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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • [1] Bayer, C., Friz, P. K., Riedel, S. and Schoenmakers, J. (2016). From rough path estimates to multilevel Monte Carlo. SIAM J. Numer. Anal. 54 1449–1483.
  • [2] Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12 1077–1098.
  • [3] Beskos, A., Peluchetti, S. and Roberts, G. (2012). $\varepsilon$-strong simulation of the Brownian path. Bernoulli 18 1223–1248.
  • [4] Beskos, A. and Roberts, G. O. (2005). Exact simulation of diffusions. Ann. Appl. Probab. 15 2422–2444.
  • [5] Blanchet, J. and Chen, X. (2015). Steady-state simulation of reflected Brownian motion and related stochastic networks. Ann. Appl. Probab. 25 3209–3250.
  • [6] Chen, N. and Huang, Z. (2013). Localization and exact simulation of Brownian motion-driven stochastic differential equations. Math. Oper. Res. 38 591–616.
  • [7] Davie, A. M. (2007). Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express. AMRX 2 Art. ID abm009, 40.
  • [8] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge.
  • [9] Gaines, J. G. and Lyons, T. J. (1994). Random generation of stochastic area integrals. SIAM J. Appl. Math. 54 1132–1146.
  • [10] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [11] Pollock, M., Johansen, A. M. and Roberts, G. O. (2016). On the exact and $\varepsilon$-strong simulation of (jump) diffusions. Bernoulli 22 794–856.
  • [12] Rhee, C. and Glynn, P. (2012). A new approach to unbiased estimation for SDE’s. In Proceedings of the Winter Simulation Conference. Winter Simulation Conference.
  • [13] Steele, J. M. (2001). Stochastic Calculus and Financial Applications. Applications of Mathematics (New York) 45. Springer, New York.