Applications des inéquations variationnelles en contrôle stochastique
01 Jan 2017-
About: The article was published on 2017-01-01 and is currently open access. It has received 274 citations till now.
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18 Dec 1992TL;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.
Abstract: This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.
3,885 citations
TL;DR: In this article, reflected solutions of one-dimensional backward stochastic differential equations are studied and the authors prove uniqueness and existence both by a fixed point argument and by approximation via penalization.
Abstract: We study reflected solutions of one-dimensional backward stochastic differential equations. The “reflection” keeps the solution above a given stochastic process. We prove uniqueness and existence both by a fixed point argument and by approximation via penalization. We show that when the coefficient has a special form, then the solution of our problem is the value function of a mixed optimal stopping–optimal stochastic control problem. We finally show that, when put in a Markovian framework, the solution of our reflected BSDE provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation.
781 citations
622 citations
TL;DR: In this paper, the authors provide a complete justification of the so-called Brennan-Schwartz algorithm for the valuation of American put options and discuss numerical methods, based on the Bensoussan-Lions methods of variational inequalities.
Abstract: This paper is devoted to the derivation of some regularity properties of pricing functions for American options and to the discussion of numerical methods, based on the Bensoussan-Lions methods of variational inequalities. In particular, we provide a complete justification of the so-called Brennan-Schwartz algorithm for the valuation of American put options.
453 citations
TL;DR: In this paper, a new class of variational inequalities is introduced and studied, and the projection technique is used to suggest an iterative algorithm for finding the approximate solution of this class.
393 citations