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Maximum likelihood estimation for small noise multiscale diffusions

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Abstract

We study the problem of parameter estimation for stochastic differential equations with small noise and fast oscillating parameters. Depending on how fast the intensity of the noise goes to zero relative to the homogenization parameter, we consider three different regimes. For each regime, we construct the maximum likelihood estimator and we study its consistency and asymptotic normality properties. A simulation study for the first order Langevin equation with a two scale potential is also provided.

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Acknowledgments

The authors would like to thank the anonymous reviewer for pointing out a gap in the proof of Theorem 5.1 in the original article, as well as all comments that lead to a significant improvement of the article. K.S. was partially supported, during revisions of this article, by the National Science Foundation (DMS 1312124).

Author information

Correspondence to Konstantinos Spiliopoulos.

Appendix

Appendix

Proof of Lemma 5.3

Let us denote \(\theta _{\epsilon ,u}=\theta _{\epsilon }+\phi (\epsilon ,\theta _{\epsilon }) u_{\epsilon }\), where \(\phi (\epsilon ,\theta )=\sqrt{\epsilon }I^{-1/2}(\theta )\). We assume that \(\theta _{\epsilon }\) belongs in a compact subset of \(\Theta \), denoted by \(\tilde{\Theta }\), and let \(u_{\epsilon }\rightarrow u\) as \(\epsilon \downarrow 0\). We start by rewriting the normed likelihood ratio as follows

$$\begin{aligned} M_{\epsilon }(\theta _{\epsilon },u)&= \frac{1}{\sqrt{\epsilon }}\int \limits _{0}^{T} \left\langle c_{\theta _{\epsilon ,u}}-c_{\theta _{\epsilon }}, \sigma dW_{s}\right\rangle {}_{\alpha }\left( x_{s},\frac{x_{s}}{\delta }\right) - \frac{1}{2\epsilon }\int \limits _{0}^{T}\left\| c_{\theta _{\epsilon ,u}}-c_{\theta _{\epsilon }}\right\| ^{2}_{\alpha }\left( x_{s},\frac{x_{s}}{\delta }\right) ds\nonumber \\&= \frac{1}{\sqrt{\epsilon }}\int \limits _{0}^{T} \left\langle c_{\theta _{\epsilon ,u}}-c_{\theta _{\epsilon }}-\left( \sqrt{\epsilon }I^{-1/2}(\theta _{\epsilon })u_{\epsilon },\nabla _{\theta }c_{\theta _{\epsilon ,u}}\right) , \sigma dW_{s}\right\rangle {}_{\alpha }\left( x_{s},\frac{x_{s}}{\delta }\right) \nonumber \\&\quad +\left( I^{-1/2}(\theta _{\epsilon ,u})u_{\epsilon },\int \limits _{0}^{T} \left\langle \nabla _{\theta }c_{\theta _{\epsilon ,u}}, \sigma dW_{s}\right\rangle {}_{\alpha }\left( x_{s},\frac{x_{s}}{\delta }\right) \right) \nonumber \\&\quad \!-\,\frac{1}{2}\!\int \limits _{0}^{T}\left[ \frac{1}{\epsilon }\left\| c_{\theta _{\epsilon ,u}}\!-\!c_{\theta _{\epsilon }}\right\| ^{2}_{\alpha }\left( \!x_{s},\frac{\!x_{s}}{\delta }\right) \!-\! \left( \!I^{-1/2}(\!\theta _{\epsilon })u_{\epsilon },q^{1/2}(\bar{X}_{s},\!\theta _{\!\epsilon })\right) ^{2}\right] \!ds \!-\!\frac{1}{2}\left( u_{\epsilon },u_{\epsilon }\right) \nonumber \\&= J^{\epsilon }_{1}(\theta _{\epsilon })+J^{\epsilon }_{2}(\theta _{\epsilon })+J^{\epsilon }_{3}(\theta _{\epsilon })+J^{\epsilon }_{4}.\nonumber \end{aligned}$$

The last line of the previous computation is easily seen to hold by the following chain of identities

$$\begin{aligned} \int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left( v,S(\theta _{\epsilon },\bar{X}_{s},y)\right) ^{2}\mu _{\theta _{\epsilon }}(dy;\bar{X}_{s})ds=\int \limits _{0}^{T}\left( v,q^{1/2}(\bar{X}_{s},\theta _{\epsilon })\right) ^{2}ds=(I(\theta _{\epsilon })v,v). \end{aligned}$$

which are applied for \(v=I^{-1/2}(\theta _{\epsilon })u_{\epsilon }\).

The goal is to prove that \(M_{\epsilon }(\theta _{\epsilon },u)=\left( u,\Phi \right) -\frac{1}{2}\left\| u\right\| ^{2}+R(\epsilon ,\theta _{\epsilon })\), where \(\Phi \) is distributed as normal \(N(0,I)\) and \(R(\epsilon ,\theta )\rightarrow 0\) as \(\epsilon \downarrow 0\) in \(\mathbb {P}_{\theta }\) probability uniformly in \(\theta \in \Theta \). This, will establish that the family \(\{{\mathbb {P}}^{\epsilon }_{\theta }: \theta \in \Theta \}\) is uniformly asymptotically normal with normalizing matrix \(\phi (\epsilon ,\theta )=\sqrt{\epsilon }I^{-1/2}(\theta )\), which then proves the lemma.

It is clear that

$$\begin{aligned} J^{\epsilon }_{4}=-\frac{1}{2}\left( u_{\epsilon },u_{\epsilon }\right) \rightarrow -\frac{1}{2}\left\| u\right\| ^{2},\quad \text {as} \quad \epsilon \downarrow 0. \end{aligned}$$

Moreover, due to averaging and the law of large numbers result Theorem 2.6, the definition of the Fisher information matrix \(I(\theta )\) implies that

$$\begin{aligned} \begin{aligned} J^{\epsilon }_{2}(\theta _{\epsilon })&=\left( I^{-1/2}(\theta _{\epsilon ,u})u_{\epsilon },\int \limits _{0}^{T} \left\langle \nabla _{\theta }c_{\theta _{\epsilon ,u}}, \sigma dW_{s}\right\rangle {}_{\alpha }\left( x_{s},\frac{x_{s}}{\delta }\right) \right) \nonumber \\&= \left( I^{-1/2}(\theta _{\epsilon ,u})u_{\epsilon },\int \limits _{0}^{T} \left\langle S\left( \theta _{\epsilon ,u},x_{s},\frac{x_{s}}{\delta }\right) , dW_{s}\right\rangle \right) \nonumber \end{aligned} \end{aligned}$$

converges in distribution with respect to \(\mathbb {P}_{\theta }\), uniformly in \(\theta \in \tilde{\Theta }\), to \(\left( u,\Phi \right) \) where \(\Phi \) is distributed as \(N(0,I)\), as \(\epsilon \downarrow 0\).

Thus it remains to consider the term \(R(\epsilon ,\theta )=J^{\epsilon }_{1}(\theta )+J^{\epsilon }_{3}(\theta )\). We shall show that both terms converge to zero in \(\mathbb {P}_{\theta }\) probability as \(\epsilon \downarrow 0\), uniformly in \(\theta \in \tilde{\Theta }\).

We start by observing that

$$\begin{aligned} c_{\theta +\ell }-c_{\theta }=\int \limits _{0}^{1}\left( \ell ,\nabla _{\theta }c_{\theta +\ell h}\right) dh. \end{aligned}$$

Then we can write

$$\begin{aligned}&\mathbb {E} \int \limits _{0}^{T}\left\| \sigma ^{-1}\left( \frac{1}{\sqrt{\epsilon }}( c_{\theta _{\epsilon ,u}}-c_{\theta _{\epsilon }})-\left( I^{-1/2}(\theta _{\epsilon ,u})u_{\epsilon },\nabla _{\theta }c_{\theta _{\epsilon ,u}}\right) \right) \left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) \right\| ^{2}ds \nonumber \\&\quad = \mathbb {E}\int \limits _{0}^{T}\left\| \left[ \sigma ^{-1}\int \limits _{0}^{1}\!\left( I^{-1/2}(\theta _{\epsilon ,u})u_{\epsilon },\nabla _{\theta }c_{\theta _{\epsilon ,u}\!+\!h\sqrt{\epsilon }I^{-1/2}(\theta _{\epsilon ,u})u_{\epsilon }}\!-\!\nabla _{\theta }c_{\theta _{\epsilon ,u}}\right) dh\right] \left( \!X^{\epsilon }_{s},\frac{\!X^{\epsilon }_{s}}{\delta }\right) \right\| ^{2}\!ds\nonumber \\&\quad \le |I^{-1/2}(\theta _{\epsilon ,u})u_{\epsilon }|^{2} \sup _{\theta \in \tilde{\Theta }}\sup _{|v|\le C\sqrt{\epsilon }} \mathbb {E}\left| \int \limits _{0}^{T}\left\| \nabla _{\theta }c_{\theta +v}-\nabla _{\theta }c_{\theta }\right\| ^{2}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds\right| \nonumber \\&\quad \le C \sup _{\theta \in \tilde{\Theta }}\sup _{|v|\le C\sqrt{\epsilon }} \mathbb {E}\left| \int \limits _{0}^{T}\left\| \nabla _{\theta }c_{\theta +v}-\nabla _{\theta }c_{\theta }\right\| ^{2}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds\right| \rightarrow 0, \mathrm as \epsilon \downarrow 0. \end{aligned}$$
(8.1)

The last convergence is true due to the uniform continuity of \(\nabla _{\theta }c_{\theta }\) in \(\theta \in \tilde{\Theta }\) and tightness of \(\left\{ X^{\epsilon },\epsilon >0\right\} \). Using Itô isometry, the last display implies that

$$\begin{aligned} \sup _{\theta \in \tilde{\Theta }}\mathbb {E}\left| J^{\epsilon }_{1}(\theta )\right| ^{2}\rightarrow 0, \quad \text {as}\quad \epsilon \downarrow 0. \end{aligned}$$
(8.2)

Lastly, it remains to consider the term \(J^{3}(\theta )\). Notice that standard averaging principle, the convergence of \(X^{\epsilon }\) to \(\bar{X}\) as \(\epsilon \downarrow 0\) by Theorem 2.6, and the continuous dependence of the involved functions on \(\theta \), imply that,

$$\begin{aligned}&\mathbb {E}\left| \int \limits _{0}^{T}\left[ \left\| \left( I^{-1/2}(\theta _{\epsilon ,u})u_{\epsilon },\nabla _{\theta }c_{\theta _{\epsilon ,u}}\right) \right\| ^{2}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) - \left( I^{-1/2}(\theta _{\epsilon ,u})u_{\epsilon },q^{1/2}(\bar{X}_{s},\theta _{\epsilon ,u})\right) ^{2}\right] ds\right| \nonumber \\&\quad \rightarrow 0, \mathrm as \epsilon \downarrow 0. \end{aligned}$$
(8.3)

By (8.1)–(8.3) and the assumptions on the dependence on \(\theta \) we obtain that

$$\begin{aligned} \sup _{\theta \in \tilde{\Theta }}\mathbb {E}\left| J^{\epsilon }_{3}(\theta )\right| ^{2}\rightarrow 0, \mathrm as \epsilon \downarrow 0. \end{aligned}$$

Therefore, we have obtained that

$$\begin{aligned} \sup _{\theta \in \tilde{\Theta }}\mathbb {E}\left| R(\epsilon ,\theta )\right| ^{2}\rightarrow 0, \quad \text {as}\quad \epsilon \downarrow 0. \end{aligned}$$

This establishes that the family \(\{{\mathbb {P}}^{\epsilon }_{\theta }: \theta \in \Theta \}\) is uniformly asymptotically normal with normalizing matrix \(\phi (\epsilon ,\theta )=\sqrt{\epsilon }I^{-1/2}(\theta )\), which concludes the proof of the lemma. \(\square \)

Proof of Lemma 5.4

The proof follows along the lines of Lemma 2.3 in Kutoyants (1994). We review it here for completeness and mention the required modifications in order to account for the extra component of averaging. Let \(\theta _{i}=\theta +\phi (\epsilon ,\theta ) u_{i}\) and define the interpolating point

$$\begin{aligned} \theta (\ell )=\theta _{1}+(\theta _{2}-\theta _{1})\ell , \quad \ell \in [0,1] \end{aligned}$$

By an absolutely continuous change of measure we have

$$\begin{aligned} \mathbb {E}_{\theta }\left| e^{\frac{1}{2m}M_{\epsilon }(\theta ,u_{2})}-e^{\frac{1}{2m}M_{\epsilon }(\theta ,u_{1})} \right| ^{2m}=\mathbb {E}_{\theta _{1}}\left| L^{\frac{1}{2m}}(\theta _{2},\theta _{1};x)-1 \right| ^{2m} \end{aligned}$$

where, we have defined \( L(\theta _{2},\theta _{1};x)=\frac{d\mathbb {P}_{\theta _{2}}}{d\mathbb {P}_{\theta _{1}}}(x)\). Then, we write

$$\begin{aligned} \log L(\theta _{2},\theta _{1};x)&= \frac{1}{\sqrt{\epsilon }}\!\int \limits _{0}^{T} \left\langle c_{\theta _{2}}\!-\!c_{\theta _{1}}, \sigma dW_{s}\right\rangle {}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) \!-\! \frac{1}{2\epsilon }\!\int \limits _{0}^{T}\left\| c_{\theta _{2}}\!-\!c_{\theta _{1}}\right\| ^{2}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds\nonumber \\&= \frac{1}{\sqrt{\epsilon }}\int \limits _{0}^{1}\int \limits _{0}^{T} \left\langle \nabla _{\theta }c_{\theta (\ell )}(\theta _{2}-\theta _{1}), \sigma dW_{s}\right\rangle {}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) d\ell \nonumber \\&- \frac{1}{2\epsilon }\int \limits _{0}^{1}\int \limits _{0}^{T}\left\langle c_{\theta _{2}}-c_{\theta _{1}}, \nabla _{\theta } c_{\theta (\ell )}(\theta _{2}-\theta _{1})\right\rangle {}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) dsd\ell \nonumber \\&= 2m\int \limits _{0}^{1}\left\langle f(\theta (\ell )),\theta _{2}-\theta _{1}\right\rangle d\ell ,\nonumber \end{aligned}$$

where

$$\begin{aligned} f(\theta (\ell ))\!=\!\frac{1}{\sqrt{\epsilon }}\!\int \limits _{0}^{T} \left\langle \nabla _{\theta }c_{\theta (\ell )}, \sigma dW_{s}\right\rangle {}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) \!-\! \frac{1}{2\epsilon }\int \limits _{0}^{T}\left\langle c_{\theta _{2}}-c_{\theta _{1}}, \nabla _{\theta } c_{\theta (\ell )}\right\rangle {}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds \end{aligned}$$

Thus, we obtain

$$\begin{aligned}&\mathbb {E}_{\theta _{1}}\left| L^{\frac{1}{2m}}(\theta _{2},\theta _{1};x)-1 \right| ^{2m}\le \mathbb {E}_{\theta _{1}}\left| \int \limits _{0}^{1}\left\langle f(\theta (\ell )),\theta _{2}-\theta _{1}\right\rangle e^{\int \limits _{0}^{\ell } \left\langle f(\theta (\kappa )),\theta _{2}-\theta _{1}\right\rangle d\kappa }d\ell \right| ^{2m}\nonumber \\&\quad \le \int \limits _{0}^{1}\mathbb {E}_{\theta _{1}}\left[ L(\theta _{2},\theta _{1};x)\left\langle f(\theta (\ell )),\theta _{2}-\theta _{1}\right\rangle ^{2m} \right] d\ell \nonumber \\&\quad = \epsilon ^{-m} (2m)^{-2m} \int \limits _{0}^{1}\mathbb {E}_{\theta _{2}}\left| \int \limits _{0}^{T}\left\langle \nabla _{\theta }c_{\theta (\ell )}(\theta _{2}-\theta _{1}),\sigma dW_{s}\right\rangle {}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) \right| ^{2m}d\ell \nonumber \\&\quad \le \epsilon ^{-m} C_{m,T} \int \limits _{0}^{1}\mathbb {E}_{\theta _{2}}\left[ \int \limits _{0}^{T}\left\langle \nabla _{\theta }c_{\theta (\ell )}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ,(\theta _{2}-\theta _{1})\right\rangle {}_{\alpha }^{2m}ds\right] d\ell \nonumber \\&\quad \le \epsilon ^{-m} C_{m,T} |\theta _{2}-\theta _{1}|^{2m}\sup _{\theta _{2},\theta \in \tilde{\Theta }}\mathbb {E}_{\theta _{2}}\left[ \int \limits _{0}^{T}\left\| \nabla _{\theta }c_{\theta }\right\| _{\alpha }^{2m}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds\right] \nonumber \\&\quad \le \epsilon ^{-m} \phi ^{2m}(\epsilon ,\theta ) C_{m,T} |u_{2}-u_{1}|^{2m}\sup _{\theta _{2},\theta \in \tilde{\Theta }}\mathbb {E}_{\theta _{2}}\left[ \int \limits _{0}^{T}\left\| \nabla _{\theta }c_{\theta }\right\| _{\alpha }^{2m}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds\right] \nonumber \\&\quad \le I^{-m}(\theta ) C_{m,T} |u_{2}-u_{1}|^{2m}\sup _{\theta _{2},\theta \in \tilde{\Theta }}\mathbb {E}_{\theta _{2}}\left[ \int \limits _{0}^{T}\left\| \nabla _{\theta }c_{\theta }\right\| _{\alpha }^{2m}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds\right] \nonumber \end{aligned}$$

and the result follows by the assumed uniform boundedness of \(\sigma ^{-1}\nabla _{\theta }c_{\theta }(x,y)\). \(\square \)

Proof of Lemma 5.5 In the absence of multiple scales, this is Lemma 2.4 in Kutoyants (1994). Here we provide the proof of the result with the additional component of multiple scales, which makes the analysis more involved. For the sake of concreteness we only present the proof for the case of Regime \(1\). The required changes for the other regimes are minimal and are mentioned below at the appropriate place.

Recall that \(\phi (\epsilon ,\theta )=\sqrt{\epsilon }I^{-1/2}(\theta )\) and set

$$\begin{aligned} \Delta c_{\theta }(x,y)=\frac{1}{\sqrt{\epsilon }}\left[ c_{\theta +\phi (\epsilon ,\theta )u}(x,y)-c_{\theta }(x,y)\right] \end{aligned}$$

We can then write

$$\begin{aligned} \mathbb {E}_{\theta }e^{pM_{\epsilon }(\theta ,u)}&= \mathbb {E}_{\theta }\left[ e^{p\int _{0}^{T}\left\langle \Delta c_{\theta },\sigma dW_{s}\right\rangle {}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) -\frac{p}{2}\int _{0}^{T}\left\| \Delta c_{\theta }\right\| _{\alpha }^{2}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds}\right] \nonumber \\&\le \left( \mathbb {E}_{\theta }e^{-p_1\cdot \frac{p-q}{2}\int _{0}^{T}\left\| \Delta c_{\theta }\right\| _{\alpha }^{2}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds}\right) ^{1/p_{1}} \nonumber \\&\quad \times \left( \mathbb {E}_{\theta }\left[ e^{pp_{2}\int _{0}^{T}\left\langle \Delta c_{\theta },\sigma dW_{s}\right\rangle {}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) -\frac{qp_{2}}{2}\int _{0}^{T}\left\| \Delta c_{\theta }\right\| _{\alpha }^{2}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds}\right] \right) ^{1/p_{2}} \end{aligned}$$

Choosing now \(p_{2}=q/p^{2}>1\), we have that

$$\begin{aligned} \mathbb {E}_{\theta }\left[ e^{pp_{2}\int _{0}^{T}\left\langle \Delta c_{\theta },\sigma dW_{s}\right\rangle {}_{\alpha }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) -\frac{qp_{2}}{2}\int _{0}^{T}\left\| \Delta c_{\theta }\right\| _{\alpha }^{2}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds}\right] \le 1 \end{aligned}$$

Setting \(\gamma =q\frac{p-q}{2(q-p^{2})}>0\), this implies

$$\begin{aligned} \begin{aligned} \mathbb {E}_{\theta }e^{pM_{\epsilon }(\theta ,u)}&\le \left( \mathbb {E}_{\theta }e^{-\gamma \int _{0}^{T}\left\| \Delta c_{\theta }\right\| _{\alpha }^{2}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds}\right) ^{(q-p^{2})/p} \end{aligned} \end{aligned}$$
(8.4)

So, the next step is to appropriately bound from above the term \(\mathbb {E}_{\theta }e^{-\gamma \int _{0}^{T}\left\| \Delta c_{\theta }\right\| _{\alpha }^{2}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds}\).

At this point, we recall the definition of \(q(x,\theta )\) from (5.1) and we write

$$\begin{aligned} q_{\epsilon ,v}(x,\theta )&= \int \limits _{\mathcal {Y}}\left( \int \limits _{0}^{1}S(\theta +\sqrt{\epsilon }v h,x,y)dh\right) \left( \int \limits _{0}^{1}S(\theta +\sqrt{\epsilon }v h,x,y)dh\right) ^{T}\mu _{\theta }(dy;x).\nonumber \end{aligned}$$

Define the operator

$$\begin{aligned} \mathcal {L}_{x} =\frac{1}{2}\sigma (x,y)\sigma ^{T}(x,y):\nabla _{y}\nabla _{y} \end{aligned}$$

and for \(v=I^{-1/2}(\theta )u\), let \(\Phi =\Phi _{\theta ,\epsilon ,v}(x,y)\) satisfy the auxiliary PDE

$$\begin{aligned} \mathcal {L}_{x}\Phi (x,y)=\left\| \Delta c_{\theta }(x,y)\right\| ^{2}_{\alpha }-\left( v,q_{\epsilon ,v}^{1/2}(x,\theta )\right) ^{2}, \quad \int \limits _{\mathcal {Y}}\Phi (x,y)\mu _{\theta }(dy;x)=0. \end{aligned}$$
(8.5)

Comparing with the case without the multiple scales, the additional difficulty here is the presence of the fast oscillating component, \(X^{\epsilon }/\delta \). The consideration of the solution to this auxiliary PDE, allows us to reduce the bound for the quantity at hand to a bound for a quantity that depends only on the slow component, \(X^{\epsilon }\).

Notice that \(\mathcal {L}_{x}\) is the operator for Regime \(1\) defined in Definition 2.4 with \(b=0\). For Regimes 2 and 3, one would need to consider the solution to the PDE governed by the corresponding operators from Definition 2.4. Since,

$$\begin{aligned} \begin{aligned} \int \limits _{\mathcal {Y}}\left\| \Delta c_{\theta }(x,y)\right\| ^{2}_{\alpha }\mu _{\theta }(dy;x)&= \int \limits _{\mathcal {Y}}\left\| \int \limits _{0}^{1}\left( v, \nabla _{\theta }c_{\theta +\sqrt{\epsilon }vh}(x,y)\right) dh\right\| ^{2}_{\alpha }\mu _{\theta }(dy;x)\nonumber \\&=\int \limits _{\mathcal {Y}} \left| \left( v, \int \limits _{0}^{1} \sigma ^{-1}\nabla _{\theta }c_{\theta +\sqrt{\epsilon }vh}(x,y)dh\right) \right| ^{2} \mu _{\theta }(dy;x)\nonumber \\&=\left( v, q^{1/2}_{\epsilon ,v}(x,\theta )\right) ^{2}\nonumber \end{aligned} \end{aligned}$$

Fredholm alternative, Theorem 3.3.4 of Bensoussan et al. (1978) guarantees that there exists a unique, smooth, periodic in \(y\) and bounded solution to the aforementioned auxiliary PDE for \(\Phi \). The boundedness of \(\Theta \) and the imposed conditions on \(\nabla _{\theta }c_{\theta }\) also guarantee that \(\Phi \) is bounded uniformly in \(\theta ,(\theta +\sqrt{\epsilon }v)\in \Theta \). Let us apply Itô formula to \(\Phi (x,x/\delta )\) with \(x=X^{\epsilon }_{s}\). Itô formula gives an expression similar to (4.6) and after some term rearrangement, we get for \(\theta \in \tilde{\Theta }\) that

$$\begin{aligned}&\int \limits _{0}^{T}\left[ \left\| {\Delta } c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) \right\| ^{2}_{\alpha }-\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}\right] ds = \int \limits _{0}^{T}\mathcal {L}_{X^{\epsilon }_{s}}\Phi \left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds\nonumber \\&\quad = (\delta ^{2}/\epsilon )\left( \Phi \left( X^{\epsilon }_{t},\frac{X^{\epsilon }_{t}}{\delta }\right) -\Phi \left( X^{\epsilon }_{0},\frac{X^{\epsilon }_{0}}{\delta }\right) \right) \nonumber \\&\quad \!-\!\int \limits _{0}^{T}\left[ \frac{\delta }{\epsilon }\left\langle c_{\theta },\nabla _{y}\Phi \right\rangle \!+\! \frac{\delta ^{2}}{\epsilon }\left\langle c_{\theta },\nabla _{x}\Phi \right\rangle \!+\!\frac{\delta ^{2}}{2}\sigma \sigma ^{T}:\nabla _{x}\nabla _{x}\Phi \!+\!\delta \sigma \sigma ^{T}:\nabla _{x}\nabla _{y}\Phi \right] \left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds\nonumber \\&\quad - \frac{\delta }{\sqrt{\epsilon }}\int \limits _{0}^{T}\left\langle \nabla _{y}\Phi ,\sigma dW_{s}\right\rangle \left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) - \frac{\delta ^{2}}{\sqrt{\epsilon }}\int \limits _{0}^{T}\left\langle \nabla _{x}\Phi ,\sigma dW_{s}\right\rangle \left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) \end{aligned}$$
(8.6)

Due to the boundedness of the involved functions, the last display gives us the existence of a constant \(C\) that may depend on \(\tilde{\Theta }\) (but not on \((\epsilon ,\delta )\in (0,1)^{2}\)), such that

$$\begin{aligned} \left| \int \limits _{0}^{T}\left[ \left\| \Delta c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) \right\| ^{2}_{\alpha }-\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}\right] ds\right| \le C \left( 1+\sup _{t\in [0,T]}|W_{t}|\right) \end{aligned}$$
(8.7)

These computations, allow us to continue the right hand side of (8.4) as follows

$$\begin{aligned} \mathbb {E}_{\theta }e^{-\gamma \int _{0}^{T}\left\| \Delta c_{\theta }\right\| _{\alpha }^{2}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds}&= \mathbb {E}_{\theta }\left\{ e^{-\gamma \left[ \int _{0}^{T}\left[ \left\| \Delta c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) \right\| ^{2}_{\alpha }-\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}\right] ds\right] }\right. \nonumber \\&\quad \left. \times \, e^{-\gamma \int _{0}^{T}\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}ds}\right\} \nonumber \\&\le \left( \mathbb {E}_{\theta }e^{-\gamma p_{3}\int _{0}^{T}\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}ds}\right) ^{1/p_{3}}\nonumber \\&\quad \times \left( \mathbb {E}_{\theta }e^{-\gamma q_{3}\left[ \int _{0}^{T}\left[ \left\| \Delta c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) \right\| ^{2}_{\alpha }-\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}\right] ds\right] }\right) ^{1/q_{3}}\nonumber \\&\le \left( \mathbb {E}_{\theta }e^{\!-\!\gamma p_{3}\!\int _{0}^{T}\left( \!v,q_{\epsilon ,v}^{1/2}\left( \!X^{\epsilon }_{s},\theta \right) \right) ^{2}ds}\right) ^{1/p_{3}}\! \left( \mathbb {E}e^{\gamma C q_{3}\left( \!1\!+\!\sup _{t\in [0,T]}|W_{t}|\right) }\right) ^{1/q_{3}} \nonumber \\ \end{aligned}$$
(8.8)

where, the first inequality in the last computation used Hölder inequality with \(1/p_{3}+1/q_{3}=1\) and the second inequality used (8.7).

So, we now need to focus on the term \(\mathbb {E}_{\theta }e^{-\gamma p_{3}\int _{0}^{T}\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}ds}\). Define the vector valued function

$$\begin{aligned} d_{\theta _{1},\theta }(x,y)=\sqrt{m_{\theta }(x,y)}\sigma ^{-1}(x,y)c_{\theta _{1}}(x,y) \end{aligned}$$

and notice that

$$\begin{aligned} \left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}&= \frac{1}{\epsilon }\int \limits _{\mathcal {Y}}\left\| c_{\theta +\sqrt{\epsilon }v}(X^{\epsilon }_{s},y)-c_{\theta }(X^{\epsilon }_{s},y)\right\| _{\alpha }^{2}\mu _{\theta }(dy;X^{\epsilon }_{s})\\&= \frac{1}{\epsilon }\int \limits _{\mathcal {Y}}\left\| d_{\theta +\sqrt{\epsilon }v,\theta }(X^{\epsilon }_{s},y)-d_{\theta ,\theta }(X^{\epsilon }_{s},y)\right\| ^{2}dy \end{aligned}$$

Using the trivial inequality \(a^{2}\ge b^{2}-2|b(a-b)|\), applied with \(a=d_{\theta +\sqrt{\epsilon }v,\theta }(X^{\epsilon }_{s},y)-d_{\theta ,\theta }(X^{\epsilon }_{s},y)\) and \(b=d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\) we can write

$$\begin{aligned} \left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}&= \frac{1}{\epsilon }\int \limits _{\mathcal {Y}}\left\| d_{\theta +\sqrt{\epsilon }v,\theta }(X^{\epsilon }_{s},y)-d_{\theta ,\theta }(X^{\epsilon }_{s},y)\right\| ^{2}dy\nonumber \\&\ge \frac{1}{\epsilon }\int \limits _{\mathcal {Y}}\left\| d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\right\| ^{2}dy\nonumber \\&\quad -\, 2\frac{1}{\epsilon }\int \limits _{\mathcal {Y}}\left[ \left( \left\| d_{\theta +\sqrt{\epsilon }v,\theta }(X^{\epsilon }_{s},y)-d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)\right\| \right. \right. \nonumber \\&\quad \left. \left. +\left\| d_{\theta ,\theta }(X^{\epsilon }_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\right\| \right) \right. \times \nonumber \\&\quad \left. \times \left\| d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y) \right\| \right] dy \end{aligned}$$

Hence, we obtain the bound

$$\begin{aligned}&\mathbb {E}_{\theta }e^{-\gamma p_{3}\int _{0}^{T}\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}ds}\le e^{-\gamma p_{3}\frac{1}{\epsilon }\int _{0}^{T}\int _{\mathcal {Y}}\Vert d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\Vert ^{2}dy ds}\nonumber \\&\quad \times \, \mathbb {E}_{\theta }\left[ e^{2\gamma p_{3}\frac{1}{\epsilon }\int _{0}^{T}\int _{\mathcal {Y}}\left( \Vert d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y) \Vert \Vert d_{\theta +\sqrt{\epsilon }v,\theta }(X^{\epsilon }_{s},y)-d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)\Vert \right) dy ds}\right. \nonumber \\&\quad \left. \times \, e^{2\gamma p_{3}\frac{1}{\epsilon }\int _{0}^{T}\int _{\mathcal {Y}}\left( \Vert d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y) \Vert \Vert d_{\theta ,\theta }(X^{\epsilon }_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\Vert \right) dy ds}\right] \end{aligned}$$
(8.9)

Notice that the assumption of uniform positive definiteness of the Fisher information matrix \(I(\theta )\) guarantees that

$$\begin{aligned} \begin{aligned} \int \limits _{0}^{T}\left( v,q^{1/2}(\bar{X}_{s},\theta )\right) ^{2}ds=(I(\theta )v,v)&\ge \left\| v\right\| ^{2}\inf _{\theta \in \Theta }\inf _{|\lambda |=1}\left( I(\theta )\lambda ,\lambda \right) \nonumber \\&\ge \left\| v\right\| ^{2}c_{0}\nonumber \end{aligned} \end{aligned}$$

So, as \(\left\| \sqrt{\epsilon }v \right\| \rightarrow 0\) we will have

$$\begin{aligned}&\frac{1}{\epsilon }\int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left\| d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\right\| ^{2}dy ds\nonumber \\&\quad =\frac{1}{\epsilon }\int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left\| c_{\theta +\sqrt{\epsilon }v}(\bar{X}_{s},y)-c_{\theta }(\bar{X}_{s},y)\right\| _{\alpha }^{2}\mu _{\theta }(dy;\bar{X}_{s}) ds\nonumber \\&\quad =\int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left\| \int \limits _{0}^{1}\left( v,\nabla _{\theta }c_{\theta +\sqrt{\epsilon }v h}(\bar{X}_{s},y)\right) dh\right\| _{\alpha }^{2}\mu _{\theta }(dy;\bar{X}_{s})ds\nonumber \\&\quad =\int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left\| \left( v,\nabla _{\theta }c_{\theta }(\bar{X}_{s},y)\right) \right\| _{\alpha }^{2}\mu _{\theta }(dy;\bar{X}_{s})ds\nonumber \\&\qquad +\int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left\| \int \limits _{0}^{1}\left( v,\nabla _{\theta }c_{\theta +\sqrt{\epsilon }v h}(\bar{X}_{s},y)-\nabla _{\theta }c_{\theta }(\bar{X}_{s},y)\right) dh\right\| _{\alpha }^{2}\mu _{\theta }(dy;\bar{X}_{s})ds+o(\left\| v \right\| ^{2} )\nonumber \\&\quad =\int \limits _{0}^{T}\left( v,q^{1/2}(\bar{X}_{s},\theta )\right) ^{2}ds+o(\left\| v \right\| ^{2} )\nonumber \\&\quad \ge \left\| v\right\| ^{2}\left( c_{0}+o(1)\right) \!. \end{aligned}$$
(8.10)

The assumed uniform boundedness of \(\sigma ^{-1}c_{\theta }\), the fact that \(m_{\theta }\) is a density and the lower bound from (8.10) mean that there exist constants \(C_{2},C_{3}\) that may depend on \(\tilde{\Theta }\) such that

$$\begin{aligned} C_{2}\left\| v\right\| ^{2}\le \frac{1}{\epsilon }\int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left\| d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\right\| ^{2}dy ds\le C_{3}\left\| v\right\| ^{2} \end{aligned}$$

Moreover, by Cauchy-Schwartz inequality, we also have that

$$\begin{aligned}&\left( \int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left( \left\| d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y) \right\| \left\| d_{\theta ,\theta }(X^{\epsilon }_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\right\| \right) dy ds\right) ^{2}\nonumber \\&\quad \le \int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left\| d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y) \right\| ^{2}dyds \int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left\| d_{\theta ,\theta }(X^{\epsilon }_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\right\| ^{2} dy ds\nonumber \\&\quad \le C_{3}\epsilon \left\| v\right\| ^{2} \int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left\| d_{\theta ,\theta }(X^{\epsilon }_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\right\| ^{2}dy ds\nonumber \\&\quad \le C_{3}C_{4}\epsilon \left\| v\right\| ^{2} \int \limits _{0}^{T} \left\| X^{\epsilon }_{s}-\bar{X}_{s}\right\| ^{2}ds\nonumber \\&\quad \le C_{3}C_{4}T\epsilon \left\| v\right\| ^{2} \sup _{t\in [0,T]} \left\| X^{\epsilon }_{t}-\bar{X}_{t}\right\| ^{2} \end{aligned}$$
(8.11)

To derive the inequality before the last one, we used the Lipschitz continuity in \(x\) of the function \(d_{\theta ,\theta }(x,y)\), with a Lipschitz constant \(C_{4}\) that may depend on \(\tilde{\Theta }\). To continue, we need to bound from above the quantity \(\sup _{t\in [0,T]} \left\| X^{\epsilon }_{t}-\bar{X}_{t}\right\| ^{2}\). For this purpose, we set \(\bar{c}_{\theta }(x)=\int _{\mathcal {Y}}c_{\theta }(x,y)\mu _{\theta }(dy;x)\) and write

$$\begin{aligned} X^{\epsilon }_{t}-\bar{X}_{t}&= \int \limits _{0}^{t}c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds-\int \limits _{0}^{t}\bar{c}_{\theta }\left( \bar{X}_{s}\right) ds+\sqrt{\epsilon } \int \limits _{0}^{t}\sigma \left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) dW_{s}\\&= \int \limits _{0}^{t}\left[ c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) -\bar{c}_{\theta }\left( X^{\epsilon }_{s}\right) \right] ds+\int \limits _{0}^{t}\left[ \bar{c}_{\theta }\left( X^{\epsilon }_{s}\right) \right. \\&\left. -\,\bar{c}_{\theta }\left( \bar{X}_{s}\right) \right] ds+\sqrt{\epsilon } \int \limits _{0}^{t}\sigma \left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) dW_{s} \end{aligned}$$

Thus, we obtain

$$\begin{aligned} \left\| X^{\epsilon }_{t}-\bar{X}_{t}\right\| ^{2}&\le 2^{3}\left\{ \left\| \int \limits _{0}^{t}\left[ c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) -\bar{c}_{\theta }\left( X^{\epsilon }_{s}\right) \right] ds\right\| ^{2}+\int \limits _{0}^{t}\left\| \bar{c}_{\theta }\left( X^{\epsilon }_{s}\right) -\bar{c}_{\theta }\left( \bar{X}_{s}\right) \right\| ^{2}ds\right. \nonumber \\&\quad \left. +\,\,\epsilon \left\| \sigma \right\| ^{2}\sup _{s\in [0,t]}\left\| W_{s}\right\| ^{2}\right\} \nonumber \\&\le 2^{3}\left\{ \left\| \int \limits _{0}^{t}\left[ c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) -\bar{c}_{\theta }\left( X^{\epsilon }_{s}\right) \right] ds\right\| ^{2}+C_{5}\int \limits _{0}^{t}\left\| X^{\epsilon }_{s}-\bar{X}_{s}\right\| ^{2}ds\right. \nonumber \\&\quad \left. +\,\,\epsilon \left\| \sigma \right\| ^{2}\sup _{s\in [0,t]}\left\| W_{s}\right\| ^{2} \right\} \end{aligned}$$
(8.12)

In the last inequality, we used the Lipschitz continuity of \(\bar{c}_{\theta }\) with a Lipschitz constant \(C_{5}\) that may depend on \(\tilde{\Theta }\). Let us explain now how the term \(\left\| \int _{0}^{t}\left[ c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) -\bar{c}_{\theta }\left( X^{\epsilon }_{s}\right) \right] ds\right\| ^{2}\) can be treated. By considering the solution to an auxiliary PDE problem analogous to (8.5) with right hand side replaced by \(c_{\theta }(x,y)-\bar{c}_{\theta }(x)\), we get (similarly to (8.6)) that

$$\begin{aligned} \left\| \int \limits _{0}^{t}\left[ c_{\theta }\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) -\bar{c}_{\theta }\left( X^{\epsilon }_{s}\right) \right] ds\right\| \le C_{6}\left( 1+\frac{\delta }{\sqrt{\epsilon }} \sup _{s\in [0,t]}\left\| W_{s}\right\| \right) \end{aligned}$$

For some constant \(C_{6}\) that may depend on \(\tilde{\Theta }\). Thus putting things together, (8.12) takes the form

$$\begin{aligned} \left\| X^{\epsilon }_{t}-\bar{X}_{t}\right\| ^{2}&\le C_{7}\left\{ 1+ \displaystyle \int \limits _{0}^{t}\left\| X^{\epsilon }_{s}-\bar{X}_{s}\right\| ^{2}ds+\left( \epsilon +\frac{\delta ^{2}}{\epsilon } \right) \sup _{s\in [0,t]}\left\| W_{s}\right\| ^{2} \right\} \end{aligned}$$
(8.13)

and by Grownwall inequality, we can conclude that there exists a constant \(C_{8}\), that may depend on \(\tilde{\Theta }\), such that

$$\begin{aligned} \sup _{t\in [0,T]}\left\| X^{\epsilon }_{t}-\bar{X}_{t}\right\|&\le C_{8}\sqrt{\epsilon +\frac{\delta ^{2}}{\epsilon }}\sup _{t\in [0,T]}\left\| W_{t}\right\| \end{aligned}$$
(8.14)

Coming back to (8.11), we have obtained

$$\begin{aligned}&\frac{1}{\epsilon }\int \limits _{0}^{T}\int \limits _{\mathcal {Y}}\left( \left\| d_{\theta +\sqrt{\epsilon }v,\theta }(\bar{X}_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y) \right\| \left\| d_{\theta ,\theta }(X^{\epsilon }_{s},y)-d_{\theta ,\theta }(\bar{X}_{s},y)\right\| \right) dy ds\nonumber \\&\quad \le \sqrt{C_{3}C_{4}}C_{8}\left\| v\right\| \sup _{t\in [0,T]}\left\| W_{t}\right\| \end{aligned}$$
(8.15)

Set \(C_{9}=\sqrt{C_{3}C_{4}T}C_{8}\). Putting (8.10) and (8.15) together and recalling that \(v=I^{-1/2}(\theta )u\), the bound (8.9) becomes

$$\begin{aligned} \mathbb {E}_{\theta }e^{-\gamma p_{3}\int _{0}^{T}\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}ds}&\le e^{-\gamma p_{3}C_{2}\left\| u \right\| ^{2}} \mathbb {E}_{\theta }\left[ e^{4\gamma p_{3}C_{9}\left\| u\right\| \sup _{t\in [0,T]}\left\| W_{t}\right\| }\right] \nonumber \\&\le e^{-\gamma p_{3}C_{2}\left\| u \right\| ^{2}} \left[ \!1\!+\!4\gamma p_{3} \!C_{9} \left\| u\right\| \sqrt{8\pi T} e^{8\gamma ^{2}\left( p_{3}C_{9}T\right) ^{2} \left\| u\right\| ^{2}}\right] , \nonumber \\ \end{aligned}$$
(8.16)

where the last inequality used Lemma 1.14 by Kutoyants, Kutoyants (1994). Now, we have all the necessary ingredients in order to continue the bound of (8.4). In particular, using (8.8), (8.4) gives

$$\begin{aligned}&\mathbb {E}_{\theta }e^{pM_{\epsilon }(\theta ,u)}\le \left( \mathbb {E}_{\theta }e^{-\gamma \int _{0}^{T}\left\| \Delta c_{\theta }\right\| _{\alpha }^{2}\left( X^{\epsilon }_{s},\frac{X^{\epsilon }_{s}}{\delta }\right) ds}\right) ^{(q-p^{2})/p}\nonumber \\&\quad \le \left( \mathbb {E}_{\theta }e^{-\gamma p_{3}\int _{0}^{T}\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}ds}\right) ^{(q-p^{2})/(pp_{3})} \left( \mathbb {E}e^{\gamma q_{3}C\left( 1+\sup _{t\in [0,T]}|W_{t}|\right) }\right) ^{(q-p^{2})/(pq_{3})}\nonumber \\ \end{aligned}$$
(8.17)

Choosing \(p,q\) such that \(\gamma =\frac{p_{3}C_{2}}{16 \left( p_{3}C_{9}T\right) ^{2}}\) and using the inequality \(1+x\le e^{x}\), we then obtain from (8.16)

$$\begin{aligned}&\left( \mathbb {E}_{\theta }e^{-\gamma p_{3}\int _{0}^{T}\left( v,q_{\epsilon ,v}^{1/2}\left( X^{\epsilon }_{s},\theta \right) \right) ^{2}ds}\right) ^{(q-p^{2})/(pp_{3})}\nonumber \\&\quad \le e^{-\frac{C_{2}}{2}\left\| u \right\| ^{2}} \left[ 1+4\gamma p_{3} C_{9} \left\| u\right\| \sqrt{8\pi T} \right] ^{(q-p^{2})/(pp_{3})}\nonumber \\&\quad \le e^{-\frac{C_{2}}{2}\left\| u \right\| ^{2} +\frac{q-p^{2}}{p}4\gamma C_{9} \left\| u\right\| \sqrt{8\pi T} } \end{aligned}$$
(8.18)

So, (8.17) and (8.18) give

$$\begin{aligned} \mathbb {E}_{\theta }e^{pM_{\epsilon }(\theta ,u)}&\le e^{-\frac{C_{2}}{2}\left\| u \right\| ^{2} +\frac{q-p^{2}}{p}4\gamma C_{9} \left\| u\right\| \sqrt{8\pi T} } \left( \mathbb {E}e^{\gamma q_{3}C\left( 1+\sup _{t\in [0,T]}|W_{t}|\right) }\right) ^{(q-p^{2})/(pq_{3})} \end{aligned}$$

The right hand side of the last inequality defines our function \(g(\left\| u\right\| )\), which certainly enjoys the property

$$\begin{aligned} \lim _{u\rightarrow \infty }u^{n}e^{-g(\left\| u\right\| )}=0,\quad \forall n\in \mathbb {N} \end{aligned}$$

This concludes the proof of the lemma. \(\square \)

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Spiliopoulos, K., Chronopoulou, A. Maximum likelihood estimation for small noise multiscale diffusions. Stat Inference Stoch Process 16, 237–266 (2013). https://doi.org/10.1007/s11203-013-9088-8

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Keywords

  • Parameter estimation
  • Central limit theorem
  • Multiscale diffusions
  • Dynamical systems
  • Rough energy landscapes

Mathematics Subject Classification

  • 62M05
  • 62M86
  • 60F05
  • 60G99