Backward doubly stochastic differential equations with polynomial growth coefficients

Previous Article
Degenerate backward SPDEs in bounded domains and applications to barrier options
DCDS Home
This Issue
Next Article
On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: The critical case

November 2015, 35(11): 5285-5315. doi: 10.3934/dcds.2015.35.5285

Backward doubly stochastic differential equations with polynomial growth coefficients

Qi Zhang ^1, and Huaizhong Zhao ^2,

1.	School of Mathematical Sciences, Fudan University, Shanghai 200433
2.	Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, United Kingdom

Received October 2013 Revised October 2014 Published May 2015

Abstract
Related Papers

In this paper we study the solvability of backward doubly stochastic differential equations (BDSDEs for short) with polynomial growth coefficients and their connections with SPDEs. The corresponding SPDE is in a very general form, which may depend on the derivative of the solution. We use Wiener-Sobolev compactness arguments to derive a strongly convergent subsequence of approximating SPDEs. For this, we prove some new estimates to the solution and its Malliavin derivative of the corresponding approximating BDSDEs. These estimates lead to the verifications of the conditions in the Wiener-Sobolev compactness theorem and the solvability of the BDSDEs and the SPDEs with polynomial growth coefficients.

Keywords: SPDEs, Backward doubly stochastic differential equations, polynomial growth coefficients, Malliavin derivative, Wiener-Sobolev compactness..

Mathematics Subject Classification: Primary: 60H15, 60H10; Secondary: 37H1.

Citation: Qi Zhang, Huaizhong Zhao. Backward doubly stochastic differential equations with polynomial growth coefficients. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5285-5315. doi: 10.3934/dcds.2015.35.5285

References:

[1]	V. Bally and B. Saussereau, A relative compactness criterion in Wiener-Sobolev spaces and application to semi-linear stochastic PDEs, J. Funct. Anal., 210 (2004), 465-515. doi: 10.1016/S0022-1236(03)00236-2.
[2]	G. Da Prato, P. Malliavin and D. Nualart, Compact families of Wiener functionals, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 1287-1291.
[3]	C. R. Feng and H. Z. Zhao, Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding, J. Funct. Anal., 262 (2012), 4377-4422. doi: 10.1016/j.jfa.2012.02.024.
[4]	D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2437-0.
[5]	E. Pardoux, BSDE's weak convergence and homogenization of semilinear PDE's, in Nonlinear Analysis, Differential Equations and Control (eds. F. Clarke and R. Stern), Kluwer Acad. Publi., 528, Dordrecht, 1999, 503-549.
[6]	E. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Rel., 98 (1994), 209-227. doi: 10.1007/BF01192514.
[7]	S. Peszat, On a Sobolev space of functions of infinite number of variables, Bull. Polish Acad. Sci. Math., 41 (1993), 55-60.
[8]	Q. Zhang and H. Z. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs, J. Funct. Anal., 252 (2007), 171-219. doi: 10.1016/j.jfa.2007.06.019.
[9]	Q. Zhang and H. Z. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients, J. Differ. Equations, 248 (2010), 953-991. doi: 10.1016/j.jde.2009.12.013.
[10]	Q. Zhang and H. Z. Zhao, Probabilistic representation of weak solutions of partial differential equations with polynomial growth coefficients, J. Theor. Probab., 25 (2012), 396-423. doi: 10.1007/s10959-011-0350-y.
[11]	Q. Zhang and H. Z. Zhao, SPDEs with polynomial growth coefficients and Malliavin calculus method, Stoch. Proc. Appl., 123 (2013), 2228-2271. doi: 10.1016/j.spa.2013.02.004.

show all references

References:

[1]	V. Bally and B. Saussereau, A relative compactness criterion in Wiener-Sobolev spaces and application to semi-linear stochastic PDEs, J. Funct. Anal., 210 (2004), 465-515. doi: 10.1016/S0022-1236(03)00236-2.
[2]	G. Da Prato, P. Malliavin and D. Nualart, Compact families of Wiener functionals, C. R. Acad. Sci. Paris Ser. I Math., 315 (1992), 1287-1291.
[3]	C. R. Feng and H. Z. Zhao, Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding, J. Funct. Anal., 262 (2012), 4377-4422. doi: 10.1016/j.jfa.2012.02.024.
[4]	D. Nualart, The Malliavin Calculus and Related Topics, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2437-0.
[5]	E. Pardoux, BSDE's weak convergence and homogenization of semilinear PDE's, in Nonlinear Analysis, Differential Equations and Control (eds. F. Clarke and R. Stern), Kluwer Acad. Publi., 528, Dordrecht, 1999, 503-549.
[6]	E. Pardoux and S. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Rel., 98 (1994), 209-227. doi: 10.1007/BF01192514.
[7]	S. Peszat, On a Sobolev space of functions of infinite number of variables, Bull. Polish Acad. Sci. Math., 41 (1993), 55-60.
[8]	Q. Zhang and H. Z. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs, J. Funct. Anal., 252 (2007), 171-219. doi: 10.1016/j.jfa.2007.06.019.
[9]	Q. Zhang and H. Z. Zhao, Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients, J. Differ. Equations, 248 (2010), 953-991. doi: 10.1016/j.jde.2009.12.013.
[10]	Q. Zhang and H. Z. Zhao, Probabilistic representation of weak solutions of partial differential equations with polynomial growth coefficients, J. Theor. Probab., 25 (2012), 396-423. doi: 10.1007/s10959-011-0350-y.
[11]	Q. Zhang and H. Z. Zhao, SPDEs with polynomial growth coefficients and Malliavin calculus method, Stoch. Proc. Appl., 123 (2013), 2228-2271. doi: 10.1016/j.spa.2013.02.004.

[1]	Renhai Wang, Yangrong Li. Backward compactness and periodicity of random attractors for stochastic wave equations with varying coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4145-4167. doi: 10.3934/dcdsb.2019054
[2]	Yufeng Shi, Qingfeng Zhu. A Kneser-type theorem for backward doubly stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1565-1579. doi: 10.3934/dcdsb.2010.14.1565
[3]	Yinggu Chen, Said HamadÈne, Tingshu Mu. Mean-field doubly reflected backward stochastic differential equations. Numerical Algebra, Control and Optimization, 2022 doi: 10.3934/naco.2022012
[4]	M. A. M. Alwash. Polynomial differential equations with small coefficients. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1129-1141. doi: 10.3934/dcds.2009.25.1129
[5]	Feng Bao, Yanzhao Cao, Weidong Zhao. A first order semi-discrete algorithm for backward doubly stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1297-1313. doi: 10.3934/dcdsb.2015.20.1297
[6]	Krzysztof Frączek. Polynomial growth of the derivative for diffeomorphisms on tori. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 489-516. doi: 10.3934/dcds.2004.11.489
[7]	Jasmina Djordjević, Svetlana Janković. Reflected backward stochastic differential equations with perturbations. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1833-1848. doi: 10.3934/dcds.2018075
[8]	Jan A. Van Casteren. On backward stochastic differential equations in infinite dimensions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 803-824. doi: 10.3934/dcdss.2013.6.803
[9]	Joscha Diehl, Jianfeng Zhang. Backward stochastic differential equations with Young drift. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 5-. doi: 10.1186/s41546-017-0016-5
[10]	Dariusz Borkowski. Forward and backward filtering based on backward stochastic differential equations. Inverse Problems and Imaging, 2016, 10 (2) : 305-325. doi: 10.3934/ipi.2016002
[11]	Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control and Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028
[12]	Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447
[13]	Renzhi Qiu, Shanjian Tang. The Cauchy problem of Backward Stochastic Super-Parabolic Equations with Quadratic Growth. Probability, Uncertainty and Quantitative Risk, 2019, 4 (0) : 3-. doi: 10.1186/s41546-019-0037-3
[14]	Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115
[15]	Yanqing Wang. A semidiscrete Galerkin scheme for backward stochastic parabolic differential equations. Mathematical Control and Related Fields, 2016, 6 (3) : 489-515. doi: 10.3934/mcrf.2016013
[16]	Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905
[17]	Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585
[18]	Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control and Related Fields, 2021, 11 (4) : 797-828. doi: 10.3934/mcrf.2020047
[19]	Jiongmin Yong. Forward-backward stochastic differential equations: Initiation, development and beyond. Numerical Algebra, Control and Optimization, 2022 doi: 10.3934/naco.2022011
[20]	Martin Hutzenthaler, Thomas Kruse, Tuan Anh Nguyen. On the speed of convergence of Picard iterations of backward stochastic differential equations. Probability, Uncertainty and Quantitative Risk, 2022, 7 (2) : 133-150. doi: 10.3934/puqr.2022009

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Qi Zhang Huaizhong Zhao
on Google Scholar
Qi Zhang Huaizhong Zhao

[Back to Top]