Convergence of Numerical Time-Averaging and Stationary Measures via Poisson Equations

Abstract

Numerical approximation of the long time behavior of a stochastic differential equation (SDE) is considered. Error estimates for time-averaging estimators are obtained and then used to show that the stationary behavior of the numerical method converges to that of the SDE. The error analysis is based on using an associated Poisson equation for the underlying SDE. The main advantage of this approach is its simplicity and universality. It works equally well for a range of explicit and implicit schemes including those with simple simulation of random variables, and for hypoelliptic SDEs. To simplify the exposition, we consider only the case where the state space of the SDE is a torus and we study only smooth test functions. However we anticipate that the approach can be applied more widely. An analogy between our approach and Stein's method is indicated. Some practical implications of the results are discussed.

High Order Numerical Approximation of the Invariant Measure of Ergodic SDEs

“…for all positive integer l, with a bound of the form (14) is to restrict (1) to E = T d as it was done in [5]. Another way is to follow the approach in [23,Lemma 2] and assume that f, g are C ∞ where derivatives of any order are bounded.…”

“…The error e(φ, h) was also the subject of study of [14]. Given an ergodic integrator of weak order p for an ergodic SDE (1), it is shown that it has order r ≥ p for the invariant measure (6).…”

Approximation of the invariant law of SPDEs: error analysis using a Poisson equation for a full-discretization scheme

“…Notice that for linear equations a specific idea simplifies the proof -so that the second tool is not used -but can not be adapted for nonlinear parabolic equations like (1): see [13], and [10] where a stochastic Schrödinger equation is discretized. Here, we are interested in another method for the approximation of the invariant measure: we want to follow the approach of [21]. There, the authors study the distance between time-averages along the realization of the numerical scheme of a test function φ, ant its expected value with respect to the invariant law µ.…”

“…The use of a Poisson equation to prove convergence results of Law of Large Numbers type is classical, as explained in [21]. In the context of SPDEs, it has been used in [2] and [6] for the study of the averaging principle for systems evolving with two separate time-scales.…”

“…Our main result is the adaptation of the approach of [21] for SPDEs, with time and space approximation procedures: we essentially obtain the following result -a more precise statement is Theorem 5.1: There exists a constant C > 0 such that for any C 2 b (H) function φ, any parameters τ ∈ (0, 1) and h ∈ (0, 1), any time N ≥ 1 and any initial condition…”