On the Acceleration of the Multi-Level Monte Carlo Method

Abstract

The multi-level Monte Carlo method proposed by M. approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper, a modified multi-level Monte Carlo estimator is proposed with significantly reduced computational costs. As the main result, it is proved that the modified estimator reduces the computational costs asymptotically by a factor (p/α) 2 if weak approximation methods of orders α and p are applied in case of computational costs growing with same order as variances decay.

Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators

“…, where L * and M * l are given by (4.9) and (4.10), respectively, achieves a complexity O ǫ −2 . In [2], Debrabant Rössler improved the multilevel Monte Carlo method by using, in the last level L, a scheme with high order of weak convergence. Although this modified method attains the same complexity, it reduces the computation time by reducing the bias.…”

“…In this paper, we propose to use the Ninomiya-Victoir scheme, which is known to exhibit weak convergence with order 2, on the finest grid at the last level L of a multilevel Monte Carlo estimator. This idea is inspired by Debrabant and Rössler [2] who suggest to use a scheme with high order of weak convergence on the finest grid at the finest level L of the multilevel Monte Carlo method. By this way, Debrabant and Rössler reduce the constant in the computational complexity by decreasing the number of discretization levels.…”

A Micro-Macro Acceleration Method for the Monte Carlo Simulation of Stochastic Differential Equations

“…Numerous methods exist to increase the efficiency of Monte Carlo simulation of SDEs. Let us mention only weak explicit [1,13,17,43,44,54,61] and implicit [2,[14][15][16]43,45] higher order schemes, which can increase the time step but might suffer from instability; various extrapolation methods [68] to obtain the precision of higher order from low-order schemes; and variance reduction techniques [18,55], including the multilevel Monte Carlo method [29,30].…”

Improved Stabilized Multilevel Monte Carlo Method for Stiff Stochastic Differential Equations

“…As mentioned in the introduction, the EM method as well as the S-ROCK1 method are both of weak order 1 and strong order 1/2. The idea is to use a numerical integrator of higher weak order for the finest time grid (see [6]), in our case S-ROCK2 [1] with weak order 2, which leads to a reduction of the bias. In fact due to the telescopic sum representation of the multilevel estimator, only the estimator based on the smallest time stepsize (which uses S-ROCK2) appears in the bias.…”