Drift estimation for discretely sampled SPDEs

Abstract

The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature, we study the asymptotic properties of the maximum likelihood (type) estimators (MLE) when both, the number of Fourier modes and the time go to infinity. In the first part of the paper we consider the usual setup of continuous time observations of the Fourier coefficients of the solutions, and show that the MLE is consistent, asymptotically normal and optimal in the mean-square sense. In the second part of the paper we investigate the natural time discretization of the MLE, by assuming that the first N Fourier modes are measured at M time grid points, uniformly spaced over the time interval [0, T]. We provide a rigorous asymptotic analysis of the proposed estimators when $N\rightarrow \infty$ and/or $T,M\rightarrow \infty$. We establish sufficient conditions on the growth rates of N, M and T, that guarantee consistency and asymptotic normality of these estimators.

Appendices

Proofs of technical Lemmas

Proof (Proof of Lemma 1)

We start by computing the Malliavin derivative of $\widehat{F}_{N,T}$. If $r\le t$ and for $1\le k\le N$, then

$$\begin{aligned} D_{r,k}u_k(t)&= \sigma \lambda _k^{-\gamma } D_{r,k} \int _0^t e^{-\theta _0 \lambda _k^{2\beta }(t-s)}d \!w_k(s)\\&= \sigma \lambda _k^{-\gamma } e^{-\theta _0 \lambda _k^{2\beta }(t-r)}. \end{aligned}$$

Moreover, one has that $D_{r,j}u_k(t)=0$ if $j\ne k$ or $r>t$. Therefore, for $r\le T$ and $1\le j\le N$, we have by (45),

$$\begin{aligned} D_{r,j}\widehat{F}_{N,T}&= -\frac{\sigma }{C_{N,T}^2} \lambda _j^{2\beta +\gamma } u_j(r)- \frac{\sigma }{C_{N,T}^2} \sum _{k=1}^N \lambda _k^{2\beta +\gamma } \int _r^{T} D_{r,j}u_k(t)d \!w_k(t) \nonumber \\&= -\frac{\sigma }{C_{N,T}^2} \lambda _j^{2\beta +\gamma } u_j(r)-\frac{\sigma }{C_{N,T}^2} \lambda _j^{2\beta +\gamma } \int _r^{T} D_{r,j}u_j(t)d \!w_j(t) \nonumber \\&= -\frac{\sigma }{C_{N,T}^2} \lambda _j^{2\beta +\gamma } u_j(r)- \frac{\sigma ^2}{C_{N,T}^2} \lambda _j^{2\beta } \int _r^{T} e^{-\theta _0 \lambda _j^{2\beta }(t-r)} d \!w_j(t). \end{aligned}$$

(33)

We continue by setting

$$\begin{aligned} A:=\left\| C_{N,T} D\widehat{F}_{N,T}\right\| _{{\mathcal {H}}}^2 = C_{N,T}^2\Vert D\widehat{F}_{N,T}\Vert _{{\mathcal {H}}}^2, \end{aligned}$$

and in view of (33), we obtain

$$\begin{aligned} A&=C_{N,T}^2\int _0^T \sum _{k=1}^N \left[ \frac{\sigma }{C_{N,T}^2} \lambda _k^{2\beta +\gamma } u_k(r)+ \frac{\sigma ^2}{C_{N,T}^2} \lambda _k^{2\beta } \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta } (t-r)} dw_k(t)\right] ^2 d \!r \nonumber \\&= \int _0^{T}\sum _{k=1}^N \Bigg [\frac{\sigma ^2}{C_{N,T}^2} \lambda _k^{4\beta +2\gamma } u_k^2(r) + 2\frac{\sigma ^3}{C_{N,T}^2} \lambda _k^{4\beta +\gamma } u_k(r) \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta }(t-r)} d \!w_k(t)\\ \nonumber&\qquad \qquad \qquad \qquad + \frac{\sigma ^4}{C_{N,T}^2} \lambda _k^{4\beta } \left( \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta } (t-r)} d \!w_k(t)\right) ^2 \Bigg ]d \!r \nonumber \\&=:A_1+ A_2+A_3. \end{aligned}$$

It is easy to see that for any process $\varPhi =\{\varPhi (s), s\in [0,t]\}$ such that $\sqrt{{\mathbb {V}}\mathrm {ar}(\varPhi (s))}$ is integrable on [0, t], it holds that $\sqrt{ {\mathbb {V}}\mathrm {ar}\left( \int _0^t \varPhi _s d \!s \right) } \le \int _0^t \sqrt{{\mathbb {V}}\mathrm {ar}(\varPhi _s)} d \!s.$ Therefore, we have

$$\begin{aligned} \sqrt{ {\mathbb {V}}\mathrm {ar}\left( \frac{1}{2} \Vert C_{N,T} D\widehat{F}_{N,T}\Vert _{{\mathcal {H}}}^2 \right) }&\le \frac{1}{2}\left( \sqrt{ {\mathbb {V}}\mathrm {ar}(A_1) } + \sqrt{ {\mathbb {V}}\mathrm {ar}(A_2) } + \sqrt{{\mathbb {V}}\mathrm {ar}(A_3) } \right) \\&\le \frac{1}{2}\left( B_1+B_2+B_3\right) , \end{aligned}$$

where

$$\begin{aligned} B_1&:=\frac{\sigma ^2}{C_{N,T}^2}\int _0^{T} \left[ {\mathbb {V}}\mathrm {ar}\left( \sum _{k=1}^N \lambda _k^{4\beta +2\gamma } u_k^2(r)\right) \right] ^{1/2} d \!r\\ B_2&:=2\frac{\sigma ^3}{C_{N,T}^2}\int _0^{T} \left[ {\mathbb {V}}\mathrm {ar}\left( \sum _{k=1}^N \lambda _k^{4\beta +\gamma } u_k(r) \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta }(t-r)} dw_k(t)\right) \right] ^{1/2} d \!r\\ B_3&:= \frac{\sigma ^4}{C_{N,T}^2} \int _0^{T} \left[ {\mathbb {V}}\mathrm {ar}\left( \sum _{k=1}^N \lambda _k^{4\beta } \left( \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta }(t-r)} dw_k(t)\right) ^2\right) \right] ^{1/2} d \!r. \end{aligned}$$

Note that by (10) and (11),

$$\begin{aligned} {\mathbb {V}}\mathrm {ar}\left( u_k^2(r)\right)&={\mathbb {E}}\left( u_k^4(r)\right) -{\mathbb {E}}^2\left( u_k^2(r)\right) =\frac{\sigma ^4 \lambda _k^{-4\beta -4\gamma }}{2\theta _0^2} \left( 1-e^{-2\theta _0 \lambda _k^{2\beta } r}\right) ^2. \end{aligned}$$

(34)

For $B_1$, by the independence of $\{u_k\}_{k\ge 1}$ and (34),

$$\begin{aligned} B_1&=\frac{\sigma ^2}{C_{N,T}^2}\int _0^{T} \left( \sum _{k=1}^N \lambda _k^{8\beta +4\gamma } {\mathbb {V}}\mathrm {ar}\left( u_k^2(r)\right) \right) ^{1/2}d \!r\nonumber \\&=\frac{\sigma ^4}{\sqrt{2}\theta _0C_{N,T}^2}\int _0^{T} \left( \sum _{k=1}^N \lambda _k^{4\beta } \left( 1-e^{-2\theta _0 \lambda _k^{2\beta } r}\right) ^2 \right) ^{1/2} d \!r \nonumber \\&\simeq \frac{\sigma ^4T}{\sqrt{2}\theta _0C_{N,T}^2}\left( \sum _{k=1}^N\lambda _k^{4\beta }\right) ^{1/2}\quad \text{ as }\ T\rightarrow \infty \nonumber \\&\sim \frac{1}{N^{1/2}}\rightarrow 0, \quad \text{ as }\ N,T \rightarrow \infty . \end{aligned}$$

(35)

For $B_2$, we note that $u_k$ and $W_l$ are independent if $k\ne l$. Therefore, we rewrite $B_2$ as

$$\begin{aligned} B_2&=\frac{2\sigma ^3}{C_{N,T}^2}\int _0^{T} \left[ \sum _{k=1}^n \lambda _k^{8\beta +2\gamma } {\mathbb {V}}\mathrm {ar}\left( u_k(r) \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta }(t-r)} d \!W_k(t) \right) \right] ^{1/2} d \!r. \end{aligned}$$

By straightforward calculations, we have that

$$\begin{aligned}&{\mathbb {V}}\mathrm {ar}\left( u_k(r) \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta } (t-r)} d \!w_k(t) \right) \\&\le {\mathbb {E}}\left[ u_k^2(r)\left[ \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta } (t-r)} d \!w_k(t)\right) ^2 \right] \nonumber \\&= {\mathbb {E}}\left[ u^2_k(r){\mathbb {E}}\left[ \left( \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta } (t-r)} d \!w_k(t)\right) ^2\Big | {\mathcal {F}}_r \right] \right] \nonumber \\&= \frac{\sigma ^2 \lambda _k^{-2\beta -2\gamma } }{2\theta _0} \left( 1-e^{-2\theta _0 \lambda _k^{2\beta }r}\right) \int _r^{T} e^{-2\theta _0 \lambda _k^{2\beta }(t-r)} d \!t\nonumber \\&\le \frac{\sigma ^2\lambda _k^{-4\beta -2\gamma } }{4\theta ^2_0}. \end{aligned}$$

Therefore, we get

$$\begin{aligned} B_2\le \frac{\sigma ^4}{\theta _0 C_{N,T}^2}\int _0^{T} \left( \sum _{k=1}^N \lambda _k^{4\beta } \right) ^{1/2} d \!r \sim \frac{1}{N^{1/2}}\rightarrow 0, \quad \text{ as } N,T\rightarrow \infty . \end{aligned}$$

(36)

Let us now consider $B_3$. Since $w_k$ and $w_j$ are independents for $k\ne j$, we have

$$\begin{aligned} B_3&= \frac{\sigma ^4}{C_{N,T}^2} \int _0^{T} \left[ {\mathbb {V}}\mathrm {ar}\left( \sum _{k=1}^N \lambda _k^{4\beta } \left( \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta }(t-r)} d \!w_k(t)\right) ^2\right) \right] ^{1/2} d \!r\nonumber \\&= \frac{\sigma ^4}{C_{N,T}^2} \int _0^{T} \left[ \sum _{k=1}^N \lambda _k^{8\beta } {\mathbb {V}}\mathrm {ar}\left( \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta }(t-r)} d \!w_k(t)\right) ^2 \right] ^{1/2} d \!r\nonumber \\&\le \frac{\sigma ^4}{C_{N,T}^2} \int _0^{T} \left[ \sum _{k=1}^N \lambda _k^{8\beta } {\mathbb {E}}\left( \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta }(t-r)} d \!w_k(t)\right) ^4 \right] ^{1/2} d \!r\nonumber \\&= \frac{\sigma ^4}{C_{N,T}^2} \int _0^{T} \left[ 3 \sum _{k=1}^N \lambda _k^{8\beta } \left[ {\mathbb {E}}\left( \int _r^{T} e^{-\theta _0 \lambda _k^{2\beta }(t-r)} d \!w_k(t)\right) ^2\right] ^2 \right] ^{1/2} d \!r\nonumber \\&\le \frac{\sqrt{3}\sigma ^4}{2\theta _0 C_{N,T}^2} T \left( \sum _{k=1}^N \lambda _k^{4\beta } \right) ^{1/2}\sim \frac{1}{N^{1/2}} \rightarrow 0, \text{ as } N,T \rightarrow \infty . \end{aligned}$$

(37)

Finally, combining (35), (36) and (37), we have that for every $\varepsilon >0$, there exist two independent constants $N_0,T_0>0$ such that for all $N\ge N_0$ and $T\ge T_0$,

$$\begin{aligned} B_1+B_2+B_3 < \varepsilon . \end{aligned}$$

This completes the proof. $\square$

Proof (Proof of Lemma 2)

Using (9), (21) follows by direct evaluations. As far as (22) and (23), since $u_k(t)-u_k(s)$ is a Gaussian random variable, it is enough to prove (22) and (23) for $l=1$. We note that for $t<s$, using (21), we deduce that

$$\begin{aligned}&{\mathbb {E}}|u_k(t)-u_k(s)|^2\\&\quad =\frac{\sigma ^2\lambda _k^{-2\gamma -2\beta }}{2\theta _0} \left[ 2(1-e^{-\theta _0 \lambda _k^{2\beta }(s-t)}) + (e^{-\theta _0 \lambda _k^{2\beta } t}+e^{-\theta _0 \lambda _k^{2\beta }s}) (e^{-\theta _0 \lambda _k^{2\beta } s}-e^{-\theta _0 \lambda _k^{2\beta } t})\right] \\&\quad \le C\sigma ^2\lambda _k^{-2\gamma }|t-s|, \end{aligned}$$

for some $C>0$, and where in the last inequality we used the fact that $e^{-x}$ is Lipschitz continuous on $[0,\infty )$. Hence, (22) is proved. Similarly, we have

$$\begin{aligned} {\mathbb {E}}|u_k(t)+u_k(s)|^2&\le C\sigma ^2\lambda _k^{-2\gamma -2\beta }, \end{aligned}$$

for some $C>0$, which implies (23). The proof is complete. $\square$

Proof (Proof of Lemma 3)

Since $u_k, \ k \ge 1$, are independent, and taking into account that

$$\begin{aligned} \int _0^T u_k(t)d \!w_k(t)=\sum _{i=1}^M \int _{t_{i-1}}^{t_i}u_k(t)d \!w_k(t), \end{aligned}$$

we have that

$$\begin{aligned}&{\mathbb {E}}|Y_{N,M,T}-Y_{N,T}|^2 \\&\quad ={\mathbb {E}}\left| \sum _{k=1}^N \lambda _k^{2\beta +\gamma } \left[ \sum _{i=1}^M u_k(t_{i-1})\left( w_k(t_i)-w_k(t_{i-1})\right) -\int _0^T u_k(t)d \!w_k(t)\right] \right| ^2\\&\quad =\sum _{k=1}^N \lambda _k^{4\beta +2\gamma }{\mathbb {E}} \left| \sum _{i=1}^M u_k(t_{i-1})\left( w_k(t_i)-w_k(t_{i-1})\right) -\int _0^T u_k(t)d \!w_k(t)\right| ^2 \\&\quad =\sum _{k=1}^N \lambda _k^{4\beta +2\gamma } {\mathbb {E}} \left| \sum _{i=1}^M \int _{t_{i-1}}^{t_i} \left( u_k(t_{i-1})-u_k(t)\right) d \!w_k(t) \right| ^2\\&\quad =\sum _{k=1}^N \lambda _k^{4\beta +2\gamma }\sum _{i=1}^M \int _{t_{i-1}}^{t_i} {\mathbb {E}}\left( u_k(t_{i-1})-u_k(t)\right) ^2d \!t \end{aligned}$$

and hence, by (22), there exist constants $C_1,C_2>0$, such that

$$\begin{aligned} {\mathbb {E}}|Y_{N,M,T}-Y_{N,T}|^2&\le C_1\sum _{k=1}^N \lambda _k^{4\beta }\sum _{i=1}^M \int _{t_{i-1}}^{t_i} |t_{i-1}-t|d \!t\\&=C_1\left( \sum _{k=1}^N \lambda _k^{4\beta }\right) \frac{T^2}{2M} \le C_2\frac{T^2 N^{\frac{4\beta }{d}+1}}{M}. \end{aligned}$$

Hence, (24) follows at once.

Next we will prove (25). We note that

$$\begin{aligned} {\mathbb {E}}|I_{N,M,T}-I_{N,T}|^2&= {\mathbb {E}}\left| \sum _{k=1}^N \lambda _k^{4\beta +2\gamma }\left( \sum _{i=1}^M u_k^2(t_{i-1})(t_i-t_{i-1}) -\int _0^T u_k^2(t)d \!t\right) \right| ^2\\&=\sum _{k=1}^N \lambda _k^{8\beta +4\gamma } {\mathbb {E}}\left| \sum _{i=1}^M \int _{t_{i-1}}^{t_i} \left( u_k^2(t_{i-1})-u_k^2(t)\right) d \!t \right| ^2. \end{aligned}$$

Consequently, letting $U_i(t):=u_k^2(t_{i-1})-u_k^2(t), \ k\ge 1$, we continue

$$\begin{aligned} {\mathbb {E}}|I_{N,M,T}-I_{N,T}|^2&=\sum _{k=1}^N \lambda _k^{8\beta +4\gamma } \sum _{i=1}^M {\mathbb {E}}\left| \int _{t_{i-1}}^{t_i} U_i(t)d \!t \right| ^2\\&\quad +\,2\sum _{k=1}^N \lambda _k^{8\beta +4\gamma } \sum _{i<j}{\mathbb {E}} \int _{t_{i-1}}^{t_i}\int _{t_{j-1}}^{t_j} U_i(t)U_j(s) d \!sd \!t =: I_1+I_2. \end{aligned}$$

Note that by Cauchy-Schwartz inequality,

$$\begin{aligned} {\mathbb {E}}|U_i^2(t)|&={\mathbb {E}}|u_k^2(t_{i-1})-u_k^2(t)|^2 ={\mathbb {E}}|u_k(t_{i-1})-u_k(t)|^2|u_k(t_{i-1})+u_k(t)|^2\\&\le \left( {\mathbb {E}}|u_k(t_{i-1})-u_k(t)|^4\right) ^{1/2} \left( {\mathbb {E}}|u_k(t_{i-1})+u_k(t)|^4\right) ^{1/2}. \end{aligned}$$

Moreover, by (22) and (23),

$$\begin{aligned} {\mathbb {E}}|U_i^2(t)|\le c_1\lambda _k^{-4\gamma -2\beta }|t-t_{i-1}|,\quad \text{ for } \text{ some }\ c_1>0. \end{aligned}$$

(38)

Again by Cauchy-Schwartz inequality and (38), we have that

$$\begin{aligned} I_1&=\sum _{k=1}^N \lambda _k^{8\beta +4\gamma } \sum _{i=1}^M {\mathbb {E}}\left| \int _{t_{i-1}}^{t_i} U_i(t)d \!t \right| ^2 \le \sum _{k=1}^N \lambda _k^{8\beta +4\gamma } \sum _{i=1}^M (t_i-t_{i-1}) \int _{t_{i-1}}^{t_i} {\mathbb {E}}|U_i^2(t)|d \!t\\&\le c_1\sum _{k=1}^N \lambda _k^{6\beta } \sum _{i=1}^M (t_i-t_{i-1}) \int _{t_{i-1}}^{t_i} (t-t_{i-1})d \!t =c_1\sum _{k=1}^N \lambda _k^{6\beta } \frac{T^3}{2M^2}. \end{aligned}$$

Turning to $I_2$, we first notice that

$$\begin{aligned}&{\mathbb {E}}|U_i(t)U_j(s)|\\&\quad ={\mathbb {E}}\Big [\left( u_k^2(t_{i-1}){-}u_k^2(t) \right) \left( u_k^2(t_{j-1}){-}u_k^2(s) \right) \Big ]\\&\quad ={\mathbb {E}}\Big [\left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{i-1}){+}u_k(t)\right) \left( u_k(t_{j-1}){-}u_k(s)\right) \left( u_k(t_{j-1}){+}u_k(s)\right) \Big ]\\&\quad ={\mathbb {E}}\Big [\left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{j-1})-u_k(s)\right) u_k(t_{i-1})u_k(t_{j-1})\Big ]\\&\qquad +{\mathbb {E}}\Big [\left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{j-1})-u_k(s)\right) (u_k(t_{i-1})u_k(s)\Big ]\\&\qquad +{\mathbb {E}}\Big [\left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{j-1})-u_k(s)\right) u_k(t)u_k(t_{j-1})\Big ]\\&\qquad +{\mathbb {E}}\Big [\left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{j-1})-u_k(s)\right) u_k(t)u_k(s)\Big ]. \end{aligned}$$

By the Wick’s Lemma [1, Lemma 3.1], we continue

$$\begin{aligned}&{\mathbb {E}}|U_i(t)U_j(s)|\\&\quad ={\mathbb {E}}\left[ \left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{i-1})+u_k(t)\right) \right] {\mathbb {E}}\left[ \left( u_k(t_{j-1})\right. \right. \\&\qquad \left. \left. -\,u_k(s)\right) \left( u_k(t_{j-1})+u_k(s)\right) \right] \\&\qquad +{\mathbb {E}}\left[ \left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{j-1})-u_k(s)\right) \right] {\mathbb {E}}\left[ \left( u_k(t_{i-1})\right. \right. \\&\qquad \left. \left. +\,u_k(t)\right) \left( u_k(t_{j-1})+u_k(s)\right) \right] \\&\qquad +{\mathbb {E}}\left[ \left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{j-1})+u_k(s)\right) \right] {\mathbb {E}}\left[ \left( u_k(t_{i-1})\right. \right. \\&\qquad \left. \left. +\,u_k(t)\right) \left( u_k(t_{j-1})-u_k(s)\right) \right] \\&\quad =:J_1+J_2+J_3. \end{aligned}$$

For $J_2$, we have

$$\begin{aligned} {\mathbb {E}}\left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{j-1})-u_k(s)\right)&= {\mathbb {E}}\left( u_k(t_{i-1})u_k(t_{j-1})\right) -{\mathbb {E}}\left( u_k(t_{i-1})u_k(s)\right) \\&\quad -{\mathbb {E}}\left( u_k(t)u_k(t_{j-1})\right) +{\mathbb {E}}\left( u_k(t)u_k(s)\right) . \end{aligned}$$

By (21), for $i<j$ and $t<s$,

$$\begin{aligned}&{\mathbb {E}}\left( u_k(t_{i-1})-u_k(t)\right) \left( u_k(t_{j-1})-u_k(s)\right) \\&\quad = \frac{\sigma ^2\lambda _k^{-2\gamma -2\beta }}{2\theta _0} \Bigg [e^{-\theta _0 \lambda _k^{2\beta }(t_{j-1}-t_{i-1})}-e^{-\theta _0 \lambda _k^{2\beta }(t_{j-1}+t_{i-1})} -e^{-\theta _0 \lambda _k^{2\beta }(s-t_{i-1})}\\&\qquad +e^{-\theta _0 \lambda _k^{2\beta }(t_{i-1}+s)} -e^{-\theta _0 \lambda _k^{2\beta }(t_{j-1}-t)}+e^{-\theta _0 \lambda _k^{2\beta }(t_{j-1}+t)} +e^{-\theta _0 \lambda _k^{2\beta }(s-t)}-e^{-\theta _0 \lambda _k^{2\beta }(s+t)} \Bigg ] \\&\quad = \frac{\sigma ^2\lambda _k^{-2\gamma -2\beta }}{2\theta _0} \Bigg [\left( e^{-\theta _0 \lambda _k^{2\beta }(t_{j-1}-t_{i-1})} -e^{-\theta _0 \lambda _k^{2\beta }(t_{j-1}-t)}\right) \\&\qquad +\left( e^{-\theta _0 \lambda _k^{2\beta }(t_{j-1}+t)} -e^{-\theta _0 \lambda _k^{2\beta }(t_{j-1}+t_{i-1})}\right) +\left( e^{-\theta _0 \lambda _k^{2\beta }(s-t)} -e^{-\theta _0 \lambda _k^{2\beta }(s-t_{i-1})}\right) \\&\qquad + \left( e^{-\theta _0 \lambda _k^{2\beta }(t_{i-1}+s)} -e^{-\theta _0 \lambda _k^{2\beta }(s+t)}\right) \Bigg ] \le c_2 \lambda _k^{-2\gamma }(t-t_{i-1}), \end{aligned}$$

for some $c_2>0$. By similar arguments, we also obtain

$$\begin{aligned} {\mathbb {E}}\left( u_k(t_{i-1})+u_k(t)\right) \left( u_k(t_{j-1})+u_k(s)\right) \le c_3\lambda _k^{-4\gamma }(s-t_{j-1}), \end{aligned}$$

for some $c_3>0$. Thus,

$$\begin{aligned} J_2 \le c_4\lambda _k^{-4\gamma }(t-t_{i-1})(s-t_{j-1}), \end{aligned}$$

for some $c_4>0$. By analogy, one can treat $J_1$ and $J_3$, and derive the following upper bounds:

$$\begin{aligned} J_1\le c_5\lambda _k^{-4\gamma }(t-t_{i-1})(s-t_{j-1}), \qquad J_3 \le c_6\lambda _k^{-4\gamma }(t-t_{i-1})(s-t_{j-1}), \end{aligned}$$

for some $c_5,c_6>0$. Finally, combining the above, we have

$$\begin{aligned} I_2\le & {} c_7\sum _{k=1}^N \lambda _k^{8\beta } \sum _{i<j} \int _{t_{j-1}}^{t_j}\int _{t_{i-1}}^{t_i}(t-t_{i-1})(s-t_{j-1})d \!t d \!s \\\le & {} c_8\sum _{k=1}^N \lambda _k^{8\beta } \frac{T^4}{M^2}, \quad \text{ for } \text{ some }\ c_8,c_9>0. \end{aligned}$$

Thus, using the estimates for $I_1,I_2$, and the fact that $\lambda _k\sim k^{1/d}$, we conclude that

$$\begin{aligned} I_1+I_2 \le c_9\sum _{k=1}^N \lambda _k^{8\beta } \frac{T^4}{M^2} \le C_2 \frac{T^4 N^{\frac{8\beta }{d}+1}}{M^2}, \end{aligned}$$

and hence (25) is proved. The estimate (26) is proved by similar arguments, and we omit the details here. This completes the proof. $\square$

1.1 Auxiliary results

For reader’s convenience, we present here some simple, or well-known, results from probability. Let X, Y, Z be random variables, and assume that $Z>0$ a.s.. For any $\varepsilon >0$ and $\delta \in (0,\varepsilon /2)$, the following inequalities hold true.

$$\begin{aligned}&{\mathbb {P}}(|Y/Z|>\varepsilon ) \le {\mathbb {P}}(|Y|>\delta ) + {\mathbb {P}}(|Z-1|> (\varepsilon -\delta )/\varepsilon ), \end{aligned}$$

(39)

$$\begin{aligned}&\sup _{x\in {\mathbb {R}}} \Big | {\mathbb {P}}(X+Y\le x) - \varPhi (x)\Big | \le \sup _{x\in {\mathbb {R}}} \Big | {\mathbb {P}}(X\le x) - \varPhi (x)\Big | + {\mathbb {P}}(|Y|>\varepsilon ) + \varepsilon , \end{aligned}$$

(40)

$$\begin{aligned}&\sup _{x\in {\mathbb {R}}} \Big | {\mathbb {P}}(Y/Z\le x) - \varPhi (x)\Big | \le \sup _{x\in {\mathbb {R}}} \Big | {\mathbb {P}}(Y\le x) - \varPhi (x)\Big | + {\mathbb {P}}(|Z-1|>\varepsilon ) + \varepsilon , \end{aligned}$$

(41)

where $\varPhi$ denotes the probability function of a standard Gaussian random variable.

Elements of Malliavin calculus

In this section, we recall some facts from Malliavin calculus associated with a Gaussian process, that we use in Sect. 3.1. For more details, we refer to [18]. Toward this end, let $T>0$ be given. We consider the space ${\mathcal {H}}=L^2\left( [0,T]\times {\mathcal {M}}\right)$, where ${\mathcal {M}}$ is the counting measure on ${\mathbb {N}}$, namely, for $v\in {\mathcal {H}}$,

$$\begin{aligned} v(t)=\sum _{k=1}^\infty v_k(t). \end{aligned}$$

We endow ${\mathcal {H}}$ with the inner product and the norm

$$\begin{aligned} \langle u,v\rangle _{{\mathcal {H}}}:= \sum _{k=1}^\infty \int _0^T u_k(t) v_k(t) d \!t,\quad \text{ and }\quad \Vert v\Vert _{{\mathcal {H}}}:= \sqrt{\langle v,v\rangle _{{\mathcal {H}}}}, \quad \ u,v\in {\mathcal {H}}. \end{aligned}$$

(42)

We fix an isonormal Gaussian process $W=\{W(h)\}_{h\in {\mathcal {H}}}$ on ${\mathcal {H}}$, defined on a suitable probability space $(\varOmega , {\mathscr {F}},{\mathbb {P}})$, such that ${\mathscr {F}}=\sigma (W)$ is the $\sigma$-algebra generated by W. Denote by $C_p^{\infty }({\mathbb {R}}^n)$, the space of all smooth functions on ${\mathbb {R}}^n$ with at most polynomial growth partial derivatives. Let ${\mathcal {S}}$ be the space of simple functionals of the form

$$\begin{aligned} F = f(W(h_1), \dots , W(h_n)),\quad f\in C_p^{\infty }({\mathbb {R}}^n), \ h_i \in {\mathcal {H}},\ 1\le i \le n. \end{aligned}$$

As usual, we define the Malliavin derivative D on ${\mathcal {S}}$ by

$$\begin{aligned} DF=\sum _{i=1}^n \frac{\partial f}{\partial x_i} (W(h_1), \dots , W(h_n)) h_i, \quad F\in {\mathcal {S}}. \end{aligned}$$

(43)

We note that the derivative operator D is a closable operator from $L^p(\varOmega )$ into $L^p(\varOmega ; {\mathcal {H}})$, for any $p \ge 1$. Let ${\mathbb {D}}^{1,p}$, $p \ge 1$, be the completion of ${\mathcal {S}}$ with respect to the norm

$$\begin{aligned} \Vert F\Vert _{1,p} = \left( {\mathbb {E}}\big [ |F|^p \big ] + {\mathbb {E}}\big [ \Vert D F\Vert ^p_{\mathcal {H}}\big ] \right) ^{1/p}. \end{aligned}$$

Also, for F of the form

$$\begin{aligned} F=f\left( W\left( \mathbb {1}_{[0,t_1]}\right) ,\dots ,W\left( \mathbb {1}_{[0,t_n]}\right) \right) , \quad t_1,\dots ,t_n\in [0,T], \end{aligned}$$

we define the Malliavin derivative of F at the point t as

$$\begin{aligned} D_t F=\sum _{i=1}^n \frac{\partial f}{\partial x_i}\left( W\left( \mathbb {1}_{[0,t_1]}\right) ,\dots ,W\left( \mathbb {1}_{[0,t_n]}\right) \right) \mathbb {1}_{[0,t_i]}(t),\quad t\in [0,T], \end{aligned}$$

where $\mathbb {1}_{A}$ denotes the indicator function of set A. For simplicity, from now on, we define $W(t):=W\left( \mathbb {1}_{[0,t]}\right)$, $t\in [0,T]$, to represent a standard Brownian motion on [0, T]. If ${\mathscr {F}}$ is generated by a collection of independent standard Brownian motions $\{W_k,\ k\ge 1\}$ on [0, T], we define the Malliavin derivative of F at the point t by

$$\begin{aligned} D_tF:=\sum _{k=1}^{\infty }D_{t,k}F:=\sum _{k=1}^{\infty }\sum _{i=1}^n \frac{\partial f}{\partial x_i}\left( W_k(t_1),\dots ,W_k(t_n)\right) \mathbb {1}_{[0,t_i]}(t),\quad t\in [0,T].\nonumber \\ \end{aligned}$$

(44)

Next, we denote by $\varvec{\delta }$, the adjoint of the Malliavin derivative D (as defined in (43)) given by the duality formula

$$\begin{aligned} {\mathbb {E}}\left( \varvec{\delta }(v) F\right) = {\mathbb {E}}\left( \langle v, DF \rangle _{\mathcal {H}}\right) , \end{aligned}$$

for $F \in {\mathbb {D}}^{1,2}$ and $v\in {\mathcal {D}}(\varvec{\delta })$, where ${\mathcal {D}}(\varvec{\delta })$ is the domain of $\varvec{\delta }$. If $v\in L^2(\varOmega ;{\mathcal {H}})\cap {\mathcal {D}}(\varvec{\delta })$ is a square integrable process, then the adjoint $\varvec{\delta }(v)$ is called the Skorokhod integral of the process v (cf. [18]), and it can be written as

$$\begin{aligned} \varvec{\delta }(v)=\int _0^T v(t) d \!W(t). \end{aligned}$$

Proposition 1

 [18, Theorem 1.3.8] Suppose that $v \in L^2(\varOmega ;{\mathcal {H}})$ is a square integrable process such that $v(t)\in {\mathbb {D}}^{1,2}$ for almost all $t\in [0,T]$. Assume that the two parameter process $\{D_tv(s)\}$ is square integrable in $L^2\left( [0,T]\times \varOmega ;{\mathcal {H}}\right)$. Then, $\varvec{\delta }(v) \in {\mathbb {D}}^{1,2}$ and

$$\begin{aligned} D_t \left( \varvec{\delta }(v)\right) = v(t) + \int _0^{T} D_tv(s) d \!W(s),\quad t\in [0,T]. \end{aligned}$$

(45)

Next, we present a connection between Malliavin calculus and Stein’s method. For symmetric functions $f\in L^2\big ([0,T]^q\big )$, $q\ge 1$, let us define the following multiple integral of order q

$$\begin{aligned} {\mathbb {I}}_q(f)=q!\int _0^T d \!W(t_1)\int _0^{t_1} d \!W(t_2)\cdots \int _0^{t_{q-1}} d \!W(t_q) f(t_1,\ldots ,t_q), \end{aligned}$$

with $0<t_1<t_2<\cdots<t_q<T$. Note that ${\mathbb {I}}_q(f)$ is also called the q-th Wiener chaos [17, Theorem 2.7.7]. Denote by $d_{TV}(F,G)$, the total variation of two random variables F and G.

Theorem 4

 [17, Corollary 5.2.8] Let $F_N={\mathbb {I}}_q(f_N)$, $N\ge 1$, be a sequence of random variables for some fixed integer $q\ge 2$. Assume that ${\mathbb {E}}\left( F_N^2\right) \rightarrow \sigma ^2>0$, as $N\rightarrow \infty$. Then, as $N \rightarrow \infty$, the following assertions are equivalent:

• 1. $F_N\overset{d}{\longrightarrow }{\mathcal {N}}:={\mathcal {N}}(0,\sigma ^2)$;

• 2. $d_{TV}\left( F_N,{\mathcal {N}}\right) \longrightarrow 0$.

We conclude this section with a result about an upper bound for the total variation of a q-th multiple integral and a Gaussian random variable.

Proposition 2

[17, Theorem 5.2.6] Let $q\ge 2$ be an integer, and let $F={\mathbb {I}}_q(f)$ be a multiple integral of order q such that ${\mathbb {E}}(F^2)=\sigma ^2>0$. Then, for ${\mathcal {N}}={\mathcal {N}}(0,\sigma ^2)$,

$$\begin{aligned} d_{TV}(F,{\mathcal {N}})\le \frac{2}{\sigma ^2}\sqrt{{\mathbb {V}}\mathrm {ar}\left( \frac{1}{q}\Vert DF \Vert _{{\mathcal {H}}}^2 \right) }. \end{aligned}$$