We consider a class of closed loop stochastic optimal control problems in finite time horizon, in which the cost is an expectation conditional on the event that the process has not exited a given bounded domain. An important difficulty is that the probability of the event that conditionates the strategy decays as time grows. The optimality conditions consist of a system of partial differential equations, including a Hamilton-Jacobi-Bellman equation (backward w.r.t. time) and a (forward w.r.t. time) Fokker-Planck equation for the law of the conditioned process. The two equations are supplemented with Dirichlet conditions. Next, we discuss the asymptotic behavior as the time horizon tends to +∞. This leads to a new kind of optimal control problem driven by an eigenvalue problem related to a continuity equation with Dirichlet conditions on the boundary. We prove existence for the latter. We also propose numerical methods and supplement the various theoretical aspects with simulations.

我们考虑了有限时间范围内的一类闭环随机最优控制问题，其中成本是对过程没有离开给定有界域的事件的期望。一个重要的困难是，随着时间的增长，决定策略的事件的概率会下降。最佳条件由偏微分方程组组成，其中包括Hamilton-Jacobi-Bellman方程（后向时间）和（前向时间）Fokker-Planck方程，用于条件过程定律。这两个方程用Dirichlet条件进行了补充。接下来，我们讨论随着时间范围趋于+∞的渐近行为。这导致了一种新的最优控制问题，该问题由与边界上具有Dirichlet条件的连续性方程有关的特征值问题驱动。我们证明后者的存在。我们还提出了数值方法，并通过仿真补充了各种理论方面。