Stochastic C-Stability and B-Consistency of Explicit and Implicit Milstein-Type Schemes

Abstract

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in [Beyn, Isaak, Kruse, J. Sci. Comp., 2015]. As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler-Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.2010 Mathematics Subject Classification. 65C30, 65L20.

A note on strong approximation of SDEs with smooth coefficients that have at most linearly growing derivatives

“…Moreover, a polynomial growth condition on the first order derivatives of the coefficients of an SDE is one of the standing assumptions in the literature when polynomial mean error rates are obtained under monotonicity conditions, see e.g. [16,13,18,35,33,34,3,21,4,6]. Therefore it is important to investigate whether a sub-polynomial rate of convergence as in (3) may also happen when the first order derivatives of the coefficients are of at most polynomial growth.…”

On explicit tamed Milstein-type scheme for stochastic differential equation with Markovian switching

“…It can be remarked that higher order approximations of stochastic differential equations (SDEs) are well developed due to the presence of Itô's-Taylor expansion, see [6] for detailed discussion. Furthermore, different variants of tamed Milstein scheme for SDE have been studied by [1,7,9,16] where authors estalish that the strong convergence of their variant of tamed Milstein schemes for SDE is 1.0 when 1 either drift or diffusion or both the coefficients satisfy non-global Lipschitz condition and can grow super-linearly. Recently, Milstein-type schemes of SDEwMS is investigated in [12] where authors establish that the rate of strong convergence of their scheme is equal to 1.0 when both drift and diffusion coefficients satisfy global Lipschitz conditions.…”

Higher order Langevin Monte Carlo algorithm

“…As for the case of superlinear ∇U , the difficulty arises from the fact that ULA is unstable (see [23]), and its Metropolis adjusted version, MALA, loses some of its appealing properties as discussed in [7] and demonstrated numerically in [2]. However, recent research has developed new types of explicit numerical schemes for SDEs with superlinear coefficients, and it has been shown in [15], [17], [16], [18], [13], [19], that the tamed Euler (Milstein) scheme converges to the true solution of the SDE (1) in L p on any given finite time horizon with optimal rate. This progress led to the creation of the tamed unadjusted Langevin algorithm (TULA) in [2], where the aforementioned convergence results are extended to an infinite time horizon and, moreover, one obtains rate of convergence results in total variation and in Wasserstein distance.…”