This paper proposes a general higher order operator splitting scheme for diffusion semigroups using the Baker-Campbell-Hausdorff type commutator expansion of non-commutative algebra and the Malliavin calculus. An accurate discretization method for the fundamental solution of heat equations or...

The paper proposes a new automatic/algorithmic differentiation for the solutions to partial differential equations of parabolic type. In particular, we provide a higher order discretization scheme which is a natural extension of the standard automatic differentiation. A Brownian polynomial...

The paper shows a new weak approximation method for stochastic differential equations as a generalization and an extension of Heath-Platen's scheme for multidimensional diffusion processes. We reformulate the Heath-Platen estimator from the viewpoint of asymptotic expansion. The proposed scheme...

This paper proposes a new third-order discretization algorithm for multidimensional Itô stochastic differential equations driven by Brownian motions. The scheme is constructed by the Euler-Maruyama scheme with a stochastic weight given by polynomials of Brownian motions, which is simply...

We show a new higher order weak approximation with Malliavin weights for multidimensional stochastic differential equations by extending the method in Takahashi and Yamada (2016). The estimate of global error of the discretization is based on a sharp small time expansion using a Malliavin...

This paper proposes a new Markov chain approach to second order weak approximation of stochastic differential equations driven by d-dimensional Brownian motion. The scheme is explicitly constructed by polynomials of Brownian motions up to second order and any discrete moment matched random...

This paper proposes an arbitrary high order weak approximation scheme for multidimensional Stratonovich stochastic differential equations using Malliavin calculus. The scheme efficiently works whether test function is smooth or not. The Malliavin Monte Carlo method, a simple numerical algorithm,...

This paper shows a higher order discretization scheme for the Bismut-Elworthy-Li formula, the differentiation of diffusion semigroups. A weak approximation type algorithm with Malliavin weights is constructed through the integration by parts on Wiener space and is efficiently implemented by a...