Stochastic Target Problems with Controlled Loss in Jump Diffusion Models

Abstract

In this paper, we consider a mixed-diffusion version of the stochastic target problem introduced in [B. Bouchard, R. Elie, and N. Touzi, SIAM J. Control Optim., 48 (2009), pp. 3123–3150]. This consists in finding the minimum initial value of a controlled process which guarantees to reach a controlled stochastic target with a given level of expected loss. It can be converted into a standard stochastic target problem by increasing both the state space and the dimension of the control. In our mixed-diffusion setting, the main difficulty comes from the presence of jumps, which leads to the introduction of a new kind of control that takes values in an unbounded set of measurable maps. This has a nontrivial technical impact on the formulation and derivation of the associated partial differential equations.