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LEVY PROCESS SIMULATION BY STOCHASTIC STEP FUNCTIONS
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DOI：10.1137/110851080.png)
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Abstract
En 中文
We study a Monte Carlo algorithm for simulation of probability distributions based on stochastic step functions, and compare it 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 drawing of uncorrelated jumps is 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 normal inverse Gaussian (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 subdivision 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, as well as the autocorrelation functions.
Keywords:
Levy process
Markov chains
Monte Carlo methods
simulation of probability distributions
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