1 Introduction and main results
Let $X_{t}$, $t\ge 0$, be a Markov process with values in ${\mathbb{R}}^{d}$. We consider the following objects:
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]).
Proposition 1.
Suppose that X is a multidimensional diffusion process with bounded Hölder continuous coefficients and that its diffusion coefficient satisfies the uniform ellipticity condition
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]):
X.
The process X possesses a transition probability density $p_{t}(x,y)$ that is differentiable with respect to t and satisfies
for some measurable function q such that for any fixed t and x, the function $q_{t,x}$ is a distribution density.
(2)
\[\big|\partial _{t}p_{t}(x,y)\big|\le C_{T}{t}^{-1}q_{t,x}(y),\hspace{1em}t\le T,\hspace{2.5pt}C_{T}\ge 1,\]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.
2 Proof of Theorem 1
Denote
\[S_{k,a,b}:=\big\{(s_{1},s_{2},\dots ,s_{k})\in {\mathbb{R}}^{k}|a\le s_{1}<s_{2}<\cdots <s_{k}\le b\big\},\hspace{1em}k\in \mathbb{N},\hspace{2.5pt}a,b\in \mathbb{R},\]
and for each $t\in [kT/n,(k+1)T/n)$, put $\eta _{n}(t):=\frac{kT}{n}$.We have:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \frac{1}{k!}\big(E_{x}\big[{\big(I_{T}(h)\big)}^{k}-{\big(I_{T,n}(h)\big)}^{k}\big]f(X_{T})\big)\\{} & \displaystyle \hspace{1em}=E_{x}\int _{S_{k,0,T}}\Bigg[\prod \limits_{i=1}^{k}h(X_{s_{i}})-\prod \limits_{i=1}^{k}h(X_{\eta _{n}(s_{i})})\Bigg]f(X_{T})\prod \limits_{i=1}^{k}ds_{i}\\{} & \displaystyle \hspace{1em}=\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\Bigg(\prod \limits_{i=1}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\times dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k}ds_{i}-\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{i=1}^{k}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)p_{T-\eta _{n}(s_{k})}(y_{k},z)dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k}ds_{i},\end{array}\]
where $s_{0}=0$, $y_{0}=x$.Rewrite the expression under the integral
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg(\prod \limits_{i=1}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{1em}-\Bigg(\prod \limits_{i=1}^{k}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)p_{T-\eta _{n}(s_{k})}(y_{k},z)\end{array}\]
in the form
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg(\prod \limits_{i=1}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\mp p_{\eta _{n}(s_{1})}(x,y_{1})\Bigg(\prod \limits_{i=2}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}-\Bigg(\prod \limits_{i=1}^{k}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)p_{T-\eta _{n}(s_{k})}(y_{k},z)\\{} & \displaystyle \hspace{1em}=\Bigg(\prod \limits_{i=1}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\mp p_{\eta _{n}(s_{1})}(x,y_{1})\Bigg(\prod \limits_{i=2}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\mp p_{\eta _{n}(s_{1})}(x,y_{1})p_{s_{2}-\eta _{n}(s_{1})}(y_{1},y_{2})\Bigg(\prod \limits_{i=3}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}-\Bigg(\prod \limits_{i=1}^{k}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)p_{T-\eta _{n}(s_{k})}(y_{k},z)\\{} & \displaystyle \hspace{1em}\hspace{2em}\hspace{2em}\hspace{2em}\hspace{2em}\hspace{2em}\hspace{2em}\vdots \\{} & \displaystyle \hspace{1em}=J_{1}+J_{2}+\cdots +J_{2k-1}+J_{2k},\end{array}\]
where
\[\begin{array}{r@{\hskip0pt}l}\displaystyle J_{1}& \displaystyle \hspace{0.1667em}=\hspace{0.1667em}\big(p_{s_{1}}(x,y_{1})-p_{\eta _{n}(s_{1})}(x,y_{1})\big)\Bigg(\prod \limits_{i=2}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z),\\{} \displaystyle J_{2}& \displaystyle \hspace{0.1667em}=\hspace{0.1667em}p_{\eta _{n}(s_{1})}(x,y_{1})\big(p_{s_{2}-s_{1}}(y_{1},y_{2})-p_{s_{2}-\eta _{n}(s_{1})}(y_{1},y_{2})\big)\\{} & \displaystyle \hspace{1em}\times \Bigg(\prod \limits_{i=3}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z),\\{} \displaystyle J_{3}& \displaystyle \hspace{0.1667em}=\hspace{0.1667em}p_{\eta _{n}(s_{1})}(x,y_{1})\big(p_{s_{2}-\eta _{n}(s_{1})}(y_{1},y_{2})-p_{\eta _{n}(s_{2})-\eta _{n}(s_{1})}(y_{1},y_{2})\big)\\{} & \displaystyle \hspace{1em}\times \Bigg(\prod \limits_{i=3}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z),\\{} \displaystyle J_{4}& \displaystyle \hspace{0.1667em}=\hspace{0.1667em}p_{\eta _{n}(s_{1})}(x,y_{1})p_{\eta _{n}(s_{2})-\eta _{n}(s_{1})}(y_{1},y_{2})\big(p_{s_{3}-s_{2}}(y_{2},y_{3})-p_{s_{3}-\eta _{n}(s_{2})}(y_{2},y_{3})\big)\\{} & \displaystyle \hspace{1em}\times \Bigg(\prod \limits_{i=4}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z),\\{} \displaystyle J_{5}& \displaystyle \hspace{0.1667em}=\hspace{0.1667em}p_{\eta _{n}(s_{1})}(x,y_{1})p_{\eta _{n}(s_{2})-\eta _{n}(s_{1})}(y_{1},y_{2})\big(p_{s_{3}-\eta _{n}(s_{2})}(y_{2},y_{3})\hspace{0.1667em}-p_{\eta _{n}(s_{3})-\eta _{n}(s_{2})}(y_{2},y_{3})\big)\\{} & \displaystyle \hspace{1em}\times \Bigg(\prod \limits_{i=4}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z),\\{} & \displaystyle \hspace{1em}\hspace{2em}\hspace{2em}\hspace{2em}\hspace{2em}\hspace{2em}\hspace{2em}\vdots \\{} \displaystyle J_{2k-1}& \displaystyle \hspace{0.1667em}=\hspace{0.1667em}\Bigg(\prod \limits_{i=1}^{k-1}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)\\{} & \displaystyle \hspace{1em}\times \big(p_{s_{k}-\eta _{n}(s_{k-1})}(y_{k-1},y_{k})-p_{\eta _{n}(s_{k})-\eta _{n}(s_{k-1})}(y_{k-1},y_{k})\big)p_{T-s_{k}}(y_{k},z),\\{} \displaystyle J_{2k}& \displaystyle \hspace{0.1667em}=\hspace{0.1667em}\Bigg(\prod \limits_{i=1}^{k}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)\big(p_{T-s_{k}}(y_{k},z)-p_{T-\eta _{n}(s_{k})}(y_{k},z)\big).\end{array}\]
Therefore,
(4)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \frac{1}{k!}\big(E_{x}\big[{\big(I_{T}(h)\big)}^{k}-{\big(I_{T,n}(h)\big)}^{k}\big]f(X_{T})\big)\\{} & \displaystyle \hspace{-0.1667em}\hspace{1em}=\hspace{-0.1667em}\hspace{-0.1667em}\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)(J_{1}\hspace{0.1667em}+\hspace{0.1667em}J_{2}\hspace{0.1667em}+\hspace{0.1667em}\cdots \hspace{0.1667em}+\hspace{0.1667em}J_{2k-1}\hspace{0.1667em}+\hspace{0.1667em}J_{2k})dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k}ds_{i}.\end{array}\]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):
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{1}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k}ds_{i}\Bigg|\\{} & \displaystyle \hspace{1em}=\Bigg|{\int _{0}^{T}}\int _{S_{k-1,s_{1},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{1}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}ds_{1}\Bigg|\\{} & \displaystyle \hspace{1em}\le \Bigg|{\int _{0}^{T/n}}\int _{S_{k-1,s_{1},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{1}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}ds_{1}\Bigg|\\{} & \displaystyle \hspace{2em}+\Bigg|{\int _{T/n}^{T}}\int _{S_{k-1,s_{1},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{1}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}ds_{1}\Bigg|\end{array}\]
Let us consider each term in detail:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|{\int _{0}^{T/n}}\int _{S_{k-1,s_{1},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{1}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}ds_{1}\Bigg|\\{} & \displaystyle \hspace{1em}\le {\int _{0}^{T/n}}\int _{S_{k-1,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg|\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{1}\Bigg|dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}ds_{1}\\{} & \displaystyle \hspace{1em}\le \| h{\| }^{k}\| f\| {\int _{0}^{T/n}}\int _{S_{k-1,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}|J_{1}|dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}ds_{1}\\{} & \displaystyle \hspace{1em}\le \frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k-1}{\int _{0}^{T/n}}\int _{{\mathbb{R}}^{d}}\big|p_{s_{1}}(x,y_{1})-p_{\eta _{n}(s_{1})}(x,y_{1})\big|dy_{1}ds_{1}\\{} & \displaystyle \hspace{1em}\le \frac{2}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{1}{n}.\end{array}\]
Next, we have
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|{\int _{T/n}^{T}}\int _{S_{k-1,s_{1},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{1}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}ds_{1}\Bigg|\\{} & \displaystyle \hspace{1em}=\Bigg|{\int _{T/n}^{T}}{\int _{\eta _{n}(s_{1})}^{s_{1}}}\int _{S_{k-1,s_{1},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\partial _{u}p_{u}(x,y_{1})\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{i=2}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z)dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}duds_{1}\Bigg|\\{} & \displaystyle \hspace{1em}\le {\int _{T/n}^{T}}{\int _{\eta _{n}(s_{1})}^{s_{1}}}\int _{S_{k-1,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg|\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\partial _{u}p_{u}(x,y_{1})\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{i=2}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)p_{T-s_{k}}(y_{k},z)\Bigg|dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}\times duds_{1}\\{} & \displaystyle \hspace{1em}\le \| h{\| }^{k}\| f\| {\int _{T/n}^{T}}{\int _{\eta _{n}(s_{1})}^{s_{1}}}\int _{S_{k-1,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\big|\partial _{u}p_{u}(x,y_{1})\big|\Bigg(\prod \limits_{i=2}^{k}p_{s_{i}-s_{i-1}}(y_{i-1},y_{i})\Bigg)\\{} & \displaystyle \hspace{2em}\times p_{T-s_{k}}(y_{k},z)dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}duds_{1}.\end{array}\]
Integrating over $z,y_{k},y_{k-1},\dots ,y_{2}$ and then over $s_{k},s_{k-1},\dots ,s_{2}$, we derive
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|{\int _{T/n}^{T}}\int _{S_{k-1,s_{1},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{1}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=2}^{k}ds_{i}ds_{1}\Bigg|\\{} & \displaystyle \hspace{1em}\le \frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k-1}{\int _{T/n}^{T}}{\int _{\eta _{n}(s_{1})}^{s_{1}}}\int _{{\mathbb{R}}^{d}}\big|\partial _{u}p_{u}(x,y_{1})\big|dy_{1}duds_{1}\\{} & \displaystyle \hspace{1em}\le C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k-1}{\int _{T/n}^{T}}{\int _{\eta _{n}(s_{1})}^{s_{1}}}\int _{{\mathbb{R}}^{d}}{u}^{-1}q_{u,x}(y_{1})dy_{1}duds_{1}\\{} & \displaystyle \hspace{1em}=C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k-1}{\int _{T/n}^{T}}{\int _{\eta _{n}(s_{1})}^{s_{1}}}{u}^{-1}duds_{1}\\{} & \displaystyle \hspace{1em}=C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k-1}\sum \limits_{i=1}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{\int _{iT/n}^{s_{1}}}{u}^{-1}duds_{1}\\{} & \displaystyle \hspace{1em}=C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k-1}\sum \limits_{i=1}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{\int _{u}^{(i+1)T/n}}{u}^{-1}ds_{1}du\\{} & \displaystyle \hspace{1em}\le C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{1}{n}\sum \limits_{i=1}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{u}^{-1}du\\{} & \displaystyle \hspace{1em}=C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{1}{n}{\int _{T/n}^{T}}{u}^{-1}du\\{} & \displaystyle \hspace{1em}=C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{\log n}{n}.\end{array}\]
Therefore,
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{1}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k}ds_{i}\Bigg|\\{} & \displaystyle \hspace{1em}\le 3C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{\log n}{n}.\end{array}\]
Now we are ready to estimate the last summand in (4):
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{2k}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k}ds_{i}\Bigg|\\{} & \displaystyle \hspace{1em}=\Bigg|{\int _{0}^{T}}\int _{S_{k-1,0,s_{k}}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{2k}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}ds_{k}\Bigg|\\{} & \displaystyle \hspace{1em}\le \Bigg|{\int _{0}^{T-T/n}}\int _{S_{k-1,0,s_{k}}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{2k}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}ds_{k}\Bigg|\\{} & \displaystyle \hspace{2em}+\Bigg|{\int _{T-T/n}^{T}}\int _{S_{k-1,0,s_{k}}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{2k}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}ds_{k}\Bigg|\end{array}\]
Let us estimate each term separately. We get
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|{\int _{T-T/n}^{T}}\int _{S_{k-1,0,s_{k}}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{2k}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}ds_{k}\Bigg|\\{} & \displaystyle \hspace{1em}\le {\int _{T-T/n}^{T}}\int _{S_{k-1,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg|\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{2k}\Bigg|dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}ds_{k}\\{} & \displaystyle \hspace{1em}\le \| h{\| }^{k}\| f\| {\int _{T-T/n}^{T}}\int _{S_{k-1,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}|J_{2k}|dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}ds_{k}\\{} & \displaystyle \hspace{1em}\le \frac{2}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{1}{n}.\end{array}\]
For the other term, we obtain:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|{\int _{0}^{T-T/n}}\int _{S_{k-1,0,s_{k}}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{2k}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}ds_{k}\Bigg|\\{} & \displaystyle \hspace{1em}=\Bigg|{\int _{0}^{T-T/n}}{\int _{\eta _{n}(s_{k})}^{s_{k}}}\int _{S_{k-1,0,s_{k}}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{i=1}^{k}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)\partial _{T-u}p_{T-u}(y_{k},z)dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}duds_{k}\Bigg|\\{} & \displaystyle \hspace{1em}\le {\int _{0}^{T-T/n}}{\int _{\eta _{n}(s_{k})}^{s_{k}}}\int _{S_{k-1,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg|\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{i=1}^{k}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)\partial _{T-u}p_{T-u}(y_{k},z)\Bigg|dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}duds_{k}\\{} & \displaystyle \hspace{1em}\le \| h{\| }^{k}\| f\| {\int _{0}^{T-T/n}}{\int _{\eta _{n}(s_{k})}^{s_{k}}}\int _{S_{k-1,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)\\{} & \displaystyle \hspace{2em}\times \big|\partial _{T-u}p_{T-u}(y_{k},z)\big|dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}duds_{k}.\end{array}\]
Let us rewrite this expression in the form
and
where $j=\overline{2,k}$.
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \| h{\| }^{k}\| f\| \int _{S_{k-1,0,T}}\int _{{({\mathbb{R}}^{d})}^{k-1}}\Bigg(\prod \limits_{i=1}^{k-1}p_{\eta _{n}(s_{i})-\eta _{n}(s_{i-1})}(y_{i-1},y_{i})\Bigg)\\{} & \displaystyle \hspace{1em}\times {\int _{0}^{T-T/n}}{\int _{\eta _{n}(s_{k})}^{s_{k}}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{\eta _{n}(s_{k})-\eta _{n}(s_{k-1})}(y_{k-1},y_{k})\\{} & \displaystyle \hspace{1em}\times \big|\partial _{T-u}p_{T-u}(y_{k},z)\big|dzdy_{k}duds_{k}\prod \limits_{j=1}^{k-1}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}\end{array}\]
and consider the inner integral
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {\int _{0}^{T-T/n}}{\int _{\eta _{n}(s_{k})}^{s_{k}}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{\eta _{n}(s_{k})-\eta _{n}(s_{k-1})}(y_{k-1},y_{k})\big|\partial _{T-u}p_{T-u}(y_{k},z)\big|dzdy_{k}duds_{k}\\{} & \displaystyle \hspace{1em}=\sum \limits_{i=0}^{n-2}{\int _{iT/n}^{(i+1)T/n}}{\int _{iT/n}^{s_{k}}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{iT/n-\eta _{n}(s_{k-1})}(y_{k-1},y_{k})\\{} & \displaystyle \hspace{2em}\times \big|\partial _{T-u}p_{T-u}(y_{k},z)\big|dzdy_{k}duds_{k}\\{} & \displaystyle \hspace{1em}=\sum \limits_{i=0}^{n-2}{\int _{iT/n}^{(i+1)T/n}}{\int _{u}^{(i+1)T/n}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{iT/n-\eta _{n}(s_{k-1})}(y_{k-1},y_{k})\\{} & \displaystyle \hspace{2em}\times \big|\partial _{T-u}p_{T-u}(y_{k},z)\big|dzdy_{k}ds_{k}du\\{} & \displaystyle \hspace{1em}\le \frac{T}{n}\sum \limits_{i=0}^{n-2}{\int _{iT/n}^{(i+1)T/n}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{iT/n-\eta _{n}(s_{k-1})}(y_{k-1},y_{k})\big|\partial _{T-u}p_{T-u}(y_{k},z)\big|dzdy_{k}du\\{} & \displaystyle \hspace{1em}\le C_{T}\frac{T}{n}\sum \limits_{i=0}^{n-2}{\int _{iT/n}^{(i+1)T/n}}\int _{{\mathbb{R}}^{d}}p_{iT/n-\eta _{n}(s_{k-1})}(y_{k-1},y_{k}){(T-u)}^{-1}dy_{k}du\\{} & \displaystyle \hspace{1em}=C_{T}\frac{T}{n}\sum \limits_{i=0}^{n-2}{\int _{iT/n}^{(i+1)T/n}}{(T\hspace{0.1667em}-\hspace{0.1667em}u)}^{-1}du\hspace{0.1667em}=\hspace{0.1667em}C_{T}\frac{T}{n}{\int _{0}^{T-T/n}}{(T\hspace{0.1667em}-\hspace{0.1667em}u)}^{-1}du\hspace{0.1667em}=\hspace{0.1667em}TC_{T}\frac{\log n}{n}.\end{array}\]
Therefore, we have:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|{\int _{0}^{T-T/n}}\int _{S_{k-1,0,s_{k}}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{2k}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k-1}ds_{i}ds_{k}\Bigg|\\{} & \displaystyle \hspace{1em}\le C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{\log n}{n}\end{array}\]
and
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)J_{2k}dz\prod \limits_{j=1}^{k}dy_{j}\prod \limits_{i=1}^{k}ds_{i}\Bigg|\\{} & \displaystyle \hspace{1em}\le 3C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{\log n}{n}.\end{array}\]
To complete the proof, we should additionally consider the following terms in (4):
(5)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\\{} & \displaystyle \hspace{1em}\times \big(p_{s_{j}-s_{j-1}}(y_{j-1},y_{j})-p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big)\\{} & \displaystyle \hspace{1em}\times \Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)p_{T-s_{k}}(y_{k},z)dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=1}^{k}ds_{r}\Bigg|\end{array}\](6)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\\{} & \displaystyle \hspace{1em}\times \big(p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})-p_{\eta _{n}(s_{j})-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big)\\{} & \displaystyle \hspace{1em}\times \Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)p_{T-s_{k}}(y_{k},z)dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=1}^{k}ds_{r}\Bigg|,\end{array}\]Consider (5) in more detail. We rewrite it in the form
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|{\int _{0}^{T}}\int _{S_{j-3,0,s_{j-2}}}{\int _{s_{j-2}}^{T}}{\int _{s_{j-2}}^{s_{j}}}\int _{S_{k-j,s_{j},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\big(p_{s_{j}-s_{j-1}}(y_{j-1},y_{j})-p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\times dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=j+1}^{k}ds_{r}ds_{j-1}ds_{j}\prod \limits_{v=1}^{j-3}ds_{v}ds_{j-2}\Bigg|\\{} & \displaystyle \hspace{1em}\le \Bigg|{\int _{0}^{T}}\int _{S_{j-3,0,s_{j-2}}}{\int _{s_{j-2}}^{T}}{\int _{s_{j-2}}^{s_{j}-T/n}}\int _{S_{k-j,s_{j},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\big(p_{s_{j}-s_{j-1}}(y_{j-1},y_{j})-p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\times dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=j+1}^{k}ds_{r}ds_{j-1}ds_{j}\prod \limits_{v=1}^{j-3}ds_{v}ds_{j-2}\Bigg|\\{} & \displaystyle \hspace{2em}+\Bigg|{\int _{0}^{T}}\int _{S_{j-3,0,s_{j-2}}}{\int _{s_{j-2}}^{T}}{\int _{s_{j}-T/n}^{s_{j}}}\int _{S_{k-j,s_{j},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\big(p_{s_{j}-s_{j-1}}(y_{j-1},y_{j})-p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\times dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=j+1}^{k}ds_{r}ds_{j-1}ds_{j}\prod \limits_{v=1}^{j-3}ds_{v}ds_{j-2}\Bigg|.\end{array}\]
We estimate each term separately:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|{\int _{0}^{T}}\int _{S_{j-3,0,s_{j-2}}}{\int _{s_{j-2}}^{T}}{\int _{s_{j}-T/n}^{s_{j}}}\int _{S_{k-j,s_{j},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\big(p_{s_{j}-s_{j-1}}(y_{j-1},y_{j})-p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\times dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=j+1}^{k}ds_{r}ds_{j-1}ds_{j}\prod \limits_{v=1}^{j-3}ds_{v}ds_{j-2}\Bigg|\\{} & \displaystyle \hspace{1em}\le \| h{\| }^{k}\| f\| \\{} & \displaystyle \hspace{2em}\times {\int _{0}^{T}}\int _{S_{j-3,0,s_{j-2}}}{\int _{s_{j-2}}^{T}}{\int _{s_{j}-T/n}^{s_{j}}}\int _{S_{k-j,s_{j},T}}\int _{{({\mathbb{R}}^{d})}^{j}}\Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\\{} & \displaystyle \hspace{2em}\times \big|p_{s_{j}-s_{j-1}}(y_{j-1},y_{j})-p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big|\\{} & \displaystyle \hspace{2em}\times \prod \limits_{q=1}^{j}dy_{q}\prod \limits_{r=j+1}^{k}ds_{r}ds_{j-1}ds_{j}\prod \limits_{v=1}^{j-3}ds_{v}ds_{j-2}\\{} & \displaystyle \hspace{1em}\le \frac{2}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{1}{n}.\end{array}\]
For the other term, we have
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|{\int _{0}^{T}}\int _{S_{j-3,0,s_{j-2}}}{\int _{s_{j-2}}^{T}}{\int _{s_{j-2}}^{s_{j}-T/n}}\int _{S_{k-j,s_{j},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\big(p_{s_{j}-s_{j-1}}(y_{j-1},y_{j})-p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\times dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=j+1}^{k}ds_{r}ds_{j-1}ds_{j}\prod \limits_{v=1}^{j-3}ds_{v}ds_{j-2}\Bigg|\\{} & \displaystyle \hspace{1em}=\Bigg|{\int _{0}^{T}}\int _{S_{j-3,0,s_{j-2}}}{\int _{s_{j-2}}^{T}}{\int _{s_{j-2}}^{s_{j}-T/n}}{\int _{\eta _{n}(s_{j-1})}^{s_{j-1}}}\int _{S_{k-j,s_{j},T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\partial _{s_{j}-u}p_{s_{j}-u}(y_{j-1},y_{j})\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)p_{T-s_{k}}(y_{k},z)\\{} & \displaystyle \hspace{2em}\times dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=j+1}^{k}ds_{r}duds_{j-1}ds_{j}\prod \limits_{v=1}^{j-3}ds_{v}ds_{j-2}\Bigg|\\{} & \displaystyle \hspace{1em}\le \| h{\| }^{k}\| f\| \\{} & \displaystyle \hspace{2em}\times {\int _{0}^{T}}\hspace{-0.1667em}\hspace{-0.1667em}\int _{S_{j-3,0,s_{j-2}}}{\int _{s_{j-2}}^{T}}{\int _{s_{j-2}}^{s_{j}-T/n}}\hspace{-0.1667em}{\int _{\eta _{n}(s_{j-1})}^{s_{j-1}}}\int _{S_{k-j,s_{j},T}}\int _{{({\mathbb{R}}^{d})}^{j}}\big|\partial _{s_{j}-u}p_{s_{j}-u}(y_{j-1},y_{j})\big|\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\prod \limits_{q=1}^{j}dy_{q}\prod \limits_{r=j+1}^{k}ds_{r}duds_{j-1}ds_{j}\prod \limits_{v=1}^{j-3}ds_{v}ds_{j-2}\\{} & \displaystyle \hspace{1em}\le \| h{\| }^{k}\| f\| \\{} & \displaystyle \hspace{2em}\times {\int _{0}^{T}}\hspace{-0.1667em}\hspace{-0.1667em}\int _{S_{j-3,0,s_{j-2}}}{\int _{s_{j-2}}^{T}}\int _{S_{k-j,s_{j},T}}{\int _{0}^{s_{j}-T/n}}\hspace{-0.1667em}{\int _{\eta _{n}(s_{j-1})}^{s_{j-1}}}\int _{{({\mathbb{R}}^{d})}^{j}}\big|\partial _{s_{j}-u}p_{s_{j}-u}(y_{j-1},y_{j})\big|\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\prod \limits_{q=1}^{j}dy_{q}duds_{j-1}\prod \limits_{r=j+1}^{k}ds_{r}ds_{j}\prod \limits_{v=1}^{j-3}ds_{r}ds_{j-2}.\end{array}\]
Again, we consider the inner integral:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {\int _{0}^{\eta _{n}(s_{j})-T/n}}{\int _{\eta _{n}(s_{j-1})}^{s_{j-1}}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{\eta _{n}(s_{j-1})-\eta _{n}(s_{j-2})}(y_{j-2},y_{j-1})\\{} & \displaystyle \hspace{2em}\times \big|\partial _{s_{j}-u}p_{s_{j}-u}(y_{j-1},y_{j})\big|dy_{j}dy_{j-1}duds_{j-1}\\{} & \displaystyle \hspace{1em}=\sum \limits_{i=0}^{\eta _{n}(s_{j})n/T-2}{\int _{iT/n}^{(i+1)T/n}}{\int _{iT/n}^{s_{j-1}}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{iT/n-\eta _{n}(s_{j-2})}(y_{j-2},y_{j-1})\\{} & \displaystyle \hspace{2em}\times \big|\partial _{s_{j}-u}p_{s_{j}-u}(y_{j-1},y_{j})\big|dy_{j}dy_{j-1}duds_{j-1}\\{} & \displaystyle \hspace{1em}=\sum \limits_{i=0}^{\eta _{n}(s_{j})n/T-2}{\int _{iT/n}^{(i+1)T/n}}{\int _{u}^{(i+1)T/n}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{iT/n-\eta _{n}(s_{j-2})}(y_{j-2},y_{j-1})\\{} & \displaystyle \hspace{2em}\times \big|\partial _{s_{j}-u}p_{s_{j}-u}(y_{j-1},y_{j})\big|dy_{j}dy_{j-1}ds_{j-1}du\\{} & \displaystyle \hspace{1em}\le \frac{T}{n}\sum \limits_{i=0}^{\eta _{n}(s_{j})n/T-2}{\int _{iT/n}^{(i+1)T/n}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{iT/n-\eta _{n}(s_{j-2})}(y_{j-2},y_{j-1})\\{} & \displaystyle \hspace{2em}\times \big|\partial _{s_{j}-u}p_{s_{j}-u}(y_{j-1},y_{j})\big|dy_{j}dy_{j-1}du\\{} & \displaystyle \hspace{1em}\le C_{T}\frac{T}{n}\sum \limits_{i=0}^{\eta _{n}(s_{j})n/T-2}{\int _{iT/n}^{(i+1)T/n}}\int _{{\mathbb{R}}^{d}}p_{iT/n-\eta _{n}(s_{j-2})}(y_{j-2},y_{j-1}){(s_{j}-u)}^{-1}dy_{j-1}du\\{} & \displaystyle \hspace{1em}=C_{T}\frac{T}{n}\sum \limits_{i=0}^{\eta _{n}(s_{j})n/T-2}{\int _{iT/n}^{(i+1)T/n}}{(s_{j}-u)}^{-1}du=C_{T}\frac{T}{n}{\int _{0}^{\eta _{n}(s_{j})-T/n}}{(s_{j}-u)}^{-1}du.\end{array}\]
We have
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {\int _{0}^{s_{j}-T/n}}{\int _{\eta _{n}(s_{j-1})}^{s_{j-1}}}\int _{{({\mathbb{R}}^{d})}^{2}}p_{\eta _{n}(s_{j-1})-\eta _{n}(s_{j-2})}(y_{j-2},y_{j-1})\\{} & \displaystyle \hspace{2em}\times \big|\partial _{s_{j}-u}p_{s_{j}-u}(y_{j-1},y_{j})\big|dy_{j}dy_{j-1}duds_{j-1}\\{} & \displaystyle \hspace{1em}\le C_{T}\frac{T}{n}{\int _{0}^{s_{j}-T/n}}{(s_{j}-u)}^{-1}du\le TC_{T}\frac{\log n}{n}.\end{array}\]
Therefore, we obtain
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\\{} & \displaystyle \hspace{2em}\times \big(p_{s_{j}-s_{j-1}}(y_{j-1},y_{j})-p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big)\Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)\\{} & \displaystyle \hspace{2em}\times p_{T-s_{k}}(y_{k},z)dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=1}^{k}ds_{r}\Bigg|\\{} & \displaystyle \hspace{1em}\le 3C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{\log n}{n}.\end{array}\]
Analogously, we also have:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \Bigg|\int _{S_{k,0,T}}\int _{{({\mathbb{R}}^{d})}^{k+1}}\Bigg(\prod \limits_{i=1}^{k}h(y_{i})\Bigg)f(z)\Bigg(\prod \limits_{l=1}^{j-1}p_{\eta _{n}(s_{l})-\eta _{n}(s_{l-1})}(y_{l-1},y_{l})\Bigg)\\{} & \displaystyle \hspace{2em}\times \big(p_{s_{j}-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})-p_{\eta _{n}(s_{j})-\eta _{n}(s_{j-1})}(y_{j-1},y_{j})\big)\\{} & \displaystyle \hspace{2em}\times \Bigg(\prod \limits_{m=j+1}^{k}p_{s_{m}-s_{m-1}}(y_{m-1},y_{m})\Bigg)p_{T-s_{k}}(y_{k},z)dz\prod \limits_{q=1}^{k}dy_{q}\prod \limits_{r=1}^{k}ds_{r}\Bigg|\\{} & \displaystyle \hspace{1em}\le 3C_{T}\frac{1}{(k-1)!}\| h{\| }^{k}\| f\| {T}^{k}\frac{\log n}{n}.\end{array}\]
Therefore, we finally obtain
which completes the proof. □3 Applications
3.1 Discrete approximation of the Feynman–Kac semigroup
Let X be a Brownian motion with values in ${\mathbb{R}}^{d}$. Then condition X holds with
where $c_{1},c_{2}$ are some positive constants.
Let h be a bounded measurable function. Then, it is known (see, e.g., [6], Chapter 1) that the family of operators
forms a semigroup on $L_{p}({\mathbb{R}}^{d}),\hspace{2.5pt}p\ge 1$, and its generator equals
This semigroup is called the Feynman–Kac semigroup.
Denote
Then, using the Taylor expansion of the exponential function and Theorem 1, we have the following statement.
Therefore, the main result of this paper provides an approximation of the Feynman–Kac semigroup with accuracy $(\log n)/n$.
3.2 Approximation of the price of an occupation-time option
Let the price of an asset $S=\{S_{t},t\ge 0\}$ be of the form
where X is a one-dimensional Markov process satisfying condition X. The time spent by S in a defined set $J\subset \mathbb{R}$ (or the time spent by X in a set ${J^{\prime }}=\{x:{e}^{x}\in J\}$) from time 0 to time T is given by
We consider an occupation-time option (see [5]) whose price depends on the time spent by the process S in a set J. In contrast to the traditional barrier options, which are activated or canceled when the process S hits a defined level (barrier), the payoff of an occupation-time option depends on the time spent by the price of the asset above/below this level.
For the strike price K, the barrier L, and the knock-out rate ρ, the payoff of a down-and-out call occupation-time option is given by
Then, for the risk-free interest rate r, its price is given by
Denote
\[\textbf{C}_{n}(T)=\exp (-rT)E\Bigg[\exp \Bigg(-\rho T/n\sum \limits_{k=0}^{n-1}\mathbb{I}_{\{S_{kT/n}\le L\}}dt\Bigg)(S_{T}-K)_{+}\Bigg].\]
We provide the following corollary of Theorem 1.Proof.
For some $N>0$, we denote
\[{\textbf{C}}^{N}(T)=\exp (-rT)E\Bigg[\exp \Bigg(-\rho {\int _{0}^{T}}\mathbb{I}_{\{S_{t}\le L\}}dt\Bigg)\big((S_{T}-K)_{+}\wedge N\big)\Bigg],\]
\[{\textbf{C}_{n}^{N}}(T)=\exp (-rT)E\Bigg[\exp \Bigg(-\rho T/n\sum \limits_{k=0}^{n-1}\mathbb{I}_{\{S_{kT/n}\le L\}}dt\Bigg)\big((S_{T}-K)_{+}\wedge N\big)\Bigg].\]
Then
\[\big|\textbf{C}_{n}(T)-\textbf{C}(T)\big|\le \big|{\textbf{C}_{n}^{N}}(T)-{\textbf{C}}^{N}(T)\big|+\big|\textbf{C}(T)-{\textbf{C}}^{N}(T)\big|+\big|\textbf{C}_{n}(T)-{\textbf{C}_{n}^{N}}(T)\big|.\]
We estimate each term separately. According to Corollary 2,
\[\big|{\textbf{C}_{n}^{N}}(T)-{\textbf{C}}^{N}(T)\big|\le NC_{T,\rho }\exp (-rT)\bigg(\frac{\log n}{n}\bigg).\]
For other terms, we have:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \big|\textbf{C}(T)-{\textbf{C}}^{N}(T)\big|+\big|\textbf{C}_{n}(T)-{\textbf{C}_{n}^{N}}(T)\big|\\{} & \displaystyle \hspace{1em}\le 2\exp (-rT)E\big[(S_{T}-K)_{+}-(S_{T}-K)_{+}\wedge N\big]\le 2\exp (-rT)E[S_{T}\mathbb{I}_{\{S_{T}>N\}}]\\{} & \displaystyle \hspace{1em}=2\exp (-rT)E\bigg[\frac{S_{T}{N}^{u-1}\mathbb{I}_{\{S_{T}>N\}}}{{N}^{u-1}}\bigg]\le \frac{2G}{{N}^{u-1}}\exp (-rT).\end{array}\]
Now, putting $N={n}^{1/u}$ completes the proof. □
Therefore, the main result of this paper provides the approximate value $\textbf{C}_{n}(T)$ of the price of an occupation-time option $\textbf{C}(T)$ with accuracy of order $(\log n)/{n}^{1-1/u}$ for the class of processes X satisfying X and the condition $E\exp (uX_{T})<+\infty $ for some $u>1$.
