×
Zhu, Qing-Feng; Shi, Yu-Feng

A class of backward doubly stochastic differential equations with discontinuous coefficients. (English) Zbl 1315.60073

Acta Math. Appl. Sin., Engl. Ser. 30, No. 4, 965-976 (2014).
The authors study a class of backward doubly stochastic differential equations (BDSDEs) under some relaxed conditions on the coefficient functions.
The main results are given in Theorem 3.1 (the existence of solutions), Theorem 3.2 (the existence of minimal solutions) and Theorem 4.1 (comparison theorem).
Their results extend some similar results given by E. Pardoux and S. Peng [Probab. Theory Relat. Fields 98, No. 2, 209–227 (1994; Zbl 0792.60050)], Q. Lin [Stat. Probab. Lett. 79, No. 20, 2223–2229 (2009; Zbl 1175.60062)], K.-H. Kim [Electron. J. Probab. 10, Paper No. 1, 1–20 (2005; Zbl 1065.60079)], M. N’zi and J.-M. Owo [Stat. Probab. Lett. 79, No. 7, 920–926 (2009; Zbl 1168.60353)], G. Jia [C. R., Math., Acad. Sci. Paris 342, No. 9, 685–688 (2006; Zbl 1119.60046)] and J. P. Lepeltier and J. San Martin [Stat. Probab. Lett. 32, No. 4, 425–430 (1997; Zbl 0904.60042)].
Reviewer: Romeo Negrea (Timişoara)

Cited in 3 Documents

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
60H05 Stochastic integrals

Keywords:

backward doubly stochastic differential equations; discontinuous coefficients; backward stochastic integral; comparison theorem

Citations:

Zbl 0792.60050; Zbl 1175.60062; Zbl 1065.60079; Zbl 1168.60353; Zbl 1119.60046; Zbl 0904.60042
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Bally, V., Matoussi, A. Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theoret. Probab., 14(1): 125-164 (2001) · Zbl 0982.60057 · doi:10.1023/A:1007825232513
[2] Duffie, D., Epstein, L. Stochastic differential utilities. Econometrica, 60(2): 354-439 (1992) · Zbl 0763.90005 · doi:10.2307/2951600
[3] Hamadene, S., Lepeltier, J.-P. Backward equations, stochastic control and zero-sum stochastic differential games. Stoch. Stoch. Rep., 54(3-4): 221-231 (1995) · Zbl 0858.60056 · doi:10.1080/17442509508834006
[4] Hu, L., Ren, Y. Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes. J. Comput. Appl. Math., 229(1): 230-239 (2009) · Zbl 1173.60023 · doi:10.1016/j.cam.2008.10.027
[5] Hu, M., Peng, S. On representation theorem of G-expectations and paths of G-Brownian motion. Acta Mathematicae Applicatae Sinica, 25(3): 539-546 (2009) · Zbl 1190.60043 · doi:10.1007/s10255-008-8831-1
[6] Jia, G. A generalized existence theorem of BSDEs. C. R. Acad. Sci. Paris, Ser. I, 342(9): 685-688 (2006) · Zbl 1119.60046 · doi:10.1016/j.crma.2006.02.020
[7] Jia, G. On existence of backward stochastic differential equations with left-Lipschitz coefficient. Chin. J. of Contemp. Math., 28(4): 345-354 (2007) (in Chinese)
[8] Jiang, L., Chen, Z. A result on the probability measures dominated by g-expectation. Acta Mathematicae Applicatae Sinica, 20(3): 1-6 (2004) · Zbl 1055.60059
[9] Kim, K. Lp estimates for SPDE with discontinuous coefficients in domains. Electron. J. Probab. 10(1): 1-20 (2005) · Zbl 1099.11070
[10] Lin, Q. A class of backward doubly stochastic differential equations with non-Lipschitz coefficients. Statist. Probab. Lett., 79(20): 2223-2229 (2009) · Zbl 1175.60062 · doi:10.1016/j.spl.2009.07.019
[11] Lepeltier, J.P., Martin, J.S. Backward stochastic differential equations with continuous coefficients. Statist. Probab. Lett., 34(4): 425-430 (1997) · Zbl 0904.60042 · doi:10.1016/S0167-7152(96)00103-4
[12] Nualart, D., Pardoux, E. Stochastic calculus with anticipating integrands. Probab. Theory Related Fields, 78(4): 535-581 (1988) · Zbl 0629.60061 · doi:10.1007/BF00353876
[13] N’zi, M., Owo, J.M. Backward doubly stochastic differential equations with discontinuous coefficients. Statist. Probab. Lett., 79(7): 920-926 (2009) · Zbl 1168.60353 · doi:10.1016/j.spl.2008.11.011
[14] Pardoux, E., Peng, S. Adapted solution of a backward stochastic differential equation. Systems Control Letters, 14(1): 55-61 (1990) · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6
[15] Pardoux, E., Peng, S. Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDE’s. Probab. Theory Related Fields, 98(2): 209-227 (1994) · Zbl 0792.60050 · doi:10.1007/BF01192514
[16] Peng, S. Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep., 37(1-2): 61-74 (1991) · Zbl 0739.60060
[17] Peng, S. Filtration consistent nonlinear expectations and evaluations of contingent claims. Acta Mathematicae Applicatae Sinica, 20(2): 1-24 (2004) · Zbl 1061.60063
[18] Ren, Y., Lin, A., Hu, L. Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes. J. Comput. Appl. Math., 223(2): 901-907 (2009) · Zbl 1154.60336 · doi:10.1016/j.cam.2008.03.008
[19] Shi, Y., Gu, Y., Liu, K. Comparison theorems of backward doubly stochastic differential equations and applications. Stoch. Anal. Appal., 23(1): 97-110 (2005) · Zbl 1067.60046 · doi:10.1081/SAP-200044444
[20] Yoo, H. Lp-estimate for stochastic PDEs with discontinuous coefficients. Stoch. Anal. Appal., 17(4): 678-711 (1999) · Zbl 0960.60056
[21] Zhang, Q., Zhao, H. Stationary solutions of SPDEs and infinite horizon BDSDEs. J. Funct. Anal., 252(1): 171-219 (2007) · Zbl 1127.60059 · doi:10.1016/j.jfa.2007.06.019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.