Duality and approximation of stochastic optimal control problems under expectation constraints
Abstract
We consider a continuous time stochastic optimal control problem under both equality and inequality constraints on the expectation of some functionals of the controlled process. Under a qualification condition, we show that the problem is in duality with an optimization problem involving the Lagrange multiplier associated with the constraints. Then by convex analysis techniques, we provide a general existence result and some a priori estimation of the dual optimizers. We further provide a necessary and sufficient optimality condition for the initial constrained control problem. The same results are also obtained for a discrete time constrained control problem. Moreover, under additional regularity conditions, it is proved that the discrete time control problem converges to the continuous time problem, possibly with a convergence rate. This convergence result can be used to obtain numerical algorithms to approximate the continuous time control problem, which we illustrate by two simple numerical examples.
Cite
@article{arxiv.2007.00561,
title = {Duality and approximation of stochastic optimal control problems under expectation constraints},
author = {Laurent Pfeiffer and Xiaolu Tan and Yulong Zhou},
journal= {arXiv preprint arXiv:2007.00561},
year = {2021}
}