On some recent aspects of stochastic control and their applications

By Huyen Pham. 2005

99 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Economics and Econometrics, Finance

View Paper: https://www.semanticscholar.org/paper/c51f6c0f98871069979b50719d3cba3b3c76496f

How researchers see this paper

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In Computational Theory and Mathematics

1. Peter A. Forsyth
2. Y. Huang
3. George Labahn
4. Lars Blackmore
5. Kenneth R. Vetzal

In Economics and Econometrics

1. Peter A. Forsyth
2. Kenneth R. Vetzal
3. Y. Huang
4. George Labahn
5. Alexander Schied
6. Steven Kou

In Finance

1. Peter A. Forsyth
2. Kenneth R. Vetzal
3. George Labahn
4. Y. Huang
5. Jari Toivanen
6. Samuli Ikonen
7. S. Clift
8. Alexander Schied

Citations in the corpus

• Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance (2007, background): “A recent survey article on the theoretical aspects of this topic is given in [35].”
• Numerical Methods for Nonlinear PDEs in Finance (2012, background): “A recent survey on the theoretical aspects of this topic is given in [24].”
• Impulse control problem on finite horizon with execution delay (2007, unknown): “8) with τ = ∞ and Itô’s formula, see [13] for the details.”
• Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB) (2012, methodology): “…condition at some191 points in ΩWmax may be irrelevant, in the sense that the boundary condition at these points does192 not influence the interior solution.193 Pham (2005) discusses another case where singular control problems cannot be continuous over194 the entire closed solution domain.”
• A Probabilistic Particle-Control Approximation of Chance-Constrained Stochastic Predictive Control (2010, background): “The problem of control under stochastic uncertainty has been researched extensively, for example, [7], [38], [41], and [54].”