随机微分方程解的期望泛函的Fréchet导数
Fréchet derivatives of expected functionals of solutions to stochastic differential equations
摘要：
在分析由随机微分方程（SDEs）描述的随机动力系统时，经常感兴趣的是分析随机微分方程解的泛函的期望值对随机微分方程参数摄动的灵敏度。在本文中，我们考虑的路径泛函的解决方案，依赖于一个停时。我们推导出公式Fréchet导数的期望值的这些泛函相对于有界扰动的漂移，使用Cameron-Martin-Girsanov定理的变化的措施。利用这些导数，我们构造了一个例子，表明将漂移的变化发送到相应的相对熵的映射一般不是凸的。然后，我们分析的存在性和唯一性的随机最优控制问题的解决方案定义在可能的随机时间间隔，以及基于梯度的数值方法解决这样的问题。
英文摘要：
In the analysis of stochastic dynamical systems described by stochastic differential equations (SDEs), it is often of interest to analyse the sensitivity of the expected value of a functional of the solution of the SDE with respect to perturbations in the SDE parameters. In this paper, we consider path functionals that depend on the solution of the SDE up to a stopping time. We derive formulas for Fréchet derivatives of the expected values of these functionals with respect to bounded perturbations of the drift, using the Cameron-Martin-Girsanov theorem for the change of measure. Using these derivatives, we construct an example to show that the map that sends the change of drift to the corresponding relative entropy is not in general convex. We then analyse the existence and uniqueness of solutions to stochastic optimal control problems defined on possibly random time intervals, as well as gradient-based numerical methods for solving such problems.
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