On some recent aspects of stochastic control and their applications
TLDR
In this article, a survey of stochastic control with respect to diffusion processes is presented, with a view towards applications and numerical issues and open questions, including degenerate singular control problems.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.read more
Citations
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Journal ArticleDOI
A Probabilistic Particle-Control Approximation of Chance-Constrained Stochastic Predictive Control
TL;DR: In this paper, the authors present a method for chance-constrained predictive stochastic control of dynamic systems, which takes into account uncertainty to ensure that the probability of failure due to collision with obstacles, for example, is below a given threshold.
Journal ArticleDOI
Numerical methods for stochastic control problems in continuous time
TL;DR: K Kushner and P.H. Dupuis as discussed by the authors have published a book called "Kushner and Duyguluis, 1992: A History of the World Wide Web".
Journal ArticleDOI
Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance
Peter Forsyth,George Labahn +1 more
TL;DR: In this paper, the authors show that the nonlinear option pricing problem can be formulated as an optimal control problem, leading to Hamilton-Jacobi-Bellman (HJB) or HJBI (HJBI-Isaacs) equations, which are very convenient for developing monotone discretization methods which ensure convergence to the financially relevant solution.
Proceedings ArticleDOI
Backward Stochastic Differential Equation, Nonlinear Expectation and Their Applications
TL;DR: A survey of the developments in the theory of Backward Stochastic Differential Equations during the last 20 years, including the solutions' existence and uniqueness, comparison theorem, nonlinear Feynman-Kac formula, g-expectation and many other important results in BSDE theory and their applications to dynamic pricing and hedging in an incomplete financial market is given in this article.
Journal ArticleDOI
A Semi-Lagrangian Approach for Natural Gas Storage Valuation and Optimal Operation
Zhuliang Chen,Peter Forsyth +1 more
TL;DR: A semi-Lagrangian method for solving the HJB equation for a typical gas storage valuation problem is presented and is able to handle a wide class of spot price models that exhibit mean-reverting seasonality dynamics and price jumps.
References
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Controlled Markov processes and viscosity solutions
Wendell H. Fleming,H. Mete Soner +1 more
TL;DR: In this paper, an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions is given, as well as a concise introduction to two-controller, zero-sum differential games.