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 Fr\'{e}chet 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.

中文翻译：

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随机微分方程解的预期泛函的 Fréchet 导数

在分析由随机微分方程 (SDE) 描述的随机动力系统时，分析 SDE 解函数的期望值相对于 SDE 参数扰动的敏感性通常很有趣。在本文中，我们考虑取决于 SDE 解直到停止时间的路径泛函。我们使用 Cameron-Martin-Girsanov 定理来推导出这些泛函的期望值相对于漂移的有界扰动的 Fr\'{e}chet 导数的公式。使用这些导数，我们构造了一个例子来表明将漂移变化发送到相应相对熵的映射通常不是凸的。