Mean Exit Times and the Multilevel Monte Carlo Method

Abstract

Abstract. Numerical methods for stochastic differential equations are relatively inefficient when used to approximate mean exit times. In particular, although the basic Euler-Maruyama method has weak order equal to one for approximating the expected value of the solution, the order reduces to one half when it is used in a straightforward manner to approximate the mean value of a (stopped) exit time. Consequently, the widely used standard approach of combining an Euler-Maruyama discretization with a Monte Carlo simulation leads to a computationally expensive procedure. 1. Background and notation. We begin with the system of stochastic differential equations (SDEs),Brownian motion, and we let (Ω, F, P, F t ) be a complete, filtered probability space satisfying the usual conditions. For a specified open set O ⊂ R d , the stopped exit time is the first time at which X(s) leaves the open set O, or T if this is smaller. Our quantity of interest is the expected value of this random variable. *

Data-driven coarse graining in action: Modeling and prediction of complex systems

“…To approximate τ one typically resorts to Monte-Carlo techniques based on numerically solving the SDE (1) [45]. For small dimensions (i.e.…”

Computing mean first exit times for stochastic processes using multi-level Monte Carlo

“…So generally, while independent Brownian paths are used at different levels l, within any particular level l, for l > 0, the pair ν l−1 comes from the same path. It is shown in Higham, Mao, Roj, Song, and Yin (2013) that a particular choice of samples-per-level, N l , given by (2) below, leads to a method with the required user-specified accuracy of TOL having complexity O(TOL −3 | log(TOL)| 1/2 ). This gives an almost O(TOL −1 ) improvement over standard Monte Carlo.…”

“…This approach has proved popular (Andersen 2000;Boyle, Broadie, and Glasserman 1997;Longstaff and Schwartz 2001;Mannella 1999) due to its simplicity and its ability to deal with high dimensions and complicated boundaries. In Higham, Mao, Roj, Song, and Yin (2013) a multi-level version of the Monte Carlo approach was proposed and analyzed, based on the ideas of Giles (2008) and Heinrich (2001). Our aim here is to give an overview of the approach and present further computational tests that back up the analysis.…”

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“…Any correspondence concerning this service should be sent to the Mao, Roj, Song, and Yin (2013), and illustrated on some scalar examples. Here, we briefly overview the algorithm and present some new computational results in higher dimensions.…”