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Trajectory fitting estimators for SPDEs driven by additive noise

Abstract

In this paper we study the problem of estimating the drift/viscosity coefficient for a large class of linear, parabolic stochastic partial differential equations (SPDEs) driven by an additive space-time noise. We propose a new class of estimators, called trajectory fitting estimators (TFEs). The estimators are constructed by fitting the observed trajectory with an artificial one, and can be viewed as an analog to the classical least squares estimators from the time-series analysis. As in the existing literature on statistical inference for SPDEs, we take a spectral approach, and assume that we observe the first N Fourier modes of the solution, and we study the consistency and the asymptotic normality of the TFE, as \(N\rightarrow \infty \).

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Notes

  1. A diagonalizable SPDE is an equation for which the first N Fourier coefficients of the solution form an N-dimensional decoupled system of ordinary stochastic differential equation. For a formal definition, in terms of the differential operators and the structure of the noise term, see for instance Lototsky (2009).

  2. Without loss of generality, we will assume that \(\nu _{k}\ge 0\), for all \(k\in \mathbb {N}\).

  3. Of course, one can consider at once just \(\lambda _{k}=\nu _{k}\). Our choice to consider m is to put the results on par with the notations from the existing literature. As mentioned later, if \(\mathcal {A}_{0}\) and \(\mathcal {A}_{1}\) are some pseudo-differential operators, then it is convenient to denote by 2m the order of the leading order operator.

  4. The terminology comes from the fact that the estimator is obtained by fitting the observed trajectory with the artificial one.

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Acknowledgements

Part of the research was performed while Igor Cialenco was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. The authors would like to thank the anonymous referees, the associate editor and the editor for their helpful comments and suggestions which improved greatly the final manuscript.

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Correspondence to Igor Cialenco.

Appendix

Appendix

Proof of Theorem 3.1

Due to the nature of desired asymptotic results, the underlying computations are somehow extensive and tedious. For simplicity of brevity, we will only provide the proof for the special case when \(u_{0}=0\), \(\gamma =0\) and \(\sigma =1\), but the genera case can be verified using similar arguments and the details can be obtained from the authors upon request. Most of the evaluations were performed using symbolic computations in Mathematica. For each \(k\in \mathbb {N}\), when \(u_{0}=0\), \(\gamma =0\) and \(\sigma =1\),

$$\begin{aligned} u_{k}(t)=e^{-\mu _{k}(\theta )t}\int _{0}^{t}e^{\mu _{k}(\theta )s}\,dw_{k}(s),\quad t\in [0,T], \end{aligned}$$

and with the notations introduced in (2.9) and (3.1), we get

$$\begin{aligned} A_{k}=\frac{1}{2}\xi _{k}^{2}-X_{k}+2\mu _{k}(\theta )Z_{k}. \end{aligned}$$

Note that for any \(t\in [0,T]\), \(u_{k}(t)\) is a centered normal random variable with variance \((1-e^{-2\mu _{k}(\theta )t})/(2\mu _{k}(\theta ))\), and thus,

$$\begin{aligned} \mathbb {E}\left( u_{k}^{2n}(t)\right) =(2n-1)!!\cdot \left( \frac{1-e^{-2\mu _{k}(\theta )t}}{2\mu _{k}(\theta )}\right) ^{n}, \quad n\in \mathbb {N}. \end{aligned}$$
(5.1)

We first verify (3.2), which now reduces to

$$\begin{aligned} \mathbb {E}(Z_{k})\asymp \frac{T^{2}}{\mu _{k}^{2}(\theta )},\quad k\rightarrow \infty , \end{aligned}$$
(5.2)

by computing

$$\begin{aligned} \mathbb {E}(Z_{k})=\mathbb {E}\left( \int _{0}^{T}\xi _{k}^{2}(t)dt\right) =\int _{0}^{T}\mathbb {E}\left( \xi _{k}^{2}(t)\right) dt =2\int _{0}^{T}\int _{0}^{t}\mathbb {E}\left( \xi _{k}(s)u_{k}^{2}(s)\right) ds\,dt, \end{aligned}$$
(5.3)

where the last equality follows from the definition of \(\xi _{k}\) in (2.9). By Itô’s formula,

$$\begin{aligned} d\xi _{k}(t)u_{k}^{2}(t)= \left( u_{k}^{4}(t)+\xi _{k}(t)\right) dt-2\mu _{k}(\theta )\xi _{k}(t)u_{k}^{2}(t)\,dt+2u_{k}(t)\xi _{k}(t)dw_{k}(t). \end{aligned}$$

Taking the expectations on both sides above, using the definition of \(\xi _{k}(t)\) in (2.9) and (5.1), we obtain that, for \(t\in [0,T]\),

$$\begin{aligned} \mathbb {E}\left( \xi _{k}(t)u_{k}^{2}(t)\right) =\frac{1-e^{-2\mu _{k}(\theta )t}}{8\mu _{k}^{3}(\theta )}-\frac{5te^{-2\mu _{k}(\theta )t}}{4\mu _{k}^{2}(\theta )}+\frac{3\left( 1-e^{-2\mu _{k}(\theta )t}\right) e^{-2\mu _{k}(\theta )t}}{8\mu _{k}^{3}(\theta )}+\frac{t}{4\mu _{k}^{2}(\theta )}. \end{aligned}$$
(5.4)

Therefore, by (5.3),

$$\begin{aligned} \mathbb {E}(Z_{k})=\frac{35-3e^{-4\mu _{k}(\theta )T}-32e^{-2\mu _{k}(\theta )T}}{64\mu _{k}^{5}(\theta )}-\frac{9T+10T\,e^{-2\mu _{k}(\theta )T}}{16\mu _{k}^{4}(\theta )}+\frac{T^{2}}{8\mu _{k}^{3}(\theta )}+\frac{T^{3}}{12\mu _{k}^{2}(\theta )}, \end{aligned}$$
(5.5)

which leads to (5.2), since by the assumption (ii), the first three terms above all have higher orders than the last term, as \(k\rightarrow \infty \), and since \(T>0\) is a fixed constant.

Next, we study the asymptotic order of \(\text {Var}(Z_{k})\), \(k\rightarrow \infty \), given by (3.3), which now reduces to

$$\begin{aligned} \text {Var}(Z_{k})\asymp \frac{T^{3}}{\mu _{k}^{5}(\theta )},\quad k\rightarrow \infty . \end{aligned}$$
(5.6)

In light of (5.5), we are left to compute \(\mathbb {E}(Z_{k}^{2})\). By Itô’s formula, and since the Itô integral terms have zero expectation, we have

$$\begin{aligned} \mathbb {E}(Z_{k}^{2})&=2\int _{0}^{T}\mathbb {E}\left( Z_{k}(t)\xi _{k}^{2}(t)\right) dt=2\int _{0}^{T}\int _{0}^{t}\mathbb {E}\left( \xi _{k}^{4}(s)\right) ds\,dt\nonumber \\&\quad +\,4\int _{0}^{T}\int _{0}^{t}\mathbb {E}\left( Z_{k}(s)\xi _{k}(s)u_{k}^{2}(s)\right) ds\,dt. \end{aligned}$$
(5.7)

To compute the first expectation in (5.7), by Itô’s formula again, we continue

$$\begin{aligned} d\xi _{k}^{4}(t)= & {} 4\xi _{k}^{3}(t)u_{k}^{2}(t)\,dt,\\ d\left( \xi _{k}^{3}(t)u_{k}^{2}(t)\right)= & {} \left( 3\xi _{k}^{2}(t)u_{k}^{4}(t)+\xi _{k}^{3}(t)-2\mu _{k}(\theta )\xi _{k}^{3}(t)u_{k}^{2}(t)\right) dt+2\xi _{k}^{3}(t)u_{k}(t)\,dw_{k}(t),\\ d\xi _{k}^{3}(t)= & {} 3\xi _{k}^{2}(t)u_{k}^{2}(t)\,dt,\\ d\left( \xi _{k}^{2}(t)u_{k}^{4}(t)\right)= & {} 2\left( \xi _{k}(t)u_{k}^{6}(t)+3\xi _{k}^{2}(t)u_{k}^{2}(t)-2\mu _{k}(\theta )\xi _{k}^{2}(t)u_{k}^{4}(t)\right) dt+4\xi _{k}^{2}(t)u_{k}^{3}(t)\,dw_{k}(t). \end{aligned}$$

Hence, we only need to compute \(\mathbb {E}(\xi _{k}(s)u_{k}^{6}(s))\) and \(\mathbb {E}(\xi _{k}^{2}(s)u_{k}^{2}(s))\). Again by Itô’s formula, we get

$$\begin{aligned} d\xi _{k}(t)u_{k}^{6}(t)= & {} \left( u_{k}^{8}(t)+15\xi _{k}(t)u_{k}^{4}(t)-6\mu _{k}(\theta )\xi _{k}(t)u_{k}^{6}(t)\right) dt+6\xi _{k}(t)u_{k}^{5}(t)\,dw_{k}(t),\\ d\xi _{k}^{2}(t)u_{k}^{2}(t)= & {} \left( 2\xi _{k}(t)u_{k}^{4}(t)+\xi _{k}^{2}(t)-2\mu _{k}(\theta )\xi _{k}^{2}(t)u_{k}^{2}(t)\right) dt+2\xi _{k}^{2}(t)u_{k}(t)\,dw_{k}(t),\\ d\xi _{k}(t)u_{k}^{4}(t)= & {} \left( u_{k}^{6}(t)+6\xi _{k}(t)u_{k}^{2}(t)-4\mu _{k}(\theta )\xi _{k}(t)u_{k}^{4}(t)\right) dt+4\xi _{k}(t)u_{k}^{3}(t)\,dw_{k}(t). \end{aligned}$$

Therefore, by (5.1) and (5.4), we can obtain first \(\mathbb {E}(\xi _{k}(s)u_{k}^{6}(s))\) and \(\mathbb {E}(\xi _{k}^{2}(s)u_{k}^{2}(s))\), and then \(\mathbb {E}(\xi _{k}^{3}(s))\), \(\mathbb {E}(\xi _{k}^{2}(s)u_{k}^{4}(s))\) and \(\mathbb {E}(\xi _{k}^{3}(s)u_{k}^{2}(s))\), and finally \(\mathbb {E}(\xi _{k}^{4}(s))\). A similar argument leads to the computation of the second expectation in (5.7).

To sum up, with the help of Mathematica, we obtain that

$$\begin{aligned} \text {Var}(Z_{k})= & {} -\frac{16917}{512\,\mu _{k}^{10}(\theta )}+\frac{3\,e^{-8\mu _{k}(\theta )T}}{128\,\mu _{k}^{10}(\theta )}+\frac{79\,e^{-6\mu _{k}(\theta )T}}{128\,\mu _{k}^{10}(\theta )}+\frac{2953\,e^{-4\mu _{k}(\theta )T}}{512\,\mu _{k}^{10}(\theta )}+\frac{3409\,e^{-2\mu _{k}(\theta )T}}{128\,\mu _{k}^{10}(\theta )}\\&\quad +\frac{1093T}{32\,\mu _{k}^{9}(\theta )}+\frac{45T\,e^{-6\mu _{k}(\theta )T}}{64\,\mu _{k}^{9}(\theta )}+\frac{1165T\,e^{-4\mu _{k}(\theta )T}}{128\,\mu _{k}^{9}(\theta )}+\frac{2321T\,e^{-2\mu _{k}(\theta )T}}{64\,\mu _{k}^{9}(\theta )}\\&\quad -\frac{659T^{2}}{64\,\mu _{k}^{8}(\theta )}+\frac{53T^{2}e^{-4\mu _{k}(\theta )T}}{16\,\mu _{k}^{8}(\theta )}+\frac{71T^{2}e^{-2\mu _{k}(\theta )T}}{8\,\mu _{k}^{8}(\theta )}-\frac{5T^{3}}{12\,\mu _{k}^{7}(\theta )}-\frac{5T^{3}e^{-4\mu _{k}(\theta )T}}{8\,\mu _{k}^{7}(\theta )}\\&\quad -\frac{113T^{3}e^{-2\mu _{k}(\theta )T}}{24\,\mu _{k}^{7}(\theta )}+\frac{23T^{4}}{48\,\mu _{k}^{6}(\theta )}-\frac{5T^{4}e^{-2\mu _{k}(\theta )T}}{2\mu _{k}^{6}(\theta )}+\frac{T^{5}}{15\mu _{k}^{5}(\theta )}, \end{aligned}$$

which clearly implies (5.6), and thus completes the proof. \(\square \)

Proof of Lemma 3.5

By (2.4) and Cauchy–Schwartz inequality, for any \(0\le s\le t\le T\),

$$\begin{aligned} u_{k}^{2}(s)= & {} e^{-2\mu _{k}(\theta )s}\left( u_{k}(0)+\sigma \lambda _{k}^{-\gamma }\int _{0}^{s}e^{\mu _{k}(\theta )r}\,dw_{k}(r)\right) ^{2}\\\le & {} e^{-2\mu _{k}(\theta )s}\left( u_{k}^{2}(0)+\sigma ^{2}\lambda _{k}^{-2\gamma }t\right) \left( 1+\frac{1}{t}\left( \int _{0}^{s}e^{\mu _{k}(\theta )r}\,dw_{k}(r)\right) ^{2}\right) . \end{aligned}$$

Hence, for any \(t\in [0,T]\), and \(n\in \mathbb {N}\),

$$\begin{aligned} \xi _{k}^{n}(t)\le & {} \left( u_{k}^{2}(0)+\sigma ^{2}\lambda _{k}^{-2\gamma }t\right) ^{n}\left\{ \int _{0}^{t}e^{-2\mu _{k}(\theta )s}\left[ 1+\frac{1}{t}\left( \int _{0}^{s}e^{\mu _{k}(\theta )r}\,dw_{k}(r)\right) ^{2}\right] ds\right\} ^{n}\\= & {} \left( u_{k}^{2}(0)+\sigma ^{2}\lambda _{k}^{-2\gamma }t\right) ^{n}\left[ \frac{1-e^{-2\mu _{k}(\theta )t}}{2\mu _{k}(\theta )}+\frac{1}{t}\int _{0}^{t}e^{-2\mu _{k}(\theta )s}\left( \int _{0}^{s}e^{\mu _{k}(\theta )r}\,dw_{k}(r)\right) ^{2}ds\right] ^{n}\\\le & {} \left( u_{k}^{2}(0)+\sigma ^{2}\lambda _{k}^{-2\gamma }t\right) ^{n}\left\{ \left( \frac{1-e^{-2\mu _{k}(\theta )t}}{\mu _{k}(\theta )}\right) ^{n}\right. \\&\left. +\,\frac{2^{n}}{t^{n}}\left[ \int _{0}^{t}e^{-2\mu _{k}(\theta )s}\left( \int _{0}^{s}e^{\mu _{k}(\theta )r}\,dw_{k}(r)\right) ^{2}ds\right] ^{n}\right\} . \end{aligned}$$

By Lototsky (2009, Theorem 2.1), there exists a constant \(\widetilde{D}_{n}=\widetilde{D}_{n}(t)>0\), such that

$$\begin{aligned} \mathbb {E}\left( \left[ \int _{0}^{t}e^{-2\mu _{k}(\theta )s}\left( \int _{0}^{s}e^{\mu _{k}(\theta )r}\,dw_{k}(r)\right) ^{2}ds\right] ^{n}\right) \le \frac{\widetilde{D}_{n}}{\mu _{k}^{n}(\theta )}. \end{aligned}$$

Therefore, for any \(t\in [0,T]\) and \(n\in \mathbb {N}\),

$$\begin{aligned} \mathbb {E}\left( \xi _{k}^{n}(t)\right) \le \left( u_{k}^{2}(0)+\sigma ^{2}\lambda _{k}^{-2\gamma }t\right) ^{n}\left( \frac{1}{\mu _{k}^{n}(\theta )}+\frac{2^{n}\widetilde{D}_{n}}{t^{n}}\frac{1}{\mu _{k}^{n}(\theta )}\right) =D_{n}\left( \frac{u_{k}^{2}(0)+\sigma \lambda _{k}^{-2\gamma }t}{\mu _{k}(\theta )}\right) ^{n}, \end{aligned}$$

where \(D_{n}=D_{n}(t):=1+2^{n}t^{-n}\widetilde{D}_{n}\). \(\square \)

We conclude the appendix by listing, for sake of completeness, a version of law of large numbers and a version of central limit theorem used in the proofs of the main results in this paper. For the detailed proofs of these results we refer the reader to Shiryaev (1996).

Theorem 5.1

(Strong Law of Large Number) Let \(\{\eta _{n}\}_{n\in \mathbb {N}}\) be a sequence of independent random variables, and let \(\{b_{n}\}_{n\in \mathbb {N}}\) be a sequence of non-decreasing positive numbers such that \(\lim _{n\rightarrow \infty }b_{n}=\infty \). If

$$\begin{aligned} \sum _{n=1}^{\infty }\frac{Var \left( \eta _{n}\right) }{b_{n}^{2}}<\infty , \end{aligned}$$

then

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{b_{n}}\sum _{k=1}^{n}\left( \eta _{k}-\mathbb {E}(\eta _{k})\right) =0,\quad \mathbb {P}-\text {a.}\,\text {s}. \end{aligned}$$

Remark 5.2

As an immediate corollary, if \(\{\eta _{n}\}_{n\in \mathbb {N}}\) is a sequence of independent non-negative random variables with

$$\begin{aligned} \sum _{n=1}^{\infty }\mathbb {E}(\eta _{n})=\infty \quad \text {and}\quad \sum _{n=1}^{\infty }\frac{\text {Var}\left( \eta _{n}\right) }{\left( \sum _{k=1}^{n}\mathbb {E}(\eta _{k})\right) ^{2}}<\infty , \end{aligned}$$

then

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\sum _{k=1}^{n}\eta _{k}}{\sum _{k=1}^{n}\mathbb {E}(\eta _{k})}=1,\quad \mathbb {P}-\text {a.}\,\text {s}. \end{aligned}$$

Theorem 5.3

(Lyapunov Central Limit Theorem) Let \(\{\eta _{n}\}_{n\in \mathbb {N}}\) be a sequence of independent random variables with finite second moments. If there exists some \(\delta >0\), such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\left( \sum _{k=1}^{n}{} Var (\eta _{k})\right) ^{2+\delta }}\sum _{k=1}^{n}\mathbb {E}\left( \left| \eta _{k}-\mathbb {E}(\eta _{k})\right| ^{2+\delta }\right) =0, \end{aligned}$$
(5.8)

then

$$\begin{aligned} \frac{\sum _{k=1}^{n}\left( \eta _{k}-\mathbb {E}(\eta _{k})\right) }{\sqrt{\sum _{k=1}^{n}{} Var (\eta _{k})}}\;{\mathop {\longrightarrow }\limits ^{d}}\;\mathcal {N}(0,1),\quad n\rightarrow \infty . \end{aligned}$$

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Cialenco, I., Gong, R. & Huang, Y. Trajectory fitting estimators for SPDEs driven by additive noise. Stat Inference Stoch Process 21, 1–19 (2018). https://doi.org/10.1007/s11203-016-9152-2

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Keywords

  • Stochastic partial differential equations
  • Trajectory fitting estimator
  • Parameter estimation
  • Inverse problems
  • Estimation of viscosity coefficient

Mathematics Subject Classification

  • 60H15
  • 35Q30
  • 65L09