Section: New Results

Discretization of stochastic differential equations

With his PhD student A. Al Gerbi and E. Clément, B. Jourdain is interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit order 2 of weak convergence. This study is aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator : indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used in the coarse and fine grids at each level. They prove strong convergence with order $1/2$ which is improved to order 1 when the vector fields corresponding to each Brownian coordinate in the SDE commute. They also check that the renormalized errors converge to affine SDEs with source terms involving the Lie brackets between these vector fields and, in the commuting case, their Lie brackets with the drift vector field. Last, they propose a modified Ninomiya-Victoir scheme, which, at the finest level of the multilevel Monte Carlo estimator, may be coupled with strong order 1 to a simpler scheme with weak order 1 recently proposed by Giles and Szpruch.

Using optimal transport tools, A. Alfonsi, B. Jourdain and A. Kohatsu-Higa have proved that the Wasserstein distance between the time marginals of an elliptic SDE and its Euler discretization with $N$ steps is not larger than $\frac{C\sqrt{log\left(N\right)}}{N}$. The logarithmic factor may be removed when the uniform time-grid is replaced by a grid still counting $N$ points but refined near the origin of times [4] . To generalize in higher dimension the result that they obtained previously in dimension one using the optimality of the explicit inverse transform, they compute the derivative of the Wasserstein distance with respect to the time variable using the theory developed by Ambrosio Gigli and Savare. The abstract properties of the optimal coupling between the time marginals then enable them to estimate this time derivative [30] .