# Two Approaches to Stochastic Optimal Control Problems with a Final-Time Expectation Constraint

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## Abstract

In this article, we study and compare two approaches to solving stochastic optimal control problems with an expectation constraint on the final state. The case of a probability constraint is included in this framework. The first approach is based on a dynamic programming principle and the second one uses Lagrange relaxation. In this article, we focus on discrete-time problems, but the two discussed approaches can be applied to discretized continuous-time problems.

## Keywords

Stochastic optimal control Expectation and probability constraints Dynamic programming Lagrange relaxation## Mathematics Subject Classification

90C15 93E20## Notes

### Acknowledgments

The author thanks J. Frédéric Bonnans for his advice and the two anonymous referees for useful remarks. The research leading to these results has received funding from the Gaspard Monge Program for Optimization and operations research (PGMO). The author gratefully acknowledges the Austrian Science Fund (FWF) for financial support under SFB F32, “Mathematical Optimization and Applications in Biomedical Sciences”.

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