Numerical evaluation of resolvents and Laplace transforms of Markov processes using linear programming

This paper uses linear programming to numerically evaluate the Laplace transform of the exit time distribution and the resolvent of the moments of various Markov processes in bounded regions. The linear programming formulation is developed from a martingale characterization of the processes and the use of occupation measures. The LP approach naturally provides both upper and lower bounds on the quantities of interest. The processes analyzed include the Poisson process, one-dimensional Brownian motion (with and without drift), an Ornstein–Uhlenbeck process and two-dimensional Brownian motion. The Laplace transform of the original Cameron–Martin formula is also numerically evaluated by reducing it to the analysis of an Ornstein–Uhlenbeck process.