# Generalized k-variations and Hurst parameter estimation for the fractional wave equation via Malliavin calculus

We analyze the generalized k-variations for the solution to the wave equation driven by an additive Gaussian noise which behaves as a fractional Brownian with Hurst parameter H>1/2 in time and which is white in space. The k-variations are defined along filters of any order p≥ 1 and of any length. We show that the sequence of generalized k-variation satisfies a Central Limit Theorem when p> H+1/4 and we estimate the rate of convergence for it via the Stein-Malliavin calculus. The results are applied to the estimation of the Hurst index. We construct several consistent estimators for H and these estimators are analyzed theoretically and numerically.

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11/26/2021

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## 1 Introduction

Since several decades, the statistical inference in stochastic (partial) differential equations (S(P)DE in the sequel) constitutes an intensive research direction in probability theory and mathematical statistics. Nowadays, a particular case of wide interest is represented by the S(P)DE driven by fractional Brownian motion (fBm) and related processes, due to the vast area of application of such stochastic models. Many recent works concern the estimation of the drift parameter for stochastic equation driven by fractional Brownian motion (we refer, among many others, to

[1], [9], [12], [24]), while fewer works deal with the estimation of the Hurst parameter in such stochastic equations.

In this paper, we consider the stochastic wave equation driven by an additive Gaussian noise which behaves as a fractional Brownian motion in time and as a Wiener process in space (we call it fractional-white noise). Our purpose is to construct and analyze an estimator for the Hurst parameter of the solution to this SPDE based on the observation of the solution at a fixed time and at a discrete number of points in space. The wave equation with fractional noise in time and/or in space has been studied in several works, such as [3], [7], [8], [11], [20] etc. We will use a standard method to construct estimators for the Hurst parameter, which is based on the -variations of the observed process. The method has been recently employed in [11] for the case of quadratic variations, i.e. . As for the fBm, it was shown that the standard quadratic variation estimator is not asymptotically normal when the Hurst index becomes bigger than and this is inconvenient for statistical applications. In order to avoid this restriction and to get an estimator which is asymptotically Gaussian for every , we will use the generalized -variations, which basically means that the usual increment of the process is replaced by a higher order increment. The idea comes from the reference [10] and since it has been used by many authors (see e.g. [6] or [5]). More precisely, if

denotes the solution to the wave equation with fractional-white noise, we define the (centered) generalized

-variation statistics ( integer),

 VN(k,α)=1N−lN∑i=l⎡⎢ ⎢ ⎢⎣∣∣Uα(iN)∣∣kE∣∣Uα(iN)∣∣k−1⎤⎥ ⎥ ⎥⎦, (1)

where represents the spatial increment of the solution at along a filter of power (order) and length (see the next section for the precise definition).

By using chaos expansion and the recent Stein-Malliavin calculus we show that the sequence (1) satisfies a Central Limit Theorem (CLT) as (in the spirit of [4]) whenever and in this way the restriction can be avoided by choosing a filter of order , i. e. by replacing, for example, the usual increment by a higher order increment. We will obtain the rate of convergence under the Wasserstein distance for this convergence in law and we also prove a multidimensional CLT. So we generalize the findings in [11] to filters of any power and to -variations of any order and in addition we show that in the special case and a non-Gaussian limit theorem occurs with limit distribution related to the Rosenblatt distribution (but more complex than it).

These theoretical results are then applied to the estimation of Hurst index of the solution the the fractional-white wave equation. Based on the behavior of the sequence (1), we prove that the associated -variation estimator for is consistent and asymptotically normal. Moreover, we provide a numerical analysis of the estimator when by analysing its performance on various filters and for several values of the Hurst parameter and confirming via simulation the theoretical results.

We organized the paper as follows. Section 2 contains some preliminaries. We present in this part the basic facts concerning the solution to the fractional-white wave equation, we introduce the filters and the increment of the solution along filters. In Section 3, we prove a CLT for the sequence (1) for any integer and we obtain the rate of convergence when is even via the Stein-Malliavin theory. In Section 4, we show a non-central limit limit in the case and for filters of order . Section 5 concerns the estimation of the Hurst parameter of the solution to the fractional-white wave equation (2.1). We included here theoretical results related to the behavior of the -variations estimators for the Hurst index as well as simulations and numerical analysis for the performance of the estimators. Section 6 (the Appendix) contains the basic tools from Malliavin calculus needed in the paper.

## 2 Preliminaries

We introduce here the fractional-white heat equation and its solution and we present the basic definitions and the notation concerning the filters used in our work.

### 2.1 The solution to the wave equation with fractional-colored noise

The object of our study will be the solution to the following stochastic wave equation

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∂2u∂t2(t,x)=Δu(t,x)+˙WH(t,x),t≥0,x∈Rd,d≥1,\omit\rm span\omit\rm span\@@LTX@noalign\vskip5.690551pt\omitu(0,x)=0,x∈Rd,\omit\rm span\omit\rm span\@@LTX@noalign\vskip5.690551pt\omit∂u∂t(0,x)=0,x∈Rd,

where is the Laplacian on , , and is a fractional-white Gaussian noise which is defined as a real valued centered Gaussian field , over a given complete filtered probability space , with covariance function given by

 E(WHt(A)WHs(B))=RH(t,s)λ(A∩B),∀A,B∈Bb(Rd), (2)

where is the covariance of the fractional brownian motion

 RH(t,s)=12(t2H+s2H−|t−s|2H),s,t≥0.

We denoted by the class of bounded Borel subsets of and we will assume throughout this work

The solution of the equation (2.1) is understood in the mild sense, that is, it is defined as a square-integrable centered field defined by

 u(t,x)=∫t0∫RdG1(t−s,x−y)WH(ds,dy),t≥0,x∈Rd, (3)

where is the fundamental solution to the wave equation and the integral in (3) is a Wiener integral with respect to the Gaussian process . Recall that for (we will later restrict to this situation in our work) we have, for and ,

 G1(t,x)=121{|x|

We know (see e.g. [3]) that the solution (3) is well-defined if and only if

 d<2H+1

and it is self-similar in time and stationary in space. Other properties of the solution can be found in [3], [7] or [23]. In particular, the spatial covariance of the solution can be expressed as follows

 E(u(t,x)u(t,y)) = 12(cH|y−x|2H+1−t|y−x|2H2+t2H+12H+1)1{|y−x|

with . When and , this expression reduces to

 E(u(t,x)u(t,y)) = 12(cH|y−x|2H+1−t|y−x|2H2+t2H+12H+1) for x,y∈[0,1]. (6)

We will fix for the rest of the work and we will associate to the process its canonical Hilbert space which is defined as the closure of the linear space generated by the indicator functions with respect to the inner product

 ⟨1[0,x],1[0,y]⟩H = E(u(t,x)u(t,y)).

We will denote by the multiple stochastic integral of order with respect to the Gaussian process and by the Malliavin derivative with respect to this process. We refer to the Appendix for the basic elements of the Malliavin calculus.

We will also use in Section 4 multiple stochastic integrals with respect to the fractional-white noise with covariance (2). We use the notation to indicate the multiple integral of order with respect to .

### 2.2 Filters

In this paragraph we will define the filters and the increments of the solution to (2.1) along filters. We start with several definitions and notations needed along this paper.

###### Definition 1

Given and

, a vector

is called a filter of length and order (or power) such that

 ⎧⎨⎩∑lq=0αqqr=0,0≤r≤p−1,∑lq=0αqqp≠0

with the convention .

For a filter of length and of order we define the space-filtered process (or the spatial increment of the process along the filter )

 Uα(iN)=l∑r=0aru(t,i−rN) for i=l,..,N. (7)

We denote for

 πα,NH(j):=E[Uα(iN)Uα(i+jN)].

From the covariance formula (5) we can write

 πα,NH(j) = l∑r1,r2=0ar1ar2E[u(t,i−r1N)u(t,i+j−r2N)] (8) = k11N2HΦH,α(j)+k21N2H+1ΦH+12,α(j)

with

 ΦH,α(j)=l∑r1,r2=0ar1ar2|j+r1−r2|2H,j≥0

and and . We write for further use

 c1(H)=−t4l∑q,r=0αqαr|q−r|2H, and c2(H)=cH2l∑q,r=0αqαr|q−r|2H+1. (9)

In particular, from (8)

 πα,NH(0) = E[Uα(iN)]2=k11N2HΦH,α(0)+k21N2H+1ΦH+12,α(0) = c1(H)1N2H+c2(H)1N2H+1

We will need the below technical lemma to etablish the asymptotical equivalent of and similar expressions. The proof of the lemma is based on a Taylor expansion, see [6] or [10].

###### Lemma 1

Let and , be filters of lengths , and of orders respectively. Then

 l1∑q=0l2∑r=0α(1)qα(2)r|q−r+k|2H∼k→∞ κHk2H−2p

with , where .

In the sequel, we write to indicate that the sequences have the same behavior as .

## 3 Central limit theorem for the spatial k-variations

In this section we focus on the asymptotic behavior in distribution of the -variation in space of the solution to the fractional-white wave equation, defined via a filter of power . In the first step we show the -variation satisfies a CLT when . Next, by taking to be an even integer, we derive a Berry-Esséen type bound for this convergence in distribution via the Stein-Malliavin calculus. Restricting ourselves in addition to , we prove a multidimensional CLT, which is needed for the estimation of the Hurst parameter.

### 3.1 Central Limit Theorem

Fix and let be a filter of length and of power as in Definition 1. Let be given by (3). For any integer we define the centered spatial -variations of the process by

 VN(k,α)=1N−lN∑i=l⎡⎢ ⎢ ⎢⎣∣∣Uα(iN)∣∣kE∣∣Uα(iN)∣∣k−1⎤⎥ ⎥ ⎥⎦ (10)

with given by (7). We will show that the sequence (10) satisfies a CLT. In order to do this we will use a criterion based on Malliavin calculus. The first step is to expand in chaos the -variation sequence . Noticing that the filtered process

as a linear combination of centered Gaussian random variables is a centered Gaussian process, we get

 E(Uα(iN)k)=EkE(Uα(iN)2)k2, (11)

where denotes the

-th absolute moment of a standard Gaussian variable given by

. We introduce the variable

 Zα(iN)=Uα(iN)(πα,NH(0))1/2. (12)

It is clear that is a standard Gaussian variable and , where denotes the correlation coefficient. Using (11) and (12) we can write as follows:

 VN(k,α) = 1N−lN∑i=l⎡⎣|Uα(iN)|kE|Uα(iN)|k−1⎤⎦=1N−lN∑i=l⎡⎣|Zα(iN)|kEk−1⎤⎦.

Recall the expansion of the development in Hermite polynomials of the function given in Lemma 2 of [6],

 Hk(t)=∞∑j=1ckjHj(t),

where for , for and denotes the j-th Hermite polynomial defined by

 Hj(t)=[j2]∑a=0(−1)aa!(j−2a)!a!2−atj−2a.

Observing that for

 Ci,α:=l∑q=0αq1[0,i−qN]

we have from (8) that we can express as an integral with respect to the process since the increment can be expressed as (recall that represents the multiple integral of order 1 with respect to the Gaussian process ) for every :

 Zα(iN) = I1(Ci,α(πα,NH(0))1/2).

Since we have for we get

 VN(k,α) = 1N−lN∑i=lHk(Zα(iN))=1N−l∑q⩾1ck2qN∑i=lH2q(Zα(iN)) = 1N−l∑q⩾1ck2qN∑i=lH2q(I1(Ci,α(πα,NH(0))1/2)) = 1N−l∑q⩾1ck2q(2q)!N∑i=lI2q⎛⎝(Ci,α(πα,NH(0))1/2)⊗2q⎞⎠.

Hence, we obtain the following chaotic expansion of the -variation statistics

 VN(k,α)=1N−lN∑i=l∞∑q=1ck2q(2q)!I2q⎛⎝C⊗2qi,α(πα,NH(0))q⎞⎠=∑q≥1I2q(fN,2q) (13)

with

 fN,2q=ck2q(2q)!1N−lN∑i=lC⊗2qi,α(πα,NH(0))q. (14)

Let us start by analyzing the asymptotic behavior of the mean square of each kernel .

###### Lemma 2

For , let be given by (14). Then

 (N−l)(2q)!∥fN,2q∥2H⊗2q→N→∞(ck2q)2(2q)!∑v∈Z(φH,α(v))2q:=σ22q

for (i.e. for and for ), where we use the notation

 φH,α(v)=ΦH,α(v)ΦH,α(0). (15)

Moreover, For ,

 N−llog(N−l)2!∥fN,2∥2H⊗2→N→∞c2:=(ck2)22limNlogN∑|v|≤N(ρH,α(v))2<∞. (16)

Proof: From (14), we get

 (2q)!∥fN,2q∥2H⊗2q = (ck2q)2(2q)!1(N−l)2N∑i,j=l⟨Ci,α,Cj,α⟩2qH(πα,NH(0))2q = 1(N−l)2(ck2q)2(2q)!N∑i,j=l(ρα,NH(j−i))2q,

where we used the notation

 ρα,NH(v)=πα,NH(v)πα,NH(0) for v∈Z. (17)

Next, we write

 1N−lN∑i,j=l(ρα,NH(j−i))2q=∑v∈Z(ρα,NH(v))2q1{|v|≤N−l}N−|v|−lN−l

and thus

 (N−l)(2q)!∥fN,2q∥2H⊗2q = (ck2q)2(2q)!∑v∈Z(ρα,NH(v))2q1{|v|≤N−l}N−|v|−lN−l. (18)

Using the expression

 ρα,NH(v)=k1ΦH,α(v)N−2H+k2ΦH+12,α(v)N−2H−1k1ΦH,α(0)N−2H+k2ΦH+12,α(0)N−2H−1=ΦH,α(v)+aN(v)ΦH,α(0)+aN(0)

with

 aN(v)=k2k1NΦH+12,α(v) (19)

we can write, with and given by (15) and (17) respectively,

 bN,H(v):=ρα,NH(v)−φH,α(v) (20)

and remark that due to Lemma 1 for large enough

 |bN,H(v)|∼|v|→∞∣∣∣aN(v)1ΦH,α(0)+aN(0)∣∣∣≤C1Nv2H+1−2p, (21)

where does not depend on .With this notation we can write

 (N−l)(2q)!∥fN,2q∥2H⊗2q = (ck2q)2(2q)!∑v∈Z(φH,α(v)+bN,H(v))2q1{|v|≤N−l}N−|v|−lN−l = (ck2q)2(2q)!∑v∈Z2q∑m=0(2qm)φH,α(v)m(bN,H(v))2q−m1{|v|≤N−l}N−|v|−lN−l = (ck2q)2(2q)!∑v∈ZφH,α(v)2q1{|v|≤N−l}N−|v|−lN−l+rN,q,1

with

 rN,q,1=(ck2q)2(2q)!∑v∈Z2q−1∑m=0(2qm)φH,α(v)m(bN,H(v))2q−m1{|v|≤N−l}N−|v|−lN−l. (22)

Clearly, by the dominated convergence theorem

 (ck2q)2(2q)!∑v∈ZφH,α(v)2q1{|v|≤N−l}N−|v|−lN−l→N→∞σ22q,

which by Lemma 1 is finite if , , and for all in the other cases. For , ,

 1log(N−l)∑v∈ZφH,α(v)21{|v|≤N−l}N−|v|−lN−l

converges to a positive constant and thus (16) is obtained.

In order to conclude it remains to show that the rest term (22) converges to as , for every . From (22), using the bound (21) and Lemma 1, we have the estimate

 |rN,q,1| ≤ C2q−1∑m=0(2qm)1N2q−m∑1≤v≤N−l|v|(2H−2)m|v|(2H+1−2p)(2q−m):=2q−1∑m=0rN,q,1,m

and for each ,

 rN,q,1,m≤CN2q−m∑1≤v≤N−l|v|(2H−2p)2q+2q−m.

When the series converges we get

 rN,q,1,m≤C1N2q−m≤C1N→N→∞0

and when the series diverges,

 rN,q,1,m≤C1N2q−mN(2H−2p)2q+2q−m+1≤CN(2H−2p)2q+1→N→∞0

if , or and . If , , the quantity

 1log(N−l)rN,q,1,m

will also converge to zero for using again (21) and Lemma 1.

The fact that the series for follows from the study of the -variations of the fractional Brownian motion, see [6] or [14].

We will consider the renormalized -variation sequence

 GN(k,α)=√N−lVN(k,α). (23)

From the above Lemma 2, it follows that

 E[GN(k,α)]2→N→∞σ2

with given in the statement of Lemma 2. We will show that the sequence (23) satisfies a central limit theorem.

###### Theorem 1

For a filter of order and of length , with , let be given by (23). Then the sequence converges in distribution, as , to the Gaussian law . Moreover, for , , the sequence converges in distribution to . The constants are those appearing in Lemma 2.

Proof: Notice that from (13), we can write

 GN(k,α)=∑q≥1I2q(gN,2q) with gN,2q=√N−lfN,2q (24)

with given by (14). Our main tool to prove the asymptotic normality of (24) is Theorem 6.3.1 from [14]. According to it, for it suffices to show that

1. and

2. for every and ,

and for , ,

1. ,

2. for every and ,

3. .

Point 1. in both cases follows from Lemma 2. Let us check point 2. By the definition of the contraction (48), we have for and

 gN,2q⊗rgN,2q = 1N−l(ck2q)2(2q)!N∑i,j=l⟨Ci<