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On some recent aspects of stochastic control and their applications. (English) Zbl 1189.93146

Summary: 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.

MSC:

93E20 Optimal stochastic control
49J20 Existence theories for optimal control problems involving partial differential equations
49L20 Dynamic programming in optimal control and differential games
60H30 Applications of stochastic analysis (to PDEs, etc.)
91G80 Financial applications of other theories
93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
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