The author's presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and generality of the multilevel Monte Carlo approach. It also offers a few original ideas and suggests areas for future research.
In 2005, Kebaier @cite_45 developed a two-level approach for path simulation which is very similar to the author's approach presented in the next section. The only differences are the use of only two levels, and the use of a general multiplicative factor as in the standard control variate approach. A similar multilevel approach was under development at the same time by Speight, but was not published until later @cite_15 @cite_27 .
@cite_27 @cite_15 @cite_45
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I present a self-contained introduction to multigrid methods with an emphasis on techniques relevant to dynamic programming and related problems. A probabilistic interpretation of the numerical principles is highlighted. Multigrid solvers are shown to be naturally matched to the challenges posed by intractable structural dynamic models routinely encountered in applied economics. I argue that multigrid techniques have potential to substantially extend the scale and complexity of models under consideration. Multigrid also provides a unified computational framework to extend model solvers to perform sensitivity analysis, calibration, estimation, and counterfactual policy experiments. We present a new variance reduction technique that naturally applies to price financial derivatives by Monte Carlo simulation. Inspired by multigrid methods for solving PDEs, the technique is based on control variates derived from a sequence of approximations that converge pathwise to a limiting model. It applies to a large class of problems, and is easy to implement. Theory and computational results show this method can substantially reduce computational time relative to crude Monte Carlo estimation and is competitive with other variance reduction techniques under Monte Carlo sampling. We study the approximation of Ef(X-T) by a Monte Carlo algorithm, where X is the solution of a stochastic differential equation and f is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps delta and delta(beta), 0 < beta < 1. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal beta (which turns out to be beta = 1 2), we establish a central limit: type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error.