The article is devoted to the estimation of the convergence rate of integral functionals of a Markov process. Under the assumption that the given Markov process admits a transition probability density differentiable in

Let

Integral functionals arise naturally in a wide class of stochastic representation formulae and applied stochastic models. It is very typical that exact calculation of the respective probabilities and/or expectations is hardly possible, which naturally suggests the usage of approximation methods. As an example of such a situation, we mention the so-called

For diffusion processes, this problem was studied in [

The transition probability

In [

Another aim of this paper is to develop tools that would allow us to get the bounds of the form (

The structure of the paper is the following. In Section

In this section, we prove two results. The first one concerns the “strong approximation rate”, that is, the control on the

This theorem extends [

The second result concerns the “weak approximation,” that is, the control on the difference between the expectations of certain terms, which involve

This theorem extends [

Using the Taylor expansion, we can directly obtain the following corollary of Theorem

Before proceeding to the proof of Theorem

Let us introduce the notation used throughout the whole section: for

We have

For

Now we finalize the argument.

1) If

2) If

Since we can obtain the required bound for

Define

Further, observe that for every

It can be easily seen that this inequality also holds if

Because

Denote

We have

Let us estimate the

Consider two cases: a)

In case a), using condition

Consider the SDE

Recall that the characteristic function of a Lévy process is of the form

Let us impose three minor conventions, which will simplify the technicalities. First, since we are mostly interested in the case

In what follows, we show how the

Let us introduce some notation and give some preliminaries. We denote the space and the space-time convolutions respectively by

Generically, the parametrix construction provides a representation of the required transition probability density in the form

Denote by

Finally, denote by

Now we are ready to formulate the main statement of this section.

First, we evaluate

By (

To get the estimate for

Our next step is to estimate the convolution powers of

Our final step is to use representation (

To show that the convolution powers

From (

In this section, we consider an

For instance, for the strike price

Assume that the price of an asset

Let us approximate the price

First, we apply Theorem

The proof is a simple corollary of Theorem

We also can control the accuracy of the approximation using the weak rate bound from Theorem

For

Then

We estimate each term separately.

Using Theorem

The second- and third-named authors gratefully acknowledge the DFG Grant Schi 419/8-1; the third-named author acknowledges the joint DFFD-RFFI project No. 09-01-14.