$W^{m,p}$-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
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DOI：
10.1016/j.jde.2013.01.013
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发表时间：
2011-05
期刊：
影响因子：
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通讯作者：
Kai Du;Shanjian Tang;Qi S. Zhang
中科院分区：
文献类型：
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作者：
Kai Du;Shanjian Tang;Qi S. Zhang
本文考虑线性退化随机偏微分方程解的倒向柯西问题。在Sobolev空间Lp(Ω；C([0，T]；Wm，p))中，当m⩾1和p⩾2都是任意的时，我们得到了解的存在唯一性结果，而没有强加Ma和Yong(1999)[21]在p=2时引入的第二未知数的梯度系数σ的对称性条件。为了说明它的应用，我们给出了退化随机偏微分方程解的最优控制的一个极大值原理。
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 Lp(Ω;C([0,T];Wm,p)) with both m⩾1 and p⩾2 being arbitrary, without imposing the symmetry condition for the coefficient σ of the gradient of the second unknown—which was introduced by Ma and Yong (1999) [21] in the case of p=2. To illustrate the application, we give a maximum principle for optimal control of degenerate stochastic partial differential equations.