Analysis of Markov Chain Approximation for Diffusion Models with Non-Smooth Coefficients

Calculation of the expected value of discounted payoffs with possible monitoring of barrier crossing under one-dimensional diffusion models is required in many applications. Markov chain approximation is a computationally efficient approach for this problem. This paper undertakes the challenge of analyzing its convergence rate when model coefficients are nonsmooth. We obtain sharp estimates of convergence rates for the value function and its first and second derivatives, which are generally first order. To improve convergence rates to second order, we propose two methods: following the midpoint rule that places all nonsmooth points midway between two neighboring grid points or applying a smoothing technique named as harmonic averaging to the model coefficients. Comparison with a widely used finite difference scheme for PDEs with nonsmooth coefficients shows the superiority of our approach. We also generalize the midpoint rule to achieve second-order convergence for two-dimensional diffusions. Numerical experiments confirm the theoretical estimates.