MICHA L KISIELEWICZ a.. for T. Therefore, up x n+1 u x n u u T [ up E u T u k(τ) up x n x n 1 dτ F u τ T [ up E k(τ) up x n x n 1 dτ F u u T τ T a.. for T and n = 1,,.... By Doob inequaliy, we obain ( [ ) E up E k(τ) up x n x n 1 dτ F u u T τ T ( 4E k(τ) up τ T ) x n x n 1 dτ for T. Therefore, for every n = 1,,... and T we have ( E up x n+1 u x n u 4E k(τ) up x n x n 1 dτ). u T τ T 4. EXISTENCE OF STRONG SOLUTIONS We hall prove ha if F and H aify condiion (A) hen BSDI(F,H) poee a lea one rong oluion. Le u oberve ha immediaely from Corollary 3. i follow ha every rong oluion o BSDI(F,H) on P IF belong o S (IF, R m ). Immediaely from he properie of mulivalued condiional expecion (ee [5, Prop. 6..) he following reul follow. Propoiion 4.1. Le F aifie condiion (A). Then for every x, y S (IF, R m ) one ha Eh ( E [ [ ) F (τ, x τ ) dτ F, E F (τ, y τ )dτ F for every T, where h i he Haudorff meric on Cl(R m ). We can prove now he following exience heorem. k(τ)e x τ y τ dτ Theorem 4.. Le P IF be given. If F : [, T R n Cl(R n ) and H : R m Cl(R m ) aify condiion (A) hen BSDI(F, H) poee a rong oluion on P IF. Proof. By virue Theorem 3.3 here i a equence (x n ) n=1 of S (IF, R m ) uch ha x n T = ξ, xn E[x n + F (τ, xn 1 τ dτ F a.. for T and E up x n+1 u x n u 4E u T ( k(τ) up x n x dτ) n 1, τ T

11 BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS 131 where ξ L (Ω, F T, R m ) i uch ha ξ H(ξ) a.. Hence i follow E up x n+1 u x n u 4T k (τ)e up u T τ u T x n u xu n 1 dτ for n = 1,,.... and T. By he properie of F and H one ha E up u T x 1 u x u IL, where IL = 4(E ξ + m (τ)dτ) + E up T x. Therefore, E up x u x 1 u 4T IL k (τ)dτ. u T Hence i follow E up x 3 u x u (4T ) IL u T = (4T ( ) IL T k (τ)dτ). ( ) k (τ) k ()d dτ τ By he inducion procedure for every n = 1,,... and T we ge E up x n+1 u x n u (4T ( )n IL n 1 T n k (τ)dτ). u T n! Then E up T x n x m a n, m. Therefore, here i a proce (x ) T S (IF, R m ) uch ha E up T x n x a n. Hence and Propoiion 4.1 i follow ( [ Edi x, E x + ( [ E x x n + Edi x n, E ( [ +Eh E x n + E x n x + E x n x + ( x n x S + F (τ, x τ )dτ F ) x n + [ F (τ, x n 1 τ )dτ F, E x + k(τ)e x n 1 τ ) 1 k (τ)dτ x n 1 x S F (τ, xn 1 τ )dτ F ) + x τ dτ F (τ, x τ )dτ F ) for every T and n = 1,,... Therefore, di(x, E[x + F (τ, x [ τ)dτ F ) = a.. for every T. Then x E x + F (τ, x τ)dτ F a.. for every T. By he definiion of (x n ) T we have x n T = ξ H(ξ) a.. for every n = 1,,.... Therefore, we alo have x T = ξ a.. Then x T H(x T ) a..

12 13 MICHA L KISIELEWICZ 5. EXISTENCE OF WEAK SOLUTIONS Aume ha F : [, T R m Cl(R m ) and H : R m Cl(R m ) aify he following condiion (B). (i) F i meaurable and uniformly quare inegrably bouned by a funcion m L ([, T, R + ), ii) H ake on convex value i meaurable and bounded by a number L >, (iii) F (, ) and H are lower emiconinuou for a.e. fixed T. We hall prove ha for F and H aifying condiion (B) here exi a coninuou weak oluion o BSDI(F,H), i.e. here exi a pair (P IF, x), wih x having a.a. coninuou rajecorie and aifying BSDI(F,H). The reul i obained by he conrucion of he Tonelli ype approximaion for a backward ochaic differenial equaion defined by ome pecial elecor of F and H on a filered probabiliy pace P IF = (Ω, F, P, IF B ) wih (Ω, F, P ) upporing an d-dimenional Browninan moion B and IF B = (F B ) T beeing a naural augmened filraion of B. The ighne of uch approximaion equence will follow from he following exenion of he claical ighne crierion (ee [1, Th..1.3). Theorem 5.1 ([9, Th. 3). A equence (x n ) n=1 of coninuou m-dimenional ochaic procee x n = (x n ()) T on a probabiliy pace (Ω, F, P ) i igh if for every ε > here i a number a > uch ha P ({ x n () > a}) ε for n 1 and here are γ, an ineger α > 1 and a coninuou nondecreaing bounded ochaic proce (Γ()) T on (Ω, F, P ) uch ha P ({ x n () x n () η}) 1 E Γ() Γ() α ηγ for every n 1, η > and, [, T. We hall alo need he following reul. Propoiion 5.. Le F : [, T R m Cl(R m ) and H : R m Cl(R m ) be meaurable and uniformly quare inegrably bounded and bounded, repecively. pair (P IF, x) i a coninuou weak oluion o BSDI(F,H) if and only if here exi ξ S(H x T ) and f S IF (cof x) uch ha x = E[ξ + f τ dτ F a.. for T and uch ha a maringale M = (M ) T defined by M = E[ξ + f τdτ F E[ξ + f τdτ F i coninuou. Proof. By virue of Theorem 3.1 a pair (P IF, x) i a weak oluion o BSDI(F,H) if and only if here are ξ S(H x T ) and f S IF (cof x) uch ha x = E[ξ + f τ dτ F a.. for T. By Corollary 3. we have x = x +M f τdτ a.. for T. Hence i follow ha x i coninuou if and only if M i coninuou. A

13 BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS 133 Propoiion 5.3. Le F : [, T R m Cl(R m ) and H : R m Cl(R m ) be meaurable and uniformly quare inegrably bounded and bounded, repecively. Aume F (, ) and H are coninuo and le (x n ) n=1 be a equence of coninuou oluion o BSDI(F,H) on a filered probabiliy pace P IF. Then (x n ) n=1 i igh. Proof. By virue of Corollary 3. we have x n = x n + M n f n τ dτ a.. for T, where f n S IF (cof x n ) and M n i an IF-maringale defined above for n = 1,,.... By properie of F and Propoiion 5., M n i for every n = 1,,... a quare inegrable coninuou maringale uch ha M n = for n 1. Furhermore M n λ a.. for T, where λ = L + m()d. Denoe by N n i for n 1 and i = 1,,..., m a real-valued IF-maringale uch ha M n = (N1 n (),..., Nm()) n for T. For every i = 1,..., m and every pariion = { = < 1 < < r = T } of [, T one ge j=1 N n i. = r 1 (Ni n ( k+1 ) Ni n ( k )) k= r 1 Ni n ( j+1 ) Ni n ( j ) max Ni n ( k+1 ) Ni n ( k ) k Ni n () max Ni n ( k+1 ) Ni n ( k ) 8λ k a.. for T. Moreover, by ([1, Th...) we have up T E Ni n r Ni n a r, where r = max k r 1 ( k+1 k ). Then here i a ubequence ( rj ) j=1 of ( r ) r=1 uch ha up T Ni n r j j. Hence i follow up Ni n up T T up T Ni n Ni n r j Ni n Ni n r j + Ni n r j + 8λ N n i a.. a a.. for every n 1 and i = 1,..., m. Then up T N n i 8λ a.. for every n 1 and i = 1,..., m. Le u oberve ha quadraic variaional proce ( N n i ) T i increaing in a.. for every n 1 and i = 1,..., m. Then for every n 1, i = 1,..., m and P -a.e. ω Ω i generae a meaure µ n i (ω) on β T = β([, T ) uch ha µ n i (ω)((, ) = N n i N n i and µ n i (ω)({}) =. Le µ n (ω)(a) = max i m µ n i (ω)(a) and µ(ω)(a) = up n 1 µ n (ω)(a) for A β T and P - a.e. ω Ω. Similarly a in he proof of ([5, Prop ) i can be verified ha µ(ω) i a meaure on β T for P -a.e. ω Ω. I can be alo verified ha for every A β T a mapping Ω µ(ω)(a) R + i a random variable uch ha µ(ω)((, T ) 8λ for P -a.e ω Ω. By Iô formula and Doob inequaliy one obain E (N n i () N n i ()) k =

14 134 MICHA L KISIELEWICZ ( k E dni (u)) n = k(k 1)E k(k 1) where C k = k(k 1) [ ( τ E up τ ( C k [E ( 4(k 1) 4k 3 in paricular for k = i follow dn n i (u) ( τ ) 4(k 1) 1 [ E ( ) (k 1) dni n (u) d Ni n τ d N n i τ ) 1 ) 4(k 1) 1 [ ( ) 1 dni n (u) E d Ni n τ, ) 4(k 1) for < T, n 1 and i = 1,..., m. Hence, E ( 1 A n i (,) (N n i () N n i ()) 4) = [ [ ( C E 1A n i (,) (Ni n () Ni n ()) 4) ( 1 ) 1 E dµ, where A n i (, ) = {ω Ω : N n i () N n i () > } for fixed < T, n 1 and i = 1,..., m. Then which implie ha [ E1A n i (,) (Ni n () Ni n ()) 4 ( 1 ) 1 C [E dµ, ( E (Ni n () Ni n ()) 4 CE ) dµ for < T, n 1 and i = 1,..., m. Hence i follow ( m 4 E M n M n 4 = E Ni n () Ni () ) n C m m i=1 i=1 ( E (Ni n () Ni n ()) 4 C m mce ) dµ for < T and n 1, where C m i a poiive number depending on m. Finally, here i a poiive number C = m ()d uch ha 4 E x n x n 4 4E M n M n 4 + 4E fτ n dτ ( 4C m mce dµ) + 4C ( ) E [C mcm = E Γ() Γ(), dµ + C( ) for < T and n 1, where Γ() = C mcm dµ+c. Hence, by Doob inequaliy i follow P ({ x n x n η}) 1 η 4 E xn x n 4 1 E Γ() Γ() η4

15 BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS 135 for η >, < T and n 1. Finally, le u recall ha x n λ a.. for T and n 1 wih λ = L + m()d. Therefore, for every N 1 one ha P ({ x n > N}) λ/n for n 1. Then up n 1 P ({ x n > N}) a N. Therefore, for every ε > here i a > uch ha P ({ x n > a}) ε for n 1. Then by virue of Theorem 5.1 a equence (x n ) n=1 i igh. We can prove now he exience of coninuou weak oluion o BSDI(F,H). Theorem 5.4. Le F : [, T R m Cl(R m ) and H : R m Cl(R m ) aify condiion (B). Then BSDI(F,H) poee a coninuou weak oluion. Proof. By Michael and Rybinki coninuou elecion heorem (ee [8, Th. II.4.1 and [15, Th. ) here exi a coninuou elecor h of H and a Carahéodory ype elecor f : [, T R m R m of cof. Le P = (Ω, F, P ) be a probabiliy pace uch ha here i a d-dimenional Brownian moion B defined on hi pace. Le IF = (F ) be a naural augmened filraion of B. Le ξ B λ be a fixed poin of h, where B λ i a cloed ball of R m cenered a he origin wih he radiu λ = L + m()d. Define on P IF = (Ω, F, P, IF) a equence (x n ) n=1 of ochaic procee x n = (x n ) T uch ha x n = ξ a.. for [T, T and x n = E[h(ξ) + f(τ, x n τ+t/n dτ F a.. for T and n = 1,.... Le u oberve ha for every n 1 a proce x n i defined ep by ep begining wih he inerval [T T/n, T. For example, for [T T/n, T we have x n = E[h(ξ) + f(τ, ξ)dτ F a.. For [T T/n, T T/n we have x n = E[h(ξ) + f(τ, x n τ+t/n dτ F a.. wih x n τ+t/n = E[h(ξ) + f(u, ξ)du F τ+t/n τ+t/n becaue τ + T/n [T T/n, T. Le u oberve ha x n i for every n 1 a coninuou proce becaue of ([14, Corollary IV.1) a proce M n = (M n ) T defined by [ [ M n = E h(ξ) + f(τ, x n τ+t/ndτ F E h(ξ) + f(τ, x n τ+t/ndτ F i a coninuou IF-maringale for every n 1. Similarly a in he proof of Propoiion 5.3 we can verify ha he equence (x n ) n=1 i igh. Then by ([6, Th. I..7) here are a probabiliy pace ( Ω, F, P ), a equence ( x n k ) k=1 of coninuou m-dimenional ochaic procee ( x n k ) T and a coninuou ochaic proce x = ( x ) T uch ha P (x n k ) 1 = P ( x n k ) 1 for k 1 and up T x n k x a.. a k. Le F = ε> σ[ x u : u + ε and le φ : C T R be a coninuou and bounded funcion uch ha φ i β (C T )-meaurable, where β (C T ) = ρ 1 (β(c T )) wih ρ (x) = x( u) for x C T and u [, T. Similarly a in he proof of Propoiion 5.3 we can verify ha x n λ a.. for T and n 1, where λ = L + m()d. Hence by he properie of x n we alo have x n λ a.. for T and n 1. Le B λ be a cloed ball of R m uch a above and le g : R m R m be a coninuou exenion of a mapping I : B λ R m defined by I(x) = x for x B λ. We have

16 136 MICHA L KISIELEWICZ g(x) λ for x R m, g(x n ) = x n and g( x n ) = x n a.. for T and n 1. Therefore, = E = E = E = Ẽ { ( φ(x n ) x n E[x n + { ( E[φ(x n ) x n x n { ( φ(x n ) g(x n ) g(x n ) { ( φ( x n ) g( x n ) g( x n ) = Ẽ { ( φ( x n ) x n x n for T and n 1. Hence, i follow { ( Ẽ φ( x) x x { = lim Ẽ φ( x) n f(τ, x n τ+t/n)dτ F f(τ, x n τ+t/n)dτ F f(τ, x n τ+t/n)dτ f(τ, x n τ+t/n)dτ f(τ, x n τ+t/n)dτ f(τ, x τ )dτ } [f(τ, x τ ) f(τ, x τ+t/n )dτ + lim Ẽ {φ( x) [( x x n ) (( x x n + n { ( lim Ẽ φ( x) [f(τ, x τ+t/n ) f(τ, x n τ+t/n)dτ n { ( + lim Ẽ (φ( x) φ( x n )) x n x n n f(τ, x n τ+t/n)dτ = for T, becaue up T x n x a.. a n, φ and a mapping C T x f(τ, x τ)dτ R m are coninuou on C T and x n x n f(τ, xn τ+t/n )dτ λ + m()d < a.. By he Monoone Cla Theorem ([14, Th. I.1.8)i follow ha { ( Ẽ ψ( x) Ẽ[ x x f(τ, x τ )dτ F = for T and every F -meaurable bounded funcion ψ on C T, which implie ha Ẽ[ x x f(τ, x τ)dτ F = a.. for every T. Hence by he properie of f, i follow ha x Ẽ[ x + F (τ, x τ)dτ F a.. for T. Finally, le u oberve ha di(g(x n T ),H(x n T )) = a.. for n 1 and a funcion R m x di(g(x), H(x)) R i coninuou. Therefore, we alo have di(g( x n T ),H( x n T )) = a.. for n 1, which implie ha x T H( x T ) a..

17 BACKWARD STOCHASTIC DIFFERENTIAL INCLUSIONS WEAK COMPACTNESS OF SOLUTIONS SET We hall conider here meaurable e-valued mapping F and H aifying condiion (B) and uch ha H and F (, ) are coninuou for a.e. fixed [, T. In wha follow we hall ay ha uch F and H aify condiion (C). Denoe by X (F, H) he e of all coninuou weak oluion o BSDI(F,H). We hall how ha if F and H aify condiion (C) hen he e X (F, H) i weakly compac wih repec o he Prohorov opology. We begin wih he following reul. Propoiion 6.1. If F and H aify condiion (C) and {(P n IF n, xn n=1 i a equence of X (F, H) hen here are a complee probabiliy pace ( Ω, F, P ) and a equence ( x n ) n=1 of m-dimenional coninuou ochaic procee x n = ( x n ) T on ( Ω, F, P ) uch ha P n (x n ) 1 = P ( x n ) 1 and uch ha [ x n Ẽ x n + F (τ, xn τ )dτ F n x n T H( xn T ) for n 1, where F n = ε> σ[ xn u : u + ε for n 1 and [, T. Proof. Similarly a in he proof of ([6, Th. I..7) we define ( Ω, F, P ) by aking Ω = [, 1), F = β([, 1)) and P = µ, where µ i a Lebegue a meaure on F. Similarly, we define a equence ( x n ) n=1 of random variable x n : Ω C T uch ha P n (x n ) 1 = P ( x n ) 1 fo r n 1. By virue of Theorem 3.1 for every n 1 here are f n S IF n(cof x n ) and ξ n S(H x n T ) uch ha xn = E n [ξ n + f n τ dτ F n a.. for T. Hence i follow ha x n λ a.. for n 1 and T, where λ i uch a above. Le g and φ be uch a in he proof of Theorem 5.4. Hence and he properie of Aumann inegral (ee [8,Th. II.3.) we obain ( E n σ(p, φ(x n )g(x n )) E {σ n p, E n [φ(x n )(g(x n ) + F (τ, x n τ )dτ F n ) ( = E {σ n p, φ(x n )(g(x n ) + F (τ, x n τ )dτ) for p R m, n 1 and T, where σ(p, ) i a uppor funcion on R m. Bu x n and x n have he ame diribuion and a funcion defined by uperpoiion of σ(p, ), φ, g and F (, ) i coninuou and bounded on C T. Then he la inequaliy implie Ẽσ(p, φ( x n )g( x n )) Ẽ { ( σ p, φ( x n )(g( x n ) + F (τ, x n τ )dτ) for p R m, n 1 and{ T. Therefore, for n 1 and T one ha Ẽ[Φ( x n ) x n Ẽ Φ( x n n ) (Ẽ[ x + F (τ, xn τ )dτ. Le f n SĨF n (cof x n ) be uch ha Ẽ { ( φ( x n ) x n x n f τ n )dτ =

18 138 MICHA L KISIELEWICZ for n 1 and T. Hence, imilarly a in he proof of Theorem 5.4, i follow ha x n = Ẽ[ xn + f n τ dτ F n for n 1 and T. Therefore x n aifie he fir condiion of (9). Similarly a in he proof of Theorem 5.4 we alo ge x n T H( xn T ) a.. for n 1. We can prove now he main reul of he ecion. Theorem 6.. If F and H aify condiion (C) hen X (F, H) i nonempy weakly compac wih repec o he convergence in diribuion. Proof. By virue of Theorem 5.4 we have X (F, H). By virue of Propoiion 6.1 for every equence {(P n IF n, xn n=1 of X (F, H) here are a complee probabiliy pace ( Ω, F, P ) and a equence ( x n ) n=1 of m-dimenional coninuou ochaic procee x n = ( x n ) T on ( Ω, F, P ) uch ha P n (x n ) 1 = P ( x n ) 1 and uch ha condiion (9) are aified. By virue of Propoiion 5.3 a equence ( x n ) n=1 i igh, which implie he ighne of a given equence (x n ) n=1. Then here i a ubequence (x n k ) k=1 converging in diribuion o a probabiliy meaure P on β(c T ) a k. By virue of ([6, Th. I..7) here are a complee probabiliy pace, denoed again by ( Ω, F, P ) and a equence ( x n k ) k=1 of m-dimenional coninuou ochaic procee x n k = ( x n k ) T and a coninuou proce x = ( x ) T on ( Ω, F, P ) uch ha P (x n k ) 1 = P ( x n k ) 1 for k 1, P ( x) 1 = P and up T x n k x a.. a k. Le F = ε> σ[ x u : u + ε and le φ : C T R be a coninuou and bounded funcion uch ha φ i β (C T )-meaurable, where β (C T ) i uch a above. Similarly a in he proof of Theorem 5.4 and Propoiion 6.1 we ge ( ) Ẽ(φ( x) x ) Ẽ φ( x)ẽ[ x + F (τ, x τ )dτ F for T. Hence by virue of Theorem 3.1 here i f SĨF (cof x) uch ha ( [ ) Ẽ Ẽ φ( x)( x x f τ dτ) F = for T, which implie ha ( ) Ẽ φ( x)( x x f τ dτ) = for T. Hence, imilarly a in he proof of Theorem 5.4 i follow ha x = Ẽ[ x + f τ dτ) F a.. for every T. Then x Ẽ[ x + F (τ, x τ)dτ) F a.. for T. Similarily a in he proof of Propoiion 5.3 we alo ge x T H( x T ) a.. Then here i a ubequence (x n k ) k=1 of a equence (x n ) n=1 converging in diribuion o a oluion x o BSDI(F,H) on a complee filered probabiliy pace PĨF = ( Ω, F, P, ĨF) wih a filraion ĨF = ( F ) T. Thu X (F, H) i weakly compac wih repec o he convergence in diribuion.

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