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 Probability Surveys > Vol. 2 (2005) open journal systems 

On some recent aspects of stochastic control and their applications

Huyên Xuan Pham, Laboratoire de Probabilités et Modèles Aléatoires, University Paris 6-Paris 7

This paper is a survey on some recent aspects and developments in stochastic control. We discuss the two main historical approaches, Bellman's optimality principle and Pontryagin's maximum principle, and their modern exposition with viscosity solutions and backward stochastic differential equations. Some original proofs are presented in a unifying context including degenerate singular control problems. We emphasize key results on characterization of optimal control for diffusion processes, with a view towards applications. Some examples in finance are detailed with their explicit solutions. We also discuss numerical issues and open questions.

AMS 2000 subject classifications: 93E20, 49J20, 49L20, 60H30.

Keywords: Controlled diffusions, dynamic programming, maximum principle, viscosity solutions, backward stochastic differential equations, finance.

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Pham, Huyên Xuan, On some recent aspects of stochastic control and their applications, Probability Surveys, 2, (2005), 506-549 (electronic).


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