Stochastic target problems with controlled loss in jump diffusion models

In this paper, we consider a mixed diffusion version of the stochastic target problem introduced by Bouchard et al. (2009). This consists in finding the minimum initial value of a controlled process which guarantees to reach a controlled stochastic target with a given lovel of expected loss. As in Bouchard et al. (2009), it can be converted into a standard stochastic target problem, as already studied by Soner and Touzi (2002) or Bouchard (2002) for the mixed diffusion case, 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 controls that take values in an unbounded set of measurable maps. This has non trivial impacts on the formulation and derivation of the associated partial differential equations.