Levy process simulation by stochastic step functions

TL;DR

This paper introduces a stochastic step function Monte Carlo algorithm for simulating Levy processes, offering uncorrelated samples and improved efficiency over traditional methods like Metropolis/Hastings.

Contribution

The paper presents a novel step function Monte Carlo method for Levy process simulation, reducing correlations and improving accuracy compared to existing algorithms.

Findings

Step function method produces uncorrelated Levy process samples.

Compared to Metropolis/Hastings, the method reduces heavy tails in distributions.

Numerical tests show superior accuracy of the step function approach.

Abstract

We study a Monte Carlo algorithm for simulation of probability distributions based on stochastic step functions, and compare to the traditional Metropolis/Hastings method. Unlike the latter, the step function algorithm can produce an uncorrelated Markov chain. We apply this method to the simulation of Levy processes, for which simulation of uncorrelated jumps are essential. We perform numerical tests consisting of simulation from probability distributions, as well as simulation of Levy process paths. The Levy processes include a jump-diffusion with a Gaussian Levy measure, as well as jump-diffusion approximations of the infinite activity NIG and CGMY processes. To increase efficiency of the step function method, and to decrease correlations in the Metropolis/Hastings method, we introduce adaptive hybrid algorithms which employ uncorrelated draws from an adaptive discrete distribution defined on a space of subdivisions of the Levy measure space. The nonzero correlations in Metropolis/Hastings simulations result in heavy tails for the Levy process distribution at any fixed time. This problem is eliminated in the step function approach. In each case of the Gaussian, NIG and CGMY processes, we compare the distribution at t=1 with exact results and note the superiority of the step function approach.