Summary: CONVERGENCE RATES FOR DIFFUSIONS ON CONTINUOUS-TIME
CLAUDIO ALBANESE AND ALEKSANDAR MIJATOVI´C
IMPERIAL COLLEGE LONDON
Abstract. 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
h 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 Ph
t (x, y) of the discretized process
converges at the rate O(h2) 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(h2).
Key Words: continuous-time Markov chains, spectral theory, functional calculus, conver-
gence estimates for probability kernels
Discretization schemes for stochastic processes are at the very core of modern mathematical
finance. Their relevance is both theoretical, as they shed light on the nature of the stochasticity
of the underlying process, and practical, since they lend themselves well to numerical methods.
Consequently there has been a plethora of publications devoted to various aspects of the topic