Levy Process Simulation by Stochastic Step Functions

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.