# On central limit theorems for power variations of the solution to the stochastic heat equation

We consider the stochastic heat equation whose solution is observed discretely in space and time. An asymptotic analysis of power variations is presented including the proof of a central limit theorem. It generalizes the theory from arXiv:1710.03519 in several directions.

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## 1 Introduction and main result

Stochastic partial differential equations (SPDEs) do not only provide key models in modern probability theory, but also become increasingly popular in applications, for instance, in neurobiology or mathematical finance. Consequently, statistical methods are required to calibrate SPDE models from given observations. However, in the statistical literature on SPDEs, see

[5] for a recent review, there are still basic questions which are not yet settled.

A natural problem is parameter estimation based on discrete observations of a solution of an SPDE which was first studied in

[10] and which has very recently attracted considerable interest. Applying similar methods the three related independent works [6, 2, 4] study parabolic SPDEs including the stochastic heat equation, consider high-frequency observations in time, construct estimators using power variations of time-increments of the solution and prove central limit theorems. As we shall see below, the marginal solution process along time at a fixed spatial point is not a (semi-)martingale such that the well-established high-frequency theory for stochastic processes from [8] cannot be (directly) applied. In view of this difficulty, different techniques are required to prove central limit theorems. Interestingly, the proof strategies in [6, 2, 4] are quite different. Cialenco and Huang [6] consider the realised fourth power variation for the stochastic heat equation with both an unbounded spatial domain , or a bounded spatial domain . In the first setting they apply the central limit theorem by Breuer and Major [3] for stationary Gaussian sequences with sufficient decay of the correlations. For

, they use Malliavin calculus instead and the fourth moment theorem from

[12]. Also in case of a bounded domain , with Dirichlet boundary conditions, Bibinger and Trabs [2] study the normalized discrete quadratic variation and establish its asymptotic normality building upon a theorem by Peligrad and Utev [14] for triangular arrays which satisfy a covariance inequality related to -mixing. Finally, Chong [4] has proved (stable) central limit theorems for power variations in the case , based on a non-obvious martingale approximation in combination with the theory from [8]. The strategy of proofs by [2] and [4] do not directly rely on a purely Gaussian model and can be transferred to more general settings. While [2] considers further nonparametric inference on a time-varying deterministic volatility, [4] already provides a proof beyond the Gaussian framework including stochastic volatility.

This note presents a concise analysis which transfers the asymptotic theory from [2] to an unbounded spatial domain , and from the normalized discrete quadratic variation to general power variations. Contrarily to [2], we do not start with the illustration of a solution as an infinite-dimensional SDE but exploit the explicit representation of the solution with the heat kernel thanks to the continuous spectrum of the Laplace operator on the whole real line. We stick here to the simplest Gaussian setting to illustrate the main aspects and deviations from the classical theory. Our findings show that the central limit theorem under a -mixing type condition used in [2] for the case with a bounded spatial domain can be used likewise for this different model with unbounded spatial domain. We moreover expect that it provides a perspective to prove central limit theorems very generally, although many approximation details, for instance, to address stochastic volatility, remain far from being obvious. We consider the stochastic heat equation in one spatial dimension

 ∂tXt(x)=ϑ2∂xxXt(x)+σ˙W(t,x),X0(x)=ξ(x),t>0,x∈R, (1)

for space-time white noise

, and with parameters , and some initial condition which is independent of . is defined as a centred Gaussian process with covariance structure , and is in terms of a distribution the space-time derivative of a Brownian sheet. Since the Laplace operator on the whole real line does not have a discrete spectrum and we do not have to discuss boundary problems, the asymptotic analysis actually simplifies compared to [2] and allows for more transparent proofs.

A mild solution of (1) is a random field that admits the representation

 Xt(x)=∫RG(t,x−y)ξ(y)dy+∫t0∫RG(t−s,x−y)σ˙W(ds,dy), (2)

for , where the integral is well-defined as the stochastic Walsh integral and with

 G(t,x):=exp(−x2/(2ϑt))√2πϑt.

is the heat kernel, the fundamental solution to the heat equation. Let us refer to [9, Ch. 2.3.1] for an introduction to the heat equation and SPDEs in general. Suppose we observe this solution on a discrete grid , at equidistant observation times . We consider infill or high-frequency asymptotics where . For statistical inference on the parameters in (1), the key quantities to study are power variations

 Vpn(x):=1nn∑i=1∣∣∣ΔiX(x)Δ1/4n∣∣∣p,ΔiX(x):=XiΔn(x)−X(i−1)Δn(x),

with . The normalization of with takes into account the (almost) -Hölder regularity in time of , see [9, Ex. 2.3.5]. By homogeneity in space, statistics to consider for volatility estimation are spatial averages

 ¯Vpn,m:=1mm∑k=1Vpn(xk)=1nmn∑i=1m∑k=1∣∣∣ΔiX(xk)Δ1/4n∣∣∣p. (3)

The main result of this note is a central limit theorem for in the double asymptotic regime where and (possibly) . An important role in our asymptotic analysis is played by the second-order increment operator for some function , being well defined on . For brevity we assume , but the result readily extends to sufficiently regular initial conditions which are independent of .

###### Theorem 1.

Consider (1) with . For assume that as . Then the power variations from (3) with satisfy as and

 √m⋅n(¯Vpn,m−(2πϑ)p4σpμp)d⟶N(0,(2πϑ)p2σ2p((μ2p−μ2p)+2∞∑r=2ρp(12D2(√⋅,r)))),

with , and with for

jointly normally distributed with expectation 0, variances 1 and correlation

.

Note the explicit formula , also referred to as for even. In particular for , that is, for the normalized discrete quadratic variation, we have and the asymptotic variance is

 ((2πϑ)1/4σ)4(2+∞∑r=2(D2(√⋅,r))2)

in analogy with Example 2.11 in [4] and with [2]. This coincides with the variance of the normalized discrete quadratic variation of a fractional Brownian motion with Hurst exponent and scale parameter , see also Theorem 6 in [1] and [11].

The above result allows for a growing time horizon and, more general than in [2], the number of spatial observations in the unbounded spatial domain can be larger than the number of observation times . The relevant condition that induces de-correlated observations in space is , tantamount to a finer observation frequency in time than in space. Based on Theorem 1, one can construct estimators and confidence statements for the parameters and , if the other one is known, see [6, 2, 4]. If no parameter is known apriori, [2, Sec. 5] show that the “viscosity-adjusted volatility” can be estimated consistently, also noted in [4, Sec. 2.3].

## 2 High-frequency asymptotic analysis of power variations

Our analysis builds upon the following result, whose proof is postponed to Section 3.

###### Proposition 2.

For with , we have that

 Cov(ΔiX(x),ΔjX(x))= √Δn√2πϑσ2(1{i=j}+12D2(√⋅,|i−j|+1)1{i≠j}+12D2(√⋅,i+j))and |Cov(ΔiX(x),ΔjX(y))|= O(Δn|x−y|(1|i−j−1|∨1+1{i=j})).

The increments thus have non-negligible covariances and is not a (semi-)martingale. The terms will turn out to be asymptotically negligible in the variance of the power variations. Since second-order differences of the square root decay as its second derivative, we observe that . This motivates an asymptotic theory exploiting -mixing arguments. From the proposition and joint normality of the increments, we readily obtain the expectation and variance of the power variations at one spatial point .

###### Corollary 3.

For any , we have that E[V_n^p(x)]=(2πϑ)^p4σ^pμ_p+O(n^-1) and
Var(V_n^p(x))=1n(2πϑ)^p2σ^2p((μ_2p-μ_p^2)+2
_r=2^ρ_p(12 D_2(,r)))+O(1n) with , and with for jointly centred Gaussian with variances 1 and correlation .

###### Proof.

For , Proposition 2 yields . Since , we obtain by a Taylor expansion that

 E[Vpn(x)]=1nn∑i=1μp∣∣σ2√2/πϑ(1+12D2(√⋅,2i))∣∣p2=(2πϑ)p4σpμp+O(n−1).

Using the joint normality of the increments , and writing , with a tight sequence , we deduce for any that

 Var(Vpn(x)) (4) =σ2p(2πϑ)p2(1n2n∑i=1Var(|Z1|p)|1+12D2(√⋅,2i)|p+2n2n∑i=1i−1∑j=1Cov(|~Zx,i|p,|~Zx,j|p)).

By the above bound, the term with is negligible such that up to this negligible term. For the covariance terms, we use Proposition 2 to obtain

 =1nn∑i=1i−1∑j=1ρp(corr(~Zx,i,~Zx,j))+O(1) =1nn∑i=1i−1∑j=1ρp(12D2(√⋅,i−j))+O(1) =∞∑r=2ρp(12D2(√⋅,r))+O(1).

The first equality comes from approximating the variances by one and the second approximation is based on the Hermite expansion of absolute power functions (15) with Hermite rank 2, see also [1, (A.6)]. The last estimate follows from

 ρp(12D2(√⋅,i−j))=O(D2(√⋅,i−j)2)=O((i−j)−3),i>j.

As we can see from the previous proof, the term in the variance would also appear for independent increments, while the additional term involving comes from the non-vanishing covariances. Proposition 2 moreover implies that the covariance of and decreases with a growing distance of the spatial observation points and . In particular, averaging over all spatial observations in (3) reduces the variance by the factor , as long as the high-frequency regime in time dominates the spatial resolution. The next corollary determines the asymptotic variance in Theorem 1.

###### Corollary 4.

Under the conditions of Theorem 1, we have that

###### Proof.

For bivariate Gaussian with correlation and variances and , we exploit the inequality

 Cov(|U1|p,|U2|p)⩽Cpσp1σp2a2,

with some constant , which is based on the Hermite expansion (15) and given in Equation (4) of [7], see also Lemma 3.3 of [13].
By this inequality and Proposition 2 for , we deduce that

 1m2∑k≠lCov(Vpn(xk),Vpn(xl)) =O(1m2n2∑i,j∑k≠lΔn|xk−xl|2(1{i=j}+1|i−j−1|∨1)2).

With the estimate

 1n2n∑i=1Δn|xk−xl|2+1n2n∑i=3i−2∑j=1Δn|xk−xl|21(i−j−1)2 ⩽Δnn|xk−xl|2+Δnn2|xk−xl|2n∑i=3i−2∑k=1k−2=O(Δnn|xk−xl|2),

we obtain in combination with Corollary 3 that

 Var(¯Vpn,m) =1m2(m∑k=1Var(Vpn(xk))+∑k≠lCov(Vpn(xk),Vpn(xl))) =1mn(2πϑ)p2σ2p((μ2p−μ2p)+2∞∑r=2ρp(12D2(√⋅,r)))+O(1mn)

under the condition , where we use that

 ∑k≠l1|xk−xl|2⩽2δ−2mm∑k=2k−1∑l=1l−2=O(log(m)δ−2m)=O(mδ−2m). (5)

We turn to the proof of the central limit theorem transferring the strategy from [2] to our model. Define the triangular array

 Zn,i:=1√mnm∑k=1Δ−p4n(|ΔiX(xk)|p−E[|ΔiX(xk)|p]).

Peligrad and Utev [14, Thm. B] established the central limit theorem , with variance , under the following conditions:

1. The variances satisfy and there is a constant , such that

 Var(b∑i=aZn,i)⩽Cb∑i=aVar(Zn,i)for all 0⩽a⩽b⩽n.
2. The Lindeberg condition is fulfilled:

 limn→∞n∑i=1E[Z2n,i1{|Zn,i|>ε}]=0for all ε>0.
3. The following covariance inequality is satisfied. For all , there is a function satisfying , such that for all integers :

 Cov(eit∑bi=aZn,i,eit∑ci=b+uZn,i)⩽ρt(u)c∑i=aVar(Zn,i).

Therefore, Theorem 1 follows if the conditions (A) to (C) are verified. (C) is a -mixing type condition generalizing the more restrictive condition from [16] that the triangular array is -mixing with a certain decay of the mixing coefficients.

Proof of Theorem 1 (A) follows from Proposition 2. More precisely, we can verify analogously to the proofs of the Corollaries 3 and 4 that

and we obtain that

 Var(b∑i=aZn,i)=(b−a+1)n(2πϑ)p2σ2p((μ2p−μ2p)+2∞∑r=2ρp(12D2(√⋅,r))) +O((b−a+1)nΔnδ2m+1n).

(B) is implied by the Lyapunov condition, since the normal distribution of yields with some constant that

 n∑i=1E[Z4n,i]⩽Cn∑i=1(E[Z2n,i])2=O(n−1)→0.

(C) Define . For a decomposition , where is independent of , an elementary estimate with the Cauchy-Schwarz inequality shows that

 |Cov(eitQba,eitQcb+u)|⩽2t2Var(Qba)1/2Var(A1)1/2, (6)

see [2, (52)]. To determine such a suitable decomposition, we write for

 ΔiX(x) :=¯¯¯¯Bbi(x)+B––bi(x),where ¯¯¯¯Bbi(x) :=∫tb0∫RΔiG(s,x−y)σ˙W(ds,dy), (7) ΔiG(s,x) :=G(ti−s,x)−G(ti−1−s,x), B––bi(x) :=∫ti−1tb∫RΔiG(s,x−y)σ˙W(ds,dy) +∫titi−1∫RG(ti−s,x−y)σ˙W(ds,dy).

Then, we set and

 A2:=1√mnc∑i=b+um∑k=1Δ−p4n(∣∣B––bi(xk)∣∣p−E[|B––bi(xk)|p]),

where is indeed independent from .

###### Lemma 5.

Under the conditions of Theorem 1, holds.

This auxiliary lemma is proved in Section 3. In combination with , with some constant , and (6), we obtain condition (C):

 |Cov(eitQba,eitQcb+u)|=O(t2u−14Var(Qca)).

This completes the proof of the central limit theorem for and Theorem 1.

## 3 Remaining proofs

In this section, we write for .

### 3.1 Proof of Proposition 2

Since , with from (7) and

 Ci(x)=∫titi−1∫RG(ti−s,x−y)σ˙W(ds,dy), (8)

with and centred and independent for , we derive for that

 Cov(ΔiX(x),ΔjX(y))=E[ΔiX(x)ΔjX(y)] (9)

Noting that is the density of , we obtain for based on the identity for the convolution that

 ∫RG(r1−s,x1−y)G(r2−s,x2−y)dy (10) =∫RG(r1−s,u)G(r2−s,(x2−x1)−u)du=G(r1+r2−2s,x2−x1).

We moreover obtain for and :

 ∫r30G(r1+r2−2s,y)ds=∫r1+r2r1+r2−2r3121√2πϑue−y2/(2ϑu)du =1√2πϑ(√r1+r2e−y2/(2ϑ(r1+r2))−√r1+r2−2r3e−y2/(2ϑ(r1+r2−2r3))) −yϑP(y√r1+r2⩽√ϑZ⩽y√r1+r2−2r3),Z∼N(0,1).

Based on that, we determine the terms in (9). Setting

 κ:=|x−y|/√Δn,gκ(s):=√se−κ2/(2ϑs) and hκ(s):=P(Z⩾κ/√ϑs), (11)

we obtain for by the generalization of Itô’s isometry for Walsh integrals

 E[¯¯¯¯Bi−1i(x)¯¯¯¯Bj−1j(y)]=σ2E[(∫ti−10∫RΔiG(s,x−z)˙W(ds,dz)) (12) ×(∫tj−10∫RΔjG(s,y−z)˙W(ds,dz))] =σ2∫tj−10∫RΔiG(s,x−z)ΔjG(s,y−z)dzds =σ2√Δn√2πϑ(12gκ(i+j)−12gκ(i−j+2)−gκ(i+j−1) +gκ(i−j+1)+12gκ(i+j−2)−12gκ(i−j)) −σ2√Δnκϑ(hκ(i+j)−hκ(i−j+2)−2hκ(i+j−1) +2hκ(i−j+1)+hκ(i+j−2)−hκ(i−j)) =√Δnσ22√2πϑ(D2(gκ,i+j)−D2(gκ,i−j+2)) −σ2√Δnκϑ(D2(hκ,i+j)−D2(hκ,i−j+2)).

Similarly, we have for that

 E[¯¯¯¯Bi−1i(x)Cj(y)]=σ2∫tjtj−1∫RΔiG(s,x−z)G(tj−s,y−z)dzds (13) =σ2∫tjtj−1(G(ti+tj−2s,x−y)−G(ti−1+tj−2s,x−y))ds =σ22√Δn√2πϑ(gκ(i−j+2)−gκ(i−j)−gκ(i−j+1)+gκ(i−j−1)) −σ2√Δnκϑ(hκ(i−j+2)−hκ(i−j)−hκ(i−j+1)+hκ(i−j−1)).

For , with , we obtain that

 E[Ci(x)Ci(y)] =σ2∫titi−1∫RG(ti−s,x−z)G(ti−s,y−z)dzds (14) =σ22√Δn√2πϑgκ(2)−σ2√Δnκϑhκ(2).

Inserting (12), (13) and (14) in (9) yields

 Cov(ΔiX(x),ΔjX(y)) =σ2√Δn√2πϑ(gκ(1)1{i=j}+12D2(gκ,|i−j|+1)1{i≠j}+12D2(gκ,i+j)) −σ2√Δnκϑ(2hκ(1)1{i=j}+D2(hκ,|i−j|+1)1{i≠j}+D2(hκ,i+j)).

For we have and obtain the result in Proposition 2. Since the second derivative of is bounded by for all , we deduce for . Similarly, implies for . With for , we conclude that for :

 |Cov(ΔiX(x),ΔjX(y))|≲