We provide strong

Let

In the paper, we establish strong

We would like to explain in this note that, in order to get the required approximation rates, one can modify some well-developed estimates from the theory of continuous additive functionals of Markov processes. An advantage of such approach is that the assumptions on the process are formulated only in terms of its transition probability density and therefore are quite flexible. The basis for the approach is given by the fact that the weak approximation rates for

In what follows,

Our standing assumption on the process

The transition probability density

The assumption

Let

Observe that, in a sense, this bound is “stable under perturbations of the process

Our principal assumption on the function

The function

Write

Our main estimate, in a shortest and most transparent form, is presented in the following theorem, which concerns the case where the only assumption on

Denote, for

Next, observe that, for every

In this section, we consider the case where

The function is Hölder continuous with index

The method of the proof remains the same as that of Theorem

The last inequality holds due to the following representation. By the Markov property of

Thus, for

Next, using (

Hence, the previous bounds for

The results of Theorem

The authors are grateful to A. Kohatsu-Higa for turning their attention to this problem and for helpful discussions. The first author was partially supported by the Leonard Euler program, DAAD project No. 57044593.