The Annals of Applied Probability

$\varepsilon$-Strong simulation for multidimensional stochastic differential equations via rough path analysis

Jose Blanchet, Xinyun Chen, and Jing Dong

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Abstract

Consider a multidimensional diffusion process $X=\{X(t):t\in [0,1]\}$. Let $\varepsilon>0$ 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 $X_{\varepsilon}$ such that

\[\sup_{0\leq t\leq1}\Vert X_{\varepsilon}(t)-X(t)\Vert_{\infty}<\varepsilon\] with probability one. Moreover, the user can adaptively choose $\varepsilon^{\prime}\in (0,\varepsilon )$ so that $X_{\varepsilon^{\prime}}$ (also piecewise constant and fully simulatable) can be constructed conditional on $X_{\varepsilon}$ to ensure an error smaller than $\varepsilon^{\prime}$ 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 $\varepsilon$ error in the underlying rough path metric.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 1 (2017), 275-336.

Dates
Received: March 2014
Revised: January 2016
First available in Project Euclid: 6 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1488790829

Digital Object Identifier
doi:10.1214/16-AAP1204

Mathematical Reviews number (MathSciNet)
MR3619789

Zentralblatt MATH identifier
06711461

Subjects
Primary: 34K50: Stochastic functional-differential equations [See also , 60Hxx] 65C05: Monte Carlo methods 82B80: Numerical methods (Monte Carlo, series resummation, etc.) [See also 65-XX, 81T80]
Secondary: 97K60: Distributions and stochastic processes

Keywords
Stochastic differential equation Monte Carlo method Brownian motion Lévy area rough path

Citation

Blanchet, Jose; Chen, Xinyun; Dong, Jing. $\varepsilon$-Strong simulation for multidimensional stochastic differential equations via rough path analysis. Ann. Appl. Probab. 27 (2017), no. 1, 275--336. doi:10.1214/16-AAP1204. https://projecteuclid.org/euclid.aoap/1488790829


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