Approximation of integro-differential equations associated with piecewise deterministic process

Aim of this paper is to present an approximation scheme for optimal control problems of piecewise deterministic processes and corresponding integro-differential Hamilton–Jacobi–Bellman equations. The method is based on a discrete dynamic programming approach. We discretize the continuous process and the cost functional obtaining a discrete time optimal control problem. The corresponding dynamic programming equation gives an approximation of the integro-differential equation. The main feature of the method is the uniform convergence to the value function of the continuous control problem, which can be characterized as the unique weak solution (in viscosity sense) of the dynamic programming equation. Moreover, under appropriate assumptions, an error estimate on the truncation error is derived. It is worth noting that the method provides approximate feedback controls at any point of the grid without extra computations. An application of the approximation scheme to the numerical solution of an optimal control problem for a storage process is also detailed.