# Optimal control of conditioned processes with feedback controls

arXiv: https://arxiv.org/abs/1912.08738 · HTML: https://mlsanity.com/paper/1912.08738/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Dec 2019.

## Built on by (most cited first, 1 of 1)

- [Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance](https://mlsanity.com/paper/2107.04568/) (2021)

## Abstract (arXiv)

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 numerical simulations.