Drift Estimation for Discretely Sampled SPDEs

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 goes to infinity and/or T, M go to infinity. We establish sufficient conditions on the growth rates of N, M and T, that guarantee consistency and asymptotic normality of these estimators.

Authors

• 7 publications
• 3 publications
• 4 publications
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1 Introduction

Undoubtedly, the stochastic partial differential equations (SPDEs) serve as a modern powerful modeling tool in describing the evolution of dynamical systems in the presence of spatial-temporal uncertainties with particular applications in fluid mechanics, oceanography, temperature anomalies, finance, economics, biological and ecological systems, and many other applied disciplines. Major breakthrough results have been established on the general analytical theory for SPDEs, such as existence, uniqueness and regularity properties of the solutions. For an in depth discussion of the theory of SPDEs and their various applications, we refer to recent monographs [LR17, LR18]. In contrast, the investigation of inverse problems for SPDEs, and in particular parameter estimation problems, are still in their emerging phase. We refer to the survey papers [Lot09, Cia18] and the monograph [LR17, Chapter 6] for an overview of the literature and existing methodologies on statistical inference for parabolic SPDEs. Most of the existing results are obtained within the so-called spectral approach, when it is assumed that the observer measures the values of one realization of the first Fourier modes of the solution continuously over a finite time interval . In such cases, usually the statistical problems are addressed via maximum likelihood estimators (MLEs), and the asymptotic properties of the estimators are studied in the large number of Fourier modes regime, , while time horizon is fixed. The large time asymptotics regime , while being fixed, usually falls in the realm of finite dimensional stochastic differential equations, which is a well-established research area. This asymptotic regime in the context of SPDEs was only briefly discussed in [LR17, CX15]. Only several works have been dedicated to parameter estimation problems for SPDEs in discrete sampling setup. In [PR97, PR02, PR03, Mar03], the authors investigate some version of the discretized MLEs for some particular equations. In [PT07], and more recently in [CH17, BT17, Cho19, BT19], using various approaches the authors study the estimation of the drift and/or volatility coefficients when the solution is sampled discretely in physical domain.

The aim of this work is to provide a rigorous and comprehensive asymptotic analysis of the time discretized MLE for the drift coefficient of a fractional heat equation driven by an additive space-time noise. The precise form of the considered equations and their well posedness are presented in Section 2. The main results of this work can be summarized as follows:

1. In Section 3 we assume the same sampling scheme as in the existing literature on spectral approach, namely continuous time observations of the Fourier modes for , and study the asymptotic properties of the MLE, when both,

. We prove that this estimator is (strongly) consistent, and asymptotically normal. We give two proofs of the asymptotic normality, one based on Malliavin calculus, which we believe can be used, with slight modifications to other similar problems. Another proof uses classical results from general probability and explodes the particular structure and properties of the underlying problem. In particular, we show that the estimator is optimal in the mean-square sense.

2. In Section 4 we consider the natural time discretization of the MLE, by assuming that the first Fourier modes are measured at time grid points, uniformly spaced over the time interval . We study the asymptotic properties of the proposed estimator when . In particular, we prove that the estimator is consistent if , or while is fixed, and if , where is the space dimension, and is the power of the Laplacian. Moreover, if , then consistency holds true, when , when are fixed. This, in particular implies that to estimate efficiently the drift parameter it is enough to observe the Fourier modes at one instant of time - a result that agrees with recent discoveries in [CH17, BT17] where the solution is sampled in physical domain. Under some additional technical assumptions on the growth rates of and , we also prove that the proposed estimator is also asymptotically normal, with the same rate of convergence as the MLE from continuous time observation setup.

Some technical proofs, auxiliary results and relevant elements of Malliavin calculus are deferred to Appendix.

Open problems and future work. A reasonable extension of the present work is to investigate similar estimators and problems given that the solution is observed discretely in the physical domain, in which case, one has to additionally approximate the Fourier modes by a sum. While analogous asymptotic properties are expected to hold true, rigourous proofs remain to be established. As already mentioned, most of the existing literature on parameter estimation for SPDEs is focused on sampling the Fourier modes in continuous time. In particular, the MLE approach was successfully applied to nonlinear equations [CGH11, PS19], and to equations driven by a fractional noise [CLP09]. Besides MLEs, in [CGH18] the authors propose an alternative class of estimators, called trajectory fitting estimators, and a Bayesian approach to estimating drift coefficients for a class of SPDEs driven by multiplicative noise is considered in [CCG19]. It is imperative, from theoretical and practical point of view, to study the asymptotic properties of the discretized versions of the estimators proposed in the above mentioned works, especially by looking at various asymptotic regimes (large time-space sampling, small mesh size, etc).

2 Setup of the problem and some auxiliary results

Let be a stochastic basis with usual assumptions, and let be a collection of independent standard Brownian motions on this basis. Assume that is a bounded and smooth domain in , and let us denote by the Laplace operator on with zero boundary conditions. The corresponding scale of Sobolev spaces will be denoted by , or simply , for . It is well known (cf. [Shu01]) that: a) the set

of eigenfunctions of

forms a complete orthonormal system in

; b) the corresponding eigenvalues

, can be arranged such that , and there exists a positive constant so that

 limk→∞|νk|k−2/d=ϖ.

In what follows, we will use the notation , and . Also, for two sequences of numbers and , we will write , if there exists a nonzero and finite number such that , and , if .

We consider the following stochastic PDE

 dU(t,x)+θ(−Δ)βU(t,x)dt=σ∑k∈Nλ−γkhk(x)dwk(t),t∈[0,T], U(0,x)=U0, x∈G, (2.1)

where , , , and for some .

Using standard arguments (cf. [Cho07, LR17, LR18]), it can be proved that if , then (2.1) has a unique solution , weak in PDE sense and strong in probability sense, such that

 U∈L2(Ω×[0,T];Hs+β)∩L2(Ω;C((0,T);Hs)).

In what follows, we will assume that , and . We denote by the Fourier coefficient of the solution of (2.1) with respect to , i.e. . Let be the finite dimensional subspace of generated by , and denote by the projection operator of into , and put , or equivalently . Clearly, the Fourier mode , follows the dynamics of an Ornstein-Uhlenbeck process given by

 duk=−θλ2βkukdt+σλ−γkdwk(t),uk(0)=(U0,hk), t≥0.

We denote by the probability measure on generated by the . The measures are equivalent for different values of the parameter , and the Radon-Nikodym derivative, or likelihood ratio, has the form

 PN,TθPN,Tθ0(UN)=exp(−(θ−θ0)σ2N∑k=1λ2β+2γk∫T0uk(t)duk(t)−(θ2−θ20)2σ2N∑k=1λ4β+2γk∫T0u2k(t)dt).

By maximizing the log likelihood ratio with the respect to the parameter of interest , we obtain the Maximum Likelihood Estimator (MLE) for given by

 ˆθN,T:=−∑Nk=1λ2β+2γk∫T0uk(t)duk(t)∑Nk=1λ4β+2γk∫T0u2k(t)dt,N∈N, T>0. (2.2)

Let us also compute the Fisher information related to . For simplicity, set . Namely,

 IN,T :=∫∣∣ ∣∣∂∂θlogdPN,TθdPN,Tθ0∣∣ ∣∣2⎛⎜⎝dPN,Tθ0dPN,Tθ⎞⎟⎠−1dPN,Tθ0 =−∫∂2∂θ2logdPN,TθdPN,Tθ0⎛⎜⎝dPN,Tθ0dPN,Tθ⎞⎟⎠−1dPN,Tθ0 =1σ2N∑k=1λ4β+2γkE[∫T0u2kdt].

By direct evaluations, we have that

 E[∫T0u2kdt]=σ2λ−2γ−2βk2θ0⎛⎝T−1−e−2θ0λ2βkT2θ0λ2βk⎞⎠,

which yields

 IN,T =12θ0N∑k=1λ2βk⎛⎝T−1−e−2θ0λ2βkT2θ0λ2βk⎞⎠≃T2θ0N∑k=1λ2βk,as T→∞ ≃ϖβdTN2βd+1(4β+2d)θ0,as N,T→∞. (2.3)

In particular, note that , when .

3 Asymptotics in large time and large number of Fourier modes

It is known that the estimator is unbiased, strongly consistent and asymptotically normal in two asymptotic regimes: and fixed, and and fixed; see for instance [CX15, Cia18] and references therein. In particular, for every fixed , , with probability one, and

 w-limN→∞Nβ/d+12(ˆθN,T−θ0) =N(0,(4β/d+2)θ0ϖβT), (3.1)

where denotes the limit in distribution111Whenever convenient, we will also use the notation ‘’ to denote the convergence in distribution of random variables., and

is a Gaussian random variable

222Throughout the text we will use the notation to denote a Gaussian random variable with mean and variance .

with mean zero and variance

. Similarly, for every fixed , , with probability one, and

 w-limT→∞√T(ˆθN,T−θ0)=N(0,2θ0/J), (3.2)

where .

To the best of our knowledge, the asymptotic properties of when (both) is not studied in the current literature. Besides this being an important question alone, the obtained results in this section will also serve as theoretical basis for investigating the statistical properties of the discretized version of the MLE studied later in this paper. In view of the above, naturally one should expect that the joint time-space consistency is satisfied. On the other hand, by (3.1)

 w-limT→∞limN→∞√TNβ/d+12(ˆθN,T−θ0)=N(0,(4β/d+2)θ0ϖβ), (3.3)

and by (3.2) same result holds for the swaped limiting order . While (3.3) does not have great statistical meaning, it leads to a reasonable ansatz that same identity (3.3) should be satisfied when both . Also note that, in view of (2.3), this estimator is also optimal in the mean-square sense, having the rate of convergence dictated by the Fisher information. Next we give a rigourous proof of these results.

Theorem 3.1.

Assume that and . Then, is strongly consistent, i.e.

 limN,T→∞ˆθN,T=θ0,  with % probability one, (3.4)

and asymptotically normal, i.e.

 w-limN,T→∞√TNβd+12(ˆθN,T−θ0)=N(0,(4β/d+2)θ0ϖβ). (3.5)
Proof.

For simplicity, we set , and hence for all . Since

 uk(t)=σλ−γk∫t0e−θ0λ2βk(t−s)dwk(s), k≥1, (3.6)

it is straightforward to show that

 Eu2k(t)= σ2λ−2β−2γk(1−e−2θ0λ2βkt)2θ0, (3.7) Eu4k(t)= 3σ4λ−4β−4γk(1−e−2θ0λ2βkt)2(2θ0)2. (3.8)

We note that

 ˆθN,T−θ0 =−σ∑Nk=1λ2β+γk∫T0uk(t)dwk(t)∑Nk=1λ4β+2γk∫T0u2k(t)dt =−σ∑Nk=1ξk,T∑Nk=1Var(ξk,T)⋅∑Nk=1Var(ξk,T)∑Nk=1λ4β+2γk∫T0u2k(t)dt, (3.9)

where

 ξk,T:=λ2β+γk∫T0uk(t)dwk(t).

To show consistency (3.4

), we will use the strong law of large numbers

[Shi96, Theorem IV.3.2]. From (3.7), we have

 Var(ξk,T)=λ4β+2γk∫T0Eu2k(t)dt=σ2λ2βk∫T01−e−2θ0λ2βkt2θ0dt≃σ2λ2βkT2θ0,as T→∞,

and thus,

 N∑k=1Var(ξk,T) ≃σ2T2θ0N∑k=1λ2βk,as T→∞ ≃σ2ϖβdTN2βd+1(4β+2d)θ0,as N,T→∞. (3.10)

Moreover, using (3.8), we get that

 Var(λ4β+2γk∫T0u2k(t)dt)≤E(λ4β+2γk∫T0u2k(t)dt)2≤λ8β+4γkT∫T0Eu4k(t)dt∼λ4βkT2,as T→∞. (3.11)

Hence, there exists such that for all ,

 ∞∑N=1Var(ξN,T)(∑Nk=1Var(ξk,T))2≤C1T∞∑N=1λ2βN(∑Nk=1λ2βk)2≤C2T∞∑N=11N2+2βd≤C3<∞, ∞∑N=1Var(λ4β+2γN∫T0u2N(t)dt)(∑Nk=1Var(ξk,T))2≤C4∞∑N=1λ4βN(∑Nk=1λ2βk)2≤C5∞∑N=11N2<∞,

where are some constants333Notoriously, we will denote by with subindexes generic constants that may change from line to line. independent of . Using the uniform boundedness of the above series, and employing the strong law of large numbers, we deduce that for every and , there exists independent of such that for ,

with probability one. Therefore,

 limN,T→∞σ∑Nk=1ξk,T∑Nk=1Var(ξk,T)=0andlimN,T→∞∑Nk=1Var(ξk,T)∑Nk=1λ4β+2γk∫T0u2k(t)dt=1 (3.12)

with probability one. From here, and using (3.9), the proof of (3.4) is complete.

Next, we will prove asymptotic normality property (3.5), starting with representation

 ˆθN,T−θ0=−σ∑Nk=1ξk,T(∑Nk=1Var(ξk,T))1/2⋅1(∑Nk=1Var(ξk,T))1/2⋅∑Nk=1Var(ξk,T)∑Nk=1λ4β+2γk∫T0u2k(t)dt. (3.13)

Let us consider the first term in (3.13). We will show that

 w-limN,T→∞σ∑Nk=1ξk,T(∑Nk=1Var(ξk,T))1/2=N(0,σ2).

By Burkholder–Davis–Gundy inequality and Cauchy–Schwartz inequality, we have

 Eξ4k,T =E(λ2β+γk∫T0uk(t)dwk(t))4≤C1λ8β+4γkE(∫T0u2k(t)dt)2 ≤C1λ8β+4γkT∫T0Eu4k(t)dt,

for some . By (3.8) and (3.11), there exists such that for all , , for some , and hence there exists independent of and such that for all and for all ,

 N∑k=1Eξ4k,T≤C3N4βd+1T2,

for some , independent of , and . We will verify the classical Lindeberg condition [Shi96, Theorem III.5.1], namely that for every ,

 limN,T→∞∑Nk=1E(ξ2k,T1{|ξk,T|>ε√∑Nk=1Var(ξk,T)})∑Nk=1Var(ξk,T)=0.

By Cauchy-Schwartz inequality and Chebyshev inequality,

 N∑k=1E(ξ2k,T1{|ξk,T|>ε√∑Nk=1Var(ξk,T)}) ≤∑Nk=1Eξ4k,Tε2∑Nk=1Var(ξk,T).

Consequently,

 ∑Nk=1E(ξ2k,T1{|ξk,T|>ε√∑Nk=1Var(ξk,T)})∑Nk=1Var(ξk,T)≤∑Nk=1Eξ4k,Tε2(∑Nk=1Var(ξk,T))2∼1ε2N,as N,T→∞.

Thus,

 ∑Nk=1ξk,T(∑Nk=1Var(ξk,T))1/2d−−−−−→N,T→∞N(0,1). (3.14)

We also note that

 (N∑k=1λ4β+2γk∫T0Eu2k(t)dt)1/2≃σ√ϖβd√TNβd+12√(4β+2d)θ0as N,T→∞.

In view of the strong law of large numbers,

 limN,T→∞∑Nk=1λ4β+2γk∫T0u2k(t)dt∑Nk=1λ4β+2γk∫T0Eu2k(t)dt=1

with probability one. Finally, combining all the above and using Slutzky’s theorem, (3.5) follows at once. This completes the proof. ∎

3.1 Asymptotic normality of the MLE by Malliavin-Stein’s approach

In this section, we give an alternative proof of (3.5) using tools and results from Malliavin calculus. While this method of proof has its stand along value, we also believe that it will serve as a theoretical base for future studies related to other discretized estimators and different asymptotic regimes.

As before, let , and for convenience, in this section we will use the following notations:

 FN,T :=ˆθN,T−θ0=−σ∑Nk=1λ2β+γk∫T0uk(t)dwk(t)∑Nk=1λ4β+2γk∫T0u2k(t)dt=:−F1(N,T)F2(N,T), ˆFN,T :=−σ∑Nk=1λ2β+γk∫T0uk(t)dwk(t)E[∑Nk=1λ4β+2γk∫T0u2k(t)dt]=−F1(N,T)C2N,T,

where . We note that can be written as a double stochastic integral and in view of [NP12, Theorem 2.7.7], belongs to the second-order chaos; see also Appendix B. Next we present a key technical result.

Lemma 3.2.

Let be the space endowed with the inner product defined in (B.1). Let be the Malliavin derivative defined in (B.3). Then, we have

 √Var(12∥CN,TDˆFN,T∥2H)⟶0, as N,T→∞.

The proof of Lemma 3.2 is deferred to Appendix A.

To prove asymptotic normality of , we will show that , as . The variance comes from the fact We split into

 CN,TFN,T=CN,T(FN,T−ˆFN,T)+CN,TˆFN,T. (3.15)

We note that

 CN,T(FN,T−^FN,T)=C2N,TF2(N,T)F1(N,T)CN,T(1−F2(N,T)C2N,T).

From (3.12),

 C2N,TF2(N,T)⟶1,1−F2(N,T)C2N,T⟶0,as N,T→∞

with probability 1. On the other hand, by Lemma 3.2, we have

 F1(N,T)CN,T=CN,TˆFN,Td⟶N(0,σ2),as N,T→∞.

Hence, by Slutzky’s theorem, we deduce that , as , which consequently implies that

 CN,T(FN,T−ˆFN,T)⟶0, as N,T→∞, in probability. (3.16)

To deal with the second term in (3.15), we note that by Lemma 3.2 and Proposition B.3, we get that

 limN,T→∞dTV(CN,TˆFN,T,N(0,σ2))=0.

Consequently, Theorem B.2 implies that

 w-limN,T→∞CN,TˆFN,T=N(0,σ2).

This, combined with (3.15) and (3.16), implies that

 w-limN,T→∞CN,TFN,T=N(0,σ2).

Finally, note that, by (3.10)

 CN,Tσ≃√ϖβd√TNβd+12(4β+2d)θ0,as N,T→∞,

which implies (3.5), and the proof is complete.

4 Asymptotic properties of the discretized MLE

In this section, we investigate statistical properties of the discretized version of MLE (2.2). Towards this end, we assume that the Fourier modes , , are observed at a uniform time grid

 0=t0

We consider the discretized MLE defined by

 ˜θN,M,T:=−∑Nk=1λ2β+2γk∑Mi=1uk(ti−1)[uk(ti)−uk(ti−1)]∑Nk=1λ4β+2γk∑Mi=1u2k(ti−1)Δt.

We are interested in studying the asymptotic properties of , as .

For simplicity of writing, we also introduce the following notations:

 YN,M,T :=N∑k=1λ2β+γkM∑i=1uk(ti−1)(wk(ti)−wk(ti−1)), YN,T :=N∑k=1λ2β+γk∫T0uk(t)dwk(t), IN,M,T :=N∑k=1λ4β+2γkM∑i=1u2k(ti−1)Δt, IN,T :=N∑k=1λ4β+2γk∫T0u2k(t)dt, VN,M,T :=N∑k=1λ4β+2γkM∑i=1uk(ti−1)∫titi−1(uk(t)−uk(ti−1))dt, Υ :=(ϖβ(4β/d+2)θ0)1/2.

A key step in the proofs of the main results is to write as

 ˜θN,M,T−θ0=θ0VN,M,TIN,M,T−σYN,M,TIN,M,T.