Adaptive Monte Carlo algorithms for stopped diffusion

Anna Dzougoutov, Kyoung Sook Moon, Erik von Schwerin, Anders Szepessy, Raúl Tempone

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

17 Scopus citations

Abstract

We present adaptive algorithms for weak approximation of stopped diffusion using the Monte Carlo Euler method. The goal is to compute an expected value E[g(X(τ), τ)] of a given function g depending on the solution X of an Itô stochastic differential equation and on the first exit time τ from a given domain. The adaptive algorithms are based on an extension of an error expansion with computable leading order term, for the approximation of E[g(X(T))] with a fixed final time T > 0 and diffusion processes X in ℝd, introduced in [17] using stochastic flows and dual backward solutions. The main steps in the extension to stopped diffusion processes are to use a conditional probability to estimate the first exit time error and introduce difference quotients to approximate the initial data of the dual solutions. Numerical results show that the adaptive algorithms achieve the time discretization error of order N-1 with N adaptive time steps, while the error is of order N-1/2 for a method with N uniform time steps.

Original languageEnglish (US)
Title of host publicationMultiscale Methods in Science and Engineering
Pages59-88
Number of pages30
Volume44
StatePublished - 2005
Externally publishedYes

Publication series

NameLecture Notes in Computational Science and Engineering
Volume44
ISSN (Print)1439-7358

Keywords

  • Adaptive mesh refinement algorithm
  • Barrier option
  • Diffusion with boundary
  • Monte Carlo method
  • Weak approximation

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering(all)
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Mathematics

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