ε-strong simulation for multidimensional stochastic differential equations via rough path analysis

Jose Blanchet, Xinyun Chen, Jing Dong

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

Consider a multidimensional diffusion process X = {X(t) : t ∈ [0, 1]}. Let ε > 0 be a deterministic, user defined, tolerance error parameter. Under standard regularity conditions on the drift and diffusion coefficients of X, we construct a probability space, supporting both X and an explicit, piecewise constant, fully simulatable process Xε such that sup ∥Xε(t) - X(t)∥ < ε 0≤t≤1 with probability one. Moreover, the user can adaptively choose ε′ ∈ (0, ε) so that Xε′ (also piecewise constant and fully simulatable) can be constructed conditional on Xε to ensure an error smaller than ε′ with probability one. Our construction requires a detailed study of continuity estimates of the Itô map using Lyons' theory of rough paths. We approximate the underlying Brownian motion, jointly with the Lévy areas with a deterministic ε error in the underlying rough path metric.

Original languageEnglish (US)
Pages (from-to)275-336
Number of pages62
JournalAnnals of Applied Probability
Volume27
Issue number1
DOIs
StatePublished - Feb 2017

Keywords

  • Brownian motion
  • Lévy area
  • Monte Carlo method
  • Rough path
  • Stochastic differential equation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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