Pathwise Stochastic Control and a Class of Stochastic Partial Differential Equations
研究了路径意义下的随机最优控制问题，分析了对应的Hamilton-Jacobi-Bellman方程，证明了值函数是该方程的唯一粘性解，并刻画了最优漂移和守恒量。
Abstract In this article, we study a stochastic optimal control problem in the pathwise sense, as initially proposed by Lions and Souganidis in [C. R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 735-741]. The corresponding Hamilton-Jacobi-Bellman (HJB) equation, which turns out to be a non-adapted stochastic partial differential equation, is analyzed. Making use of the viscosity solution framework, we show that the value function of the optimal control problem is the unique solution of the HJB equation. When the optimal drift is defined, we provide its characterization. Finally, we describe the associated conserved quantities, namely the space-time transformations leaving our pathwise action invariant.