Probability Surveys

On some recent aspects of stochastic control and their applications

Huyên Pham

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Abstract

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.

Article information

Source
Probab. Surveys, Volume 2 (2005), 506-549.

Dates
First available in Project Euclid: 29 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.ps/1135911648

Digital Object Identifier
doi:10.1214/154957805100000195

Mathematical Reviews number (MathSciNet)
MR2203679

Zentralblatt MATH identifier
1189.93146

Subjects
Primary: 93E20, 49J20, 49L20, 60H30

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

Citation

Pham, Huyên. On some recent aspects of stochastic control and their applications. Probab. Surveys 2 (2005), 506--549. doi:10.1214/154957805100000195. https://projecteuclid.org/euclid.ps/1135911648


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