Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups

Abstract

We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires (i) exponential decay in time of the space-derivatives of the semigroup associated with the SDE and (ii) bounds on (some) moments of the Euler approximation. We show by means of examples (and counterexamples) how both (i) and (ii) are needed to obtain the desired result. If the weak error converges to zero uniformly in time, then convergence of ergodic averages follows as well. We also show that Lyapunov-type conditions are neither sufficient nor necessary in order for the weak error of the Euler approximation to converge uniformly in time and clarify relations between the validity of Lyapunov conditions, (i) and (ii). Conditions for (ii) to hold are studied in the literature. Here we produce sufficient conditions for (i) to hold. The study of derivative estimates has attracted a lot of attention, however not many results are known in order to guarantee exponentially fast decay of the derivatives. Exponential decay of derivatives typically follows from coercive-type conditions involving the vector fields appearing in the equation and their commutators; here we focus on the case in which such coercive-type conditions are non-uniform in space. To the best of our knowledge, this situation is unexplored in the literature, at least on a systematic level. To obtain results under such space-inhomogeneous conditions we initiate a pathwise approach to the study of derivative estimates for diffusion semigroups and combine this pathwise method with the use of Large Deviation Principles.

Poisson Equations with locally-Lipschitz coefficients and Uniform in Time Averaging for Stochastic Differential Equations via Strong Exponential Stability

“…Since we are testing a result that is UiT, it is important that the discretisation of the system that we use to solve it is itself UiT. For this, we refer the reader to [10]. Using the results of [10], one can see that both the coupled system and the averaged equation are approximated uniformly in time by the Euler-Maruyama scheme.…”

“…For this, we refer the reader to [10]. Using the results of [10], one can see that both the coupled system and the averaged equation are approximated uniformly in time by the Euler-Maruyama scheme.…”

“…In Section 6.1 we will explain how this Poisson problem emerges in the context of averaging. For the time being it suffices to say that the operator L x is the generator of the process Y x,y t := Y 1,x,y t which solves the SDE dY x,y t = g(x, Y x,y t )dt + √ 2a(x, Y x,y t ) dB t , (9) and is obtained from (3) by setting = 1; clearly, for each x fixed, the asymptotic behaviour as t → ∞ of ( 9) is the same as the asymptotic behaviour as → 0 of (3) and the invariant measure µ x (dy) (of either processes) is precisely the only probability measure such that R d (L x f )(x, y)µ x (dy) = 0 , (10) for every f in the domain of L x . Under our assumptions (see Section 2) such an invariant measure exists and, since we will take L x to be elliptic, it is unique and it has a smooth density (for every x ∈ R n ); with abuse of notation, we still denote the density by µ x (y).…”

“…We specify that, in contexts different from averaging, there are important works in the literature which deal with UiT results, e.g. [10,13,16,19,35] to mention just a few, and the first, second and fourth authors of this paper have been pushing a programme to show how SES is key in proving uniform in time results, in a variety of settings, including convergence of particle systems and numerical methods for SDEs, see e.g. [3,9].…”

Approximations of Piecewise Deterministic Markov Processes and their convergence properties

“…The strategy of this proof is inspired by [18], which uses derivative estimates to obtain uniform in time convergence of an Euler Scheme for an SDE. In that case the authors rely on having exponential decay of the derivatives of the semigroup for the SDE of interest, for which conditions are given in [19].…”