Exact Simulation of Multi-Dimensional Stochastic Differential Equations

We develop a weak exact simulation technique for a process X defined by a multi-dimensional stochastic differential equation (SDE). Namely, for a Lipschitz function g, we propose a simulation based approximation of the expectation E[g(X_{t_1}, \cdots, X_{t_n})], which by-passes the discretization error. The main idea is to start instead from a well-chosen simulatable SDE whose coefficients are up-dated 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 Elworthy's formula from Malliavin calculus, as exploited by Fournie et al. for the simulation of the Greeks in financial applications.Unlike the exact simulation algorithm of Beskos and Roberts, our algorithm is suitable for the multi-dimensional case. Moreover, its implementation is a straightforward combination of the standard discretization techniques and the above mentioned automatic differentiation method.