Journal
PROBABILITY THEORY AND RELATED FIELDS
Volume 123, Issue 3, Pages 381-411Publisher
SPRINGER-VERLAG
DOI: 10.1007/s004400100193
Keywords
degenerate backward stochastic partial differential equations; adapted solutions; non-linear Feynman-Kac formula
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In this paper we study a class of one-dimensional, degenerate, semilinear backward stochastic partial differential equations (BSPDEs, for short) of parabolic type. By establishing some new a priori estimates for both linear and semilinear BSPDEs, we show that the regularity and uniform boundedness of the adapted solution to the semilinear BSPDE can be determined by those of the coefficients, a special feature that one usually does not expect from a stochastic differential equation. The proof follows the idea of the so-called bootstrap method, which enables us to analyze each,of the derivatives of the solution under consideration. Some related results, including some comparison theorems of the adapted solutions for semilinear BSPDEs, as well as a nonlinear stochastic Feynman-Kac formula, are also given.
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