Backward doubly stochastic differential equations with polynomial growth coefficients

Qi  Zhang; Huaizhong  Zhao

doi:10.3934/dcds.2015.35.5285

Article Contents

2015, Volume 35, Issue 11: 5285-5315. Doi: 10.3934/dcds.2015.35.5285

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

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

Received: September 30, 2013

Revised: September 30, 2014

Published: October 2015

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Abstract

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:

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

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

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