A Positive and Monotone Numerical Scheme for Volterra-Renewal Equations with Space Fluxes (Journal of Computational Mathematics, 2018)
We study a numerical method for solving a system of Volterra-renewal integral equationswith space fluxes, that represents the Chapman-Kolmogorov equation for a class ofpiecewise deterministic stochastic processes. The solution of this equation is related tothe time dependent distribution function of the stochastic process and it is a non-negativeand non-decreasing function of the space. Based on the Bernstein polynomials, we buildup and prove a non-negative and non-decreasing numerical method to solve that equation,with quadratic convergence order in space.
Keywords: Volterra renewal; Piecewise deterministic process; Monotone positive numerical scheme; Bernstein polynomials.