Multilevel Monte Carlo for stochastic differential equations with small noise

David F. Anderson, Desmond J. Higham, Yu Sun

Research output: Working paper

Abstract

We consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by Brownian motions in the small noise regime. We consider (i) standard Monte Carlo methods combined with numerical discretization algorithms tailored to the small noise setting, and (ii) a multilevel Monte Carlo method combined with a standard Euler-Maruyama implementation. The multilevel method combined with Euler-Maruyama is found to be the most efficient option under the assumptions we make on the underlying model. Further, under a wide range of scalings the multilevel method is found to be optimal in the sense that it has the same asymptotic computational complexity that arises from Monte Carlo with direct sampling from the exact distribution --- something that is typically impossible to do. The variance between two coupled paths, as opposed to the L2 distance, is directly analyzed in order to provide sharp estimates in the multilevel setting.
LanguageEnglish
Place of PublicationIthaca, N.Y.
Number of pages26
Publication statusPublished - 4 Jun 2015

Fingerprint

Multilevel Methods
Stochastic Equations
Differential equations
Monte Carlo methods
Differential equation
Monte Carlo method
Euler
Brownian movement
Computational complexity
Exact Distribution
Sampling
Brownian motion
Computational Complexity
Discretization
Scaling
Path
Estimate
Range of data
Standards
Model

Keywords

  • stochastic differential equation
  • Monte Carlo method
  • multilevel settings

Cite this

Anderson, D. F., Higham, D. J., & Sun, Y. (2015). Multilevel Monte Carlo for stochastic differential equations with small noise. Ithaca, N.Y.
Anderson, David F. ; Higham, Desmond J. ; Sun, Yu. / Multilevel Monte Carlo for stochastic differential equations with small noise. Ithaca, N.Y., 2015.
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Anderson, DF, Higham, DJ & Sun, Y 2015 'Multilevel Monte Carlo for stochastic differential equations with small noise' Ithaca, N.Y.

Multilevel Monte Carlo for stochastic differential equations with small noise. / Anderson, David F.; Higham, Desmond J.; Sun, Yu.

Ithaca, N.Y., 2015.

Research output: Working paper

TY - UNPB

T1 - Multilevel Monte Carlo for stochastic differential equations with small noise

AU - Anderson, David F.

AU - Higham, Desmond J.

AU - Sun, Yu

PY - 2015/6/4

Y1 - 2015/6/4

N2 - We consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by Brownian motions in the small noise regime. We consider (i) standard Monte Carlo methods combined with numerical discretization algorithms tailored to the small noise setting, and (ii) a multilevel Monte Carlo method combined with a standard Euler-Maruyama implementation. The multilevel method combined with Euler-Maruyama is found to be the most efficient option under the assumptions we make on the underlying model. Further, under a wide range of scalings the multilevel method is found to be optimal in the sense that it has the same asymptotic computational complexity that arises from Monte Carlo with direct sampling from the exact distribution --- something that is typically impossible to do. The variance between two coupled paths, as opposed to the L2 distance, is directly analyzed in order to provide sharp estimates in the multilevel setting.

AB - We consider the problem of numerically estimating expectations of solutions to stochastic differential equations driven by Brownian motions in the small noise regime. We consider (i) standard Monte Carlo methods combined with numerical discretization algorithms tailored to the small noise setting, and (ii) a multilevel Monte Carlo method combined with a standard Euler-Maruyama implementation. The multilevel method combined with Euler-Maruyama is found to be the most efficient option under the assumptions we make on the underlying model. Further, under a wide range of scalings the multilevel method is found to be optimal in the sense that it has the same asymptotic computational complexity that arises from Monte Carlo with direct sampling from the exact distribution --- something that is typically impossible to do. The variance between two coupled paths, as opposed to the L2 distance, is directly analyzed in order to provide sharp estimates in the multilevel setting.

KW - stochastic differential equation

KW - Monte Carlo method

KW - multilevel settings

UR - http://arxiv.org/abs/1412.3039v1

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BT - Multilevel Monte Carlo for stochastic differential equations with small noise

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Anderson DF, Higham DJ, Sun Y. Multilevel Monte Carlo for stochastic differential equations with small noise. Ithaca, N.Y. 2015 Jun 4.