BFS 2002

Contributed Talk

Stochastic targets with mixed diffusion processes

Bruno Bouchard

Let $Z_{t,z}^\nu$ be a ${\R}^{d+1}$-valued mixed diffusion process controlled by $\nu$ with initial condition $Z_{t,z}^\nu(t)$ $=$ $z$. In this paper, we characterize the set of initial conditions such that $Z_{t,z}^\nu$ 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.