Controlled stochastic hybrid processes with discounted cost

The authors consider a rather general stochastic hybrid system formed by a switching diffusion (a pair consisting of a diffusion and a pure jump process that affects the coefficients of the diffusion) that may jump between different regimes whenever the diffusive part hits some predetermined subsets of the state space. It is shown how the switching diffusion can be represented by a system driven by a Wiener process and a Poisson random measure and this system is controlled by a continuous nonanticipative control. The jumps between regimes can in turn be controlled by a discrete set of controls affecting a predetermined impulsive-type map. The control (continuous and discrete) is chosen so as to minimize an infinite horizon discounted cost criterion. Hamilton-Jacobi-Bellman type equations are derived, existence and uniqueness of a classical solution is discussed and a verification theorem is proved. Computational aspects are not considered.