Stochastic targets with mixed diffusion processes and viscosity solutions

Let Zt,zν be a Rd+1-valued mixed diffusion process controlled by ν with initial condition Zt,zν(t)=z. In this paper, we characterize the set of initial conditions such that Zt,zν can be driven above a given stochastic target at time T by proving that the corresponding value function is a discontinuous viscosity solution of a variational partial differential equation. As applications of our main result, we study two examples: a problem of optimal insurance under self-protection and a problem of option hedging under jumping stochastic volatility where the underlying stock pays a random dividend at a fixed date.