论文信息 - Stochastic viscositysolutions for nonlinear stochastic partial di#erential equations. Part II

Stochastic viscositysolutions for nonlinear stochastic partial di#erential equations. Part II

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205–228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial di#erential equations. We introduce a de7nition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration given to the stochastic integrals. We show that a stochastic PDE can be converted to a PDE with random coe9cients via a Doss–Sussmann-type transformation, so that a stochastic viscosity solution can be de7ned in a “point-wise” manner. Using the recentlydeveloped theoryon backward=backward doubly stochastic di#erential equations, we prove the existence of the stochastic viscositysolution, and further extend the nonlinear Feynman–Kac formula. Some properties of the stochastic viscosity solution will also be studied in this paper. The uniqueness of the stochastic viscositysolution will be addressed separatelyin Part II where the relation between the stochastic viscositysolution and the !-wise, “deterministic” viscositysolution to the PDE with random coe9cients will be established. c 2001 Elsevier Science B.V. All rights reserved. MSC: 60H15; 30; 60G46; 60; 35R60

Rainer Buckdahn | Jin Ma | Jin Ma | R. Buckdahn

[1] D. Ocone,et al. A generalized Itô-Ventzell formula. Application to a class of anticipating stochastic differential equations , 1989 .

[2] A. Bensoussan. Stochastic Control of Partially Observable Systems , 1992 .

[3] P. Souganidis,et al. Fully nonlinear stochastic partial differential equations: non-smooth equations and applications , 1998 .

[4] Shige Peng,et al. Backward doubly stochastic differential equations and systems of quasilinear SPDEs , 1994 .

[5] Hani J. Doss,et al. Liens entre equations di erentielles stochastiques et ordinaires , 1977 .

[6] P. Protter. Stochastic integration and differential equations : a new approach , 1990 .

[7] Luciano Tubaro,et al. Fully nonlinear stochastic partial differential equations , 1996 .

[8] P. Meyer,et al. Probabilities and potential C , 1978 .

[9] Mtw,et al. Stochastic flows and stochastic differential equations , 1990 .

[10] W. Fleming,et al. Controlled Markov processes and viscosity solutions , 1992 .

[11] P. Lions,et al. User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[12] Ioannis Karatzas,et al. Brownian Motion and Stochastic Calculus , 1987 .

[13] David Nualart,et al. Stochastic calculus with anticipating integrands , 1988 .

[14] S. Peng,et al. Backward stochastic differential equations and quasilinear parabolic partial differential equations , 1992 .