On some recent aspects of stochastic control and their applications

Details On some recent aspects of stochastic control and their applications On some recent aspects of stochastic control and their applications

2005-09-30

arxiv.org/abs/math/0509711v3

Probability Surveys 2005, Vol. 2, 506-549

Mathematics

Probability

Published at http://dx.doi.org/10.1214/154957805100000195 in the Probability Surveys (http://www.i-journals.org/ps/) by the In

Scientific paper

10.1214/154957805100000195

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.

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