Convergence Rates for Diffusions on Continuous-Time Lattices

In this paper we introduce a discretization scheme based on a continuous-time Markov chain for the Black-Scholes diffusion process. Our principal aim is to find the optimal convergence rate for the probability density function of the discretized process as the distance between the nodes of the state-space of the Markov chain goes to zero. The main theorem of the paper (theorem 4.1) states that the probability kernel of the discretized process converges at the rate O(h^2) to the probability density function pt(x, y) of the diffusion process. We also show that this convergence is uniform in the state variables x and y and that the proposed discretization scheme converges at a rate which is no faster than O(h^2).