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| SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN $R^n$ |
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Citation： |
TANG Shanjian.SEMI-LINEAR SYSTEMS OF BACKWARD STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS IN $R^n$[J].Chinese Annals of Mathematics B,2005,26(3):437~456 |
| Page view： 1056
Net amount： 787 |
Authors： |
TANG Shanjian; |
Foundation： |
Project supported by the National Natural Science Foundation of China (No.10325101, No.101310310) and the Science Foundation of the Ministry of Education of China (No.20030246004). |
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| Abstract： |
This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs. The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions. |
Keywords： |
Semi-linear system of backward stochastic partial differential equation, Backward stochastic differential equation, Stochastic differential equation, Probabilistic representation, Stochastic flow |
Classification： |
60H15, 35R60, 34F05, 93E20 |
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