Abstract
In this paper we establish a probabilistic representation for the spatial gradient ofthe viscosity solution to a quasilinear parabolic stochastic partial differential equations(SPDE, for short) in the spirit of the Feynman-Kac formula, without using thederivatives of the coefficients of the corresponding backward doubly stochastic differentialequations (FBDSDE, for short).
Highlights
Linear backward stochastic differential equations (BSDEs, for short) have been considered by Bismut [1, 2] in the context of optimal stochastic control
In this paper we establish a probabilistic representation for the spatial gradient of the viscosity solution to a quasilinear parabolic stochastic partial differential equations (SPDE, for short) in the spirit of the Feynman-Kac formula, without using the derivatives of the coefficients of the corresponding backward doubly stochastic differential equations (FBDSDE, for short)
BSDEs provide a probabilistic interpretation for solutions to elliptic or parabolic nonlinear partial differential equations generalizing the classical Feynman-Kac formula [16, 18]
Summary
Linear backward stochastic differential equations (BSDEs, for short) have been considered by Bismut [1, 2] in the context of optimal stochastic control. Yst,x = l(X0t,x) + f (r, Xrt,x, Yrt,x, Zrt,x) dr + g(r, Xrt,x, Yrt,x) dBr − Zrt,xdWr, which are a time reversal of that considered by Pardoux and Peng [17] In this framework, the stochastic integral whith respect to dB is a forward Itô integral and the stochastic integral driving by dW is a backward Itô integral. Where ”Dg” denotes the classical differential of the function g and N t is some process defined on [0, t], for each t ∈ [0, T ], depending only on the forward diffusion and its variational equation Such a relation in a sense could be viewed as an extension of the nonlinear Feynman-Kac formula to stochastic PDEs, which, to our best knowledge, is new. For the rest of this paper, we give all the necessary preliminaries in section 2, our main results are stated in section 3 while section 4 is devoted to its proof
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