Unbiased simulation of stochastic differential equations

We propose an unbiased Monte 3 estimator for $\mathrm{\mathbb{E}}\left[\mathrm{g}\right({\mathrm{X}}_{{\mathrm{t}}_{1}},\mathrm{\ldots},{\mathrm{X}}_{{\mathrm{t}}_{\mathrm{n}}}\left)\right]$, 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.