Asymptotics for the normalized error of the Ninomiya–Victoir scheme

Abstract

In [1] we proved strong convergence with order 1/2 of the Ninomiya-Victoir scheme X N V,η with time step T /N to the solution X of the limiting SDE. In this paper we check that the normalized error defined by √ N X − X N V,η converges to an affine SDE with source terms involving the Lie brackets between the Brownian vector fields. The limit does not depend on the Rademacher random variables η. This result can be seen as a first step to adapt to the Ninomiya-Victoir scheme the central limit theorem of Lindeberg Feller type, derived in [2] for the multilevel Monte Carlo 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 strong convergence with order 1.

Ninomiya-Victoir scheme : Multilevel Monte Carlo estimators and discretization of the involved Ordinary Differential Equations

“…Under some regularity assumptions, we proved, in [3], strong convergence with order 1 of the Ninomiya-Victoir scheme when the commutativity condition (C) holds. More precisely, we showed the following result.…”

“…In [3], we checked that the normalized error V N converges to the solution of an affine SDE with source terms :…”

“…Our motivation comes from the multilevel Monte Carlo method introduced by Giles [10], the complexity of which is more influenced by the order of strong convergence of the scheme than its order of weak convergence. In the first section of 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 obtained in [2,3,4]. The results significantly differ depending on whether the Brownian vector fields σ j , j ∈ {1, .…”

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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 obtained in [2,3,4]. We then recall the properties of the multilevel Monte Carlo estimators involving this scheme that we introduced and studied in [2].

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