Finite approximation schemes for Lévy processes, and their application to optimal stopping problems

This paper proposes two related approximation schemes, based on a discrete grid on a finite time interval [0,T], and having a finite number of states, for a pure jump Lévy process Lt. The sequences of discrete processes converge to the original process, as the time interval becomes finer and the number of states grows larger, in various modes of weak and strong convergence, according to the way they are constructed. An important feature is that the filtrations generated at each stage by the approximations are sub-filtrations of the filtration generated by the continuous time Lévy process. This property is useful for applications of these results, especially to optimal stopping problems, as we illustrate with an application to American option pricing. The rates of convergence of the discrete approximations to the underlying continuous time process are assessed in terms of a "complexity" measure for the option pricing algorithm. By adding in a construction for a discrete approximation to Brownian motion, we also extend the approximation results to a general Lévy process.