Reflected Discontinuous Backward Doubly Stochastic Differential Equations With Poisson Jumps

Transcription

1 Journal of Numerical Mahemaics and Sochasics, () : 73-93, 8 hp:// JNM@S uclidean Press, LLC Online: ISSN 5-3 Refleced Disconinuous Backward Doubly Sochasic Differenial quaions Wih Poisson Jumps B. MANSOURI, and M. A.. SAOULI Universiy Mohamed Khider, POBox 45, 7, Biskra, Algeria, -mail: mansouri.badreddine@gmail.com Absrac. In his paper we prove he exisence of a soluion for refleced backward doubly sochasic differenial equaions wih poisson jumps (RBDSDPs) wih one coninuous barrier where he generaor is coninuous and also we sudy he RBDSDPs wih a linear growh condiion and lef coninuiy in y on he generaor. By a comparison heorem esablished here for his ype of equaion we provide a minimal or a maximal soluion o RBDSDPs. Key words : Refleced Backward Doubly Sochasic Differenial quaions, Random Poisson Measure, Minimal Soluion, Comparison heorem, Disconinuous Generaor. AMS Subjec Classificaions : 6H5, 6H, 6H3. Inroducion A new kind of backward sochasic differenial equaions was inroduced by Pardoux and Peng [] in 994, which is a class of backward doubly sochasic differenial equaions (BDSDs for shor) Y fs,ys,z s ds gs,ys,z s db s Zs dw s,, where is a random variable ermed he erminal condiion, f :, d, g :, d are wo joinly measurable processes, W and B are wo muually independen sandard Brownian moion, wih values, respecively in d and. Several auhors ineresed in weakening his assumpion see Bahlali e al [3], Boufoussi e al. [5], Lin. Q [8] and[9],n zielal.[],shieal.[3],wue al. [5], Zhu e al. [7]. A class of backward doubly sochasic differenial equaions wih jumps was sudy by Sun el al. [4], Zhu e al. [6] hey have proved he exisence and uniqueness of soluions for his ype of BDSDs under uniformly Lipschiz condiions. In addiion, Bahlali e al [] prove he exisence and uniqueness of soluions o refleced 73

2 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps 74 backward doubly sochasic differenial equaions (RBDSDs) wih one coninuous barrier and uniformly Lipschiz coefficiens. he exisence of a maximal and a minimal soluion for RBDSDs wih coninuous generaor is also esablished. In his paper, we sudy he now well-know refleced backward doubly sochasic differenial equaions wih jumps (RBDSDPs for shor): Y fs, sds gs, sdbs dk s Z sdws Use ds,de,, where s Y s,z s,u s. Moivaed by he above resuls and by he resul inroduced by Fan. X, Ren. Y [6] and Zhu, Q., Shi, Y [6, 7], we esablish firsly he exisence of he soluion of he refleced BDSD wih Poisson jumps (RBDSDP in shor) under he coninuous coefficien, also we prove he exisence soluion of a RBDSDP where he coefficien f saisfy a linear growh and lef coninuiy in y condiions on he generaor of his ype of equaion.f he organizaion of he paper is as follows. In secion, we give some preliminaires and we consider he spaces of processus also we define he Iô s formula. In secion 3, we proof a comparison heorem, secion 4 under a coninuous condiions on f we obain he exisence of a minimal soluion of RBDSDP, and finally in secion 5, we sudy RBDSDP where he generaor f saisfied a lef coninuiy in y and linear growh condiions.. Noaions, Assumpions and Definiions Le,F,P be a complee probabiliy space. For, We suppose ha F is generaed by he following hree muually independen processes: (i) Le W, and B, be wo sandard Brownian moion defined on,f,p wih values in d and, respecively. (ii) Le random Poisson measure on wih compensaor d,de ded, where he space is equipped wih is Borel field such ha, A, A is a maringale for any A saisfying A. is a finie measure on and saisfies e de. Le F W : W s ; s, F : s ; s and F, : B s B ; s, compleed wih P-null ses. We pu, F : F W F B, F. I should be noed ha F is no an increasing family of sub fields, and hence i is no a filraion. For d N, sands for he uclidian norm in d,. We consider he following spaces of processes: We denoe by S,, d, he se of coninuous F measurable processes ;,, which saisfy sup. Le M,, d denoe he se of d dimensional, F measurable processes ;,, such ha d. A se of coninuous, increasing, F measurable process K :,, wih K, K. L se of F - measurable random variables : wih. We denoe by L,,, d, he space of mappings U :, d which are measurable such ha B

3 75 B. MANSOURI, and M. A.. SAOULI U L,,, d U L,,, d d, where denoed he algebra of F predecable ses of, and U L,,, d U e de. Noice also he space D S,,R M,,R d L,,,R A endowed wih he norm Y,Z,U,K D Y S Z M U L K A. is a Banach space. Definiion.. wich saisfies A soluion of a refleced BDSDPs is a quadruple of processes Y,Z,K,U i Y S,,, Z M,, d,k A,U L,,,, ii Y fs,ys,z s,u s ds gs,ys,z s,u s db s dks Zs dw s U s e ds,de,, iii S Y, and Y S dk. We give he following (H) assumpions on he daa, f, g, S : (H.) f :, d L,,, ; g :, d L,,, be joinly measurable such ha for any y,z,u d L,,, f,,y,z,u M,,, g,,y,z,u M,,. (H.) here exis consan C and a consan such ha for every,, and y,y, z,z d, u,u L,,, i f,,y,z,u f,,y,z,u C y y z z u u, ii g,,y,z,u g,,y,z,u C y y z z u u. (H.3) he erminal value be a given random variable in L. (H.4) S, is a coninuous progressively measurable real valued process saisfying

4 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps 76 sup S, where S : maxs,. (H.5) S, P-almos surely. heorem [6].. Assume ha H. H.5 hold. hen equaion admis a unique soluion Y,Z,U,K D. he resul depends on he following exension of he well-krown Iô s formula. Is proof follows he same way as Lemma.3 of []. Lemma.. Le S,, k,, M k, M kd and L,,, k such ha: hen i ii s ds s db s s dw s dks s e ds,de, s, s ds s, s db s s, s dw s s,dk s s,e ds,de s ds s ds Δ s, s e deds s ds s e deds s, s ds s,dk s 3. A Comparison heorem s ds. Given wo parameers,f,g, and,f,g,, we considere he refleced BDSDPs, i,, Yi i f i s,y s i,zs i,us i ds gs,y s i,zs i,us i dbs dk s i Z s i dws Us i e ds,de,. heorem 3.. Assume ha he refleced BDSDP associaed wih daes,f,g,, resp,f,g, has a soluion Y,Z,K,U,, resp Y,Z,K,U,. ach one saisfying he assumpion (H). Assume moreover ha:

5 77 B. MANSOURI, and M. A.. SAOULI,, S S, f,y,z,u f,y,z,u. hen we have P a.s., Y Y. Proof. Le us show ha Y Y, using he equaion (), we ge Ȳ Y Y f s,y s,z s,u s f s,y s,z s,u s ds gs,ys,z s,u s gs,y s,z s,u s db s dks dk s Z sdw s Ū s eds,de, where, Z Z Z and Ū U U. Since Ȳs gs,y s,z s,u s gs,y s,z s,u s db s and Ȳs Z sdw s are a uniformly inegrable maringale hen aking expecaion, we ge by applying lemma. Ȳ Ȳs Z s ds Ȳs Ū s e deds Ȳs f s,y s,z s,u s f s,y s,z s,u s ds Ȳs dk s dk s Ȳ s gs,y s,z s,u s gs,y s,z s,u s ds. Since, we ge Ȳs dk s dk s Ys Y s dk s, Ȳ Ȳs Z s ds Ȳs Ū se deds Ȳs f s,y s,z s,u s f s,y s,z s,u s ds gs,y Ȳ s s,z s,u s gs,y s,z s,u s ds, o obain, by hypohesis H. and Young s inequaliy, he following inequaliy Ȳs f s,y s,z s,u s f s,y s,z s,u s ds C C Ȳs ds Z s Ū s de ds.

6 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps 78 Also, we apply assumpion H. for g o arrive a gs,y s,z s,u s gs,y s,z s,u s C Ȳ s ds Z s Ū s. hen, we have he following inequaliy Ȳ Ȳ s Z ds Ȳ s Ū se deds C C Ȳs ds Z s Ū s de ds C Ȳ s Ȳ s ds Ȳ s Z s Ȳ s Ū s de ds, C 3C Ȳs ds Ȳ s Z s ds Ȳ s Ū s deds. By choosing such ha, we have Ȳ 3C C Ȳs ds. hen using Gronwall s lemma implies ha Ȳ. Finally, we have Y Y. 4. Refleced BDSDPs Wih Coninuous Coefficiens In his secion we are ineresed in weakening he condiions on f. We assume ha f and g saisfy he following assumpions: (H.6) here exiss andc s.. for all,,y,z,u, d L,,,, f,,y,z,u C y z u, g,,y,z,u g,,y,z,u C y y z z u u. (H.7) For fixed and, f,,,, is coninuous. he nex hree lemmas will be useful in he sequel. Before saing hem, we recall he following classical lemma, ha can be proved by adaping he proof given by J. J. Aliber and K. Bahlali in []. Lemma 4.. Lef :, d L,,, be a mesurable funcion such ha: Lemma.. For a.s. every,,, f,,y,z,u is a coninuous. 3. here exiss a consan C such ha for every,,y,z,u, d L,,,, f,,y,z,u C y z u. hen exiss he sequence of funcions f n

7 79 B. MANSOURI, and M. A.. SAOULI f n,,y,z,u y, z, u inf f,,y,z,u n y y z z u u, B is well defined for each n C, and i saisfies, dp d a.s. i Linear growh: n, y,z,u d L, f n,,y,z,u C y z u. ii Monooniciy in n : y,z,u, f n,,y,z,u is increases in n. iii Convergence:,,y,z,u, B, if,,y n,z n,u n,,y,z,u, hen f n,,y n,z n,u n f,,y,z,u. iv Lipschiz condiion: n,,,, y,z,u B and y,z,u B, wehave f n,,y,z,u f n,,y,z,u n y y z z u u. Now given L, n N, we consider Y n,z n,k n,u n and resp V,N,K,M be soluions of he following refleced BDSDPs: Y n f ns,ys n,zs n,us n ds gs,y s n,zs n,us n dbs dk s n Z s n dws Us n e ds,de,, 3 S Y n,, and Y n S dk n. respecively V Hs,Vs,N s,m s ds gs,vs,n s,m s db s dks Ns dw s M s e ds,de,, S V,, and V S dk, where Hs,,V,N,M C V N M. Lemma 4.. i a.s. for all, and n m, Y n Y m V, ii assume ha H., H.3 H.7 is in force. hen here exiss a consan A depending only on C,, and such ha: U n L,,, A, Zn M,, d A. Proof.he prove of he i follow from comparison heorem. I remains o prove ii, bylemma., we have

8 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps 8 Y n Zs n ds U s n e deds Ys n f n s,y s n,z s n,u s n ds Ys n dk s n gs,ys n,z s n,u s n ds. By i in lemma 4., we have Ys n f n s,y s n,z s n,u s n ds C Ys n Y s n Z s n U s n ds Ys n ds C C Ys n ds C Ys n ds Zs n ds C Ys n ds U s n e deds, C C C Ys n ds C Also by he hypohesis associaed wih g, wege Zs n ds U s n e deds. gs,y s n,z s n,u s n gs,y s n,z s n,u s n gs,,, gs,,,, C Y s n Z s n U s n gs,,,. Chossing. hen, we obain he following inequaliy Y n Zs n ds U s n e deds C C 4C C Ys n ds Ys n dk s n Zs n ds U s n e deds Consequenly, we have gs,,, ds. Zs n U s n e de ds where C n Zs ds U n s e deds K n K n, gs,,, ds sup S s s sup Y n. Now chossing and such ha, we obain C 4C

9 8 B. MANSOURI, and M. A.. SAOULI Z s n ds Us n e deds K n K n. On he oher hand, we have from q.3, 4 K n K n Y n fn s,y s n,z s n,u s n ds gs,ys n,z s n,u s n db s Zs n dw s U s n e ds,de. Using he Hölder s inequaliy and assupmion H.6, wehave K n K n C C Zs n ds U n deds, From inequaliy 4, wege Zs n U s n e de ds C C Zs n U n de ds. Finally chossing such ha C, we obain Zs n ds U s n e deds C. hus he prove of his lemma is comple. Lemma 4.3. Assume ha H., H.3 H.7 is in force. hen he sequence Z n,u n converges a.s. in M,, d L,,,. Proof. Le n C. Fromq.4., we deduce ha here exiss a process Y S,, such ha Y n Y a.s., as n. Applying lemma. o Y n Y m,forn,m n Y n Y m Zs n Z s m ds U s n e U s m e deds Ys n Y s m f n s,y s n,z s n,u s n f m s,y s m,z s m,u s m ds Ys n Y s m dk s n dk s m gs,ys n,z s n,u s n gs,y s m,z s m,u s m ds. Since Ys n Y s m dk s n dk s m, we deduce ha Z n Z m ds U s n e U s m e deds Ys n Y s m f n s,y s n,z s n,u s n f m s,y s m,z s m,u s m ds gs,ys n,z s n,u s n gs,y s m,z s m,u s m ds. Using Hölder s inequaliy and assumpion H.6 for g, we deduce ha Z n Z m ds U s n e U s m e deds fn s,y s n,z s n,u s n f m s,y s m,z s m,u s m ds Ys n Y s m ds C Ys n Y s m ds. Applying assumpion H.6 for f and he boundedness of he sequence Y n,z n,u n, we deduce

10 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps 8 ha Z n Z m ds U s n e U s m e deds C e Ys n Y s m ds, where he consan C e depend only, C, and. Which yields ha Z n n respecively U n n is a Cauchy sequence in M,, d, respecively in L,,,. hen here exiss Z,U M,, d L,,, such ha Zs n Z s ds U s n e U s e deds, as n. heorem 4.. Assume ha H., H.3 H.7 holds. hen q admis a soluion Y,Z,U,K D. Moreover here is a minimal soluion Y,Z,U of RBDSDP in he sense ha for any oher soluion Y,Z,U of q., we have Y Y. Proof. From q.4., i s readily seen ha Y n converges in S,,, d dp a.s. o Y S,,. Oherwise hanks o lemma 4.3 here exiss wo subsequences sill noed as he whole sequence Z n n respecively U n n such ha Zs n Z s ds asn, and U s n e U s e deds, as n. Applying lemma 4., we have f n,y n,z n,u n f,y,z,u and he linear growh of f n implies f n,y n,z n,u n C sup Y n Z n U n n L,;d. hus by Lebesgue s dominaed convergence heorem, we deduce ha for almos all and uniformly in, wehave fn s,y s n,z s n,u s n ds fs,ys,z s,u s ds. We have by H.6 he following esimaion n gs,ys,z n s,u n s gs,y s,z s,u s ds C Ys n Y s ds Zs n Z s ds U s n e U s e deds, as n, Using Burkholder-Davis-Gundy inequaliy, we have

11 83 B. MANSOURI, and M. A.. SAOULI sup n Zs dw s Zs dw s, sup U n s e ds,de U s e ds,de, sup n gs,ys,z n s,u n s db s gs,ys,z s,u s db s, in probabiliy as, n. Le he following refleced BDSDPs wih daa,f,g,s Ŷ S,,, Ẑ M,, d, K A, Û L,,,, Ŷ fs,ys,z s,u s ds gs,ys,z s,u s db s dks Ẑs dw s Û s e ds,de, S Ŷ, and Ŷ S dk. By Iô s formula, we obain Y n Ŷ n Ys Ŷ s f n s,y n s,z n s,u n s fs,y s,z s,u s ds Ys n Ŷ s dk s n dk s gs,ys n,z s n,u s n gs,y s,z s,u s ds U s n e Û s e deds Zs n Ẑ s ds. Using he fac ha Ys n Ŷ s dk s n dk s, we ge Y n Ŷ U s n e Û s e deds Zs n Ẑ s ds Ys n Ŷ s f n s,y s n,z s n,u s n fs,y s,z s,u s ds gs,ys n,z s n,u s n gs,y s,z s,u s ds. he by leing n, wehavey Ŷ, U Û and Z Ẑ dp d a.e. Le Y,Z,U,K be a soluion of. hen by heorem 3., we have for any n N, Y n Y. herefore, Y is a minimal soluion of.

12 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps RBDSDPs Wih Disconinuous Coefficiens In his secion we are ineresed in weakening he condiions on f. We assume ha f saisfy he following assumpions: (H.8) here exiss a nonnegaive process f M,, and consan C, such ha,y,z,u, B, f,y,z,u f C y z u. (H.9) f,,z,u : is a lef coninuous and f,y,, is a coninuous. (H.) here exiss a coninuous foncion :, B saisfying for y y, z,u d L,,,,y,z,u C y z u, f,,y,z,u f,,y,z,u,y y,z z,u u. (H.) g saisfies H.,ii. 5.. xisence resul he wo nex lemmas will be useful in he sequel. Lemma 5.. Assume ha saisfies H., g saisfies H. and h belongs in M,,. For a coninuous processes of finie variaion A belong in A we consider he processes Ȳ,Z,Ū S,, M,, d L,,, such ha: i Ȳ s,ȳ s,z s,ūs hsds gs,ȳ s,z s,ūsdbs da s Z sdws Ūse ds,de,, 5 ii Ȳ da s. hen we have, he RBDSDPs 5. admis a minimal soluion Y,Z,A,U D. If h and, we have Ȳ, dp d a.s. Proof. Has been obained from a previous par. Applying lemma. o Ȳ, leads o Ȳ Ȳs Z s ds Ȳs Ū se deds Ȳs s,ȳ s,z s,ū s hsds Ȳs da s Ȳs gs,ȳ s,z s,ū s ds. Since h,, and using he fac ha Ȳ da s, we obain

13 85 B. MANSOURI, and M. A.. SAOULI Ȳ Ȳs Z s ds Ȳs Ū se deds Ȳs s,ȳ s,z s,ū s ds Ȳs gs,ȳ s,z s,ū s ds. AccordingoassumpionsH., wehave Ȳ Ȳs Z s ds Ȳs Ū se deds Ȳs s,ȳ s,z s,ū s ds C Ȳs Ȳ s ds Ȳs Z s ds Ȳs Ū se deds. hen by applying assumpion H. and using Young s inequaliy, we can wrie Ȳs s,ȳ s,z s,ū s ds C Ȳs ds Ȳs ds C Z s ds Ȳs ds C Ū s e deds. hen Ȳ Ȳs Z s ds Ȳs Ū se deds 3C Ȳs ds C Ȳs Z s Ū s e de ds. herefore, choosing, and C such ha C and using Gronwall s inequaliy, we have Ȳ, P a.s. for all,. his implies ha Ȳ, P a.s. for all,. Now by heorem 4., we consider he processes Ỹ,Z,K,Ũ, Y,Z,K,U and he sequence of processes Ỹ n,z n,k n,ũ n n respecively as minimal soluion, for all,, o he following RBDSDPs iỹ C Ỹ s Z s Ũ s fs ds gs,ỹ s,z s,ũ s dbs dk s Z s dw s Ũs e ds,de,, 6 iiỹ S, iii Ỹ s Ss dk s,

14 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps 86 i Y C Y s Zs Us fs ds gs,y s,zs,us dbs dk s Z s dws Us e ds,de,, ii Y S, 7 iii Y s Ss dks, and i Ỹ n fs,ỹ s n,z sn,ũ s n ds s,ỹ n Ỹ n,z n Z n,ũ n Ũ n ds gs,ỹ s n,z sn,ũ s n dbs dk sn Z sn dw s Ũs n e ds,de,, iiỹ n S, iii Ỹ s n SsdK sn. 8 Lemma 5.. Under he assumpions H., H.3, H.5 and H.8 H., and for any n wih,,here holds Ỹ Ỹ n Ỹ n Y. Proof. For any n, we se n n n, and Δ n s,ỹ n s,z sn,ũ n s s,ỹ n s Ỹ n s,z sn Z sn,ũ n s Ũ n s s,ỹ n s,z sn,ũ n s. Invoke 8 o wrie where Ỹ n s,ỹs n,z sn,ũ s n s n ds Δg n s,ỹ s n,z sn,ũ s n db s dk sn Z sn dw s Ũ s n e ds,de,

15 87 B. MANSOURI, and M. A.. SAOULI n s Δf n s,ỹ n s,z sn,ũ n s s,ỹ n s,z sn,ũ n s, and s fs,ỹ s,z s,ũ s C Ỹ s Z s Ũ s f s, n. According o is definiion, one can show ha s and Δg n, n saisfy all assumpion of lemma 5.. Moreover, since K n is a coninuous and increasing process, for all n, hen K sn is a coninuous process of finie variaion. Using he same argumens as in firs par, i is possible o show ha Ỹ n Ỹ n dk n Ỹ n Ỹ n dk n Ỹ n Ỹ n dk n. By applying lemma 5., we deduce ha Ỹ n, i.e. Ỹ n Ỹ n, for all,.so we have Ỹ n Ỹ n Ỹ. Now we shall prove ha Ỹ n Y. By definiion, we have Y Ỹ n C Ys Ỹ s n Z s Z sn U s Ũ s n s n ds gs,ys,z s,u s gs,ỹ s n,z sn,ũ s n db s dks dk sn Zs Z sn dw s U s e Ũ s n e ds,de, where n s C Y n s Ỹ s Z s Z sn U n s Ũ s Y s Z s U s f s fs,ỹ n s,z sn,ũ n s s,ỹ n s,z sn,ũ n s. Also by repeaed use of lemma 5., we deduce ha Y Ỹ n, i.e. Y Ỹ n, for all,. hus, we have for all n, Y Ỹ n Ỹ n Ỹ, dp d a.s.,,. Now we can sae our main resul. heorem 5.. Under assumpion H., H.3, H.5 and H.8 H., he RBDSDPs has a minimal soluion Y,Z,K,U, D. Proof. Since Ỹ n maxỹ,y Ỹ Y for all,, wehave sup n sup Ỹ n sup Ỹ sup Y. herefore, we deduce from he Lebesgue s dominaed convergence heorem ha Ỹ n n converges in S,, o a limi Y. On he oher hand, by 8 we can wrie

16 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps 88 Ỹ n Ỹ n fs,ỹs n,z sn,ũ s n ds s,ỹ n,z n,ũ n ds gs,ỹs n,z sn,ũ s n db s dk sn Z sn dw s Ũ s n e ds,de. Apply hen lemma. o obain Ỹ n Z sn ds Ũ s n e deds Ỹs n fs,ỹ s n,z sn,ũ s n s,ỹ s n,z sn,ũ s n ds Ỹs n dk sn gs,ỹ s n,z sn,ũ s n ds. By H.8 and H., i follows ha Ỹ n f,ỹ n,z n,ũ n,ỹ n,z n,ũ n Ỹ n n f C Ỹ Z n Ũ n n C Ỹ Z n Ũ n Ỹ n C Ỹ n f C C C Ỹ n Ỹ n C Ỹ n Z n C Ỹ n Ũ n C 3 Ỹ n 3 Z n C 4 Ỹ n 4 Ũ n C C 3 C 4 C Ỹ n 3 Z n 4 n Ũ n Ỹ Z n n Ũ f. Also applying H. leads o he following inequaliy gs,ỹ n s,z sn,ũ n s gs,ỹ n s,z sn,ũ n s gs,,, gs,,, hen using Young s inequaliy, allows wriing Ỹs n dk sn Ss dk sn C Ỹ s n Z sn Ũ s n gs,,,. sup S K n. herefore, here exiss a consan C independen of n such ha for any i,wherei :4,we have Z sn ds Ũs n e deds C 3 Z sn ds 4 Ũs n deds 9 Z sn ds Ũs n deds K n. Moreover, since

17 89 B. MANSOURI, and M. A.. SAOULI K n Ỹ n fs,ỹs n,z sn,ũ s n ds s,ỹ s n,z sn,ũ s n ds gs,ỹs n,z sn,ũ s n db s Z sn dw s Ũ s n e ds,de, we may use Hölder s inequaliy and assumpions H.8 and H.. Consequenly, here exiss wo consans C and C depending on,c,, i,i,...,4, and we have ha K n C C Z sn Z sn ds Ũ s n Ũ s n deds. We reurn back o inequaliy 9 and wrie Z sn ds Ũ s n e deds C C C Z sn ds C Ũ s n deds 3 C Z sn ds 4 C Ũ s n deds. hen by aking and 3 4, wehave Z sn ds Ũ s n e deds C C C Z sn ds Ũ s n deds C Z sn Ũ s n e de ds. A furher choise of, and such ha C, allows for Z sn ds Ũ s n e deds C C C Z sn ds Ũ s n deds i n C C C i C n Z s ds Ũ s deds. i Now choosing, and C such ha C, and noing ha Z s Ũ s de ds,allows for sup Z sn ds and n N sup Ũ s n e deds. n N Consequenly, we deduce ha K n. Now we shall prove ha Z n,k n,ũ n is a Cauchy sequence in M,, d A L,,,, seγ n s fs,ỹ n s,z sn,ũ n s s,ỹ n s,z sn,ũ n s o arrive

18 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps 9 a Ỹ n Ỹ m Γs n Γ s m ds gs,ỹs n,z sn,ũ s n gs,ỹ s m,z sm,ũ s m db s dk sn dk sm Z sn Z sm dw s Ũ s n e Ũ s m e ds,de. Furher applicaion of lemma. o Ỹs n,m Ỹ s n Ỹ s m, resuls wih Ỹ n Ỹ m Z sn Z sm ds Ũ s n Ũ s m deds Ỹs n Ỹ s m Γ s n Γ s m ds Ỹs n Ỹ s n dk sn dk sm gs,ỹ s n,z sn,ũ s n gs,ỹ s m,z sm,ũ s m ds. Since Ỹs n Ỹ s n dk sn dk sm, we obain Z sn Z sm ds Ũ s n Ũ s m deds Ỹs n Ỹ s m Γ s n Γ s m ds gs,ỹ s n,z sn,ũ s n gs,ỹ s m,z sm,ũ s m ds. Apply nex Hölder s inequaliy and assumpion H. o wrie Z sn Z sm ds Ũ s n Ũ s m deds Ỹs n Ỹ s m ds Γs n Γ s m ds C Ỹs n Ỹ s m ds. he boundedness of he sequence Ỹ n,z n,k n,ũ n implies ha sup n Γs ds, o allow for Z sn Z sm ds Ũ s n Ũ s m deds n N 4 Ỹs n Ỹ s m ds C Ỹs n Ỹ s m ds, which implies ha each of Z n n and Ũ n n, is a Cauchy sequence in M,, d and in L,,, respecively. hen here exiss Z,U M,, d L,,, such ha, Z sn Z s ds Ũs n Us de, as n. On he oher hand, applying he Burkholder-Davis-Gundy inequaliy and, ends up wih

19 9 B. MANSOURI, and M. A.. SAOULI sup Z sn dw s Z sdws Z sn Z s ds, as n, sup Ũs n e ds,de Use ds,de Ũs n Us deds, as n, sup gs,ỹ s n,z sn,ũ s n dbs gs,y s,zs,usdbs C Ỹ s n Ys ds Z sn Z s ds Ũs n Us deds, as n. herefore, from he properieies of f,, wehave Γ n s fs,ỹ n s,z sn,ũ n s s,ỹ n s,z sn,ũ n s fs,y s,z s,u s, P a.s., for all, as n. I follows hen, by he dominaed convergence heorem, ha n Γs fs,y s,z s,u s ds. Since Ỹ s n,z sn,ũ s n,γ s n converges in B M,, and sup K n K m Ỹ n Ỹ m sup sup Ỹ n Ỹ m Γs n Γ s m ds gs,ỹs n,z sn,ũ s n gs,ỹ s m,z sm,ũ s m db sup for any n,m, we deduce ha Z sn Z sm dw s sup Ũ s n e Ũ s m e ds,de, sup K n K m, as n,m. Consequenly, here exiss a F measurable process K wich value in such ha sup K n K Finally, we have, as n.

20 Refleced Disconinuous Doubly Sochasic Ds Wih Poisson Jumps 9 sup Ỹ n Y Z sn Z s ds Ũ s n U s deds sup K n K, as n. Obviously, K andk ; is a increasing and coninuous process. From 8, we have for all n, Ỹ n S,,.hen Y S,,. On he oher hand, from he resul of Saisho [], i follows ha Ỹs n S s dk sn Ys S s dk s, P a.s., as n. Using he ideniy Ỹs n S s dk sn for all n, we conclude ha Ys S s dk s. Leing hen n in equaion, proves ha Y,Z,K,U, is a soluion o i. Assume ulimaely Y,Z,U,K o be a soluion o o invoke heorem 3., and observe ha for any n N, Y n Y. herefore, Y is a minimal. Remark 5.. Using he same argumens and he following approximaing sequence f n,,y,z,u sup f,,y,z,u n y y z z u u, y, z, u B one can prove ha he RBDSD () has a maximal soluion. References [] J. J. Aliber, and K. Bahlali, Genericiy in deerminisic and sochasic differenial equaions, Séminaire de Probabiliés XXXV, Lecure Noes in Mahemaics 755, Springer Verlag, Berlin-Heidelberg, (), -4. [] K. Bahlali, M. Hassani, B. Mansouri, and N. Mrhardy, One barrier refleced backward doubly sochasic differenial equaions wih coninuous generaor, Compes Rendus Mahemaique 347(9), (9), -6. [3] K. Bahlali, R. Ga, and B. Mansouri, Backward doubly sochasic differenial equaions wih a superlinear growh generaor, Compes Rendus Mahemaique 353(), (5), 5-3. [4] J. M. Bismu, Conjugae convex funcions in opimal sochasic conrol, Journal of Mahemaical Analysis and Applicaions 44, (973), [5] B. Boufoussi, J. Van Caseren, and N. Mrhardy, Generalized backward doubly sochasic differenial equaions and SPDs wih nonlinear Neumann boundary condiions, Bernoulli 3(), (7), [6] X. Fan, and Y. Ren, Refleced backward doubly sochasic differenial equaion wih jumps, Mahemaica Applicaa (4), (9), [7] G. Jia, A class of backward sochasic differenial equaions wih disconinuous

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