$W^{m,p}$-Solution ($p\geq2$) of Linear Degenerate Backward Stochastic Partial Differential Equations in the Whole Space

Details $W^{m,p}$-Solution ($p\geq2$) of Linear Degenerate Backward Stochastic Partial Differential Equations in the Whole Space $W^{m,p}$-Solution ($p\geq2$) of Linear Degenerate Backward Stochastic Partial Differential Equations in the Whole Space

2011-05-07

arxiv.org/abs/1105.1428v1

Mathematics

Probability

29 pages

Scientific paper

In this paper, we consider the backward Cauchy problem of linear degenerate stochastic partial differential equations. We obtain the existence and uniqueness results in Sobolev space $L^p(\Omega; C([0,T];W^{m,p}))$ with both $m\geq 1$ and $p\geq 2$ being arbitrary, without imposing the symmetry condition for the coefficient $\sigma$ of the gradient of the second unknown---which was introduced by Ma and Yong [Prob. Theor. Relat. Fields 113 (1999)] in the case of $p=2$. To illustrate the application, we give a maximum principle for optimal control of degenerate stochastic partial differential equations.

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