Optimal control of direction of reflection for jump diffusions
Résumé
We extend the optimal control of direction of reflection problem introduced in Bouchard [4] to the jump diffusion case. In a Brownian diffusion framework with jumps, the controlled process is defined as the solution of a stochastic differential equation (SDE) reflected at the boundary of a domain along oblique directions of reflection which are controlled by a predictable process which may have jumps. We also provide a version of the weak dynamic programming principle of Bouchard and Touzi [5] adapted to our context and which is sufficient to provide a viscosity characterization of the associated value function without requiring the usual heavy measurable selection arguments nor the a-priori continuity of the value function.
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