1 Introduction
Let $X_{t}$, $t\ge 0$, be a Markov process with values in ${\mathbb{R}}^{d}$. Consider the following objects:
In this paper, we establish strong $L_{2}$-approximation rates, that is, the bounds for
The current research is mainly motivated by the recent papers [2] and [3].
In [3], strong $L_{p}$-approximation rates are considered for an important particular case where X is a one-dimensional diffusion. The approach developed in this paper contains both the Malliavin calculus tools and the Gaussian bounds for the transition probability density of the process X, and relies substantially on the structure of the process.
Another approach to that problem has been developed in [2]. This approach is, in a sense, a modification of Dynkin’s theory of continuous additive functionals (see [1], Chap. 6) and also involves the technique similar to that used in the proof of the classical Khasminskii lemma (see, e.g., [4, Lemma 2.1]). This approach allows us to obtain strong $L_{p}$-approximation rates under assumptions on the process X formulated only in terms of its transition probability density.
For a bounded function h, the strong $L_{p}$-rates of approximation of the integral functional $I_{T}(h)$ obtained in [2] essentially coincide with those established in [3]. However, under additional regularity assumptions on the function h (e.g., when h is Hölder continuous), the rates obtained in [3] are sharper (see [2, Thm. 2.2] and [3, Thm. 2.3]).
In this note, we improve the method developed in [2], so that under the assumption of the Hölder continuity of h, the strong $L_{2}$-approximation rates coincide with those obtained in [3], preserving at the same time the advantage of the method that the assumptions on the process X are quite general and do not essentially rely on the structure of the process.
2 Main result
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 $|\cdot |$.
We make the following assumption on the process X.
A. The process X possesses a transition probability density $p_{t}(x,y)$ that is differentiable with respect to t and satisfies the following estimates:
for some fixed $\alpha \in (0,2]$ and some distribution density Q such that $\int _{{\mathbb{R}}^{d}}|z{|}^{2\gamma }Q(z)\hspace{0.1667em}dz<\infty $. Without loss of generality, we assume that in (1)–(3) $C_{T}\ge 1$.
(2)
\[\big|\partial _{t}p_{t}(x,y)\big|\le C_{T}{t}^{-1-d/\alpha }Q\big({t}^{-1/\alpha }(x-y)\big),\hspace{1em}t\le T,\](3)
\[\big|{\partial _{tt}^{2}}p_{t}(x,y)\big|\le C_{T}{t}^{-2-d/\alpha }Q\big({t}^{-1/\alpha }(x-y)\big),\hspace{1em}t\le T,\]We assume that the function h satisfies the Hölder condition with exponent $\gamma \in (0,\alpha /2]$, that is,
Now we formulate the main result of the paper.
Theorem 1.
Suppose that Assumption A holds. Then
\[\mathbb{E}_{x}{\big|I_{T}(h)-I_{T,n}(h)\big|}^{2}\le \left\{\begin{array}{l@{\hskip10.0pt}l}D_{T,\gamma ,\alpha ,Q}C_{\gamma ,\alpha }\| h{\| _{\gamma }^{2}}{n}^{-(1+2\gamma /\alpha )},\hspace{1em}& \gamma \ne \alpha /2,\\{} D_{T,\gamma ,\alpha ,Q}\| h{\| _{\gamma }^{2}}{n}^{-2}\ln n,\hspace{1em}& \gamma =\alpha /2,\end{array}\right.\hspace{2.5pt}\]
where
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle D_{T,\gamma ,\alpha ,Q}=8{C_{T}^{2}}{T}^{2+2\gamma /\alpha }\int _{{\mathbb{R}}^{d}}|z{|}^{2\gamma }Q(z)\hspace{0.1667em}dz,\\{} & \displaystyle C_{\gamma ,\alpha }=\max \bigg\{{(1-2\gamma /\alpha )}^{-1}{(2\gamma /\alpha )}^{-1},\hspace{0.1667em}\underset{n\ge 1}{\max }\bigg(\frac{{(\ln n)}^{2}}{{n}^{1-2\gamma /\alpha }}\bigg)\bigg\}.\end{array}\]
We provide the proof of Theorem 1 in Section 3.
Remark 1.
Any diffusion process satisfies conditions (1)–(3) with $\alpha =2$, $Q(x)=c_{1}{e}^{-c_{2}|x{|}^{2}}$, and properly chosen $c_{1},c_{2}$ (see [2]). In the case where X is a one-dimensional diffusion, Theorem 1 provides the same rates of convergence as those obtained in [3] (see Theorem 2.3 in [3]).
3 Proof of Theorem 1
Proof.
For $t\in [kT/n,(k+1)T/n)$, denote
and put $\Delta _{n}(s):=h(X_{s})-h(X_{\eta _{n}(s)})$, $s\in [0,T]$.
By the Markov property of X, for any $r<s$, we have
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \mathbb{E}_{x}|X_{s}-X_{r}{|}^{2\gamma }& \displaystyle =\mathbb{E}_{x}\int _{{\mathbb{R}}^{d}}p_{s-r}(X_{r},z)|X_{r}-z{|}^{2\gamma }\hspace{0.1667em}dz\\{} & \displaystyle \le C_{T}\mathbb{E}_{x}\int _{{\mathbb{R}}^{d}}{(s-r)}^{-d/\alpha }Q\big({(s-r)}^{-1/\alpha }(X_{r}-z)\big)|X_{r}-z{|}^{2\gamma }\hspace{0.1667em}dz\\{} & \displaystyle =C_{T}{(s-r)}^{2\gamma /\alpha }\int _{{\mathbb{R}}^{d}}|z{|}^{2\gamma }Q(z)\hspace{0.1667em}dz.\end{array}\]
Therefore, using the inequality $s-\eta _{n}(s)\le T/n$, $s\in [0,T]$ and the Hölder continuity of the function h, we obtain:
Split
where
where in (6) we used that $\int _{{\mathbb{R}}^{d}}p_{r}(y,z)\hspace{0.1667em}dz=1$, $r>0$, $y\in {\mathbb{R}}^{d}$, and in (7) we used the Chapman–Kolmogorov equation.
(5)
\[\mathbb{E}_{x}{\big|I_{T}(h)-I_{T,n}(h)\big|}^{2}=2\mathbb{E}_{x}{\int _{0}^{T}}{\int _{s}^{T}}\Delta _{n}(s)\Delta _{n}(t)\hspace{0.1667em}dt\hspace{0.1667em}ds=J_{1}+J_{2}+J_{3},\]
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle J_{1}=2\mathbb{E}_{x}{\int _{0}^{T}}{\int _{s}^{\zeta _{n}(s)+T/n}}\Delta _{n}(s)\Delta _{n}(t)\hspace{0.1667em}dt\hspace{0.1667em}ds,\\{} & \displaystyle J_{2}=2\mathbb{E}_{x}{\int _{0}^{T/n}}{\int _{\zeta _{n}(s)+T/n}^{T}}\Delta _{n}(s)\Delta _{n}(t)\hspace{0.1667em}dt\hspace{0.1667em}ds,\\{} & \displaystyle J_{3}=2\mathbb{E}_{x}{\int _{T/n}^{T}}{\int _{\zeta _{n}(s)+T/n}^{T}}\Delta _{n}(s)\Delta _{n}(t)\hspace{0.1667em}dt\hspace{0.1667em}ds.\end{array}\]
For $|J_{1}|$ and $|J_{2}|$, the estimates can be obtained in the same way. Indeed, using the Cauchy inequality and (4), we get
\[\begin{array}{r@{\hskip0pt}l}\displaystyle |J_{1}|& \displaystyle \le 2{\int _{0}^{T}}{\int _{s}^{\zeta _{n}(s)+T/n}}{\big(\mathbb{E}_{x}{\big|\Delta _{n}(s)\big|}^{2}\big)}^{1/2}{\big(\mathbb{E}_{x}{\big|\Delta _{n}(t)\big|}^{2}\big)}^{1/2}\hspace{0.1667em}dt\hspace{0.1667em}ds\\{} & \displaystyle \le 2C_{T}{T}^{2\gamma /\alpha }\| h{\| _{\gamma }^{2}}\bigg(\int _{{\mathbb{R}}^{d}}|z{|}^{2\gamma }Q(z)\hspace{0.1667em}dz\bigg){n}^{-2\gamma /\alpha }{\int _{0}^{T}}\big(T/n+\zeta _{n}(s)-s\big)\hspace{0.1667em}ds\\{} & \displaystyle \le 4C_{T}{T}^{2+2\gamma /\alpha }\| h{\| _{\gamma }^{2}}\bigg(\int _{{\mathbb{R}}^{d}}|z{|}^{2\gamma }Q(z)\hspace{0.1667em}dz\bigg){n}^{-(1+2\gamma /\alpha )}.\end{array}\]
In the last inequality, we have used the inequality $\zeta _{n}(s)-s\le T/n$, $s\in [0,T]$. Similarly,
\[|J_{2}|\le 2C_{T}{T}^{2+2\gamma /\alpha }\| h{\| _{\gamma }^{2}}\bigg(\int _{{\mathbb{R}}^{d}}|z{|}^{2\gamma }Q(z)\hspace{0.1667em}dz\bigg){n}^{-(1+2\gamma /\alpha )}.\]
Now we proceed to the estimation of $|J_{3}|$, which is the main part of the proof. Observe that the following identities hold: (6)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \int _{{\mathbb{R}}^{d}}{\partial _{uv}^{2}}p_{u}(x,y)p_{v-u}(y,z)\hspace{0.1667em}dz& \displaystyle ={\partial _{uv}^{2}}p_{u}(x,y)\int _{{\mathbb{R}}^{d}}p_{v-u}(y,z)\hspace{0.1667em}dz\\{} & \displaystyle ={\partial _{uv}^{2}}p_{u}(x,y)=0,\hspace{1em}y\in {\mathbb{R}}^{d},\end{array}\](7)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \int _{{\mathbb{R}}^{d}}{\partial _{uv}^{2}}p_{u}(x,y)p_{v-u}(y,z)\hspace{0.1667em}dy& \displaystyle ={\partial _{uv}^{2}}\int _{{\mathbb{R}}^{d}}p_{u}(x,y)p_{v-u}(y,z)\hspace{0.1667em}dy\\{} & \displaystyle ={\partial _{uv}^{2}}p_{v}(x,z)=0,\hspace{1em}z\in {\mathbb{R}}^{d},\end{array}\]We have:
where in the last identity we have used (6) and (7).
(8)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle J_{3}& \displaystyle =2{\int _{T/n}^{T}}{\int _{\zeta _{n}(s)+T/n}^{T}}\int _{{\mathbb{R}}^{d}}\int _{{\mathbb{R}}^{d}}h(y)h(z)\big[p_{s}(x,y)p_{t-s}(y,z)\\{} & \displaystyle \hspace{1em}-p_{\eta _{n}(s)}(x,y)p_{t-\eta _{n}(s)}(y,z)-p_{s}(x,y)p_{\eta _{n}(t)-s}(y,z)\\{} & \displaystyle \hspace{1em}+p_{\eta _{n}(s)}(x,y)p_{\eta _{n}(t)-\eta _{n}(s)}(y,z)\big]\hspace{0.1667em}dz\hspace{0.1667em}dy\hspace{0.1667em}dt\hspace{0.1667em}ds\\{} & \displaystyle =2{\int _{T/n}^{T}}{\int _{\zeta _{n}(s)+T/n}^{T}}\int _{{\mathbb{R}}^{d}}\int _{{\mathbb{R}}^{d}}{\int _{\eta _{n}(s)}^{s}}{\int _{\eta _{n}(t)}^{t}}h(y)h(z){\partial _{uv}^{2}}\big(p_{u}(x,y)\\{} & \displaystyle \hspace{1em}\times p_{v-u}(y,z)\big)\hspace{0.1667em}dv\hspace{0.1667em}du\hspace{0.1667em}dz\hspace{0.1667em}dy\hspace{0.1667em}dt\hspace{0.1667em}ds\\{} & \displaystyle =-{\int _{T/n}^{T}}{\int _{\zeta _{n}(s)+T/n}^{T}}\int _{{\mathbb{R}}^{d}}\int _{{\mathbb{R}}^{d}}{\int _{\eta _{n}(s)}^{s}}{\int _{\eta _{n}(t)}^{t}}{\big(h(y)-h(z)\big)}^{2}{\partial _{uv}^{2}}\big(p_{u}(x,y)\\{} & \displaystyle \hspace{1em}\times p_{v-u}(y,z)\big)\hspace{0.1667em}dv\hspace{0.1667em}du\hspace{0.1667em}dz\hspace{0.1667em}dy\hspace{0.1667em}dt\hspace{0.1667em}ds,\end{array}\]Further, we have
\[{\partial _{uv}^{2}}p_{u}(x,y)p_{v-u}(y,z)=p_{u}(x,y){\partial _{rr}^{2}}p_{r}(y,z)\big|_{r=v-u}+\partial _{u}p_{u}(x,y)\partial _{r}p_{r}(y,z)\big|_{r=v-u}.\]
Then, using condition A and the Hölder continuity of the function h, we obtain
(9)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \int _{{\mathbb{R}}^{d}}\int _{{\mathbb{R}}^{d}}{\big(h(y)-h(z)\big)}^{2}|{\partial _{uv}^{2}}\big(p_{u}(x,y)p_{v-u}(y,z)\big)|\hspace{0.1667em}dz\hspace{0.1667em}dy\\{} & \displaystyle \hspace{1em}\le {C_{T}^{2}}\| h{\| _{\gamma }^{2}}\bigg(\int _{{\mathbb{R}}^{d}}|z{|}^{2\gamma }Q(z)\hspace{0.1667em}dz\bigg)\big({(v-u)}^{2\gamma /\alpha -2}+{(v-u)}^{2\gamma /\alpha -1}{u}^{-1}\big).\end{array}\]Therefore, according to (8) and (9),
(10)
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle |J_{3}|\le {C_{T}^{2}}\| h{\| _{\gamma }^{2}}\bigg(\int _{{\mathbb{R}}^{d}}|z{|}^{2\gamma }Q(z)\hspace{0.1667em}dz\bigg)\\{} & \displaystyle \hspace{1em}\times {\int _{T/n}^{T}}{\int _{\zeta _{n}(s)+T/n}^{T}}{\int _{\eta _{n}(s)}^{s}}{\int _{\eta _{n}(t)}^{t}}\big({(v-u)}^{2\gamma /\alpha -2}+{(v-u)}^{2\gamma /\alpha -1}{u}^{-1}\big)\hspace{0.1667em}dv\hspace{0.1667em}du\hspace{0.1667em}dt\hspace{0.1667em}ds.\end{array}\]Denote $a_{\alpha ,\gamma }(u,v):={(v-u)}^{2\gamma /\alpha -2}+{(v-u)}^{2\gamma /\alpha -1}{u}^{-1}$. Then
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {\int _{T/n}^{T}}{\int _{\zeta _{n}(s)+T/n}^{T}}{\int _{\eta _{n}(s)}^{s}}{\int _{\eta _{n}(t)}^{t}}a_{\alpha ,\gamma }(u,v)\hspace{0.1667em}dv\hspace{0.1667em}du\hspace{0.1667em}dt\hspace{0.1667em}ds\\{} & \displaystyle \hspace{1em}=\sum \limits_{i=1}^{n-1}\sum \limits_{j=i+2}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{\int _{jT/n}^{(j+1)T/n}}{\int _{iT/n}^{s}}{\int _{jT/n}^{t}}a_{\alpha ,\gamma }(u,v)\hspace{0.1667em}dv\hspace{0.1667em}du\hspace{0.1667em}dt\hspace{0.1667em}ds\\{} & \displaystyle \hspace{1em}=\sum \limits_{i=1}^{n-1}\sum \limits_{j=i+2}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{\int _{jT/n}^{(j+1)T/n}}{\int _{u}^{(i+1)T/n}}{\int _{v}^{(j+1)T/n}}a_{\alpha ,\gamma }(u,v)\hspace{0.1667em}dt\hspace{0.1667em}ds\hspace{0.1667em}dv\hspace{0.1667em}du\\{} & \displaystyle \hspace{1em}\le {T}^{2}{n}^{-2}\sum \limits_{i=1}^{n-1}\sum \limits_{j=i+2}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{\int _{jT/n}^{(j+1)T/n}}a_{\alpha ,\gamma }(u,v)\hspace{0.1667em}dv\hspace{0.1667em}du\\{} & \displaystyle \hspace{1em}={T}^{2}{n}^{-2}\sum \limits_{i=1}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{\int _{(i+2)T/n}^{T}}a_{\alpha ,\gamma }(u,v)\hspace{0.1667em}dv\hspace{0.1667em}du,\end{array}\]
where in the fourth line we used that, for $u\in [iT/n,(i+1)T/n)$ and $v\in [jT/n,(j+1)T/n)$, we always have $(i+1)T/n-u\le T/n$ and $(j+1)T/n-v\le T/n$.Thus, from (10) we obtain
where
(11)
\[|J_{3}|\le {C_{T}^{2}}{T}^{2}\| h{\| _{\gamma }^{2}}\bigg(\int _{{\mathbb{R}}^{d}}|z{|}^{2\gamma }Q(z)\hspace{0.1667em}dz\bigg){n}^{-2}(S_{1}+S_{2}),\]
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle S_{1}=\sum \limits_{i=1}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{\int _{(i+1)T/n}^{T}}{(v-u)}^{2\gamma /\alpha -2}\hspace{0.1667em}dv\hspace{0.1667em}du,\\{} & \displaystyle S_{2}=\sum \limits_{i=1}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{\int _{(i+2)T/n}^{T}}{(v-u)}^{2\gamma /\alpha -1}{u}^{-1}\hspace{0.1667em}dv\hspace{0.1667em}du.\end{array}\]
We estimate each term separately. In what follows, we consider the case $\gamma <\alpha /2$; the case of $\gamma =\alpha /2$ is similar and therefore omitted. We have
(12)
\[\begin{array}{r@{\hskip0pt}l}\displaystyle S_{1}& \displaystyle \le {(1-2\gamma /\alpha )}^{-1}\sum \limits_{i=1}^{n-1}{\int _{iT/n}^{(i+1)T/n}}{\big((i+1)T/n-u\big)}^{2\gamma /\alpha -1}\hspace{0.1667em}du\\{} & \displaystyle ={(1-2\gamma /\alpha )}^{-1}{(2\gamma /\alpha )}^{-1}\sum \limits_{i=1}^{n-1}{\big((i+1)T/n-iT/n\big)}^{2\gamma /\alpha }\\{} & \displaystyle \le {(1-2\gamma /\alpha )}^{-1}{(2\gamma /\alpha )}^{-1}{T}^{2\gamma /\alpha }{n}^{1-2\gamma /\alpha }\le C_{\gamma ,\alpha }{T}^{2\gamma /\alpha }{n}^{1-2\gamma /\alpha }.\end{array}\]Finally, since $v-u\le T$ for $0\le u<v\le T$, we have
