带跳平均场随机偏微分方程的随机控制
Stochastic control of mean-field SPDEs with jumps
摘要：
在\textit{partial information}控制的情形下，研究了由布朗运动和独立Poisson随机测度驱动的平均场随机偏微分方程（随机发展方程）的最优控制问题.我们的问题的一个重要的新颖性是由\textit{general mean-field}运营商的介绍，作用于受控状态过程和控制过程。首先，我们制定了充分和必要的最大值原理这种类型的控制。然后证明了这类一般的正、倒向平均场随机偏微分方程解的存在唯一性。最后，我们应用我们的结果，找到明确的最优控制的最优收获问题。
英文摘要：
We study the problem of optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in the case of \textit{partial information} control. One important novelty of our problem is represented by the introduction of \textit{general mean-field} operators, acting on both the controlled state process and the control process. We first formulate a sufficient and a necessary maximum principle for this type of control. We then prove existence and uniqueness of the solution of such general forward and backward mean-field stochastic partial differential equations. We finally apply our results to find the explicit optimal control for an optimal harvesting problem.
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