Unbiased simulation of stochastic differential equations *

Pierre Henry-Labordère (1) , Xiaolu Tan (2) , Nizar Touzi (3)

1 SOCIETE GENERALE - Equity Derivatives Research Societe Generale
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We propose an unbiased Monte-Carlo estimator for E[g(X t 1 , · · · , X tn)], where X is a diffusion process defined by a multi-dimensional 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 Bismu-Elworthy-Li formula from Malliavin calculus, as exploited by Fournié et al. [14] 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 [3, Section 6.1] as an application of the parametrix method. MSC2010. Primary 65C05, 60J60; secondary 60J85, 35K10.

Mots clés: Unbiased simulation of SDEs; regime switching diffusion; linear parabolic PDEs

Domaines: Probabilités [math.PR]

Pierre Henry-Labordère, Xiaolu Tan, Nizar Touzi. Unbiased simulation of stochastic differential equations *. 2017. ⟨hal-01429548⟩