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International Journal of Stochastic Analysis
Volume 2014, Article ID 201491, 17 pages
http://dx.doi.org/10.1155/2014/201491
Research Article

The Relationship between the Stochastic Maximum Principle and the Dynamic Programming in Singular Control of Jump Diffusions

Laboratory of Applied Mathematics, University Mohamed Khider, P.O. Box 145, 07000 Biskra, Algeria

Received 7 September 2013; Revised 28 November 2013; Accepted 3 December 2013; Published 9 January 2014

Academic Editor: Agnès Sulem

Copyright © 2014 Farid Chighoub and Brahim Mezerdi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main objective of this paper is to explore the relationship between the stochastic maximum principle (SMP in short) and dynamic programming principle (DPP in short), for singular control problems of jump diffusions. First, we establish necessary as well as sufficient conditions for optimality by using the stochastic calculus of jump diffusions and some properties of singular controls. Then, we give, under smoothness conditions, a useful verification theorem and we show that the solution of the adjoint equation coincides with the spatial gradient of the value function, evaluated along the optimal trajectory of the state equation. Finally, using these theoretical results, we solve explicitly an example, on optimal harvesting strategy, for a geometric Brownian motion with jumps.