Modern Stochastics: Theory and Applications

Let $X_{t}$, $t\ge 0$, be a Markov process with values in ${\mathbb{R}}^{d}$. We consider the following objects:

1) the integral functional of this process;

2) the sequence of integral sums The problem we are focused on is obtaining upper bounds on the accuracy of approximation of the integral functional $I_{T}(h)$ by the integral sums $I_{T,n}(h)$ without any regularity assumption on the function h. The function h is only assumed to be measurable and bounded. Therefore, the class of functionals $I_{T}(h)$ contains, for example, the occupation time of the process X in a set $A\subset \mathcal{B}({\mathbb{R}}^{d})$ (in this case, $h=\mathbb{I}_{A}$).

The problem of estimating the expectation of expressions that contain both the value of a process and the value of an integral functional of this process arises naturally in a wide range of probabilistic problems. Two of them related with the Feynman–Kac semigroup and the price of an occupation-time option are discussed in Section 3. An exact calculation of such expressions, if possible, can be performed only under substantial assumptions on the structure of functionals and processes; see, for example, [3], where the price of an occupation-time option is precisely calculated for a Lévy process with only negative jumps. For more complicated models, it is natural to use approximative methods, which naturally require estimates of approximation errors. This motivates the main aim of the paper to evaluate the error bounds for discrete approximations of the integral functional $I_{T}(h)$.

In what follows, $P_{x}$ denotes the law of the Markov process X conditioned by $X_{0}=x$, and $\mathbb{E}_{x}$ denotes the expectation with respect to this law. Both the absolute value of a real number and the Euclidean norm in ${\mathbb{R}}^{d}$ are denoted by $|\cdot |$; $\| \cdot \| $ denotes the sup-norm in $L_{\infty }$.

The following result was obtained in [2] as a part of the proof of a more general statement (see Theorem 2.5 in [2]).

The scheme of the proof of this result can be extended straightforwardly to the case of arbitrary Markov process that satisfies the following assumption (see Proposition 2.1 [1]):

Under assumption X, Proposition 1 and Proposition 2.1 in [1] give bounds for the rate of approximation of expectations of the integral functionals of the process X. Such approximation rates are called weak. Strong $L_{p}$-rates, that is, the bounds for have been recently obtained in [4] for diffusion processes and in [1] without restrictions on the structure of the processes. In this paper, we provide another generalization of the weak rate (1), namely, the rates of approximation for expectations of more complicated functionals. Let us formulate the main result of this paper.

Clearly, Proposition 2.1 in [1] is a particular case of Theorem 1. The latter statement is a substantial extension of the former one: it contains both the moments of any order of the integral functional and the value of the process in the final time moment. Using the Taylor expansion, we obtain the following corollary of Theorem 1.

Consider any analytic function g defined in a neighborhood of 0 and constants $D_{g},R_{g}>0$ such that $|\frac{{g}^{(m)}(0)}{m!}|\le D_{g}{(\frac{1}{R_{g}})}^{m}$ for any natural m.

We provide the proof of Theorem 1 in Section 2. In Section 3, we give an application of Theorem 1 to estimates of the rates of approximation of the Feynman–Kac semigroup and of the price of an occupation-time option.

Denote and for each $t\in [kT/n,(k+1)T/n)$, put $\eta _{n}(t):=\frac{kT}{n}$.

We have: where $s_{0}=0$, $y_{0}=x$.

Rewrite the expression under the integral in the form where

Therefore,

Our way to estimate each of $2k$ terms in (4) is mostly the same, but its realization is different for the first, the last, and the intermediate terms. Let us estimate the first term in (4): Let us consider each term in detail: Next, we have

Integrating over $z,y_{k},y_{k-1},\dots ,y_{2}$ and then over $s_{k},s_{k-1},\dots ,s_{2}$, we derive Therefore, Now we are ready to estimate the last summand in (4): Let us estimate each term separately. We get For the other term, we obtain:

Let us rewrite this expression in the form and consider the inner integral Therefore, we have: and To complete the proof, we should additionally consider the following terms in (4): and where $j=\overline{2,k}$.

Consider (5) in more detail. We rewrite it in the form We estimate each term separately: For the other term, we have Again, we consider the inner integral: We have Therefore, we obtain Analogously, we also have: Therefore, we finally obtain which completes the proof.  □