Approximation of occupation time functionals

TL;DR

This paper investigates the strong $L^2$-approximation of occupation time functionals for $d$-dimensional cdlg processes, providing general bounds that extend to non-Markovian processes like fractional Brownian motion.

Contribution

It offers new upper bounds on approximation errors under weak assumptions, applicable to a broad class of processes including non-Markovian ones.

Findings

Bounds are sharp up to a log-factor for Brownian motion.

Results generalize previous literature significantly.

Applicable to non-Markovian processes like fractional Brownian motion.

Abstract

The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional c\`adl\`ag process. Upper bounds on the error are obtained under weak assumptions, generalizing previous results in the literature considerably. The approach relies on regularity for the marginals of the process and applies also to non-Markovian processes, such as fractional Brownian motion. The results are used to approximate occupation times and local times. For Brownian motion, the upper bounds are shown to be sharp up to a log-factor.

Equations

$${\Gamma }_T\left(f\right)={\int }_0^{T}f\left(X_t\right)dt$$

$${\hat{\Gamma }}_{T,n}(f)={\Delta }_n\sum _{k=1}^{n}f(X_{t_{k-1}}).$$

$$C^s({\mathbb{R}}^d)$$

$$H^s({\mathbb{R}}^d)$$

$${\mathbb{T}}_{T,n}$$

$$\Vert {\Gamma }_T(f)-{\hat{\Gamma }}_{T,n}(f){\Vert }_{L^2(\mathbb{P})}^2$$

$$\text{with}{w}_{T,n}^{(x,y)}={\Delta }_n$$

$$v_{T,n}^{(x,y)}={\Delta }_n$$

$${\tilde{v}}_{T,n}^{(x,y)}=$$

$$\Vert {\Gamma }_T(f(\cdot +\xi ))-{\hat{\Gamma }}_{T,n}(f(\cdot +\xi )){\Vert }_{L^2(\mathbb{P})}^2\le C\Vert \mu {\Vert }_{\infty }{\int }_{{\mathbb{R}}^d}|\mathcal{F}f(u){|}^2(w_{T,n}^{(u)}+v_{T,n}^{(u)})du,$$

$$\text{with}{w}_{T,n}^{(u)}={\Delta }_n{\int }_{{\Delta }_n}^T|g(u,t)|dt+T{\int }_0^{{\Delta }_n}|g(u,t)|dt,$$

$$v_{T,n}^{(u)}={\Delta }_n{\int }_{{\mathbb{T}}_{T,n}}|{\partial }_{t^{\prime }}\phi (u,t;-u,t^{\prime })-{\partial }_{t^{\prime }}\phi (u,\lfloor t{\rfloor }_{{\Delta }_n};-u,t^{\prime })|d(t,t^{\prime }),$$

$${\tilde{v}}_{T,n}^{(u)}$$

$${\int }_{{\mathbb{R}}^{2d}}(f(x)-f(y){)}^2\rho (x,y)d(x,y),$$

$${\int }_{{\mathbb{R}}^d}|\mathcal{F}f(u){|}^2\bar{\rho }(u)du\le \Vert f{\Vert }_{H^s}^2\underset{u}{\sup }|u{|}^{-2s}\bar{\rho }(u).$$

$${\int }_{{\mathbb{R}}^{2d}}(f(x)-f(y){)}^2\rho (y-x)d(x,y)=(2\pi {)}^{-2d}{\int }_{{\mathbb{R}}^{2d}}|\mathcal{F}f(u)(1-e^{-i⟨u,y⟩}){|}^2\rho (y)d(u,y)$$

$$\le (2\pi {)}^{-2d}\Vert f{\Vert }_{H^s}^2{\int }_{{\mathbb{R}}^d}|y{|}^{2s}\rho (y)dy,$$

$$c(t,t^{\prime })=\mathbb{E}[X_t^{(m)}{X}_{t^{\prime }}^{(m)}]=\frac{1}{2}((t^{\prime }{)}^{2H}+t^{2H}-|t^{\prime }-t{|}^{2H}),0\le t,t^{\prime }\le T.$$

$$\Vert {\Gamma }_T(f)-{\hat{\Gamma }}_{T,n}(f){\Vert }_{L^2(\mathbb{P})}$$

$$\Vert {\Gamma }_T(f)-{\hat{\Gamma }}_{T,n}(f){\Vert }_{L^2(\mathbb{P})}\le C(\Vert f{\Vert }_{\infty }{\Delta }_n+\Vert f{\Vert }_{H^s}^{1/2}\max (\Vert f{\Vert }_{H^s}^{1/2},\Vert f{\Vert }_{L^2}^{1/2})$$

$$\max (1,T^{1/4})T^{1/4-H/2}{\Delta }_n^{1/2+sH}).$$

$$\Vert {\Gamma }_T(1_{[a,b]})-{\hat{\Gamma }}_{T,n}(1_{[a,b]}){\Vert }_{L^2(\mathbb{P})}$$

$$\Vert L_T(a)-{\hat{\Gamma }}_{T,n}(f_{a,n}){\Vert }_{L^2(\mathbb{P})}$$

$$\mathbb{E}[f(X_{t^{\prime }})|X_t=x]={\int }_{{\mathbb{R}}^d}f(y)p_{t,t^{\prime }}(x,y)dy,x\in {\mathbb{R}}^d,$$

$$p_{t,t^{\prime }}(x,y)\le C{q}_{t^{\prime }-t}(y-x),$$

$$|{\partial }_{t{t}^{\prime }}^2{p}_{t,t^{\prime }}(x,y)|\le C\frac{q_{t^{\prime }-t}(y-x)}{(t^{\prime }-t{)}^2}.$$

$${\partial }_{t^{\prime }}{p}_{t,t^{\prime }}(x,\cdot )=L^{\ast }{p}_{t,t^{\prime }}(x,\cdot ),t<t^{\prime },x\in {\mathbb{R}}^d,$$

$$\Vert {\Gamma }_T(f)-{\hat{\Gamma }}_{T,n}(f){\Vert }_{L^2\left(\mathbb{P}\right)}\le C\Vert \mu {\Vert }_{\infty }^{1/2}\Vert f{\Vert }_{L^2}{T}^{1/2}{\Delta }_n^{1/2}(\log n{)}^{1/2},$$

$$\Vert {\Gamma }_T(f)-{\hat{\Gamma }}_{T,n}(f){\Vert }_{L^2(\mathbb{P})}\le C\Vert \mu {\Vert }_{\infty }^{1/2}\Vert f{\Vert }_{H^s}\{\begin{array}{ll}{T}^{1/2}{\Delta }_n^{1/2+s/\alpha },& s<\alpha /2,\\ T^{1/2}{\Delta }_n(\log n{)}^{1/2},& s=\alpha /2.\end{array}$$

$$\Vert {\Gamma }_T(f)-{\hat{\Gamma }}_{T,n}(f){\Vert }_{L^2({\mathbb{P}}_{x_0})}\le C\Vert f{\Vert }_{\infty }{T}^{1/2}{\Delta }_n^{1/2}(\log n{)}^{1/2},$$

$$\Vert {\Gamma }_T(f)-{\hat{\Gamma }}_{T,n}(f){\Vert }_{L^2({\mathbb{P}}_{x_0})}\le C\Vert f{\Vert }_{C^s}\{\begin{array}{ll}{T}^{1/2}{\Delta }_n^{1/2+s/\alpha },& s<\alpha /2,\\ T^{1/2}{\Delta }_n(\log n{)}^{1/2},& s=\alpha /2.\end{array}$$

$$i{\int }_t^{t^{\prime }}⟨u,b_s⟩ds-\frac{1}{2}{\int }_t^{t^{\prime }}|{\sigma }_s^{\top }u{|}^{2}ds+{\int }_t^{t^{\prime }}{\int }_{{\mathbb{R}}^d}(e^{i⟨u,x⟩}-1-i⟨u,x⟩1_{\left\{|x|\le 1\right\}})d{F}_s(x)ds.$$

$$|e^{{\Psi }_{t,t^{\prime }}(u)}|$$

Full text

MSC 2000 subject classification: Primary: 62M99, 60G99; Secondary: 65D32

Keywords: occupation time; local time; integral functional; heat kernel bounds; fractional Brownian motion; lower bound.

1 Introduction

The approximation of integral-type functionals for random integrands is a classical problem. It appears in the study of numerical approximation schemes for stochastic differential equations ([14, 21, 24]) and in the analysis of statistical methods for stochastic processes ([5, 8, 16]). Early works focused on choosing optimal sampling times ([28]) or on random integrands as a tool for Bayesian numerical analysis (cf. [7] or [27] for an overview). Recently, there has been growing interest in estimating integral functionals of the form

[TABLE]

for a known measurable function $f$ and an $\mathbb{R}^{d}$-valued stochastic process $X=(X_{t})_{0\leq t\leq T}$, $T>0$. Such functionals are called occupation time functionals, as they generalize the occupation time $\Gamma_{T}(\mathbf{1}_{A})$ of a set $A\subset\mathbb{R}^{d}$.

Suppose we have access to $X_{t_{k}}$ at discrete time points $t_{k}=k\Delta_{n}$, where $\Delta_{n}=T/n$ and $k\in\{0,\dots,n\}$. The paths of $f(X)$ are typically rough, even for smooth $f$, allowing only for lower order quadrature rules to approximate $\Gamma_{T}(f)$ ([6]). A natural estimation scheme is given by the Riemann estimator

[TABLE]

Its theoretical properties have been considered systematically only in few works and only for rather specific processes $X$ and functions $f$. The goal of this paper is to study in a general setting the strong $L^{2}(\mathbb{P})$-approximation of $\Gamma_{T}(f)$ by $\widehat{\Gamma}_{T,n}(f)$ and to derive upper bounds on the error, which are explicit in terms of $f$, $T$ and $\Delta_{n}$, and depend on the dimension $d$ only through $f$. The function $f$ is considered in Hölder or fractional Sobolev spaces. In this way, our results unify and generalize, to the best of our knowledge, all previous results in the literature and explain previous results for indicator functions by their Sobolev regularity. In particular, arguing by Sobolev regularity instead of Hölder regularity yields approximations for occupation times and local times, if they exist.

The central idea is to expand the $L^{2}(\mathbb{P})$-norm of $\Gamma_{T}(f)-\widehat{\Gamma}_{T,n}(f)$ in terms of the bivariate distributions $(X_{t},X_{t^{\prime}})$, $0\leq t,t^{\prime}\leq T$, and to derive upper bounds in terms of either their Lebesgue densities or their characteristic functions. This approach is therefore generic and not restricted to Markov processes, and covers fractional Brownian motion and other non-Markovian processes.

For the $L^{2}(\mathbb{P})$-error, lower bounds can be derived by the conditional expectation of $\Gamma_{T}(f)$ with respect to the data. For Brownian motion and functions with fractional Sobolev regularity, this idea is used to prove that the upper bounds are sharp with respect to $\Delta_{n}$ and $T$. In particular, no other quadrature rule can achieve a faster rate of convergence than the Riemann estimator uniformly over the considered function class. Deriving similar upper and lower bounds for strong $L^{p}(\mathbb{P})$-approximations and $p>2$ for processes different from Brownian motion is a challenging problem left for future research.

Let us shortly review related findings in the literature. Central limit theorems for $\Gamma_{T}(f)-\widehat{\Gamma}_{T,n}(f)$ were studied in the case of semimartingales in Chapter 6 of [17] for $f\in C^{2}(\mathbb{R}^{d})$ and by [1] for weakly differentiable functions. The weak error $\mathbb{E}[\Gamma_{T}(f)-\widehat{\Gamma}_{T,n}(f)]$ was considered by [14, 21] for bounded $f$, and here higher regularity of $f$ does not improve the result. Using different techniques, [10, 11, 12, 21] study the $L^{p}(\mathbb{P})$-error for Hölder functions and Markov processes satisfying heat kernel bounds, in particular, scalar diffusions. [25, 21, 16, 15] approximate occupation times $\Gamma_{T}(\mathbf{1}_{[a,b]})$, $a,b\in\mathbb{R}$ and local times for scalar diffusions, while [18] estimate derivatives of local times for fractional Brownian motion. Surprisingly, for occupation times and Brownian motion the rate of convergence corresponds to the one obtained for occupation time functionals and $1/2$-Hölder-continuous functions, which cannot be explained by the specific analysis for indicator functions. For stationary diffusion processes with infinitesimal generator in divergence form this is achieved by [2], who consider fractional Sobolev spaces. Since the proof relies heavily on stationarity and semigroup theory, it is not clear how this can be generalized. We want to emphasize that none of the results above applies to general occupation time functionals and non-Markovian processes.

This paper is organized as follows. Section 2 derives general upper bounds for bounded or square integrable functions $f$. In Section 3, several concrete processes $X$ are studied, namely fractional Brownian motion, Markov processes and processes with independent increments. This does not cover all possible examples, by far, but hints at how to derive similar results for other processes. The reader interested in scalar fractional Brownian motion only may skip to Theorem 5 below. The approximation of occupation and local times is proven only for fractional Brownian motion (Corollaries 6, 7), but the proofs are generic and apply to all other examples. Finally, Section 4 shows that the upper bounds are sharp for Brownian motion.

Proofs are deferred to the appendix. Let us introduce some notation. In the following, $C$ always denotes a positive absolute constant, which may change from line to line. We write $a\lesssim b$ for $a\leq Cb$ and set $\lfloor t\rfloor_{\Delta_{n}}:=\lfloor t/\Delta_{n}\rfloor\Delta_{n}$ for $t\geq 0$, $\lVert\cdot\rVert_{\infty}$ is the usual sup-norm on $\mathbb{R}^{d}$ and $\lVert\cdot\rVert_{L^{2}}$ the $L^{2}$-norm. For $0\leq s\leq 1$ denote by $C^{s}(\mathbb{R}^{d})$ and $H^{s}(\mathbb{R}^{d})$ the Hölder and fractional Sobolev spaces of order $s$, that is,

[TABLE]

where $\mathcal{F}f$ is the Fourier transform of $f$, which for $f\in L^{1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d})$ is given by $\mathcal{F}f(u)=\int_{\mathbb{R}^{d}}f(x)e^{i\langle u,x\rangle}dx$, $u\in\mathbb{R}^{d}$.

2 General $L^{2}(\mathbb{P})$-upper bounds

Let $X=(X_{t})_{0\leq t\leq T}$ be a càdlàg process on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F})_{0\leq t\leq T},\mathbb{P})$ and let $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be measurable. We want to explicitly allow discontinuous functions $f$. For this we make the standing assumption that the compositions $f(X_{t})$ are well-defined random variables. In particular, $f$ bounded or $f\in L^{2}(\mathbb{R}^{d})$ means that $f$ is a function and not only an equivalence class. On the other hand, if the distribution of $X_{t}$ has a Lebesgue density and $f=\tilde{f}$ almost everywhere, then $f(X_{t})=\tilde{f}(X_{t})$ $\mathbb{P}$-almost surely.

Let $(u,v)\mapsto\varphi(u,t;v,t^{\prime})=\mathbb{E}[e^{i\langle u,X_{t}\rangle+i\langle v,X_{t^{\prime}}\rangle}]$ denote the characteristic function of the bivariate random variable $(X_{t},X_{t^{\prime}})\in\mathbb{R}^{2d}$, and $0\leq t,t^{\prime}\leq T$, for $t\neq t^{\prime}$ let $(x,y)\mapsto p(x,t;y,t^{\prime})$ denote its Lebesgue density, if it exists. We always assume that $(x,t,y,t^{\prime})\mapsto p(x,t;y,t^{\prime})$ and $(u,t,v,t^{\prime})\mapsto\varphi(u,t;v,t^{\prime})$ are jointly measurable. In order to handle singularities for the distribution of $(X_{t},X_{t^{\prime}})$ at $t=t^{\prime}$ and near $t=0$ (e.g., when $X_{0}$ is constant), introduce the set

[TABLE]

After these preliminaries we obtain the following general upper bounds.

Proposition 1**.**

Assume that the distributions of the bivariate random variables $(X_{t},X_{t^{\prime}})$ have Lebesgue densities $p(\cdot,t;\cdot,t^{\prime})$ for all $t\neq t^{\prime}$, $t,t^{\prime}>0$. Then the following holds for bounded $f$:

1. (i)

If $t^{\prime}\mapsto p(x,t;y,t^{\prime})$ is differentiable for all $x,y\in\mathbb{R}^{d}$, $0<t<t^{\prime}<T$ with $\partial_{t^{\prime}}p(x,\cdot;y,\cdot)\in L^{1}(\mathbb{T}_{T,n})$, then

[TABLE] 2. (ii)

If also $t\mapsto\partial_{t^{\prime}}p(x,t;y,t^{\prime})$ is differentiable for $t<t^{\prime}$ with $\partial_{tt^{\prime}}^{2}p(x,\cdot;y,\cdot)\in L^{1}(\mathbb{T}_{T,n})$, then the upper bound in (i) holds with $v_{T,n}^{(x,y)}$ replaced by

[TABLE]

The proof of this proposition is inspired by Theorem 1 of [10], but applies also to non-Markovian processes. In order to formulate a similar result with respect to the characteristic functions, we include an additional smoothing by an independent random variable $\xi$.

Proposition 2**.**

Let $f\in L^{2}(\mathbb{R}^{d})$ and let $\xi$ be an $\mathbb{R}^{d}$-valued random variable, independent of $X$ with bounded Lebesgue density $\mu$.

1. (i)

If $t^{\prime}\mapsto\varphi(u,t;v,t^{\prime})$ is differentiable for all $u,v\in\mathbb{R}^{d}$, $0<t<t^{\prime}<T$ with $t^{\prime}\mapsto\partial_{t^{\prime}}\varphi(u,t;v,t^{\prime})\in L^{1}(\mathbb{T}_{T,n})$, then

[TABLE]

where $g(u,t)=2-2\text{Re}\varphi(u,\lfloor t\rfloor_{\Delta_{n}};-u,t)$. 2. (ii)

If also $t\mapsto\partial_{t^{\prime}}\varphi(u,t;v,t^{\prime})$ is differentiable for $t<t^{\prime}$ with $t\mapsto\partial_{tt^{\prime}}^{2}\varphi(u,t;v,t^{\prime})\in L^{1}(\mathbb{T}_{T,n})$, then the upper bound in (i) holds with $v_{T,n}^{(u)}$ replaced by

[TABLE]

Due to these two results the $L^{2}(\mathbb{P})$-error is controlled by upper bounding

[TABLE]

with respect to certain weight functions $\rho$, $\bar{\rho}$. This naturally suggests to consider $f\in C^{s}(\mathbb{R}^{d}),H^{s}(\mathbb{R}^{d})$ and to prove that the weights decay sufficiently fast. For example, if $f\in H^{s}(\mathbb{R}^{d})$, then

[TABLE]

If also $\rho(x,y)=\rho(y-x)$, then the Plancherel theorem shows

[TABLE]

and so we are left with with studying the weights $\rho$, $\bar{\rho}$.

Remark 3*.*

1. (i)

The regularization with $\xi$ in Proposition 2 has also been used in Theorem 2 of [1]. The independence of $\xi$ allows for using $L^{2}$-arguments in the proofs, cf. inequality (12) below. We will use in Theorem 5 below a different argument involving Fourier transforms, that demonstrates how we can argue, in principle, without this additional smoothing. Moreover, no smoothing is necessary for deriving the lower bounds in Section 4. 2. (ii)

The upper bounds in Proposition 2 depend only on $\varphi(u,t;-u,t^{\prime})=\mathbb{E}[e^{i\langle u,X_{t}-X_{t^{\prime}}\rangle}]$, that is, on increments $X_{t^{\prime}}-X_{t}$.

3 Application to examples

3.1 Fractional Brownian motion

Let $X$ be a fractional Brownian motion in $\mathbb{R}^{d}$ with Hurst index $0<H<1$. The $d$ component processes $(X_{t}^{(m)})_{0\leq t\leq T}$ for $m=1,\dots,d$ are independent centered Gaussian processes with covariance function

[TABLE]

For $H=1/2$, $X$ is a Brownian motion. For $H\neq 1/2$, fractional Brownian motion is an important example of a non-Markovian process, which is also not a semimartingale.

Both the densities and the characteristic functions of $(X_{t},X_{t^{\prime}})$, $0<t<t^{\prime}\leq T$, are explicit by Gaussianity, but it is much easier to upper bound the time derivatives of the latter one. In the setting of Proposition 2 we have:

Theorem 4**.**

Let $X$ be a $d$-dimensional fractional Brownian motion with $0<H<1$, and let $X_{0}=\xi$ be as in Proposition 2. If $f\in H^{s}(\mathbb{R}^{d})$, $0\leq s\leq\min(1,1/(2H))$, then

[TABLE]

For $H=1/2$, this extends Theorem 3.9 of [2] to general dimension $d$. We see that the rate $\Delta_{n}^{1/2+sH}$ improves for more regular $f$, but never exceeds $\Delta_{n}$. In Theorem 16 we show for $H=1/2$ that it is optimal.

In the scalar case, we demonstrate now how the random variable $\xi$ in Proposition 2 can be avoided. The proof is not based on Proposition 2 and replaces the role of the density $\mu$ by the characteristic function $\varphi(u,t;v,t^{\prime})$ to achieve sufficient integrability, cf. Remark 3.

Theorem 5**.**

Let $X$ be a scalar fractional Brownian motion with $0<H\leq 1/2$. If $f\in H^{s}(\mathbb{R})$, $0\leq s\leq 1$, is bounded, then

[TABLE]

While the rate $\Delta_{n}^{1/2+sH}$ remains unchanged, the factor $T^{1/2}$ from Theorem 4 changes to $T^{1/2-H/2}$ for $T\geq 1$. The proof of the lower bound in Theorem 16 reveals that this is necessary for $H=1/2$ due to the singularity of the distribution of $X_{t}$ near $t=0$. Moreover, for $H>1/2$, the proof of Theorem 5 yields a slower rate than $\Delta_{n}^{1/2+sH}$.

Our interest in deriving convergence rates for $f$ with fractional Sobolev regularity is the possibility to approximate occupation times and local times of a scalar process $X$. Such rates will now be derived as corollaries from Theorem 5. With respect to the occupation time of a set $[a,b]$, $-\infty<a<b<\infty$, note that $f=\mathbf{1}_{[a,b]}\in H^{1/2-}(\mathbb{R})$. Theorem 5 therefore suggests the rate $\Delta_{n}^{1/2+H/2-}$, but by interpolation we obtain even an exact result.

Corollary 6**.**

Let $X$ be a scalar fractional Brownian motion with $0<H<1$ and let $-\infty\leq a<b\leq\infty$. If $X_{0}=\xi$ is as in Proposition 2, then the Riemann estimator for the occupation time of the set $[a,b]$ satisfies

[TABLE]

with a constant $C$ independent of $a,b$. If $0<H\leq 1/2$ and $-\infty<a<b<\infty$, then also with $X_{0}=0$

[TABLE]

but this time the constant $C_{a,b}$ depends on $a,b$.

On the other hand, let $(L_{T}(a))_{a\in\mathbb{R}}$ be the family of *local times *of $X$ until $T$, cf. Chapter 5 of [26]. Formally, $L_{T}(a)=\Gamma_{T}(\delta_{a})$, where $\delta_{a}$ is the Dirac delta function. If we use for $H^{s}(\mathbb{R})$ the norm ${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|f\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}{}_{H^{s}}=(\int_{\mathbb{R}}|\mathcal{F}f(u)|^{2}(1+|u|^{2})^{s}du)^{1/2}$, then this allows for defining Sobolev spaces of negative regularity $s<0$, which is a space of distributions. In particular, the Fourier transform (in the sense of distributions) of $\delta_{a}$ is $\mathcal{F}\delta_{a}(u)=e^{-iua}$, implying $\delta_{a}\in H^{s}(\mathbb{R})$ for $s<-1/2$, and so Theorem 5 formally suggests the rate $\Delta_{n}^{1/2-H/2-}$. For simplicity, we only consider $0<H\leq 1/2$, the case $1/2<H<1$ follows as in Corollary 6 assuming $X_{0}=\xi$.

Corollary 7**.**

Let $X$ be a scalar fractional Brownian motion with $0<H\leq 1/2$ and set $f_{a,n}(x)=(2\Delta_{n}^{H})^{-1}\mathbf{1}_{[a-\Delta_{n}^{H},a+\Delta_{n}^{H}]}(x)$ for $x,a\in\mathbb{R}$. Then for any $\varepsilon>0$

[TABLE]

For $H=1/2$, the results of Corollaries 6, 7 have been obtained before ([25], [21]), but are new for general $H$. The loss in the rate $\Delta_{n}^{1/2-H/2-\varepsilon}$ in Corollary 7 for arbitrarily small $\varepsilon>0$ is due to a suboptimal moment bound for local times. In case of a Brownian motion, we can instead use [23] to remove $\varepsilon$, and recover the rate $\Delta_{n}^{1/4}$ from [16], improving on Theorem 2.6 of [21].

3.2 Markov processes with heat kernel

bounds

Let $X$ be a continuous-time Markov process on $\mathbb{R}^{d}$ with transition densities $p_{t,t^{\prime}}$, $0\leq t<t^{\prime}\leq T$, such that

[TABLE]

for continuous and bounded $f$. The distribution of $(X_{t},X_{t^{\prime}})$, $t<t^{\prime}$, conditional on $X_{0}=x_{0}\in\mathbb{R}^{d}$, has the density $p(x,t;y,t^{\prime};x_{0})=p_{0,t}(x_{0},x)p_{t,t^{\prime}}(x,y)$. Suppose the following:

Assumption 8**.**

$X$* is a $d$-dimensional Markov process such that $(x,t,y,t^{\prime})\mapsto p_{t,t^{\prime}}(x,y)$ is jointly measurable and $t^{\prime}\mapsto p_{t,t^{\prime}}(x,y)$ is continuously differentiable for all $x,y\in\mathbb{R}^{d}$, $0<t<t^{\prime}<T$, and such that the heat kernel bounds*

[TABLE]

are satisfied, where the $(q_{t})_{0<t\leq T}$ are probability densities and $(t,x)\mapsto q_{t}(x)$ is jointly measurable.

Assumption 9**.**

In addition to Assumption 8, $t\mapsto\partial_{t^{\prime}}p_{t,t^{\prime}}(x,y)$ is continuously differentiable for all $x,y\in\mathbb{R}^{d}$, $0<t<t^{\prime}<T$, $\int_{\mathbb{R}^{d}}|x|^{\alpha^{\prime}}q_{t}(x)dx\leq Ct^{\alpha^{\prime}/\alpha}$ for some $0<\alpha\leq 2$ and all $0<\alpha^{\prime}\leq\alpha$, and

[TABLE]

Such heat kernel bounds are satisfied for elliptic diffusion processes with sufficiently regular coefficients. In this case the transition densities satisfy the Kolmogorov forward equation

[TABLE]

where $L^{*}$ is the adjoint of the infinitesimal generator of $X$, cf. Chapter 5.7 of [19]. Upper bounds on $\partial_{t^{\prime}}p_{t,t^{\prime}}$, $\partial_{tt^{\prime}}^{2}p_{t,t^{\prime}}$ follow therefore from bounds on the partial derivatives of $(x,y)\mapsto p_{t,t^{\prime}}(x,y)$ with $q_{t}(x)=Ct^{-d/2}e^{-C|xt^{-1/2}|^{2}}$, $\alpha=2$, cf. Theorem 9.4.2 of [9]. Important examples for Markov processes satisfying Assumptions 8 and 9 are Lévy driven SDEs ( [20, 22]), in particular, $\alpha$-stable processes. For slightly more general heat kernel bounds see [11].

Plugging the heat kernel bounds into the abstract bounds of Proposition 1 yields the following. We write $\mathbb{P}_{x_{0}}$ to indicate the initial value $X_{0}=x_{0}$.

Theorem 10**.**

Let $X_{0}$ have a bounded Lebesgue density $\mu$. Under Assumption 8 we have for all $f\in L^{2}(\mathbb{R}^{d})$

[TABLE]

while under Assumption 9 for $f\in H^{s}(\mathbb{R}^{d})$, $0\leq s\leq\alpha/2$,

[TABLE]

Theorem 11**.**

Under Assumption 8 we have for bounded $f$

[TABLE]

while under Assumption 9 for $f\in C^{s}(\mathbb{R}^{d}),0\leq s\leq\alpha/2$,

[TABLE]

These two results unify and generalize previously obtained results for the strong $L^{2}(\mathbb{P})$-error for Markov diffusion processes in arbitrary dimension $d$, up to log-terms. These are Theorem 2.14 of [12], Theorem 2.1 [11] for bounded $f$, [10], Theorem 2.3 of [21] for $f\in C^{s}(\mathbb{R}^{d})$ and Theorem 3.7 of [2] for $f\in H^{s}(\mathbb{R}^{d})$. The additional $\log$-terms are not optimal, in general. They are not present for $f\in H^{s}(\mathbb{R}^{d})$, for example, when $X$ is a Brownian motion (Theorem 4) or a stationary reversible diffusion process (Theorem 3.7 of [2]).

Results for occupation or local times can be obtained exactly as in Corollaries 6, 7 by interpolation in Theorem 10, and as long as moment bounds for the local times exist. We want to emphasize that this interpolation does not work using Theorem 11, since indicator functions are not well approximated in Hölder spaces.

Remark 12*.*

There are different ways to relax the assumption on $X_{0}$ in Theorem 10.

1. (i)

If we are estimating $\int_{T_{0}}^{T}f(X_{t})dt$ for $T_{0}>0$ using the corresponding Riemann estimator, then Theorem 10 remains valid by the Markov property, if $X_{0}$ is replaced by $X_{T_{0}}$ and $q$ from Assumption 8 is bounded. 2. (ii)

It is enough to upper bound $\sup_{x\in\mathbb{R}^{d}}q_{t}(x)$ in the proof (cf. Equations (30) to (32) and (26) to (28)). Since $q_{t}$ typically has a singularity near $t=0$, this will yield a slower rate. 3. (iii)

If $X_{0}\in L^{2}(\mathbb{R}^{d})$ is not bounded, then we obtain again slower rates, cf. [24].

Remark 13*.*

The assumption on $f$ being bounded in Theorem 11 can be relaxed by considering weighted Hölder norms, cf. [11, 12].

3.3 Processes with independent increments

Let $X$ be an additive process on $\mathbb{R}^{d}$ with local characteristics $(\sigma_{t},F_{t},b_{t})_{t\geq 0}$, where $t\mapsto\sigma_{t}$ is a continuous $\mathbb{R}^{d\times d}$-valued function, $t\mapsto b_{t}$ is a locally integrable $\mathbb{R}^{d}$-valued function and $(F_{t})_{t\geq 0}$ is a family of positive measures on $\mathbb{R}^{d}$ with $F_{t}(\{0\})=0$ and $\sup_{0\leq t\leq T}\{\int(|x|^{2}\wedge 1)dF_{t}(x)\}<\infty$, cf. Chapter 14 of [30].

$X$ is an inhomogeneous Markov process with independent increments, in particular every Lévy process is an additive process. We can therefore apply the results from Section 3.2 as soon as heat kernel bounds are available. In general, however, it is rather difficult to compute or even upper bound the marginal densities. When $\sigma_{t}\sigma_{t}^{\top}$ is not invertible at some $t$, the densities might not even exist. On the other hand, the characteristic functions are known explicitly by the Lévy-Khintchine formula, cf. Theorem 14.1 of [30]. The characteristic function of $(X_{t},X_{t^{\prime}})$, $0<t<t^{\prime}$, is $\varphi(u,t;v,t^{\prime})=e^{\Psi_{t,t^{\prime}}(v)+\Psi_{0,t}(u+v)}$, $u,v\in\mathbb{R}^{d}$, with characteristic exponents $\Psi_{t,t^{\prime}}(u)$ equal to

[TABLE]

For concrete bounds suppose the following:

Assumption 14**.**

$X$* is a $d$-dimensional additive process with characteristic exponents satisfying for $0\leq t<t^{\prime}\leq T$, $u\in\mathbb{R}^{d}$*

[TABLE]

where $0\leq\alpha\leq 2$, $\beta\geq 0$ are such that $0\leq\alpha(1+\beta)\leq 2$, $\alpha^{*}=\max(1,\alpha(1+\beta))$.

This assumption holds, for example, if $X$ is a generalized $\alpha$-stable process with $\alpha^{*}=\alpha$ or with time varying stability index $t\mapsto\alpha(1+\beta_{t})$, $0\leq\beta_{t}\leq\beta$. On the other hand, if $\sigma_{t}\sigma_{t}^{\top}$ is non-degenerate for all $t$, then $\alpha^{*}=\alpha=2$ (use Equation 8.9 of Sato, [29]).

Theorem 15**.**

Grant Assumption 14 and let $X_{0}$ have a bounded Lebesgue density $\mu$. Then:

1. (i)

If $f\in H^{s}(\mathbb{R}^{d})$, $0\leq s\leq\alpha^{*}/2$, then

[TABLE] 2. (ii)

If $\alpha^{*}=0$, then for $f\in L^{2}(\mathbb{R}^{d})$ the upper bound is $C\lVert\mu\rVert_{\infty}^{1/2}\lVert f\rVert_{L^{2}}(T^{1/2}+T)\Delta_{n}$. If $-C(t^{\prime}-t)\leq\Psi_{t,t^{\prime}}(u)\leq 0$ for $t^{\prime}>t$, then $T^{1/2}+T$ can be replaced by $T^{1/2}$.

If $\alpha^{*}=\alpha$, then the rate in (i) is $\Delta_{n}^{1/2+s/\alpha}$ as in Theorem 10, but without the $(\log n)^{1/2}$-term. $\alpha^{*}=0$ holds for a compound Poisson process. In this case, $\Psi_{t,t^{\prime}}(u)=|t^{\prime}-t|\int_{\mathbb{R}^{d}}(e^{i\langle u,x\rangle}-1)dF(x)$ for a finite measure $F$, and so $\Psi_{t,t^{\prime}}(u)$ is bounded. The improved bound in (ii) applies, if $F$ is symmetric. For stationary $X$ this has been shown also in Section 3.1 of [2]. As above, we can argue as in Corollaries 6 and 7 to obtain estimates for occupation times and local times of $X$, as long as the latter exist.

4 Sharpness of upper bounds for Brownian

motion

Recall from Theorems 4 and 5 that the upper bound for the $L^{2}(\mathbb{P})$-error with respect to the Riemann estimator $\widehat{\Gamma}_{T,n}(f)$ is of order $\Delta_{n}^{(1+s)/2}$ as $n\rightarrow\infty$ for $f\in H^{s}(\mathbb{R}^{d})$ and when $X$ is a Brownian motion. We prove now that this rate is sharp up to a log-factor uniformly in $H^{s}(\mathbb{R}^{d})$.

The key idea of the following proof, which extends Theorem 5 of [1], is that the minimal $L^{2}(\mathbb{P})$-error is achieved by the conditional expectation,

[TABLE]

where $\widehat{\Gamma}$ is any square integrable estimator of $\Gamma_{T}(f)$, being measurable with respect to the sigma field $\mathcal{G}_{n}=\sigma(X_{t_{k}}:k\in\{0,\dots,n\})$. When $X$ is a Brownian motion, the conditional expectation can be computed explicitly. The only other explicit lower bound in the literature is Proposition 2.3 of [25] for $f=\mathbf{1}_{[0,\infty)}$ and a scalar Brownian motion.

Theorem 16**.**

Let $X$ be a $d$-dimensional Brownian motion and define for $0\leq s\leq 1$

[TABLE]

Then $f^{*}\in H^{s}(\mathbb{R}^{d})$ and we have the asymptotic lower bound

[TABLE]

where $\hat{\Gamma}$ is any square integrable estimator for $\Gamma_{T}(f^{*})$ based on $(X_{t_{k}})_{k\in\{0,\dots,n\}}$. In particular, the rate $\Delta_{n}^{(1+s)/2}$, achieved by the Riemann estimator $\hat{\Gamma}_{T,n}(f)$, is sharp up to a log-factor uniformly for $f\in H^{s}(\mathbb{R}^{d})$.

If $X_{0}$ is not constant, then the time dependency in the error necessarily increases.

Corollary 17**.**

Consider the setting of Theorem 16. If $X_{0}$ is independent of $X-X_{0}$ and has a Lebesgue density $\mu$ with nonnegative and Fourier transform $\mathcal{F}\mu\in L^{1}(\mathbb{R}^{d})$, then

[TABLE]

Appendix A Proofs

A.1 Proof of Proposition 1

Proof.

(i). Recall the definition of $\mathbb{T}_{T,n}$ from (1) and set

[TABLE]

For $0\leq t,t^{\prime}\leq T$ let

[TABLE]

Using symmetry decompose

[TABLE]

For the result it is enough to show with

[TABLE]

that

[TABLE]

For the first part a rough argument suffices. Observe that $|E_{t,t^{\prime}}|\leq\frac{1}{2}E_{t,t}+\frac{1}{2}E_{t^{\prime},t^{\prime}}$. The claim in (8) follows therefore from $|A_{1}|\lesssim\Delta_{n}\int_{\Delta_{n}}^{T}E_{t,t}dt$,

[TABLE]

and $E_{t,t}=\int_{\mathbb{R}^{2d}}(f(x)-f(y))^{2}p(x,\lfloor t\rfloor_{\Delta_{n}};y,t)d(x,y)$. With respect to (9), the regularity assumptions on the joint densities are crucial. Consider $(t,t^{\prime})\in\tilde{\mathbb{T}}_{n}$. Clearly,

[TABLE]

Here comes the main insight: If $f(x)$ is replaced in this equality by $f(y)$, then the $d(x,y)$-integral vanishes. The same holds with $f(y)$ replaced by $f(x)$. This allows two modifications in the last display. First, replace $f(x)f(y)$ by $-1/2(f(x)-f(y))^{2}$, and second, use differentiability of the joint density. Then the last display reduces to

[TABLE]

If $(t,t^{\prime})\in\mathbb{\tilde{T}}_{n}$ and $\lfloor t^{\prime}\rfloor_{\Delta_{n}}\leq r<t^{\prime}$, then $(t,r)\in\mathbb{T}_{T,n}$ and $|t^{\prime}-r|\leq\Delta_{n}$. Integrating in the last display over $t,t^{\prime}\in\tilde{\mathbb{T}}_{n}$, it can therefore be upper bounded by a double integral over $(t,r)\in\mathbb{T}_{T,n}$, yielding an additional $\Delta_{n}$. This implies (9).

(ii). It is enough to prove (9) with $\tilde{v}_{T,n}^{(x,y)}=\Delta_{n}^{2}\int_{\mathbb{T}_{T,n}}|\partial_{rr^{\prime}}^{2}p(x,t;y,t^{\prime})|d(t,t^{\prime})$ instead of $v_{T,n}^{(x,y)}$. As in (i), $(t,t^{\prime})\in\mathbb{\tilde{T}}_{n}$ and $\lfloor t\rfloor_{\Delta_{n}}\leq r<t$, $\lfloor t^{\prime}\rfloor_{\Delta_{n}}\leq r^{\prime}<t^{\prime}$ imply $(r,r^{\prime})\in\mathbb{T}_{T,n}$ and $|t-r|,|t^{\prime}-r^{\prime}|\leq\Delta_{n}$. Since $t\mapsto\partial_{r}p(x,t;y,r)$ is differentiable, the $dr$-integral in (11) equals $\int_{\lfloor t^{\prime}\rfloor_{\Delta_{n}}}^{t^{\prime}}\int_{\lfloor t\rfloor_{\Delta_{n}}}^{t}\partial_{rr^{\prime}}^{2}p(x,r;y,r^{\prime})drdr^{\prime}$, which can be upper bounded by a double integral, incurring in all an additional $\Delta_{n}^{2}$. From this the result is obtained. ∎

A.2 Proof of Proposition 2

Proof.

(i). Let $\mathbb{T}_{T,n}$ and $\tilde{\mathbb{T}}_{n}$ be as in the proof of Proposition 1. By independence of $\xi$ we have

[TABLE]

using the Plancherel Theorem in the last line. For $u\in\mathbb{R}^{d}$, $0\leq t,t^{\prime}\leq T$, set

[TABLE]

Write $\lVert\Gamma_{T}(e^{i\langle u,\cdot\rangle})-\widehat{\Gamma}_{T,n}(e^{i\langle u,\cdot\rangle})\rVert_{L^{2}(\mathbb{P})}^{2}$ as $\int_{[0,T]^{2}}E_{t,t^{\prime}}^{u}d(t,t^{\prime})=A_{1}^{u}+2A_{2}^{u}+2A_{3}^{u}$, with $A_{i}^{u}$, $i=1,2,3$ as in (7) above, but with $E_{t,t^{\prime}}$ replaced by $E_{t,t^{\prime}}^{u}$. For the result it is enough to show, with

[TABLE]

that

[TABLE]

For $t=t^{\prime}$, $\overline{E_{t,t}^{u}}=E_{t,t}^{u}$ and so $E_{t,t}^{u}=2-2\text{Re}(\varphi(u,\lfloor t\rfloor_{\Delta_{n}};-u,t))=g(u,t)$. (14) follows immediately as for $A_{1}$, $A_{2}$ in the proof of Proposition 1, again with $E_{t,t^{\prime}}^{u}$ instead of $E_{t,t^{\prime}}$. On the other hand, differentiability of the characteristic functions for $(t,t^{\prime})\in\mathbb{\tilde{T}}_{n}$ shows

[TABLE]

Arguing as after (11) above yields (15) and thus the result.

(ii). It is enough to prove (15) with $\tilde{v}_{T,n}^{(u)}=\Delta_{n}^{2}\int_{\mathbb{T}_{T,n}}|\partial_{tt^{\prime}}^{2}\varphi(u,t;-u,t^{\prime})|d(t,t^{\prime})$ instead of $v_{T,n}^{(u)}$. As in the proof of Proposition 1, for this it suffices to note by differentiability of $r\mapsto\partial_{r}\varphi(u,r;-u,r^{\prime})$ that $E_{t,t^{\prime}}^{u}=\int_{\lfloor t^{\prime}\rfloor_{\Delta_{n}}}^{t^{\prime}}\int_{\lfloor t\rfloor_{\Delta_{n}}}^{t}\partial_{rr^{\prime}}^{2}\varphi(u,r;-u,r^{\prime})drdr^{\prime}$. ∎

A.3 Proofs of Section 3.1

Observe first the following elementary lemma, which will be used frequently.

Lemma 18**.**

We have for $\alpha,\beta,\gamma\in\mathbb{R}$:

[TABLE]

Proof.

By the changes of variables $t=Tr$, $t^{\prime}=Tr^{\prime}$ we have

[TABLE]

For fixed $r$ the $dr^{\prime}$-integral equals

[TABLE]

and the same upper bound applies to the $dr$-integral with $\beta$ instead of $\alpha+\gamma$. ∎

If $X$ is a fractional Brownian motion, then the distribution of $(X_{t},X_{t^{\prime}})$, $t\neq t^{\prime}$, is Gaussian and its characteristic function is $\varphi(u,t;v,t^{\prime})=e^{-\frac{1}{2}\Phi_{t,t^{\prime}}(u,v)}$ with

[TABLE]

where we recall the covariance function $c(t,t^{\prime})$ from (4). A simple computation shows

[TABLE]

Self-similarity of a fractional Brownian motion also implies $(X_{t})_{0\leq t\leq T}\overset{d}{\sim}(T^{H}X_{t/T})_{0\leq t\leq T}$, and therefore

[TABLE]

and similarly for $\partial_{t^{\prime}}\Phi_{t,t^{\prime}}(u,v)$, $\partial_{t}\Phi_{t,t^{\prime}}(u,v)$. In order to upper bound the characteristic function, we use that a fractional Brownian motion is locally nondeterministic, cf. [31, 3], that is, for $r^{\prime}>r$

[TABLE]

In particular,

[TABLE]

With this let us prove the two theorems and Corollary 7.

Proof of Theorem 4.

It is easily checked by a look at (17) that the assumptions of Proposition 2(i,ii) are satisfied with $X_{0}=\xi$ $X-X_{0}$ independent of $X_{0}$. For $T=1$ and $v=-u$ we have in (17)

[TABLE]

Consequently, with $g(u,t)=2(1-e^{-\frac{1}{2}|u|^{2}|t-\lfloor t\rfloor_{\Delta_{n}}|^{2H}})$,

[TABLE]

On the other hand, by (17) and $\sup_{x\geq 0}(x{}^{2}e^{-x})\lesssim 1$, we find

[TABLE]

concluding by the definition of the set $\mathbb{T}_{1,n}$ in (1) for the last two inequalities. If $2sH<1$, then using $\sup_{x\geq 0}(x{}^{\gamma}e^{-x})\lesssim 1$ for $\gamma=2sH$ gives

[TABLE]

while if $2sH=1$, then using in (21) the change of variables $|u|^{2}(t^{\prime})^{1/s}\mapsto z$ shows

[TABLE]

implying (LABEL:eq:upperBoundH) for all $0\leq s\leq\min(1,1/(2H))$. This proves $w_{1,n}^{(u)}+\tilde{v}_{1,n}^{(u)}\lesssim\Delta_{n}^{1+2sH}|u|^{2s}$ and therefore the result for $T=1$. For general $T$ we only note by the self-similarity identities in (18), (19) that $w_{T,n}^{(u)}+\tilde{v}_{T,n}^{(u)}=T^{2}(w_{1,n}^{(T^{H}u)}+\tilde{v}_{1,n}^{(T^{H}u)})$. ∎

Proof of Theorem 5.

Since $X_{0}$ may not have a density, the error of the Riemann estimator on $[0,2\Delta_{n}]$ can generally not be controlled by the $L^{2}$-norm of $f$, but for bounded $f$ we have

[TABLE]

with $R_{n}(f)=\lVert\int_{2\Delta_{n}}^{T}(f(X_{t})-f(X_{\lfloor t\rfloor_{\Delta_{n}}}))dt\rVert_{L^{2}(\mathbb{P})}$. In order to prove

[TABLE]

note that smooth and compactly supported functions are dense in $H^{s}(\mathbb{R}^{d})$. We can therefore always find a sequence of such functions $f^{(\varepsilon)}$ with $\lVert f^{(\varepsilon)}-f\rVert_{L^{2}},\lVert f^{(\varepsilon)}-f\rVert_{H^{s}}\rightarrow 0$ as $\varepsilon\rightarrow 0$. Moreover, as the distribution of each $X_{t}$, $t>0$, has a Lebesgue density, this means that also the error terms converge

[TABLE]

in $L^{2}(\mathbb{P})$ and so it is enough to show (22) for a smooth and compactly supported $f$. For such $f$ we have by Fourier inversion $f=(2\pi)^{-1}\int_{\mathbb{R}}\mathcal{F}f(u)e^{-iu\cdot}du$ and thus

[TABLE]

where for $u,v\in\mathbb{R}$ and (cf. $E_{t,t^{\prime}}^{u}$ in the proof of Proposition 2)

[TABLE]

By symmetry in $u,v$ and $t,t^{\prime}$ we decompose the previous display similar to (7) into

[TABLE]

If $t\geq 2\Delta_{n}$ and $t^{\prime}>t+4\Delta_{n}$, then by (17)

[TABLE]

with

[TABLE]

and with

[TABLE]

On the other hand, when $t,t^{\prime}\geq 2\Delta_{n}$, $|t-t^{\prime}|\leq 3\Delta_{n}$ and $t\leq\lfloor t^{\prime}\rfloor_{\Delta_{n}}$, noting that the characteristic functions $\varphi(u,t;v,t^{\prime})$ are bounded by $1$ and using (20) with $t\leq\lfloor t^{\prime}\rfloor_{\Delta_{n}}\leq r^{\prime}$

[TABLE]

The same upper bound holds up to a constant when $t\geq t^{\prime}$, and when $\lfloor t^{\prime}\rfloor_{\Delta_{n}}<t<t^{\prime}$ by arguing as above on each of the summands in

[TABLE]

and consequently,

[TABLE]

Combining this with (24), (23) we conclude by the Cauchy-Schwarz inequality

[TABLE]

In order to obtain (22) we are therefore left with showing

[TABLE]

The first line follows from integrating with respect to $u$. With respect to the second one, using the self-similarity relation (18), we have

[TABLE]

and similarly when integrating with respect to $u$. It is therefore enough to consider only $T=1$. Observe first from (20) and from $\sup_{x\geq 0}x^{\beta}e^{-|x|}\lesssim 1$ for $\beta\geq 0$ and $u,v\in\mathbb{R}^{d}$, using also $|u|\leq|u+v|+|v|$, the inequalities

[TABLE]

Together with $\int_{\mathbb{R}}e^{-C(u+v)^{2}r^{2H}}du$, $\int_{\mathbb{R}}e^{-C(u+v)^{2}r^{2H}}dv$, $\int_{\mathbb{R}}e^{-Cv^{2}r^{2H}}dv\lesssim r^{-H}$, this implies

[TABLE]

which also holds when integrating with respect to $du$ instead of $dv$, as well as

[TABLE]

Plugging these estimates into the definitions of the $U_{1,n}^{(i)}(u,v)$ with $sH\leq H$, $H\leq 1/2$ and using Lemma 18 yields (25), and thus finishes the proof. ∎

Proof of Corollary 6.

We use an approximation argument similar to Theorem 3.6 of [2] for $f=\mathbf{1}_{[a,b]}$. Let first $H>1/2$ and let $a,b$ be finite. Consider the smooth convolution approximation $f_{\varepsilon}=f*\varphi_{\varepsilon}$, where $\varphi_{\varepsilon}(x)=\varepsilon^{-1}\varphi(\varepsilon^{-1}x)$ for $\varepsilon>0$ and a smooth $\varphi$ with support in $[-1,1]$, $\int_{\mathbb{R}}\varphi(x)dx=1$. Then $f_{\varepsilon}\in H^{1}(\mathbb{R}^{d})$ and for sufficiently small $\varepsilon$, $f-f_{\varepsilon}$ is supported in \text{\mathcal{A}=}[a-\varepsilon,a+\varepsilon]\cup[b-\varepsilon,b+\varepsilon]. Consequently, $\lVert f-f_{\varepsilon}\rVert_{L^{2}}^{2}\lesssim\varepsilon$ and

[TABLE]

and the implied constants are independent of $a,b$. By Theorem 4 with $s=0$ and $s=1$ this means

[TABLE]

The claim for finite $a,b$ follows with $\varepsilon=\Delta_{n}^{H/2}$. Since the upper bound is independent of $a,b$, letting $a\rightarrow-\infty$ and $b\rightarrow\infty$ yields the claim for all $a,b$ and $H>1/2$. The proof for $H\leq 1/2$ follows in the same way from Theorem 5, noting $\lVert f\rVert_{\infty},\lVert f_{\varepsilon}\rVert_{\infty}\lesssim 1$, but this time $\lVert f\rVert_{L^{2}}$ and thus the implied constants above depend on $a,b$. ∎

Proof of Corollary 7.

We have from Corollary 6

[TABLE]

Self-similarity of $X$ implies that $L_{T}$ has the same distribution as $T^{1-H}L_{1}(T^{-H}\cdot)$. By the occupation time formula, cf. [13], this means also $\int_{0}^{T}f_{a,n}(X_{t})dt=\int_{\mathbb{R}}f_{a,n}(x)L_{T}(x)dx$ and $T^{1-H}\int_{\mathbb{R}}f_{a,n}(x)L_{1}(T^{-H}x)dx$ have the same distribution. With $\int_{\mathbb{R}}f_{a,n}(x)dx=1$,

[TABLE]

uniformly in $a\in\mathbb{R}$ with $\gamma<(1-H)/(2H)$, where the last line follows from moment bounds for the local time (e.g., [4] or Equation 4.18 of [31]). Choosing $\gamma$ arbitrarily close to $(1-H)/(2H)$ gives the result. ∎

A.4 Proof of Theorem 10

Proof.

Without loss of generality we can assume $f$ to be smooth and compactly supported. Indeed, we can always find a sequence of smooth and compactly supported functions $f^{(\varepsilon)}$ with $\lVert f^{(\varepsilon)}-f\rVert_{L^{2}},\lVert f^{(\varepsilon)}-f\rVert_{H^{s}}\rightarrow 0$ as $\varepsilon\rightarrow 0$. Since the distribution of each $X_{t}$, $t\geq 0$, has a Lebesgue density, this means $\Gamma_{T}(f^{(\varepsilon)})\rightarrow\Gamma_{T}(f)$ and $\widehat{\Gamma}_{T,n}(f^{(\varepsilon)})\rightarrow\widehat{\Gamma}_{T,n}(f)$ in $L^{2}(\mathbb{P})$ and so the claim of the theorem transfers from $f^{(\varepsilon)}$ to $f$.

The density of $(X_{t},X_{t^{\prime}})$ for $t<t^{\prime}$, $x,y\in\mathbb{R}^{d}$, is $p(t,x;t^{\prime},y)=\int p(t,x;t^{\prime},y;x_{0})\mu(x_{0})dx_{0}$. As $\mu$ is bounded,

[TABLE]

The respective heat kernel bounds from Assumptions 8 and 9 yield then, using (30), (31), (32) above,

[TABLE]

This means

[TABLE]

Moreover, (3) and Assumption 9 with $\rho=q_{t^{\prime}-t}$ show for $0\leq s\leq\alpha/2$

[TABLE]

Proposition 1(i,ii) then yields for $s=0$

[TABLE]

while for $s>0$ we get instead the upper bound

[TABLE]

The result follows from Lemma 18. ∎

A.5 Proof of Theorem 11

Proof of Theorem 11.

Under Assumptions 8 and 9, respectively, the required integrabilities of $\partial_{t^{\prime}}p(x,t;y,t^{\prime};x_{0})$ and $\partial_{tt^{\prime}}^{2}p(x,t;y,t^{\prime};x_{0})$ follow from continuity of $\partial_{t^{\prime}}p_{t,t^{\prime}}(x,y)$ and $\partial_{tt^{\prime}}^{2}p_{t,t^{\prime}}(x,y)$ on $\mathbb{T}_{T,n}$. Formally, we have for $t<t^{\prime}$

[TABLE]

Proposition 1(i,ii) yields with $\mathbb{P}_{x_{0}}$ and $p(x,t;y,t^{\prime};x_{0})$ instead of $\mathbb{P}$ and $p(x,t;y,t^{\prime})$ for bounded $f$ or $f\in C^{s}(\mathbb{R}^{d})$, respectively, that

[TABLE]

The heat kernel bounds on $p_{t,t^{\prime}}$ and the formal derivatives of $p(x,t;y,t^{\prime};x_{0})$ above show

[TABLE]

Recall that the $q_{t}$ are probability densities. Using (30) and (31) we have

[TABLE]

concluding by Lemma 18 in the last inequality. This proves the first part of the result, that is, when $f$ is only bounded. For the second part set

[TABLE]

Under Assumption 9 we have $h(t,t^{\prime})\leq\int_{\mathbb{R}^{2d}}|y|^{2s}q_{t^{\prime}-t}(y)dy\lesssim|t^{\prime}-t|^{2s/\alpha}$. Combining this with (30), (31) and Lemma 18 yields

[TABLE]

from which we obtain the second part of the result. ∎

A.6 Proof of Theorem 15

Proof.

(i). The characteristic functions $\varphi(\cdot,t;\cdot,t^{\prime})$ of $X$ and $X-X_{0}$ evaluated at $(u,-u)$ coincide and so the assumptions of Proposition 2(ii) are satisfied with $\xi=X_{0}$. For $0<t<t^{\prime}$ we have $\varphi(u,t;-u,t^{\prime})=e^{\Psi_{t,t^{\prime}}(-u)}$,

[TABLE]

and $\partial_{tt^{\prime}}^{2}\Psi_{t,t^{\prime}}(-u)=0$. With $g(u,t)$ from Proposition 2(i) and $\gamma=2s/\alpha^{*}\leq 1$, Assumption 14 shows

[TABLE]

This yields

[TABLE]

For $|u|\geq 1$ with $\gamma^{\prime}=2(\alpha^{*}-s)/\alpha$ we have $(|u|^{\alpha}|t^{\prime}-t|)^{\gamma^{\prime}}e^{-C|u|^{\alpha}(t^{\prime}-t)}\lesssim 1$ and thus

[TABLE]

using in the last two lines Lemma 18 and because always $2(\alpha^{*}-s)\alpha-1\geq 0$ for $s\leq\alpha^{*}/2$. By a different argument for $2(\alpha^{*}-s)=\alpha$ the $\log$-term can be removed. Indeed, upper bounding $\tilde{v}_{T,n}^{(u)}$ and integrating over $t^{\prime}$ in that case yields

[TABLE]

The same estimates show for $|u|\leq 1$ that $\tilde{v}_{T,n}^{(u)}\lesssim T\Delta_{n}^{2}$, because $2\alpha^{*}-\alpha\geq 0$. The result follows from this, the upper bound on $w_{T,n}^{(u)}$ and (2) with $\bar{\rho}=w_{T,n}^{(u)}+v_{T,n}^{(u)}$ such that by Proposition 2(ii)

[TABLE]

(ii). With $|\Psi_{t,t^{\prime}}(-u)|\lesssim|t^{\prime}-t|$ we have this time $|g(u,t)|\lesssim\Delta_{n}$ and

[TABLE]

Since this is bounded, we immediately find as in (i), $w_{T,n}^{(u)}\lesssim T\Delta_{n}^{2}$, $\tilde{v}_{T,n}^{(u)}\lesssim T^{2}\Delta_{n}^{2}$. For the supplement it is enough to note that $|e^{\Psi_{t,t^{\prime}}(-u)}|\leq e^{-C(t^{\prime}-t)}$ such that $\tilde{v}_{T,n}^{(u)}\lesssim\Delta_{n}^{2}\int_{\mathbb{T}_{T,n}}e^{-C(t^{\prime}-t)}d(t,t^{\prime})\lesssim T\Delta_{n}^{2}$. ∎

A.7 Proofs of Section 4

Proof of Theorem 16.

By definition of $g$ we have $g(1+|u|)^{s/2}\in L^{2}(\mathbb{R}^{d})$ and therefore $f=f^{*}\in H^{s}(\mathbb{R}^{d})$. Assume $T>0$. The sigma field $\mathcal{G}_{n}$ is generated by $X_{0}$ and the increments $X_{t_{k}}-X_{t_{k-1}}$ for $k\in\{1,\dots,n\}$. Since they are independent, the Markov property shows $\mathbb{E}[f(X_{t})|\mathcal{G}_{n}]=\mathbb{E}[f(X_{t})|X_{t_{k-1}},X_{t_{k}}]$. This also shows that the random variables $Y_{k}=\int_{t_{k-1}}^{t_{k}}(f(X_{t})-\mathbb{E}[f(X_{t})|\mathcal{G}_{n}])dt$ are uncorrelated, implying

[TABLE]

where $\text{Var}_{k}(Z)$ denotes the variance of a random variable $Z$, conditional on the sigma algebra $\sigma(X_{t_{k-1}},X_{t_{k}})=\sigma(X_{t_{k-1},}X_{t_{k}}-X_{t_{k-1}})$. Stationarity and independence of increments yield

[TABLE]

for another independent $d$-dimensional Brownian motion $B=(B_{t})_{0\leq t\leq 1}$, with $p_{t}$ denoting the marginal density of $X_{t}$ and with a random function

[TABLE]

For $k\geq 1$ and $0<T_{0}<t_{1}$ we have in all

[TABLE]

For $u\in\mathbb{R}^{d}$ set $\kappa(u)=\int_{0}^{1}(e^{-i\langle u,B_{t}\rangle}-\mathbb{E}[e^{-i\langle u,B_{t}\rangle}|B_{1}])dt$. We will show below

[TABLE]

with $\bar{h}=\mathcal{F}^{-1}(|\cdot|^{-s-d/2}\kappa(\cdot))$, which is therefore $\mathbb{P}$-almost surely well-defined in $L^{2}(\mathbb{R}^{d})$. Recall that $\mathcal{F}f=g$ with $g$ from the statement of the Theorem. By properties of the Fourier transform it follows

[TABLE]

Writing $\log(e+\Delta_{n}^{-1/2}|u|)=\log(\Delta_{n}^{1/2}e+|u|)+\log(\Delta_{n}^{-1/2})$, we see that $\log(\Delta_{n}^{-1/2})/\log(e+\Delta_{n}^{-1/2}|u|)$ converges to $1$ as $n\rightarrow\infty$ for fixed $u\neq 0$, and is for $|u|>1$ also upper bounded by $1$, while for $|u|\leq 1$ this follows from $\log(\Delta_{n}^{-1/2})=\log(\Delta_{n}^{-1/2}|u|)-\log(|u|)\leq\log(e+\Delta_{n}^{-1/2}|u|)$. In particular, we have $\mathbb{P}$-almost surely for all $u\in\mathbb{R}^{d}$

[TABLE]

The dominated convergence theorem implies therefore $\mathbb{P}$-almost surely the $L^{2}$-convergence $\mathcal{F}\bar{h}_{n}\rightarrow\mathcal{F}\bar{h}$ or equivalently $\bar{h}_{n}\rightarrow\bar{h}$. Moreover, $\bar{p}_{n}\xrightarrow{}\int_{T/2}^{T}p_{t}dt$ in $L^{1}(\mathbb{R}^{d})$ and $\sup_{x\in\mathbb{R}^{d}}\bar{p}_{n}(x)$ is bounded. By a change of variables and with $V_{n}$ from (35) this means $\mathbb{P}$-almost surely

[TABLE]

concluding by the Plancherel theorem and (38) in the last line. The result follows from using the dominated convergence theorem in (34) and from (36), $\int_{T/2}^{T}p_{t}(0)\gtrsim T^{1-d/2}$.

We are left with proving (36). The lower bound is clearly true, since $\kappa$ does not vanish identically. For the upper bound it is enough by Fubinis Theorem to show

[TABLE]

Since the complex exponentials $e^{-i\langle u,B_{t}\rangle}$ are bounded in absolute value, we clearly have $\mathbb{E}[|\kappa(u)|^{2}]\lesssim 1$ for all $u\in\mathbb{R}^{d}$. On the other hand, if $|u|\leq 1$, then

[TABLE]

implying the wanted upper bound in (39). ∎

Proof of Corollary 17.

The proof follows along the lines of Theorem 16. Since $X_{0}$ is independent of $(X_{t}-X_{0})_{t\geq 0}$, we can argue in (33) with $X_{t}-X_{0}$ instead of $X_{t}$, such that

[TABLE]

Then (34) follows as above, but with $\bar{p}_{n}(x)=\Delta_{n}\sum_{k=2}^{n}\left(\int_{\mathbb{R}^{d}}p_{t_{k-1}}(x-y)\mu(y)dy\right)$. By the Plancherel Theorem and $\mathcal{F}p_{t_{k-1}}(u)=\mathcal{F}p(t_{k-1}^{1/2}u)=e^{-\frac{1}{2}|u|^{2}t_{k-1}}$ this equals

[TABLE]

We find $\bar{p}_{n}\rightarrow\bar{p}_{\infty}=(2\pi)^{-d}\int_{\mathbb{R}^{d}}(\int_{0}^{T}e^{-\frac{1}{2}|u|^{2}t}dt)\mathcal{F}\mu(u)e^{i\langle u,\cdot\rangle}du$ $L^{1}(\mathbb{R}^{d})$ and that $\sup_{x\in\mathbb{R}^{d}}\bar{p}_{n}(x)$ is bounded. As in the proof of Theorem 16 conclude by the dominated convergence theorem that $\mathbb{E}\left[V_{n}\right]\rightarrow\bar{p}_{\infty}(0)\mathbb{E}\left[\int_{\mathbb{R}^{d}}\bar{h}(x)^{2}dx\right]\gtrsim\bar{p}_{\infty}(0)$. The result follows from applying

[TABLE]

to $x=-\frac{1}{2}|u|^{2}T$ such that $\int_{0}^{T}e^{-\frac{1}{2}|u|^{2}t}dt=\frac{2}{\left|u\right|^{2}}(1-e^{-\frac{1}{2}|u|^{2}T})\geq\frac{T}{2}$. ∎

Acknowledgement

Support by the DFG Research Training Group 1845 “Stochastic Analysis with Applications in Biology, Finance and Physics” is gratefully acknowledged.