AL GERBI Anis

n Gerbi et al. (2016) we proved strong convergence with order 1/2 of the Ninomiya–Victoir schemeXN V,ηwith time stepT/Nto the solutionXof the limiting SDE. In this paper we check that thenormalized error defined by√N(X−XN V,η)converges to an affine SDE with source terms involvingthe Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher randomvariablesη. This result can be seen as a first step to adapt to the Ninomiya–Victoir scheme the centrallimit theorem of Lindeberg Feller type, derived in Ben Alaya and Kebaier (2015) for the multilevel MonteCarlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes.This suggests that the rate of convergence is greater than 1/2 in this case and we actually prove strongconvergence with order 1 and study the limit of the normalized errorN(X−XN V,η). The limiting SDEinvolves the Lie brackets between the Brownian vector fields and the Stratonovich drift vector field. Whenall the vector fields commute, the limit vanishes, which is consistent with the fact that the Ninomiya–Victoirscheme coincides with the solution to the SDE on the discretization grid.

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In this paper, we summarize the results about the strong convergence rate of the Ninomiya-Victoir scheme and the stable convergence in law of its normalized error that we obtained in previous papers. We then recall the properties of the multilevel Monte Carlo estimators involving this scheme that we introduced and studied before. Last, we are interested in the error introduced by discretizing the ordinary differential equations involved in the Ninomiya-Victoir scheme. We prove that this error converges with strong order 2 when an explicit Runge-Kutta method with order 4 (resp. 2) is used for the ODEs corresponding to the Brownian (resp. Stratonovich drift) vector fields. We thus relax the order 5 for the Brownian ODEs needed by Ninomiya and Ninomiya (2009) to obtain the same order of strong convergence. Moreover, the properties of our multilevel Monte-Carlo estimators are preserved when these Runge-Kutta methods are used.

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.

This thesis is dedicated to the study of the strong convergence properties of the Ninomiya-Victoir scheme, which is based on the resolution of $d+1$ ordinary differential equations (ODEs) at each time step, to approximate the solution to a stochastic differential equation (SDE), where $d$ is the dimension of the Brownian. This study is aimed at analysing the use of this scheme in a multilevel Monte Carlo estimator. Indeed, the optimal complexity of this method is driven by the order of convergence to zero of the variance between the two schemes used on the coarse and fine grids at each level, which is related to the strong convergence order between the two schemes. In the second chapter, we prove strong convergence with order $1/2$ of the Ninomiya-Victoir scheme $X^{NV,eta}$, with time step $T/N$, to the solution $X$ of the limiting SDE. Recently, Giles and Szpruch proposed a modified Milstein scheme and its antithetic version, based on the swapping of each successive pair of Brownian increments in the scheme, permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity $Oleft(epsilon^{-2}right)$ for the precision $epsilon$, as in a simple Monte Carlo method with independent and identically distributed unbiased random variables. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order $1$ to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. This idea is inspired by Debrabant and R"ossler who suggest to use a scheme with high order of weak convergence on the finest grid at the finest level of the multilevel Monte Carlo method. As the optimal number of discretization levels is related to the weak order of the scheme used in the finest grid at the finest level, Debrabant and R"ossler manage to reduce the computational time, by decreasing the number of discretization levels. The coupling with the Giles-Szpruch scheme allows us to combine both ideas. By this way, we preserve the optimal complexity $Oleft(epsilon^{-2}right)$ and we reduce the computational time, since the Ninomiya-Victoir scheme is known to exhibit weak convergence with order 2. In the third chapter, we check that the normalized error defined by $sqrt{N}left(X - X^{NV,eta}right)$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. This result ensures that the strong convergence rate is actually $1/2$ when at least two Brownian vector fields do not commute. To link this result to the multilevel Monte Carlo estimator, it can be seen as a first step to adapt to the Ninomiya-Victoir scheme the central limit theorem of Lindeberg Feller type, derived recently by Ben Alaya and Kebaier for the multilevel Monte Carlo estimator based on the Euler scheme. When the Brownian vector fields commute, the limit vanishes. We then prove strong convergence with order $1$ in this case. The fourth chapter deals with the convergence of the normalized error process $Nleft(X - X^{NV}right)$, where $X^{NV}$ is the Ninomiya-Victoir in the commutative case. We prove its stable convergence in law to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually $1$ when the Brownian vector fields commute, but at least one of them does not commute with the Stratonovich drift vector field Cette thèse est consacrée à l'étude des propriétés de convergence forte du schéma de Ninomiya et Victoir.

In this paper, we are interested in the strong convergence properties of the Ninomiya-Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2. 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 on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity ${\mathcal O}(\epsilon^{-2})$ for the precision $\epsilon$. In the same spirit, we propose a modified Ninomiya-Victoir scheme, which may be strongly coupled with order 1 to the Giles-Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya-Victoir scheme permits to reduce the number of discretization levels.

In this paper, we are interested in the strong convergence properties of the Ninomiya–Victoir scheme which is known to exhibit weak convergence with order 2. We prove strong convergence with order 1/2. 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 on the coarse and fine grids at each level. Recently, Giles and Szpruch proposed a scheme permitting to construct a multilevel Monte Carlo estimator achieving the optimal complexity O(ϵ−2) for the precision ϵ. In the same spirit, we propose a modified Ninomiya–Victoir scheme, which may be strongly coupled with order 1 to the Giles–Szpruch scheme at the finest level of a multilevel Monte Carlo estimator. Numerical experiments show that this choice improves the efficiency, since the order 2 of weak convergence of the Ninomiya–Victoir scheme permits to reduce the number of discretisation levels.

In a previous work, we proved strong convergence with order 1 of the Ninomiya-Victoir scheme $X^{\rm NV}$ with time step $T/N$ to the solution $X$ of the limiting SDE when the Brownian vector fields commute. In this paper, we prove that the normalized error process $N(X−X^{\rm NV})$ converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields and the drift vector field. This result ensures that the strong convergence rate is actually 1 when the Brownian vector fields commute, but at least one of them does not commute with the drift vector field. When all the vector fields commute the limit vanishes. Our result is consistent with the fact that the Ninomiya-Victoir scheme solves the SDE in this case.