Abstract

Consider a marked point process or a jump-diffusion, from which we want to simulate a trajectory of the process or a functional of it. In both cases the magnitude of noise contributors is controlled by a small parameter epsilon. Raw Monte Carlo methods produce estimators with a large relative error, which increases even more as N increases or epsilon decreases. Using viscosity sub-solutions of Hamilton-Jacobi equations, we were able to produce importance sampling algorithms with optimal asymptotic behaviour and low relative error across a variety of small values of noise contribution. Some basic stochastic knowledge and means to produce the discretization of a trajectory of the jump-diffusion are needed, both of which are provided in this text. Furthermore, we applied the algorithm we developed to model the bistability in the concentration of certain molecular species.