Simulation of a Lévy process by PCA sampling to reduce the effective dimension

Abstract

We consider a Levy process monitored at s (fixed) observation times. The goal is to estimate the expected value of some function of these s observations by (randomized) quasi-Monte Carlo. For the case where the process is a Brownian motion, clever techniques such as Brownian bridge sampling and PCA sampling have been proposed to reduce the effective dimension of the problem. The PCA method uses an eigen-decomposition of the covariance matrix of the vector of observations so that a larger fraction of the variance depends on the first few (quasi)random numbers that are generated. We show how this method can be applied to other Levy processes, and we examine its effectiveness in improving the quasi-Monte Carlo efficiency on some examples. The basic idea is to simulate a Brownian motion at s observation points using PCA, transform its increments into independent uniforms over (0, 1), then transform these uniforms again by applying the inverse distribution function of the increments of the Levy process. This PCA sampling technique is quite effective in improving the quasi-Monte Carlo performance when the sampled increments of the Levy process have a distribution that is not too far from normal, which typically happens when the process is observed at a large time scale, but may turn out to be ineffective in cases where the increments are far from normal.