Backward doubly stochastic differential equations (BDSDEs): Existence and Uniqueness

Asma Alwasm

doi:10.24297/jam.v21i.9211

Authors

Asma Alwasm Mathematics department, Qassim University Qassim, Saudi Arabia

DOI:

https://doi.org/10.24297/jam.v21i.9211

Keywords:

stochastic partial differential-integral equations, Backward doubly stochastic differential equations, Brownian motion/Wiener process, Itô Formula

Abstract

In this paper, we present a class of stochastic differential equations with terminal condition, called backward doubly stochastic differential equations (BDSDEs). Precisely, we will prove the existence and uniqueness of the solutions of FBDSDEs but under weaker conditions

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References

A. Al-Hussein and B. Gherbal, Stochastic maximum principle for Hilbert space valued forward-backward doubly SDEs with Poisson jumps. 26th IFIP TC 7 Conference, CSMO 2013, Klagenfurt, Austria, September 9–13, 2013. System modeling and optimization, (2014), 1–10.

A. Al-Hussein and B. Gherbal, Existence and uniqueness of the solutions of forward-backward doubly stochasticdifferential equations with Poisson jumps, arXiv:1809.01875, 2018.

F. Antonelli, Backward-forward stochastic differential equations, Ann. Appl. Probab., 3 (1993), 777–793.

J. M. Bismut, Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications, 1973.

F. Delarue and S. Menozzi, A forward-backward stochastic algorithm for quasi-linear PDEs, Ann. Appl. Probab., 16 (2006), 140–184.

Y. Hu, D. Nualart and X. Song, An implicit numerical scheme for a class of backward doubly stochastic differential equations, arXiv:1702.00910v1 [math.PR], 2017.

Y. Hu and S. Peng, Solution of forward-backward stochastic differential equations, Prob. Th. & Rel. Fields, 103 (1995), 273–283.

Y. Hu, D. Nualart and X. Song, Malliavin calculus for backward stochastic differential equations and application to numerical solutions, Ann. Appl. Probab., 21, 6 (2011), 2379–2423.

J. Ma, P. Protter and J. Yong, Solving forward-backward stochastic differential equations explicitlya four step scheme, Prob. Th. and Rel. Fields, 98 (1994), 339–359.

J. Ma and J. Yong, Forward-backward stochastic differential equations and their applications. Number 1702. Springer Science Business Media, 1999.

J. Ma, J. Shen and Y. Zhao, On numerical approximations of forward-backward stochastic differential equations, Siam J. Numer. Anal, vol. 46, no. 5 (2008), 2636–2661.

E. Pardoux and S. Peng, Adapted solution of backward stochastic differential equations, System Control Lett, no. 14, (1990), 55-61.

E. Pardoux and S. Peng, Backward doubly stochastic differential equations and system of quasilinear SPDEs, Prob. Th. & Rel. Fields, 98, 2 (1994), 209–227.

E. Pardoux and S. Tang, Forward-backward stochastic differential equations and quasilinear parabolic PDEs, Prob. Th. & Rel. Fields, 114 (1999), 123–150.

S. Peng and Z. Wu, Fully coupled forward-backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim, 37 (1999), 825–843.

S. Peng and Y. Shi, A type-symmetric forward-backward stochastic differential equations, C. R. Acad. Sci. Paris Ser. I, 336, 1 (2003), 773–778.

X. Sun and Y. Lu, The property for solutions of the multi-dimensional backward doubly stochastic differential equations with jumps, Chin. J. Appl. Probab. Stat., 24 (2008), 73–82.

J. Yong, Finding adapted solutions of forward-backward stochastic differential equations–method of continuation, Prob. Th. & Rel. Fields, 107 (1997), 537–572.