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

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. 1. Introduction In this paper, we consider a mixed classical-singular control problem, in which the state evolves according to a stochastic differential equation, driven by a Poisson random measure and an independent multidimensional Brownian motion, of the following form: where , , , and are given deterministic functions and is the initial state. The control variable is a suitable process , where is the usual classical absolutely continuous control and is the singular control, which is an increasing process, continuous on the right with limits on the left, with . The performance functional has the form The objective of the controller is to choose a couple of adapted processes, in order to maximize the performance functional. In the first part of our present work, we investigate the question of necessary as well as sufficient optimality conditions, in the form of a Pontryagin stochastic maximum principle. In the second part, we give under regularity assumptions, a useful verification theorem. Then, we show that the adjoint process 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. Note that our results improve those in [1, 2] to the jump diffusion setting. Moreover we generalize results in [3, 4], by allowing both classical and singular controls, at least in the complete information setting. Note that in our control problem, there are two types of jumps for the state process, the inaccessible ones which come from the Poisson martingale part and the predictable ones which come from the singular control part. The inclusion of these jump terms introduces a major