Approximation of integro-differential equations associated with piecewise deterministic process

Abstract

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. © 1997 John Wiley & Sons, Ltd.

Finite element scheme for integro-partial differential equations

“…In the first order case and when the Levy measure is bounded, convergence rate 1/2 for semi-Lagrangian schemes like (3.8) have previously be obtained in [16]. However the integral term in [16] has a different form compared to the one we consider here.…”

“…In [11] it is proved that u is the unique viscosity solution of equation (1.1). Following the approach of [16,17] we construct an approximation scheme for the equation (1.1) by discretizing the associated control problem. We fix a discretization step h > 0 and consider two stochastic processes N n and Z n , n ∈ N, taking values in N and in R N and representing the n-th jump time and the corresponding z-jump (size and direction) of the Poisson measure μ.…”

“…Semi-Lagrangian schemes for second order HJB equations with no integral terms have been considered in [37,17,5,1]. Moreover in [16] such schemes were derived for HJB equations associated to piecewise deterministic processes (compound Poisson processes with drift). One advantage of this type of schemes is that in general they produce also an approximation of the optimal control law in feedback form and the optimal trajectories (this is a key point in the study of a control problem).…”

Limited Commercial Licensing Strategies: A Piecewise Deterministic Differential Game

“…where we set the discount factor β = e −ωh . Camilli (1997) shows that V h (x, i) satisfies the following dynamic programming equation…”