Analysis of Markov Chain Approximation for Diffusion Models with Nonsmooth Coefficients

Abstract

In this study, we analyze the convergence of continuous-time Markov chain approximation for one-dimensional diffusions with nonsmooth coefficients. We obtain a sharp estimate of the convergence rate for the value function and its first and second derivatives, which is generally first order. To improve it to second order, we propose two methods: applying the midpoint rule that places all nonsmooth points midway between two neighboring grid points or applying harmonic averaging to smooth the model coefficients. We conduct numerical experiments for various financial applications to confirm the theoretical estimates. We also show that the midpoint rule can be applied to achieve second-order convergence for some jump-diffusion and two-factor short rate models.