论文标题
在随机最大原理上：向后的随机部分微分方程的观点
On Stochastic Maximum Principle: A Backward Stochastic Partial Differential Equations Point of View
论文作者
论文摘要
在本文中，我们考虑了具有随机系数的随机微分方程的一类随机控制问题。控制域不必为凸，但是不允许以扩散项输入控制过程。此外，终端成本涉及终端状态期望值的非线性项。我们的目的是通过采用彭的作品来得出庞特里金随机原理的新版本[S. S. Peng，NonConvex Control域的随机最佳控制的最大原理，Control＆Information Sciences中的讲义，114，（1990），第724-732页]。更具体地说，我们表明，如果我们将最佳控制的尖峰扰动与线性后向后随机偏微分方程的随机Feynman-kac表示（BSPDE，简称BSPDE），则可以得出一个新版本的随机最大原理。我们还研究了足够的最佳条件。在本文的最后一部分中，以我们的SMP版本的启发，自然而然地引入了一类有趣的向后随机偏微分方程，并简要介绍了这种方程式的可溶性。
In this paper, we consider a class of stochastic control problems for stochastic differential equations with random coefficients. The control domain need not to be convex but the control process is not allowed to enter in diffusion term. Moreover, the terminal cost involves a non linear term of the expected value of terminal state. Our purpose is to derive a new version of the Pontryagin's stochastic maximum principle by adopting an idea inspired from the work of Peng [S. Peng, Maximum Principle for Stochastic Optimal Control with Nonconvex Control Domain, Lecture Notes in Control & Information Sciences, 114, (1990), pp. 724-732]. More specifically, we show that if we combine the spike perturbation of the optimal control combined with the stochastic Feynman-Kac representation of linear backward stochastic partial differential equations (BSPDE, for short), a new version of the stochastic maximum principle can be derived. We also investigate sufficient conditions of optimality. In the last part of this paper, motivated by our version of SMP, an interesting class of forward backward stochastic partial differential equations is naturally introduced and the solvability of such kind of equations is briefly presented.
