Estimation of the drift parameter for the fractional stochastic heat equation via power variation
aa r X i v : . [ m a t h . P R ] D ec Modern Stochastics: Theory and Applications 6 (4) (2019) 397–417https://doi.org/10.15559/19-VMSTA141
Estimation of the drift parameter for the fractionalstochastic heat equation via power variation
Zeina Mahdi Khalil, Ciprian Tudor ∗ Laboratoire Paul Painlevé, Université de Lille, CNRS, UMR 8524,F-59655 Villeneuve d’Ascq, France [email protected] (Z. Mahdi Khalil), [email protected] (C. Tudor)Received: 8 April 2019, Revised: 22 July 2019, Accepted: 11 September 2019,Published online: 3 October 2019
Abstract
We define power variation estimators for the drift parameter of the stochastic heatequation with the fractional Laplacian and an additive Gaussian noise which is white in timeand white or correlated in space. We prove that these estimators are consistent and asymptoti-cally normal and we derive their rate of convergence under the Wasserstein metric.
Keywords
Stochastic heat equation, fractional Brownian motion, fractional Laplacian, q variation, drift parameter estimation The purpose of this work is to estimate the drift parameter θ > of the fractionalstochastic heat equation ∂u θ ∂t ( t, x ) = − θ ( − ∆) α u θ ( t, x ) + ˙ W ( t, x ) , t ≥ , x ∈ R , (1)with vanishing initial conditions, where ( − ∆) α denotes the fractional Laplacian oforder α ∈ (1 , , θ > and W is a Gaussian noise which is white in time and whiteor correlated in space. ∗ Corresponding author.
Preprint submitted to VTeX / Modern Stochastics: Theory andApplications
Z. Mahdi Khalil, C. Tudor
The parameter estimation for stochastic partial differential equations (SPDEs inthe sequel) constitutes a research direction of wide interest in probability theory andmathematical statistics. We refer, among many others, to the recents surveys [16]and [4]. On the other side, there are relatively few works that consider the solu-tion to a SPDE observed at discrete points in time and/or in space. Among the firstworks in this direction, we refer to [18] and [17] for the maximum likelihood andleast square estimators for parabolic, respectively elliptic type SPDEs driven by aspace-time white noise. The study in [17] has been then extended in [2], by adding atime-varying volatility in the noise term and by using power variation techniques toestimate the parameter of the model. Other recent works on parameter estimates fordiscretely sampled SPDEs via power variations are [5, 3, 1, 21] and [24].In this paper, we extend the above results into two directions. Firstly, we replacethe standard Laplacian operator used in all the above references by a fractional Lapla-cian. On the other hand, we consider a simpler form, comparing to [2, 17], of thedifferential operator. Secondly, we also consider a noise term which is correlated inspace. Our purpose is to propose power variation type estimators for the drift param-eter in the stochastic model (1), based on discrete observations of the solution in timeor in space, and to analyze the consistency and the limit distribution of the estima-tors by taking advantage of the link between the solution and the fractional Brownianmotion. Our approach to construct and analyze the estimators for the drift parameteris based on the asymptotic behavior of the q -variations of the mild solution to (1). Itis well known (see, e.g., [7, 13, 23]) that there exists a strong link between the lawof this mild solution with θ = 1 and the fractional Brownian motion and related pro-cesses. We will use this connection in order to deduce the behavior of the q -variations(of suitable order q ) of the solutions to (1) and to prove the consistency, asymptoticnormality and Berry–Esséen bounds under the Wasserstein distance for the associatedestimators. For the situation when W is a space-time white noise, we will obtain twoestimators for the drift parameter: one based on the temporal variations and one basedof the spatial variations of the mild solution u θ . Similarly, two estimators are definedwhen the Gaussian noise W is white in time and colored in space (with the spatial co-variance given by the Riesz kernel). Even if the order of the variations which appearin the definition of the estimator is different in the four cases (this order may dependon the parameter α of the fractional Laplacian and/or on the spatial correlation), allthe estimators are asymptotically normal, they have the same rate of convergence oforder n − and they have the same distance to the Gaussian distribution. The case ofthe standard Laplacian (i.e., α = 2 ) has been studied in [21].We organize the paper as follows. In Section 2 we present general facts on thestochastic heat equation with the fractional Laplacian and the behavior of the varia-tions of the perturbed fractional Brownian motion. In Section 3 we discuss the driftparameter estimation for the fractional heat equation with a space-time white noisewhile in Section 4 we treat the case when the noise is correlated in space.We will denote by c , C a generic positive constant that may change from lineto line (or even inside of the the same line). By → ( d ) we denote the convergence indistribution while ≡ ( d ) stands for the equivalence of finite dimensional distributions. stimation of the drift parameter for the fractional stochastic heat equation via power variation We start by treating the fractional stochastic heat equation with a space-time whitenoise. We recall the basic properties of the solution, its relation with the fractionalBrownian motion and then we discuss the estimation of the drift parameter θ via the q -variations. On the standard probability space (Ω , F , P ) , we consider a centered Gaussian field ( W ( t, A ) , t ≥ , A ∈ B b ( R )) with covariance E W ( t, A ) W ( s, B ) = ( s ∧ t ) λ ( A ∩ B ) for every s, t ≥ , A, B ∈ B b ( R ) , (2)where λ denotes the Lebesgue measure on R and B b ( R ) is the class of bounded Borelsubsets of R . The Gaussian field W is usually called the space-time white noise.We will consider the stochastic heat equation ∂u θ ∂t ( t, x ) = − θ ( − ∆) α u θ ( t, x ) + ˙ W ( t, x ) , t ≥ , x ∈ R , (3)with vanishing initial condition u (0 , x ) = 0 for every x ∈ R . In the above equation, ( − ∆) α represents the fractional Laplacian of order α . We will assume in the sequelthat α ∈ (1 , . We refer to [6, 11, 10, 12] for the precise definition and other prop-erties of the fractional Laplacian operator. We will denote its Green kernel (or thefundamental solution) by G α , which represents the deterministic kernel that solvesthe heat equation without noise ∂∂t u ( t, x ) = − ( − ∆) α u ( t, x ) . It is know from theabove references that for t > , x ∈ R G α ( t, x ) = Z R e itξ − t | ξ | α dξ. (4)It is an immediate conclusion that the fundamental solution associated to the op-erator − θ ( − ∆) α u θ ( t, x ) is G α ( θt, x ) .The solution to (3) is understood in the mild sense, i.e., u θ ( t, x ) = Z t Z R G α ( θ ( t − s ) , x − y ) W ( ds, dy ) , (5)where the stochastic integral W ( ds, dy ) is the usual Wiener integral with respect tothe space-time white noise, which satisfies the isometry E Z T Z R H ( s, y ) W ( ds, dy ) ! = Z T Z R H ( s, y ) dyds for every T > and for every measurable square integrable function H .For θ = 1 , the solution to the heat equation (3) has been studied in [13]. Thissolution exists only if the spatial dimension is d = 1 , and it is connected to thebifractional Brownian motion. Recall that (see [9, 22]), given constants H ∈ (0 , Z. Mahdi Khalil, C. Tudor and K ∈ (0 , , the bifractional Brownian motion (bi-fBm for short) ( B H,Kt ) t ≥ is acentered Gaussian process with covariance R H,K ( t, s ) := R ( t, s ) = 12 K (cid:16)(cid:0) t H + s H (cid:1) K − | t − s | HK (cid:17) , s, t ≥ . (6)In particular, for K = 1 , B H, := B H, is the fractional Brownian motion (fBm in thesequel) with the Hurst parameter H ∈ (0 , .Let us recall some of the results in [13] which will be needed in the sequel.• The mild solution (5) is well-defined. For every x ∈ R , the process ( u ( t, x ) , t ≥ ) coincides in distribution, modulo a constant, with the bifractional Brow-nian motion, i.e., ( u ( t, x ) , t ≥ ≡ ( d ) (cid:16) c ,α B , − α t , t ≥ (cid:17) , where B , − α is a bifractional Brownian motion with the Hurst parameters H = and K = 1 − α and c ,α = c ,α − α with c ,α = 12 π ( α −
1) Γ (cid:18) α (cid:19) . (7)• For every t ≥ , we have (see Proposition 3.1 in [7]) ( u ( t, x ) , x ∈ R ) ≡ ( d ) (cid:16) m α B α − ( x ) + S t ( x ) , x ∈ R (cid:17) , (8)where B α − is a fractional Brownian motion with the Hurst parameter α − ∈ [0 , ] , ( S t ( x )) x ∈ R is a centered Gaussian process with C ∞ sample paths and m α is an explicit numerical constant.The above facts, combined with the decomposition (18) of the bifractional Brow-nian motion, show that the solution to the heat equation can be expressed as the sumof a fBm and a smooth process (we will call this sum as a perturbed fractional Brow-nian motion). Since the process (5) is connected to the perturbed fBm (i.e., the sum of a fBm anda smooth Gaussian process), let us recall some facts concerning the asymptotic be-havior of the variation of the perturbed fBm. Some of the below results are directlytaken from [13] while those concerning the rate of convergence under the Wassersteindistance are deduced from [19].We first define the notion of (exact) q -variation for stochastic processes. Definition 1.
Let A < A , and for n ≥ , let t i = A + in ( A − A ) for i = 0 , . . . , n .A continuous stochastic process ( X t ) t ≥ admits a q -variation (or a variation of order q ) over the interval [ A , A ] if the sequence S n,q [ A ,A ] ( X ) := n − X i =0 (cid:12)(cid:12) X t i +1 − X t i (cid:12)(cid:12) q stimation of the drift parameter for the fractional stochastic heat equation via power variation converges in probability as n → ∞ . The limit, when it exists, is called the exact q -variation of X over the interval [ A , A ] .If [ A , A ] = [0 , t ] , we will simply denote S n,qt ( X ) := S n,q [0 ,t ] ( X ) . Moreover, if t = 1 , we denote S q,n ( X ) := S n,qt ( X ) . In the case q = 2 the limit of S ,n is calledthe quadratic variation, while for q = 3 we have the cubic variation.Let us recall the following result (see [13]) concerning the exact variation of theperturbed fractional Brownian motion, i.e., the sum of a fBm and a smooth Gaussianprocess. In the rest of this section, we will fix an interval [ A , A ] with A < A anda partition t j = A + jn ( A − A ) , n ≥ , j = 0 , . . . , n , of this interval. Also, wedenote by Z a standard normal random variable, and µ q = E Z q for q ≥ . Define σ H,q = q ! P v ∈ Z ρ H ( v ) q , with ρ H ( v ) = (cid:0) | v + 1 | H + | v − | H − | v | H (cid:1) for v ∈ Z . Lemma 1.
Let ( B Ht ) t ≥ be a fBm with H ∈ (0 , ] and consider a centered Gaussianprocess ( X t ) t ≥ such that E | X t − X s | ≤ C | t − s | for every s, t ≥ . (9) Define Y Ht = aB Ht + X t for every t ≥ with a = 0 .1. The process Y has H -variation over the interval [ A , A ] which is equal to a − H E | Z | /H ( A − A ) .
2. Let V q,n ( Y H ) := n − X i =0 (cid:20) n Hq ( A − A ) qH a q ( Y Ht i +1 − Y Ht i ) q − µ q (cid:21) . (10) Then, if H ∈ (0 , ) and q ≥ is an integer, √ n V q,n ( Y H ) = 1 √ n n − X i =0 (cid:20) n Hq ( A − A ) qH a q ( Y Ht i +1 − Y Ht i ) q − µ q (cid:21) → ( d ) N (0 , σ H,q ) . (11) If H = , q = 2 and the process ( X t ) t ≥ is adapted to the filtration generatedby B , then √ n V ,n ( Y H ) = 1 √ n n − X i =0 (cid:20) n ( A − A ) a ( Y t i +1 − Y t i ) − (cid:21) → ( d ) N (0 , σ , ) . (12)Using the recent Stein–Malliavin theory, it is also possible to deduce the rateof convergence in the above Central Limit Theorem (CLT in the sequel) under theWasserstein distance. Before stating and proving the result, let us briefly recall the Z. Mahdi Khalil, C. Tudor definition of the Wasserstein distance. The Wasserstein distance between the laws oftwo R d -valued random variables F and G is defined as d W ( F, G ) = sup h ∈A | E h ( F ) − E h ( G ) | (13)where A is the class of Lipschitz continuous function h : R d → R such that k h k Lip ≤ , where k h k Lip = sup x,y ∈ R d ,x = y | h ( x ) − h ( y ) |k x − y k R d . Proposition 1.
Assume H ≤ . Let Y H be as in Lemma 1 and let V q,n ( Y H ) be givenby (10). Then for n large and with σ H,q from (11), d W (cid:18) √ n V q,n ( Y H ) , N (0 , σ H,q ) (cid:19) ≤ C √ n . Proof.
From the proof of Lemma 2.1 in [13], we can express the variation of Y H and the variation of the fBm B H plus a rest term, i.e., √ n V q,n ( Y H ) = 1 √ n V q,n ( B H ) + R n , where R n satisfies, for every n ≥ , E | R n | ≤ cn H − . (14)By the definition of the Wasserstein distance, we can write d W (cid:18) √ n V q,n ( Y H ) , N (0 , σ H,q ) (cid:19) ≤ d W (cid:18) √ n V q,n ( B H ) , N (0 , σ H,q ) (cid:19) + d W (cid:18) √ n V q,n ( Y H ) , √ n V q,n ( B H ) (cid:19) ≤ d W (cid:18) √ n V q,n ( B H ) , N (0 , σ H,q ) (cid:19) + E | R n | . In order to estimate d W ( √ n V q,n ( B H ) , N (0 , σ H,q )) , we will use the chaos expansionof the random variable V q,n ( B H ) and several results in [19]. Notice that (see, e.g.,the proof of Corollary 3 in [20]), V q,n ( B H ) = q X k =1 k ! C kq µ q − k n − X i =0 H k (cid:18) n HK ( A − A ) HK (cid:16) B Ht i +1 − B Ht i (cid:17)(cid:19) , where H k is the k -th probabilists’ Hermite polynomial H k ( x ) = ( − k e − x d n dx n (cid:16) e − x (cid:17) stimation of the drift parameter for the fractional stochastic heat equation via power variation for k ≥ with H ( x ) = 1 . We know from [19] that the vector ( F ,n , F ,n , . . . , F q,n ) := √ n n − X i =0 H k (cid:18) n HK ( A − A ) HK (cid:16) B Ht i +1 − B Ht i (cid:17)(cid:19)! k =1 ,...,q converges in distribution to a centered Gaussian vector with diagonal covariance ma-trix C (the explicit expression of C can be found in [19], it is not needed in our work).Moreover, Proposition 6.2.2 and Corollary 7.4.3 in [19] imply that d W (( F k,n ) k =1 ,...,q , N (0 , C )) ≤ c √ n . This will easily lead to d W (cid:18) √ n V q,n ( B H ) , N (0 , σ H,q ) (cid:19) ≤ c √ n . (15)Since H ≤ , we obtain the conclusion via (14) and (15). Our purpose is to estimate the parameter θ > based on the observations of the pro-cess u θ . We will define two estimators: the first is based on the temporal variations ofthe process u θ while the second is constructed via its variation in space. Their behav-ior is strongly related to the law of the process u θ , therefore we start by analyzing thedistribution of this Gaussian process. Let G α ( t, x ) be the Green kernel associated to the operator − ( − ∆) α . Then the Greenkernel associated to the operator operator − θ ( − ∆) α is G α ( θt, x ) . Lemma 2.
Suppose that the process ( u θ ( t, x ) , t ≥ , x ∈ R ) satisfies (3). Define v θ ( t, x ) := u θ (cid:18) tθ , x (cid:19) , t ≥ , x ∈ R . (16) Then the process ( v θ ( t, x ) , t ≥ , x ∈ R ) satisfies ∂v θ ∂t ( t, x ) = − ( − ∆) α v θ ( t, x ) + ( θ ) − ˙ f W ( t, x ) , t ≥ , x ∈ R , (17) with v θ (0 , x ) = 0 for every x ∈ R , where ˙ f W is a space-time white noise, i.e., acentered Gaussian random field with covariance (2). Proof.
From (5), we have for every t ≥ , x ∈ R , v θ ( t, x ) = u θ (cid:18) tθ , x (cid:19) = Z tθ Z R G α ( t − θs, x − y ) W ( ds, dy ) Z. Mahdi Khalil, C. Tudor = Z t Z R G α ( t − s, x − y ) W ( d sθ , dy )= θ − Z t Z R G α ( t − s, x − y ) ˜ W ( ds, dy ) , where, for t ≥ , A ∈ B ( R ) , we denoted ˜ W ( t, A ) := θ W (cid:0) tθ , A (cid:1) . Notice that ˜ W has the same finite dimensional distributions as W , due to the scaling property of thewhite noise.We can deduce the law of the process u θ in time and space. Proposition 2.
For every x ∈ R and θ > , we have ( u θ ( t, x ) , t ≥ ≡ ( d ) (cid:16) θ − α c ,α B , − α t , t ≥ (cid:17) , where B , − α is a bifractional Brownian motion with parameters H = and K =1 − α and c ,α is given by (7). Proof.
Fix x ∈ R and θ > . Then for every s, t ≥ , we have E u θ ( t, x ) u θ ( s, x ) = E v θ ( θt, x ) v θ ( θs, x )= θ − E u ( θt, x ) u ( θs, x ) = θ − c ,α h ( θt + θs ) − α − | θt − θs | − α i = θ − α c ,α E B , − α t B , − α s . Proposition 3.
For every t ≥ , θ > , we have the following equality in distribution ( u θ ( t, x ) , x ∈ R ) ≡ ( d ) (cid:16) θ − m α B α − ( x ) + S θt ( x ) , x ∈ R (cid:17) , where B α − is a fractional Brownian motion with the Hurst parameter α − ∈ (0 , ] , ( S θt ( x )) x ∈ R is a centered Gaussian process with C ∞ sample paths and m α from (8). Proof.
The result is immediate since for every t > , θ > u θ ( t, x ) , x ∈ R ) = ( v θ ( θt, x ) , x ∈ R ) ≡ ( d ) θ − ( u ( θt, x ) , x ∈ R ) ≡ ( d ) (cid:16) θ − m α B α − ( x ) + S θt ( x ) , x ∈ R (cid:17) , where we used (8).Notice that the Hurst parameter of the fBm in Proposition 3 may be if α = 2 . Proposition 2 indicates that the process u θ behaves as a bi-fBm in time. Recall thefollowing connection between the fBm and the bi-fBm (see [14]): Let H ∈ (0 , , K ∈ (0 , . If ( B HKt ) t ≥ is a fBm with the Hurst parameter HK and ( B H,Kt ) t ≥ isa bi-fBm, then (cid:16) C X H,Kt + B H,Kt , t ≥ (cid:17) ≡ ( d ) (cid:0) C B HKt , t ≥ (cid:1) , (18) stimation of the drift parameter for the fractional stochastic heat equation via power variation with C > and C = 2 − K . In (18), X H,K is a Gaussian process, independentof B H,K with C ∞ sample paths. In particular, it satisfies (9). Therefore, the bi-fBmis a perturbed fBm and the same holds true for the solution ( u θ ( t, x ) , t ≥ , byProposition 2. Therefore, we obtain, by using the notation t j = A + jn ( A − A ) , n ≥ , j = 0 , . . . , n , the following lemma. Lemma 3.
Let u θ be the solution to (3). Then for every x ∈ R , S n, αα − [ A ,A ] := n − X i =0 | u θ ( t j +1 , x ) − u θ ( t j , x ) | αα − → n →∞ c αα − ,α α − µ αα − ( A − A ) | ( θ ) | − α − (19) in probability. Relation (19) motivates the definition of the following estimator for the parameter θ > of the model (3): b θ n, = (cid:18) c αα − ,α α − µ αα − ( A − A ) (cid:19) − n − X i =0 | u θ ( t j +1 , x ) − u θ ( t j , x ) | αα − ! − α = (cid:18) c αα − ,α α − µ αα − ( A − A ) (cid:19) α − (cid:16) S n, αα − ( u θ ( · , x )) (cid:17) − α , (20)and so b θ − α n, = 1 c αα − ,α α − µ αα − ( A − A ) n − X i =0 | u θ ( t j +1 , x ) − u θ ( t j , x ) | αα − . (21)We will prove the consistency and the asymptotic normality of the above estima-tor. Proposition 4.
Assume q := αα − is an even integer and consider the estimator b θ n, defined by (20). Then b θ n, → n →∞ θ in probability and √ n (cid:20)b θ − α n, − θ − α (cid:21) → ( d ) N (0 , s ,θ,α ) with s ,θ,α = σ q ,q θ − α µ − αα − . (22) Moreover, for n large enough d W (cid:18) √ n (cid:20)b θ − α n, − θ − α (cid:21) , N (0 , s θ,α ) (cid:19) ≤ c √ n . Proof.
From Proposition 2 and the relation between the fBm and the bi-fBm (18),we obtain that (cid:16) u θ ( t, x ) + c ,α θ − α X t (cid:17) ≡ ( d ) c ,α θ − α α B α − α , Z. Mahdi Khalil, C. Tudor where B α − α is a fBm with the Hurst parameter α − α ∈ (0 , ) . Therefore, u θ is aperturbed fBm and we obtain, by taking H = α − α and q = H = αα − , √ n n − X i =0 nθ α − c αα − ,α α − ( A − A ) ( u θ ( t j +1 , x ) − u θ ( t j , x )) αα − − θ − α → ( d ) N (0 , σ q ,q ) . This means √ nµ αα − θ α − (cid:20)b θ − α n, − θ − α (cid:21) → ( d ) N (0 , σ q ,q ) which is equivalent to (22).Using the so-called delta-method, we can get the asymptotic behavior of the esti-mator b θ n . Recall that if ( X n ) n ≥ is a sequence of random variables such that √ n ( X n − γ ) → ( d ) N (0 , σ ) and g is a function such that g ′ ( γ ) exists and does not vanish, then √ n ( g ( X n ) − g ( γ )) → ( d ) N (0 , σ g ′ ( γ ) ) . (23) Proposition 5.
Consider the estimator (20) and let s ,θ,α be given by (22). Then as n → ∞ , √ n ( b θ n, − θ ) → N (0 , s ,θ,α (1 − α ) θ αα − ) , (24) and for n large enough, d W (cid:16) √ n ( b θ n, − θ ) , N (0 , s ,θ,α (1 − α ) θ αα − ) (cid:17) ≤ c √ n . Proof.
By applying the delta-method for the function g ( x ) = x − α , X n = b θ − α n, and γ = θ − α , we immediately obtain the convergence (24). Concerning the rate ofconvergence, we can write, with e γ a random point located between X n and γ , √ n ( g ( X n ) − g ( γ )) = √ ng ′ ( e γ )( X n − γ )= g ′ ( γ ) √ n ( X n − γ ) + √ n ( X n − γ )( g ′ ( e γ ) − g ′ ( γ ))=: g ′ ( γ ) √ n ( X n − γ ) + T n . We have, for n large, E | T n | = E (cid:12)(cid:12) √ n ( X n − γ )( g ′ ( e γ ) − g ′ ( γ )) (cid:12)(cid:12) ≤ (cid:16) E (cid:0) √ n ( X n − γ ) (cid:1) (cid:17) (cid:0) E ( g ′ ( e γ ) − g ′ ( γ )) (cid:1) ≤ c (cid:0) E ( g ′ ( e γ ) − g ′ ( γ )) (cid:1) ≤ c (cid:18) E (cid:16)b θ αα − n, − θ αα − (cid:17) (cid:19) stimation of the drift parameter for the fractional stochastic heat equation via power variation ≤ c E (cid:18)b θ α − n, − θ α − (cid:19) ! ≤ c √ n where we used the assumption α > for the first inequality of the line above andrelation (22) (which gives in particular the L (Ω) -convergence of b θ α − n, to θ αα − as n → ∞ ) for the second inequality on the same line. Therefore, by the triangle in-equality and Proposition 4, for n large enough, d W (cid:16) √ n ( b θ n, − θ ) , N (0 , s ,θ,α (1 − α ) θ αα − ) (cid:17) ≤ cd W (cid:0) √ n ( X n − γ ) , N (0 , s ,θ,α ) (cid:1) + E | T n | ≤ c √ n . It is possible to define an estimator for the parameter θ based on the spatial variationsof the solution (5). The result in Proposition 3 says that the process ( u θ ( t, x ) , x ∈ R ) is a perturbed fBm, so we know its exact variation in space. Below x j = A + jn ( A − A ) , j = 0 , . . . , n , will denote a partition of the interval [ A , A ] . Proposition 6.
Let u θ be given by (3). Then n − X i =0 | u θ ( t, x j +1 ) − u θ ( t, x j ) | α − → n →∞ m α − α µ α − ( A − A ) | θ | − α − and if q := α − is an integer, √ n n − X i =0 " nm α − α ( A − A ) ! θ α − ( u θ ( t, x i +1 ) − u θ ( t, x i )) α − − µ α − → ( d ) N (0 , σ α − , α − ) . Proposition 6 leads to the definition of the estimator b θ n, = " ( m α − α µ α − ( A − A )) − n − X i =0 | u θ ( t, x j +1 ) − u θ ( t, x j ) | α − − α , (25)and we can immediately deduce from Proposition 3 its asymptotic proprieties. Proposition 7.
The estimator (25) converges in probability as n → ∞ to the param-eter θ . Moreover, if q := α − is an even integer, √ n (cid:20)b θ − α n, − θ − α (cid:21) → ( d ) N (0 , s ,θ,α ) with s ,θ,α = σ α − , α − µ − α − θ − α . (26) Moreover, for n large, d W (cid:18) √ n (cid:20)b θ − α n, − θ − α (cid:21) , N (0 , s ,θ,α ) (cid:19) ≤ c √ n . Z. Mahdi Khalil, C. Tudor
Proof.
Using the law of the process ( u θ ( t, x ) , x ∈ R ) obtained in Proposition 3, wededuce that the Gaussian process (cid:16) θ m − α u θ ( t, x ) , x ∈ R (cid:17) is a perturbed fractionalBrownian motion. Therefore, by relation (11) in Lemma 1, √ n n − X i =0 nθ α − ( A − A ) m α − α ( u θ ( t, x j +1 ) − u θ ( t, x j )) α − − µ α − ! = √ nµ α − θ α − (cid:20)b θ − α n, − θ − α (cid:21) → ( d ) n →∞ N (cid:16) , σ α − , α − (cid:17) . Moreover, Proposition 1 implies that d W (cid:18) √ nµ α − θ α − (cid:20)b θ − α n, − θ − α (cid:21) , N (0 , σ α − , α − ) (cid:19) ≤ c √ n and this obviously leads to the desired conclusion.By using the delta-method, we can obtain the asymptotic distribution of b θ n, . Proposition 8.
Let b θ n, be given by (25). Then, with s ,θ,α from (26), as n → ∞ , √ n ( b θ n, − θ ) → ( d ) N (cid:16) , s ,θ,α (1 − α ) θ αα − (cid:17) , and for n large enough, d W (cid:16) √ n ( b θ n, − θ ) , N (0 , s ,θ,α (1 − α ) θ αα − ) (cid:17) ≤ c √ n . Proof.
It suffices to apply (23) to the function g ( x ) = x − α and γ = θ − α and tofollow the proof of Proposition 5. Remark 1. • The estimators (20) and (25) coincide with the estimators in [21] in the case ofthe standard Laplacian α = 2 .• The distance of the estimators (20) and (25) to their limit distribution is of thesame order, although they involve q -variations with different q . In this section, we will consider the stochastic heat equation with an additive Gaus-sian noise which behaves as a Wiener process in time and as a fractional Brownianmotion in space, i.e. its spatial covariance is given by the so-called Riesz kernel. Wewill again study the distribution of the solution, its connection with the fractionaland bifractional Brownian motion and we apply the q -variation method to obtain anasymptotically normal estimator for the drift parameter. stimation of the drift parameter for the fractional stochastic heat equation via power variation We will consider the stochastic heat equation ∂∂t u θ ( t, x ) = − θ ( − ∆) α u θ ( t, x ) + ˙ W γ ( t, x ) , t ≥ , x ∈ R d , (27)with u θ (0 , x ) = 0 for every x ∈ R d . In (27), − ( − ∆) α denotes the fractional Lapla-cian with exponent α , α ∈ (1 , , and W γ is the so-called white-colored noise, i.e. W γ ( t, A ) , t ≥ , A ∈ B ( R d ) , is a centered Gaussian field with covariance E W γ ( t, A ) W γ ( s, B ) = ( t ∧ s ) Z A Z B f ( x − y ) d x d y , (28)where f is the so-called Riesz kernel of order γ given by f ( x ) = R γ ( x ) := g γ,d k x k − d + γ , < γ < d, (29)where g γ,d = 2 d − γ π d/ Γ(( d − γ ) / / Γ( γ/ . As usual, the mild solution to (27) isgiven by u θ ( t, x ) = Z t Z R d G α ( θ ( t − s ) , x − z ) W γ ( ds, d z ) , (30)where the above integral W γ ( ds, d z ) is a Wiener integral with respect to the Gaussiannoise W γ .We know the following facts concerning the mild solution (30) when θ = 1 .• The mild solution (27) is well-defined as a square integrable process satisfying sup t ∈ [0 ,T ] , x ∈ R d E | u ( t, x ) | < ∞ if and only if d < γ + α. (31)In particular, condition (31) shows that the solution exists in any spatial dimen-sion d , via suitable choice of the parameter γ .• Assume (31) is satisfied. Then for every x ∈ R d , we have the following equiv-alence in distribution ( u ( t, x ) , t ≥ ≡ ( d ) (cid:18) c ,α,γ B , − d − γα t , t ≥ (cid:19) , (32)where B , − d − γα is a bifractional Brownian motion with the Hurst parameters H = and K = 1 − d − γα and c ,α,γ = c ,α,γ − d − γα (33)with c ,α,γ = (2 π ) − d Z R d dξ k ξ k − γ e −k ξ k α − d − γα ) . Z. Mahdi Khalil, C. Tudor • For every t ≥ , we have (see Proposition 4.6 in [13]) (cid:0) u ( t, x ) , x ∈ R d (cid:1) ≡ ( d ) (cid:16) m α,γ B α + γ − d ( x ) + S t ( x ) , x ∈ R d (cid:17) , (34)where B α + γ − d is an isotropic d -dimensional fractional Brownian motion (seethe next section) with the Hurst parameter α + γ − d , ( S t ( x )) x ∈ R d is a centeredGaussian process with C ∞ sample paths and m α,γ is an explicit numericalconstant. Since the law of the solution (30) is related to the isotropic fBm, let us recall thedefinition of this process. The isotropic d -parameter fBm (also known as the LévyfBm) ( B Hd ( x ) , x ∈ R d ) with the Hurst parameter H ∈ (0 , is defined as a centeredGaussian process, starting from zero, with covariance function E ( B Hd ( x ) B Hd ( y )) = 12 (cid:0) k x k H + k y k H − k x − y k H (cid:1) for every x , y ∈ R d , (35)where k · k denotes the Euclidean norm in R d . It can be also represented as a Wienerintegral with respect to the Wiener sheet, see [8, 15].As in the one-parameter case, we define the q -variation of the isotropic fBm asthe limit in probability as n → ∞ of the sequence S n,q [ A ,A ] ( B H ) = n − X i =0 (cid:12)(cid:12) B Hd ( x i +1 ) − B Hd ( x i ) (cid:12)(cid:12) q , where x i = ( x (1) i , . . . , x ( d ) i ) with x ( j ) i = A + in ( A − A ) for i = 0 , . . . , n and j = 1 , . . . , d . And from [13] we know that the isotropic fBm ( B H ( x )) x ∈ R d has H -variation over [ A , A ] which is equal to ( A − A ) E | B Hd ( ) | /H = ( A − A ) √ d E | Z | /H . The q -variation of the isotropic fBm perturbed by a regular multiparameter pro-cess has been obtained in [13], Lemma 4.1. Lemma 4.
Let ( B H ( x )) x ∈ R d be a d -parameter isotropic fBm and consider a d -pa-rameter stochastic process ( X ( x )) x ∈ R d , independent of B H , that satisfies E (cid:12)(cid:12) X ( x ) − X ( y ) (cid:12)(cid:12) ≤ C k x − y k , for every x , y ∈ R d . (36) Define Y ( x ) = B Hd ( x ) + X ( x ) for every x ∈ R d . Then:1. The process ( Y ( x )) x ∈ R d has H -variation which is equal to ( A − A ) √ d E | Z | /H . stimation of the drift parameter for the fractional stochastic heat equation via power variation
2. If H ∈ (0 , ) and q ≥ , √ n V q,n ( Y H ) := 1 √ n n − X i =0 (cid:20) n Hq d − Hq/ ( A − A ) qH ( Y H ( x i +1 ) − Y H ( x i )) q − µ q (cid:21) → ( d ) N (0 , σ H,q ) . (37)It is immediate to deduce the rate of convergence in the above central limit theo-rem. Recall that we denoted by d W the Wasserstein distance. Proposition 9.
Let Y H be as in the statement of Lemma 4. Then for n large, d W (cid:18) √ n V q,n ( Y H ) , N (0 , σ H,q ) (cid:19) ≤ C √ n . Proof.
We notice that the Gaussian vector (cid:0) B Hd ( x i +1 ) − B Hd ( x i ) (cid:1) , ,...,n − has thesame law as d H/ ( B H ( x j +1 ) − B H ( x j )) , ,...,n − where B is a one-parameter fBmwith the Hurst parameter H and we then apply Lemma 1. Therefore, the distributionof the sequence √ n V q,n ( B Hd ) is independent of d ≥ and we can use the sameargument as in the proof of Proposition 1 above. Throughout this section we will assume (31). As in the previous section, we willconstruct and analyze estimators for the drift parameter θ by using the limit behaviorof the variations (in time and in space) of the process (30). Let us start by analyzing the distribution of the solution to (27) and its link with the(bi)fractional Brownian motion.
Proposition 10.
For every x ∈ R d and θ > , we have ( u θ ( t, x ) , t ≥ ≡ ( d ) (cid:18) θ − d − γ α c ,α,γ B , − d − γα t , t ≥ (cid:19) , where B , − d − γα is a bifractional Brownian motion with parameters H = and K = 1 − d − γα and the constant c ,α,γ is defined by (33). Proof.
Denote v θ ( t, x ) = u θ (cid:18) tθ , x (cid:19) for every t ≥ , x ∈ R d . Then, as in Lemma 2, v θ solves the equation ∂v θ ∂t ( t, x ) = − ( − ∆) α v θ ( t, x ) + ( θ ) − ˙ f W γ ( t, x ) , t ≥ , x ∈ R d , (38)with v θ (0 , x ) = 0 for every x ∈ R d , where ˙ f W γ is a white colored Gaussian noise(i.e. a Gaussian process with zero mean and covariance (28)). Z. Mahdi Khalil, C. Tudor
Fix x ∈ R d and θ > . For every s, t ≥ , we have E u θ ( t, x ) u θ ( s, x ) = E v θ ( θt, x ) v θ ( θs, x )= θ − E u ( θt, x ) u ( θs, x )= θ − c ,α,γ h ( θt + θs ) − d − γα − | θt − θs | − d − γα i = θ − d − γα c ,α,γ E B , − d − γα t B , − d − γα s . For the behavior with respect to the space variable, we obtain the following result.
Proposition 11.
For every t ≥ , θ > , we have the following equality in distribu-tion (cid:0) u θ ( t, x ) , x ∈ R d (cid:1) ≡ ( d ) (cid:16) θ − m α,γ B α + γ − d ( x ) + S θt ( x ) , x ∈ R d (cid:17) , where B α + γ −− d is a fractional Brownian motion with the Hurst parameter α + γ − d ∈ (0 , ] , ( S θt ( x )) x ∈ R d is a centered Gaussian process with C ∞ sample paths and m α,γ from (34). Proof.
The result is immediate since for a fixed time t > (cid:0) u θ ( t, x ) , x ∈ R d (cid:1) = (cid:0) v θ ( θt, x ) , x ∈ R d (cid:1) ≡ ( d ) θ − (cid:0) u ( θt, x ) , x ∈ R d (cid:1) ≡ ( d ) (cid:16) θ − m α,γ B α + γ − d ( x ) + S θt ( x ) , x ∈ R d (cid:17) . Again t j = A + jn ( A − A ) , j = 0 , . . . , n , will denote a partition of the interval [ A , A ] . Lemma 5.
Assume (31). Let u θ be the solution to (27). Then for every x ∈ R d , theprocess ( u θ ( t, x ) , t ≥ admits αα + γ − d -variation over the interval [ A , A ] , i.e. S n, αα + γ − d [ A ,A ] := n − X i =0 | u θ ( t j +1 , x ) − u θ ( t j , x ) | αα + γ − d → n →∞ c αα + γ − d ,α,γ d − γα + γ − d µ αα + γ − d ( A − A ) | θ | γ − dα + γ − d (39) in probability. Proof.
Clearly, for fixed x ∈ R d , n − X i =0 | u θ ( t j +1 , x ) − u θ ( t j , x ) | αα + γ − d = n − X i =0 | v ( θt j +1 , x ) − v ( θt j , x ) | αα + γ − d , where ( v θ ( t, x ) , t ≥ ≡ ( d ) ( θ − u ( t, x ) , t ≥ . And from Proposition . in [13]we know that u admits a variation of order αα + γ − d which is equal to c αα + γ − d ,α,γ C , − d − γα ( A − A ) with C , − d − γα = 2 d − γα + γ − d µ αα + γ − d and it means that n − X i =0 | u θ ( t j +1 , x ) − u θ ( t j , x ) | αα + γ − d stimation of the drift parameter for the fractional stochastic heat equation via power variation → n →∞ c αα + γ − d ,α,γ d − γα + γ − d µ αα + γ − d ( θA − θA ) | θ − | αα + γ − d c αα + γ − d ,α,γ d − γα + γ − d µ αα + γ − d ( A − A ) | θ | γ − dα + γ − d . From relation (39) we can naturally define the following estimator for the param-eter θ > of the stochastic partial differential equation (27) b θ n, = (cid:18) c αα + γ − d ,α,γ d − γα + γ − d µ αα + γ − d ( A − A ) (cid:19) − × n − X i =0 | u θ ( t j +1 , x ) − u θ ( t j , x ) | αα + γ − d ! α + γ − dγ − d = (cid:18) c αα + γ − d ,α,γ d − γα + γ − d µ αα + γ − d ( A − A ) (cid:19) d − γα + γ − d × (cid:16) S n, αα + γ − d ( u θ ( · , x )) (cid:17) α + γ − dγ − d , (40)and so b θ γ − dα + γ − d n, = 1 c αα + γ − d ,α,γ d − γα + γ − d µ αα + γ − d ( A − A ) n − X i =0 | u θ ( t j +1 , x ) − u θ ( t j , x ) | αα + γ − d . (41)We have the following asymptotic behavior. Proposition 12.
Assume αα + γ − d := q is an even integer and consider the estimator b θ n, in (40). Then b θ n, → n ∞ θ in probability and √ n (cid:20)b θ γ − dα + γ − d n, − θ γ − dα + γ − d (cid:21) → ( d ) N (0 , s ,θ,α,γ ) with s ,θ,α,γ = σ q ,q θ γ − d ) α + γ − d µ − αα + γ − d , (42) and for n large enough, d W (cid:18) √ n (cid:20)b θ γ − dα + γ − d n, − θ γ − dα + γ − d (cid:21) , N (0 , s θ,α ) (cid:19) ≤ c √ n . (43) Proof.
From Proposition 10 and the relation between the fractional and bifractionalBrownian motion (see (18)), we can see that, as n → ∞ , (cid:16) c − ,α,γ d − γ α θ d − γ α u θ ( t, x ) , t ≥ (cid:17) converges to a perturbed fBm with Hurst parameter H = α − d + γ α . By taking H = α + γ − d α and q = H = αα + γ − d in Lemma 1, we get √ n n − X i =0 nθ d − γα + γ − d c αα + γ − d ,α,γ d − γα + γ − d ( A − A ) ( u θ ( t j +1 , x ) − u θ ( t j , x )) αα + γ − d − µ αα + γ − d Z. Mahdi Khalil, C. Tudor → N (0 , σ q ,q ) or, equivalently √ nµ αα + γ − d θ d − γα + γ − d (cid:20)b θ γ − dα + γ − d n, − θ γ − dα + γ − d (cid:21) → N (0 , σ q ,q ) , which is equivalent to (22). The bound (43) follows easily from Proposition 1.We finally obtain the asymptotic normality and the rate of convergence for theestimator b θ n, . Proposition 13.
Let b θ n, be given by (40) and s ,θ,α,γ be given by (42). Then as n → ∞ , √ n (cid:16)b θ n, − θ (cid:17) → ( d ) N , s ,θ,α,γ (cid:18) α + γ − dγ − d (cid:19) θ αα + γ − d ! and d W √ n (cid:16)b θ n, − θ (cid:17) , N , s ,θ,α,γ (cid:18) α + γ − dγ − d (cid:19) θ αα + γ − d !! ≤ c √ n . Proof.
It suffices to apply (23) with g ( x ) = x α + γ − dγ − d and γ = θ γ − dγ + α − d and to followthe proof of Proposition 5. We will repeat the method employed in the previous parts of our work in order todefine an estimator expressed in terms of the variations in space of the process (30)for the parameter θ in (27).Recall that we proved in Proposition 11 that for every fixed time t > , (cid:16) θ m − α,γ u θ ( t, x ) , x ∈ R d (cid:17) is a perturbed multiparameter isotropic fractional Brownian motion as defined inLemma 4. Then we can deduce the variation in space of u θ recalling that x i =( x (1) i , . . . , x ( d ) i ) with x ( j ) i = A + in ( A − A ) for i = 0 , . . . , n and j = 1 , . . . , d . Proposition 14.
Let u θ be given by (30). Then n − X i =0 | u θ ( t, x j +1 ) − u ( θ t, x j ) | α + γ − d → n →∞ m α + γ − d α,γ ( A − A ) √ dµ α + γ − d | θ | − α + γ − d Proof.
We use Lemma 4, point 1. stimation of the drift parameter for the fractional stochastic heat equation via power variation
For every n ≥ , define b θ n, = " ( m α + γ − d α,γ µ α + γ − d √ d ( A − A )) − × n − X i =0 | u θ ( t, x j +1 ) − u θ ( t, x j ) | α + γ − d − ( α + γ − d ) , (44)and so b θ − α + γ − d n, = 1 m α + γ − d α,γ µ α + γ − d √ d ( A − A ) n − X i =0 | u θ ( t, x j +1 ) − u θ ( t, x j ) | α + γ − d . (45)We can deduce the asymptotic properties of the estimator by using Lemma 4 andProposition 9. Proposition 15.
The estimator (44) converges in probability as n → ∞ to the pa-rameter θ . Moreover, if α + γ − d is an even integer, then √ n (cid:20)b θ − α + γ − d n, − θ − α + γ − d (cid:21) → N (0 , s ,θ,α,γ ) with s ,θ,α,γ = σ α + γ − , α + γ − d µ − α + γ − d θ − α + γ − d . We also have, for n large enough, d W (cid:18) √ n (cid:20)b θ − α + γ − d n, − θ − α + γ − d (cid:21) , N (0 , s ,θ,α,γ ) (cid:19) ≤ c √ n . Finally, we get the following proposition.
Proposition 16.
With b θ n, from (44), as n → ∞ , √ n (cid:16)b θ n, − θ (cid:17) → ( d ) N , s ,θ,α,γ (cid:18) α + γ − dγ − d (cid:19) θ αα + γ − d ! and d W p n, (cid:16)b θ n − θ (cid:17) , N , s ,θ,α,γ (cid:18) α + γ − dγ − d (cid:19) θ αα + γ − d !! ≤ c √ n . Proof.
Apply again (23) with g ( x ) = x α + γ − dγ − d and γ = θ γ − dγ + α − d . Remark 2.
Notice that in the case γ = 1 (i.e., there is no spatial correlation and inthis case d has to be 1), we retrieve the results of Section 2. Observe, as in Section2, that the distance of the estimators (40) and (44) to their limit distribution is of thesame order, although they involve q -variations of different orders. Z. Mahdi Khalil, C. Tudor
To conclude, in this paper we provide estimators based on power variation for the driftparameter θ of the solution to the fractional stochastic heat equation (3). The noveltyof our approach is that it allows, comparing with the literature on statistical inferencefor SPDEs (see [4, 17, 2], etc.), to consider the case of a Gaussian noise with non-trivial spatial correlation and to treat the situation when the differential operator in theheat equation (3) is the fractional Laplacian instead of the standard Laplacian. Theproofs of the asymptotic behavior of the estimators are relatively simple and they arebased on the link between the law of the solution and the fractional Brownian motion,using known results on the behavior of the power variations of the fBm. Our approachalso gives the rate of convergence of the estimators under the Wasserstein distance viasome recent results in Stein–Malliavin calculus (see [19]). We assumed for simplicitya vanishing initial condition in (3) but the case of a notrivial initial value, whose powervariations are dominated by those of the fBm, can be also treated by our approach.Another open problem of interest that could by treated via our techniques is adding anunknown volatility parameter in the disturbance term and jointly estimating the driftand the volatility parameters. The case of the fractional heat equation on boundeddomains is also interesting but in this case the fundamental solution and implicitlythe law of the mild solution changes. Consequently, the relation between the law ofthe solution and the fBm is not obvious and therefore new techniques are needed. Funding
C. Tudor is partially supported by Labex Cempi (ANR-11-LABX-0007-01) andMATHAMSUD Project SARC (19-MATH-06).
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