A new extrapolation method for weak approximation schemes with applications

Abstract

We review Fujiwara's scheme, a sixth order weak approximation scheme for the numerical approximation of SDEs, and embed it into a general method to construct weak approximation schemes of order 2m for m ∈ N. Those schemes cannot be seen as cubature schemes, but rather as universal ways how to extrapolate from a lower order weak approximation scheme, namely the Ninomiya-Victoir scheme, for higher orders.

Ninomiya–Victoir scheme: Strong convergence, antithetic version and application to multilevel estimators

“…In this section we present numerical tests in which we compare the multilevel Monte Carlo and the multilevel Richardson-Romberg estimators. Although we have not proved a theoretical expansion of the bias like (4.26) for the Ninomiya-Victoir and the Giles-Szpruch schemes, we will use these schemes in the multilevel Richardson-Romberg estimators (see [4] and [9] for extrapolation methods based on the Ninomiya-Victoir scheme). More precisely, we compare the following estimators:…”

On two numerical problems in applied probability : discretization of Stochastic Differential Equations and optimization of an expectation depending on a parameter

“…In fact, as we will see, the construction of these schemes requires to better approximate the law of the SDE increments, which explains the better approximation of the path distributions observed in practice. Besides, it is in general still possible to use again extrapolation techniques as it has been shown by Fujiwara [11] and Oshima, Teichmann and Velušček [27] for the case of the Ninomiya and Victoir scheme. Second, for SDEs that are defined on a strict subdomain of R d (i.e.…”

Cubature methods for stochastic (partial) differential equations in weighted spaces

“…This is efficient, as (6.12) corresponds to entirely separated ODEs at the discretisation points, whence the non-linear equations to be solved are onedimensional. Finally, the split problems are concatenated using the symmetrically weighted sequential splitting, see Oshima et al [27]. Hence, we expect to observe second order weak convergence.…”

“…We extend the results of Bayer and Teichmann [2], where strong conditions are imposed on the vector fields, to more general coefficients and test functions. This allows us to obtain methods of order higher than 2 without having to resort to extrapolation, see Blanes and Casas [3] and Oshima et al [27]. The weighted spaces developed originally in Röckner and Sobol [29] and used for the numerical analysis of weak approximation methods in Dörsek and Teichmann [11] are a suitable tool for our needs, and we provide a refined analysis of the vector fields defined on these spaces.…”