Multilevel Monte Carlo in the small noise setting: from jump to diffusion processes
Monte Carlo methods are used ubiquitously in the study of stochastic processes. The multilevel Monte Carlo method of Mike Giles (2008), with related earlier work by Stefan Heinrich (1998), can dramatically increase the speed of Monte Carlo estimation by generating multiple samples from coupled pairs of processes (with different discretization parameters) and combining them in an appropriate manner. The computational complexity of multilevel methods depends sensitively upon the size of the variance between the coupled pairs of processes. In order to estimate this variance, at least asymptotically, it is often sufficient to simply analyze the $L^2$ distance between the pairs.
In this talk, I will consider multilevel Monte Carlo in the small noise setting. Stochastic models with small noise structure arise in a number of settings including biochemistry and cell biology, finance, computational fluid dynamics, ecology, neuroscience, and population dynamics. In this setting, I will show that the $L^2$ distance between coupled pairs of paths is sometimes asymptotically larger than the variance between the pairs, and, therefore, a direct analysis of the variance is necessary in order to obtain optimal complexity estimates. I will focus on the cases of (i) continuous time Markov chain models, as arise in biochemistry and cellular biology, and (ii) diffusion processes with small noise structure. In the diffusive setting, multilevel Monte Carlo combined with an Euler-Maruyama discretization is shown to work optimally without the need for discretization schemes that are customized to the small noise setting. Perhaps surprisingly, I will show that even though diffusion processes with small noise structure often arise as natural approximations to continuous time Markov chain models under an appropriate scaling, the variance of the coupled diffusion processes can not be inferred from the variance of the analogous coupled jump processes.