The Annals of Applied Probability

Unbiased simulation of stochastic differential equations

Pierre Henry-Labordère, Xiaolu Tan, and Nizar Touzi

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We propose an unbiased Monte 3 estimator for $\mathbb{E}[g(X_{t_{1}},\ldots,X_{t_{n}})]$, where $X$ is a diffusion process defined by a multidimensional stochastic differential equation (SDE). The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are updated at independent exponential times. Such a simulatable process can be viewed as a regime-switching SDE, or as a branching diffusion process with one single living particle at all times. In order to compensate for the change of the coefficients of the SDE, our main representation result relies on the automatic differentiation technique induced by the Bismut–Elworthy–Li formula from Malliavin calculus, as exploited by Fournié et al. [Finance Stoch. 3 (1999) 391–412] for the simulation of the Greeks in financial applications. In particular, this algorithm can be considered as a variation of the (infinite variance) estimator obtained in Bally and Kohatsu-Higa [Ann. Appl. Probab. 25 (2015) 3095–3138, Section 6.1] as an application of the parametrix method.

Article information

Ann. Appl. Probab., Volume 27, Number 6 (2017), 3305-3341.

Received: March 2016
Revised: November 2016
First available in Project Euclid: 15 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 65C05: Monte Carlo methods 60J60: Diffusion processes [See also 58J65]
Secondary: 60J85: Applications of branching processes [See also 92Dxx] 35K10: Second-order parabolic equations

Unbiased simulation of SDEs regime switching diffusion linear parabolic PDEs


Henry-Labordère, Pierre; Tan, Xiaolu; Touzi, Nizar. Unbiased simulation of stochastic differential equations. Ann. Appl. Probab. 27 (2017), no. 6, 3305--3341. doi:10.1214/17-AAP1281.

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