Ninomiya-Victoir scheme : strong convergence, asymptotics for the normalized error and multilevel Monte Carlo methods.

Summary This thesis is devoted to the study of the strong convergence properties of the Ninomiya and Victoir scheme. The authors of this scheme propose to approximate the solution of a stochastic differential equation (SDE), denoted $X$, by solving $d+1$ ordinary differential equations (ODE) on each time step, where $d$ is the dimension of the Brownian motion. The aim of this study is to analyze the use of this scheme in a multi-step Monte-Carlo method. Indeed, the optimal complexity of this method is directed by the order of convergence towards $0$ of the variance between the schemes used on the coarse and on the fine grid. This order of convergence is itself related to the strong order of convergence between the two schemes. We then show in chapter $2$, that the strong order of the Ninomiya-Victor scheme, denoted $X^{NV,eta}$ and of time step $T/N$, is $1/2$. Recently, Giles and Szpruch proposed a multi-step Monte Carlo estimator realizing $Oleft(epsilon^{-2}right)$ complexity using a modified Milstein scheme. In the same spirit, we propose a modified Ninomiya-Victoir scheme that can be coupled at high order $1$ with the Giles and Szpruch scheme at the last level of a multi-step Monte Carlo method. This idea is inspired by Debrabant and Rossler. These authors suggest using a high low order scheme at the finest discretization level. Since the optimal number of discretization levels of a multi-step Monte Carlo method is directed by the low error of the scheme used on the fine grid of the last discretization level, this technique allows to accelerate the convergence of the multi-step Monte Carlo method by obtaining a high low order approximation. The use of the $1$ coupling with the Giles-Szpruch scheme allows us to keep a multi-step Monte-Carlo estimator realizing an optimal complexity $Oleft( epsilon^{-2} right)$ while taking advantage of the $2$ low order error of the Ninomiya-Victoir scheme. In the third chapter, we are interested in the renormalized error defined by $sqrt{N}left(X - X^{NV,eta}right)$. We show the stable law convergence to the solution of an affine SDE, whose source term is formed by the Lie brackets between the Brownian vector fields. Thus, when at least two Brownian vector fields do not commute, the limit is non-trivial. This ensures that the strong order $1/2$ is optimal. On the other hand, this result can be seen as a first step towards proving a central limit theorem for multi-step Monte-Carlo estimators. To do so, we need to analyze the stable law error of the scheme between two successive discretization levels. Ben Alaya and Kebaier proved such a result for the Euler scheme. When the Brownian vector fields commute, the limit process is zero. We show that in this particular case, the strong order is $1$. In chapter 4, we study the convergence to a stable law of the renormalized error $Nleft(X - X^{NV}right)$ where $X^{NV}$ is the Ninomiya-Victor scheme when the Brownian vector fields commute. We demonstrate the convergence of the renormalized error process to the solution of an affine SDE. When the dritf vector field does not commute with at least one of the Brownian vector fields, the strong convergence speed obtained previously is optimal.