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

This paper uses linear programming to numerically evaluate the Laplace tran sform of the exit time distribution and the resolvent of the moments of var ious Markov processes in bounded regions. The linear programming formulatio n is developed from a martingale characterization of the processes and the use of occupation measures. The LP approach naturally provides both upper a nd lower bounds on the quantities of interest. The processes analyzed inclu de the Poisson process, one-dimensional Brownian motion (with and without d rift), an Ornstein-Uhlenbeck process and two-dimensional Brownian motion. T he Laplace transform of the original Cameron-Martin formula is also numeric ally evaluated by reducing it to the analysis of an Ornstein-Uhlenbeck proc ess.