Central limit theorem and self-normalized Cramér-type moderate deviation for Euler-Maruyama scheme

Abstract

We consider a stochastic differential equation and its Euler-Maruyama (EM) scheme, under some appropriate conditions, they both admit a unique invariant measure, denoted by π and π η respectively (η is the step size of the EM scheme). We construct an empirical measure η of the EM scheme as a statistic of π η , and use Stein's method developed in Fang, Shao and Xu (Probab. Theory Related Fields 174 (2019) 945-979) to prove a central limit theorem of η . The proof of the self-normalized Cramér-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chain, splitting η −1/2 ( η (.) − π(.)) into a martingale difference series sum H η and a negligible remainder R η . We handle H η by the time-change technique for martingale, while prove that R η is exponentially negligible by concentration inequalities, which have their independent interest. Moreover, we show that SNCMD holds for x = o(η −1/6 ), which has the same order as that of the classical result in Shao (J.

Central Limit Theorem for Temporal Average of Backward Euler-Maruyama Method

“…In [26], the CLT of the temporal average of the Euler-Maruyama (EM) method with decreasing step-size for ergodic stochastic ordinary differential equations (SODEs) is given. In addition, [23] proves the CLT and moderate deviation of the EM method 2 D. JIN with a fixed step-size for SODEs. For a class of ergodic stochastic partial differential equations (SPDEs), [6] shows that the temporal average of a full discretization with fixed temporal and spatial step-sizes satisfies the CLT.…”

“…In this section, we will prove Theorem 3.2. The main strategy follows from [23] (see also [6]), whose main idea is to use the Poisson equation (4.25) to split (Π τ,2 (h)−π(h))/ √ τ into a martingale difference series sum and a negligible remainder. Then, we will prove that the martingale difference series sum converges to a normal distribution in Lemma 4.3, and that the remainder converges to zero in probability in Lemma 4.4.…”

“…It should be mentioned that although the main idea is similar to that of [6] and [23], we still need to overcome the difficulties brought by the super-linearly growing drift coefficient and the multiplicative noise. The first difficulty is to give the p-th (p > 2) moment boundedness of the BEM method, which is overcome by a careful control of the iteration coefficient (see also the detailed statement below the proof of Theorem 4.1).…”

“…The second difficulty lies in the regularity estimate of the Poisson equation. Note that [23,Lemma 3.1] is not applicable to our case due to the super-linearly growing coefficients. In addition, the argument of [6, Lemma 4.1] relies on the a.s. differentiability of the exact solution w.r.t.…”

“…The key of this proof lies in that the deviation order (α − 1)/2 is smaller than the optimal strong order 1/2 for α ∈ (1, 2), which does not apply to the case α = 2. In order to tackle the more subtle case α = 2, we follow the arguments in [23] and [6] to obtain the CLT for Π τ,2 (h). The main idea is to reformulate the normalized temporal average (Π τ,α (h) − π(h))/τ (α−1)/2 by means of the Poisson equation.…”