TY - JOUR

T1 - Unbiased simulation of stochastic differential equations

AU - Henry-Labordère, Pierre

AU - Tan, Xiaolu

AU - Touzi, Nizar

N1 - Funding Information:
ERC 321111 Rofirm, the ANR Isotace, and the Chairs Financial Risks (Risk Foundation, sponsored by Société Générale) and Finance and Sustainable Development (IEF sponsored by EDF and CA).
Funding Information:
Received March 2016; revised November 2016. 1Supported by ERC 321111 Rofirm, the ANR Isotace, and the Chairs Financial Risks (Risk Foundation, sponsored by Société Générale) and Finance and Sustainable Development (IEF sponsored by EDF and CA). MSC2010 subject classifications. Primary 65C05, 60J60; secondary 60J85, 35K10. Key words and phrases. Unbiased simulation of SDEs, regime switching diffusion, linear parabolic PDEs.
Publisher Copyright:
© Institute of Mathematical Statistics, 2017.

PY - 2017/12

Y1 - 2017/12

N2 - We propose an unbiased Monte 3 estimator for E[g(Xt1, . . . , Xtn )], 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 fromMalliavin 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.

AB - We propose an unbiased Monte 3 estimator for E[g(Xt1, . . . , Xtn )], 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 fromMalliavin 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.

KW - Linear parabolic PDEs

KW - Regime switching diffusion

KW - Unbiased simulation of SDEs

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U2 - 10.1214/17-AAP1281

DO - 10.1214/17-AAP1281

M3 - Article

AN - SCOPUS:85038854568

SN - 1050-5164

VL - 27

SP - 3305

EP - 3341

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 6

ER -