Duality and approximation of stochastic optimal control problems under expectation constraints

Laurent Pfeiffer; Xiaolu Tan; Yulong Zhou

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arXiv · 2007.00561

Duality and approximation of stochastic optimal control problems under expectation constraints

Laurent Pfeiffer·Xiaolu Tan·Yulong Zhou

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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.

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BibTeX RIS

Laurent Pfeiffer, Xiaolu Tan, Yulong Zhou. 2020-07-01. Duality and approximation of stochastic optimal control problems under expectation constraints. https://arxiv.org/abs/2007.00561

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