Local linearization for estimating the diffusion parameter of nonlinear stochastic wave equations with spatially correlated noise
October 02, 2025
We study the bi-parameter local linearization of the one-dimensional nonlinear stochastic wave equation driven by a Gaussian noise, which is white in time and has a spatially homogeneous covariance structure of Riesz-kernel type. We establish that the second-order increments of the solution can be approximated by those of the corresponding linearized wave equation, modulated by the diffusion coefficient. These findings extend the previous results of Huang et al. [1], which addressed the case of space-time white noise. As applications, we analyze the quadratic variation of the solution and construct a consistent estimator for the diffusion parameter. Stochastic wave equation; local linearization; quadratic variation; parameter estimation.
60H15; 60G17; 60G22.
1 Introduction↩︎
Consider the non-linear stochastic wave equation (SWE, for short) in one spatial dimension: \[\label{SWE} \begin{cases}\displaystyle{\frac{\partial^2}{\partial t^2} u(t, x) =\frac{\partial^2}{\partial x^2} u(t, x) + F(u(t,x))\dot{W}(t, x), \quad t\ge 0, \, x \in \mathbb{R},}\\ \displaystyle{u(0, x) = 0, \quad \frac{\partial}{\partial t} u(0, x) =0.} \end{cases}\tag{1}\] The diffusion coefficient \(F:\mathbb{R}\rightarrow\mathbb{R}\) is assumed to be globally Lipschitz continuous. The term \(\dot{W}\) is a centered Gaussian field on a probability space \((\Omega, \mathcal{F}, \mathbb{P})\), which is white in time and has a spatially homogeneous correlation of fractional type. More precisely, the noise \(\dot{W}\) is defined by a family of centered Gaussian random variables \(\{W(\varphi), \, \varphi\in \mathcal{D}\}\), where \(\mathcal{D}:=C_0^{\infty}([0,\infty\times \mathbb{R}))\) is the space of infinitely differentiable functions with compact support, with the covariance function \[\begin{align} \begin{aligned}\label{Eq:noise95cov} \mathbb{E}[W(\varphi) W(\psi)] & = \int_{\mathbb{R}_+} dt \int_\mathbb{R}\mu(d\xi)\, \mathscr{F}(\varphi(t, \cdot))(\xi) \overline{\mathscr{F}(\psi(t, \cdot))(\xi)}, \end{aligned} \end{align}\tag{2}\] for all \(\varphi, \psi \in\mathcal{D}\). Here, \(\mathscr{F}(\varphi(s, \cdot))(\xi)\) denotes the Fourier transform of the function \(y \mapsto \varphi(s, y)\), \[\mathscr{F}(\varphi(s, \cdot))(\xi): = \int_\mathbb{R}e^{-i\xi y} \varphi(s, y) dy.\] For any \(H\in [1/2,1)\), the spectral measure \(\mu_H\) is given by \[\begin{align} \label{eq32constant} \mu(d\xi) :=c_H |\xi|^{1-2H} d\xi, \;\; \; \;\text{with }\; c_H = \frac{\Gamma(2H+1)\sin (\pi H)}{2\pi}. \end{align}\tag{3}\]
When \(H=\frac{1}{2}\), that is, \(\dot{W}\) is the space-time Gaussian white noise, the existence of a real-valued solution to 1 was studied in Walsh [2]. When \(H\in (\frac{1}{2},1)\), the solution to 1 was studied in Dalang [3] and the Hölder continuity was studied in Dalang and Sanz-Solé [4]. Huang et al. [1] recently established a local linearization property for solutions to the SWE 1 with space-time white noise. Following this research, we generalize the approximation scheme developed in [1] to the case of spatially colored noise in this paper.
To present the local linearization result, we first introduce the following stochastic heat equation (SHE, for short): \[\label{SHE} \begin{cases}\displaystyle{\frac{\partial}{\partial t} X(t, x) =\frac{\partial^2}{\partial x^2} X(t, x) + F(X(t,x)) \dot{W}(t, x), \quad t \ge 0, x \in \mathbb{R},}\\ \displaystyle{X(0, x) = 0.} \end{cases}\tag{4}\] It is well known that under the Lipschitz continuity of \(F\), the solution to 4 exhibits a local linearization property; see, e.g., [5], [6]. Specifically, let \(Y\) denote the linearized version of \(X\); that is, \(Y\) is the solution to the SHE 4 with \(F\equiv1\). Then, \[\label{SHE32appro} X(t, x + \varepsilon) - X(t, x) = F(X(t, x)) \left\{Y(t, x + \varepsilon) - Y(t, x)\right\} + R_{\varepsilon}(t, x),\tag{5}\] where the remainder term \(R_{\varepsilon}(t, x)\) tends to \(0\) as \(\varepsilon\to 0\) at a rate faster than \(Y(t, x + \varepsilon) -Y(t, x)\). The relation 5 implies that, for fixed \(t>0\), the local spatial fluctuations of \(X(t, \cdot)\) are essentially governed by those of \(Y(t, \cdot)\). In other words, ignoring precise regularity conditions, 5 indicates that \(X(t, \cdot)\) is controlled by \(Y(t, \cdot)\) in the sense of Gubinelli’s theory of controlled paths [7].
A similar local linearization behavior holds for temporal increments: for fixed \(t>0\) and \(x\in \mathbb{R}\), as \(\varepsilon \downarrow 0\), the increment \(X(t+\varepsilon, x)-X(t, x)\) admits an analogous structure. See [8]–[11]. Those results provide a quantitative framework for analyzing the local structure of sample paths of the solution, including properties such as Khinchin’s law of the iterated logarithm, Chung’s law of the iterated logarithm, quadratic variation of the process, and small-ball probability estimates. See, e.g., [12]–[16]. The local linearization of spatio-temporal increments of solutions to nonlinear SHEs has been further studied by Hu and Lee in [17].
In [18], Huang and Khoshnevisan investigated an analogous problem for the SWE with space-time white noise. They showed that, in contrast to the case of the SHE, the spatial increment of the solution to the SWE with initial data \((u_0, u_1) \equiv (0, 1)\) does not exhibit local linearization for fixed \(t\). Remarkably, by adopting a bi-parameter perspective, Huang et al. [1] revisited the local linearization problem for the SWE. Specifically, they introduced a new coordinate system \((\tau, \lambda)\) obtained by rotating the \((t, x)\)-coordinates by \(-45^\circ\). That is, \[\label{eq32transform} (\tau, \lambda): = \Big(\frac{t-x}{\sqrt{2}}, \frac{t+x}{\sqrt{2}}\Big) \quad \text{and} \quad (t, x) = \Big( \frac{\tau+\lambda}{\sqrt{2}}, \frac{-\tau+\lambda}{\sqrt{2}} \Big).\tag{6}\] For \(\tau>0\) and \(\lambda\ge -\tau\), define \[\label{SWE322} v(\tau, \lambda) := u\left(\frac{\tau+\lambda}{\sqrt{2}}, \frac{-\tau+\lambda}{\sqrt{2}}\right).\tag{7}\] Let \(U:=\{U(t,x)\}_{t\ge0, x\in \mathbb{R}}\) be the solution of the following linear SWE in one spatial dimension: \[\label{SWE32linear} \begin{cases}\displaystyle{\frac{\partial^2}{\partial t^2}U(t, x) =\frac{\partial^2}{\partial x^2} U(t, x) + \dot{W}(t, x), \quad t\ge 0, \, x \in \mathbb{R},}\\ \displaystyle{U(0, x) = 0, \quad \frac{\partial}{\partial t} U(0, x) = 0.} \end{cases}\tag{8}\] For any \(\tau\ge 0\) and \(\lambda\ge -\tau\), denote \[\label{SWE323} V(\tau, \lambda) := U\left(\frac{\tau+\lambda}{\sqrt{2}}, \frac{-\tau+\lambda}{\sqrt{2}}\right).\tag{9}\] For \(\varepsilon \in \mathbb{R}\), define the difference operators \(\delta_{\varepsilon}^{(j)}, j=1, 2\) by \[\label{eq32diff} \begin{align} \delta_{\varepsilon}^{(1)}f(\tau, \lambda):=&\, f(\tau+\varepsilon, \lambda)-f(\tau, \lambda),\\ \delta_{\varepsilon}^{(2)}f(\tau, \lambda):=&\, f(\tau, \lambda+\varepsilon)-f(\tau, \lambda). \end{align}\tag{10}\] Define the remainder terms \(R_{\varepsilon}^+(\tau, \lambda)\) and \(R_{\varepsilon}^-(\tau, \lambda)\) as follows: \[\label{eq32R} \begin{align} R_{\varepsilon}^{\pm}(\tau, \lambda):=& \,\delta_{\pm\varepsilon}^{(1)}\delta_{\varepsilon}^{(2)}v(\tau, \lambda)-F(v(\tau, \lambda))\delta_{\pm\varepsilon}^{(1)}\delta_{\varepsilon}^{(2)}V(\tau, \lambda)\\ =&\, \left\{ v(\tau\pm\varepsilon, \lambda+\varepsilon)-v(\tau\pm\varepsilon, \lambda) - v(\tau, \lambda+\varepsilon)+v(\tau, \lambda)\right\}\\ & \,\, -F(v(\tau, \lambda))\left\{V(\tau\pm\varepsilon, \lambda+\varepsilon)-V(\tau\pm\varepsilon, \lambda)-V(\tau, \lambda+\varepsilon)+V(\tau, \lambda) \right\}. \end{align}\tag{11}\] When \(H=\frac{1}{2}\), it is established in Huang et al. [1] that \[\left\| R_{\varepsilon}^{\pm}(\tau, \lambda)\right\|_{L^p(\Omega)}\le c(p)|\varepsilon|^{\frac{3}{2}},\] which converges to \(0\) as \(\varepsilon\to 0\) at a rate faster than \[\left\|Y(t, x + \varepsilon) -Y(t, x)\right\|_{L^p(\Omega)}\le c(p) |\varepsilon|,\] where \(c(p)\in(0,\infty)\).
The following local linearization result extends the previous work by Huang et al. in [1].
Theorem 1. Assume that \(H \in \left(\frac{1}{2}, 1\right)\). For any \(p \ge 1\), there exists a constant \(c(p)>0\) such that \[\begin{align} \label{eq32error} \left\| R_{\varepsilon}^{\pm}(\tau, \lambda)\right\|_{L^p(\Omega)} \le c(p)\varepsilon^{2H+\frac{1}{2}}, \end{align}\qquad{(1)}\] holds uniformly for all \(\tau > 0\), \(\lambda \ge -\tau\), and for all sufficiently small \(\varepsilon > 0\).
As an immediate consequence of Theorem 1, we are thus led to the following bi-parameter local linearization result for the solution of Eq. 1 in the original coordinates.
Theorem 2. Assume that \(H \in \left(\frac{1}{2}, 1\right)\), and let \(u\) and \(U\) be the solutions to equations 7 and 8 , respectively. Then, for any \((t,x) \in \mathbb{R}_+ \times \mathbb{R}\) and any \(p \ge 1\), it holds that \[\label{eq32error322} \begin{align} &\left\| \Delta_{\varepsilon}^{(1)} u(t, x) - F(u(t,x)) \Delta_{\varepsilon}^{(1)} U(t, x) \right\|_{L^p(\Omega)} \\ &\quad + \left\| \Delta_{\varepsilon}^{(2)} u(t, x) - F(u(t,x)) \Delta_{\varepsilon}^{(2)} U(t, x) \right\|_{L^p(\Omega)} \le c(p, t, x) \varepsilon^{2H + \frac{1}{2}}, \end{align}\qquad{(2)}\] uniformly for all sufficiently small \(\varepsilon > 0\). Here, the difference operators \(\Delta{\varepsilon}^{(1)}\) and \(\Delta_{\varepsilon}^{(2)}\) are defined for a function \(f(t, x)\) as follows: \[\begin{align} \Delta_{\varepsilon}^{(1)}f(t,x)=&\, f(t,x+2\varepsilon)-f(t-\varepsilon, x+\varepsilon)-f(t+\varepsilon, x+\varepsilon)+f(t,x),\\ \Delta_{\varepsilon}^{(2)}f(t,x)=&\, f(t+2\varepsilon, x)-f(t+\varepsilon, x-\varepsilon)-f(t+\varepsilon, x+\varepsilon)+f(t,x). \end{align}\]
For every \(N\ge1\) and \(0\le i\le N\), let \(\tau_i:=\frac{i}{N}, \lambda_j:=\frac{j}{N}\). Define the rescaled quadratic variation of the process \(v\) given in 7 by \[\begin{align} \label{eq32V32v} Q_{N}(v):= N^{2H-1}\sum_{i=0}^{N-1}\sum_{j=0}^{N-1}\Delta_{i,j}(v)^2, \end{align}\tag{12}\] with \[\begin{align} \label{eq32V32ij} \Delta_{i,j}(v) := v\left(\tau_{i+1}, \lambda_{j+1} \right)-v\left(\tau_{i+1}, \lambda_{j}\right)-v\left(\tau_{i}, \lambda_{j+1} \right) +v\left(\tau_{i}, \lambda_{j} \right). \end{align}\tag{13}\]
We have the following asymptotic behavior of the sequence \(\{Q_{N}(v)\}_{ N\ge 1}\), as \(N\rightarrow\infty\). The idea is to approximate the increment \(\Delta_{i,j}(v)\) by \(F(v(\tau_i, \lambda_j))\Delta_{i,j}(V)\), where \(V\) is given in 9 .
Proposition 3. Assume \(H\in \left[\frac{1}{2}, 1\right)\). There exists a constant \(c>0\) such that for every fixed \(N\ge 2\), \[\begin{align} \label{eq32quad} \mathbb{E}\left[\left| Q_{N}(v)- 2^{H-\frac{5}{2}} \int_0^1\int_0^1 F^2\big(v(\tau, \lambda) \big) d\tau d\lambda \right| \right]\le c N^{- H}. \end{align}\qquad{(3)}\]
The asymptotic limit of the quadratic variation \(Q_N(v)\), characterized in Proposition 3, paves the way for a direct estimation of the diffusion parameter. A comprehensive construction of the estimator and its consistency is provided in Section 4.
We structure the remainder of this paper as follows. In Section 2, we present preliminary results on stochastic integration and properties of solutions to Equation 1 , adapted from [3], [19]. In Section 3, we adapt techniques from [1] to prove Theorem 1 concerning estimation error bounds. Finally, we develop the quadratic variation analysis for the nonlinear stochastic wave equation and construct the corresponding parameter estimation framework in Section 4.
2 The noise and stochastic integral↩︎
Following [3], [19], when \(H\in (\frac{1}{2},1)\), the covariance 2 is used to construct an inner product on the space \(\mathcal{D}\) defined by \[\begin{align} \begin{aligned}\label{Eq:noise95cov2} \langle \varphi, \psi\rangle := & \, \frac{1}{2\pi}\int_{\mathbb{R}_+} ds \int_\mathbb{R}\mu(d\xi)\, \mathscr{F}(\varphi(s, \cdot))(\xi) \overline{\mathscr{F}(\psi(s, \cdot))(\xi)}\\ = & \, \int_{\mathbb{R}_+} ds \int_\mathbb{R}dy \int_\mathbb{R}dy' \, \varphi(s, y) |y-y'|^{2H-2} \psi(s, y'). \end{aligned} \end{align}\tag{14}\] Let \(\mathcal{H}\) be the completion of \(\mathcal{D}\) with respect to the inner product \(\langle \cdot, \cdot\rangle\). This space \(\mathcal{H}\) will be the natural space of deterministic integrands with respect to \(W\). For any \(g\in \mathcal{H}\), we say that \(W(\varphi)\) is the Wiener integral of \(\varphi\), denoted by \[\int_0^{\infty}\int_{\mathbb{R}} g(t, x) W(dt, dx):= W(g).\] The space \(\mathcal{H}\) contains all functions \(g\) such that its Fourier transform in the space variable satisfies \[\int_{0}^{\infty} \int_\mathbb{R}\left| \mathscr{F}(g(t, \cdot))(\xi) \right|^2 |\xi|^{1-2H} d\xi dt <\infty.\] In particular, the space \(\mathcal{H}\) contains all elements of the form \({\boldsymbol{1}}_{[0,t]\times [0,x]}\), with \(t>0\) and \(x\in \mathbb{R}\). As a consequence of the representation of the fractional Brownian motion as a Wiener type integral with respect to a complex Brownian motion (see, instance, [20]), we have \[\begin{align} & \mathbb{E}\left[W\left(\mathbf{1}_{[0,t]\times [0,x]}\right) W\left(\mathbf{1}_{[0,s]\times [0,y]}\right)\right] \\ = & \, \frac{1}{2} (t \wedge s) \left(|x|^{2H} + |y|^{2H} - |x-y|^{2H} \right). \end{align}\] This covariance corresponds to that of a standard Brownian motion in the time variable, while in the space variable it matches the covariance of a fractional Brownian motion with Hurst parameter \(H\). The underlying filtration \(\{\mathcal{F}_t, t\ge0\}\) generated by \(W\) is \[\begin{align} \label{eq32sigma32filed} \mathcal{F}_t=\sigma\left\{W({\boldsymbol{1}}_{[0,s]} \varphi), s\in [0, t], \varphi\in C_0^{\infty}\right\} \vee \mathcal{N}, \end{align}\tag{15}\] where \(\mathcal{N}\) denotes the class of \(\mathbb{P}\)-null sets in \(\mathcal{F}\).
The solution to equation 1 is understood in the mild sense. Specifically, for any \(T>0\), we say that an adapted and jointly measurable process \(u=\{u(t,x)\}_{t\ge0, x\in \mathbb{R}}\) is a solution to 1 if, for all \((t,x)\in [0,T]\times \mathbb{R}\), the following holds: \[\label{eq32solution1} \begin{align} u(t,x) = \, \int_0^t\int_{\mathbb{R}} G(t-s, x-y) F(u(s,y))W(ds,dy), \end{align}\tag{16}\] where, \(G(t, x)\) is the fundamental solution of the wave equation in \(\mathbb{R}\), given by \[\begin{align} \label{eq32fund} G(t,x)=\frac{1}{2}\mathbf{1}_{\{|x|\le t\}}. \end{align}\tag{17}\] As a result, the last term in 16 can be expressed as \[\label{eq32solution} \begin{align} \, \frac{1}{2}\iint_{\Delta(t,x)} F(u(s,y))W(ds,dy), \end{align}\tag{18}\] where \[\begin{align} \Delta(t, x):=\left\{(s,y)\in \mathbb{R}_+\times \mathbb{R}: 0\le s\le t, \, |x-y|\le t-s \right\}, \end{align}\] as illustrated by the shaded area in Figure 1.
The existence, uniqueness, and regularity of the solution to 1 are guaranteed by the following results from [3], [4].
Proposition 4. ([3], [4]) Assume that \(H\in (\frac{1}{2}, 1)\) and that \(F\) is Lipschitz continuous. Then equation 1 admits a unique mild solution \(u=\{u(t,x)\}_{t\ge0, x\in \mathbb{R}}\). Moreover, the following properties hold:
For every \(T>0\) and \(1 \le p<\infty\), \[\begin{align} \label{eq32moment32bound} \sup_{0\le t\le T}\sup_{x\in \mathbb{R}}\mathbb{E}\left[|u(t,x)|^p\right]<\infty. \end{align}\qquad{(4)}\]
For every \(T>0\) and \(1 \le p<\infty\), there exists a constant \(c(p)>0\) such that \[\begin{align} \label{eq32Holder} \mathbb{E}\left[\left|u(t, x)-u(s, y)\right|^p \right]\le c(p)\left[|t-s|^{H}+|x-y|^{H} \right] ^{p}, \end{align}\qquad{(5)}\] for all \(t, s\in [0,T]\) and \(x, y\in \mathbb{R}\).
3 Local linearization↩︎
The following lemma is taken from Lee and Xiao [21].
Lemma 1. [21] Assume that \(H\in \left(\frac{1}{2}, 1\right)\). Let \(V\) be given by 9 . For every \(\tau>0, \lambda \ge-\tau\), and \(0 < \varepsilon< \tau\), \[\label{Eq:rec-inc} \begin{align} \mathbb{E}\left[\left(V(\tau\pm\varepsilon, \lambda + \varepsilon) - V(\tau\pm \varepsilon, \lambda) -V(\tau, \lambda + \varepsilon) + V(\tau, \lambda)\right)^2\right] = 2^{H-\frac{5}{2}}\varepsilon^{2H+1}. \end{align}\qquad{(6)}\]
Here are two elementary facts that will be used in the proof of Theorem 1.
Lemma 2. For any \(a<b\) and \(H\in \left(\frac{1}{2}, 1\right)\), we have
\[\label{eq32ab1} \int_a^bdx \int_a^bdy |x-y|^{2H-2} = \frac{1}{(2H-1)H} (b-a)^{2H}.\qquad{(7)}\]
\[\label{eq32ab2} \int_{a}^{\frac{a+b}{2}}dx\int_{\frac{a+b}{2}}^bdy |x-y|^{2H-2} =\frac{ 2^{2H-1} - 1}{H(2H-1)} \left(\frac{b-a}{2}\right)^{2H}.\qquad{(8)}\]
Proof. The identity ?? follows by direct computation. We now prove ?? . Using the change of variables \[x= \left(\frac{b-a}{2}\right)u+a,\, y=\left(\frac{b-a}{2}\right)v +\frac{a+b}{2},\] we obtain \[\begin{align} \int_{a}^{\frac{a+b}{2}}dx\int_{\frac{a+b}{2}}^bdy |x-y|^{2H-2} = \, \left(\frac{b-a}{2}\right)^{2H}\int_{0}^{1}dv\int_{0}^1du (1+v-u)^{2H-2}. \end{align}\] Note that \[\begin{align} \int_{0}^{1} dv \int_{0}^{1} du (1 + v - u)^{2H-2} =&\, \frac{1}{2H-1} \int_{0}^{1} \left((1+v)^{2H-1} - v^{2H-1}\right)dv \\ =&\, \frac{1}{2H-1} \left( \frac{2^{2H} - 1}{2H} - \frac{1}{2H}\right)\\ =&\, \frac{ 2^{2H-1} - 1}{H(2H-1)}. \end{align}\]
The proof is complete. ◻
Proof of Theorem 1. The proof follows a strategy similar to that of Theorem 1.3 in [1]. By symmetry, we only need to prove ?? for \(R_{\varepsilon}(\tau, \lambda)\). By 7 and Proposition 4, it follows that for any \(p\ge1\), there exists a constant \(c(p)>0\) such that \[\begin{align} \label{eq32Holder32v} \|v(\tau_1, \lambda_1)-v(\tau_2, \lambda_2)\|_{L^p(\Omega)}\le c(p)\left[|\tau_1-\tau_2|^{H}+|\lambda_1-\lambda_2|^{H}\right], \end{align}\tag{19}\] for all \(\tau_1, \tau_2>0, \lambda_1\ge -\tau_1, \lambda_2\ge -\tau_2\). To ensure that the integrand is adapted, we decompose the domain of integration into two triangular regions. A similar decomposition technique to recover adaptedness is employed in [12]. For every fixed \(\tau>0, \lambda\ge -\tau\), and \(\varepsilon>0\), define the regions \(D_1\) and \(D_2\) as follows: \[\begin{align} D_1:= & \, \Bigg\{(s, y)\in \mathbb{R}_+\times \mathbb{R}:\, \frac{\tau+\lambda}{\sqrt 2} < \, s\le \frac{\tau+\lambda+\varepsilon}{\sqrt 2},\\ & \,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\, -s+\sqrt{2}\lambda < y < s-\sqrt{2}\tau \Bigg\},\\ D_2:=& \, \Bigg\{(s,y)\in \mathbb{R}_+\times \mathbb{R} :\, \frac{\tau+\lambda+\varepsilon}{\sqrt 2} < \, s \le \frac{\tau+\lambda+2\varepsilon}{\sqrt 2},\\ & \,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\, s-\sqrt{2}(\tau+\varepsilon)\le \, y\le -s+\sqrt 2(\lambda+ \varepsilon) \Bigg\}. \end{align}\] The sets \(D_1\) and \(D_2\) are illustrated in Figure 2. Specifically, \(D_1\) is the triangular region with vertices: \[\label{eq32P} \begin{align} P_1 :=&\, \left(\frac{\tau+\lambda}{\sqrt 2}, \frac{-\tau+\lambda}{\sqrt 2}\right),\\ P_2:=&\, \left(\frac{\tau+\lambda+\varepsilon}{\sqrt 2}, \frac{-\tau+\lambda-\varepsilon}{\sqrt 2}\right),\\ P_3:=&\, \left(\frac{\tau+\lambda+\varepsilon}{\sqrt 2}, \frac{-\tau+\lambda+\varepsilon}{\sqrt 2}\right),\\ \end{align}\tag{20}\] and \(D_2\) is the triangular region with vertices: \[\begin{align} P_2 , \, P_3, \text{ and }\; P_4 :=&\, \left(\frac{\tau+\lambda+2\varepsilon}{\sqrt 2}, \frac{-\tau+\lambda}{\sqrt 2}\right). \end{align}\]
Using equations 6 and 18 , we decompose the integral as follows: \[\begin{align} & \frac{1}{2}\left(\iint_{D_1}+\iint_{D_2} \right)\left\{F(u(s,y))-F(u(P_1)) \right\}W(ds,dy)\\ =&\, \frac{1}{2} \iint_{D_1} \left\{F(u(s, y))-F(u(P_1))\right\}W(ds,dy)\\ &\, +\frac{1}{2} \left\{F(u(P_2))- F(u(P_1)) \right\} \iint_{D_2} W(ds,dy)\\ &\, + \frac{1}{2} \iint_{D_2} \left\{F(u(s,y))-F(u(P_2)) \right\}W(ds,dy)\\ =:&\, I_1+I_2+I_3. \end{align}\]
By the Burkholder–Davis–Gundy (BDG, for short) inequality, 2 , ?? , and the Lipschitz continuity of \(F\), we have \[\begin{align} \mathbb{E}\left[\left|I_1\right|^p\right]^{\frac{2}{p}} \lesssim &\, \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \int_{-s+ \sqrt{2}\lambda}^{s -\sqrt{2}\tau } dy \int_{-s + \sqrt{2}\lambda}^{s -\sqrt{2}\tau} d\tilde{y} |y - \tilde{y}|^{2H-2}\\ & \;\;\;\cdot \left\| F(u(s,y)) - F(u(P_1)) \right\|_{L^p(\Omega)} \left\| F(u(s,\tilde{y})) - F(u(P_1)) \right\|_{L^p(\Omega)} \\ \lesssim &\, \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds\int_{-s + \sqrt{2}\lambda}^{s -\sqrt{2}\tau } dy \int_{-s + \sqrt{2}\lambda}^{s -\sqrt{2}\tau} d\tilde{y} |y - \tilde{y}|^{2H-2}\\ &\, \cdot \left( \left| s - \frac{\tau+\lambda}{\sqrt{2}} \right| + \left| y - \frac{-\tau+\lambda}{\sqrt{2}} \right| \right)^{H} \cdot \left( \left| s- \frac{\tau+\lambda}{\sqrt{2}} \right| + \left| \tilde{y} - \frac{-\tau+\lambda}{\sqrt{2}} \right| \right)^{H}. \end{align}\] Here and below, \(a\lesssim b\) means that there exists a constant \(c>0\) such that \(a\le c b\).
We further divide the domain of integration into four parts: \[\begin{align} \left( \int_{-s + \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } + \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s-\sqrt{2}\tau } \right) dy \cdot \left( \int_{-s + \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } + \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s -\sqrt{2}\tau } \right) d\tilde{y}. \end{align}\]
For the first term, we have \[\begin{align} & \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \int_{-s + \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } dy \int_{-s + \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } d\tilde{y} |y - \tilde{y}|^{2H-2} \\ &\, \cdot \left( \left|s - \frac{\tau+\lambda}{\sqrt{2}} \right| + \left| y - \frac{-\tau+\lambda}{\sqrt{2}} \right| \right)^{H} \cdot \left( \left| s - \frac{\tau+\lambda}{\sqrt{2}} \right| + \left| \tilde{y} - \frac{-\tau+\lambda}{\sqrt{2}} \right| \right)^{H}\\ =&\, \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \int_{-s+ \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } dy \int_{-s + \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } d\tilde{y} |y - \tilde{y}|^{2H-2} \cdot \left(s- y-\sqrt{2}\tau \right)^{H} \cdot \left( s- \tilde{y}-\sqrt{2}\tau \right)^{H} \\ \lesssim &\, \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \left (2s-\sqrt{2}(\tau+\lambda)\right)^{2H} \int_{-s + \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } dy \int_{-s + \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } d\tilde{y} |y - \tilde{y}|^{2H-2} \\ \lesssim &\, \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \left ( s- \frac{\tau+\lambda}{\sqrt{2}}\right)^{4H} \\ =&\, \frac{1}{(1+4H) 2^{\frac{1}{2}+2H}} \varepsilon^{1+4H}, \end{align}\] where ?? is used in the last second step.
For the second term, changing variables yields that \[\begin{align} & \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \int_{-s + \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } dy \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s -\sqrt{2}\tau } d\tilde{y} |y - \tilde{y}|^{2H-2} \\ &\, \cdot \left( \left|s - \frac{\tau+\lambda}{\sqrt{2}} \right| + \left| y - \frac{-\tau+\lambda}{\sqrt{2}} \right| \right)^{H} \cdot \left( \left| s - \frac{\tau+\lambda}{\sqrt{2}} \right| + \left| \tilde{y}- \frac{-\tau+\lambda}{\sqrt{2}} \right| \right)^{H}\\ =&\, \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \int_{-s+ \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } dy \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s -\sqrt{2}\tau } d\tilde{y} |y - \tilde{y}|^{2H-2} \cdot \left( s - y-\sqrt{2}\tau \right)^{H} \cdot \left( s+\tilde{y}-\sqrt{2} \lambda \right)^{H}\\ \le & \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \left (2s-\sqrt{2}(\tau+\lambda)\right)^{2H} \int_{-s + \sqrt{2}\lambda}^{\frac{\lambda-\tau}{\sqrt 2} } dy \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s -\sqrt{2}\tau } d\tilde{y} |y - \tilde{y}|^{2H-2} \\ \lesssim &\, \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \left (s- \frac{\tau+\lambda}{\sqrt{2}}\right)^{4H} \\ =&\, \frac{1}{(1+4H) 2^{\frac{1}{2}+2H}} \varepsilon^{1+4H}, \end{align}\] where ?? is used in the last second step.
By symmetry, the third term equals to the second. For the fourth term, we have \[\begin{align} & \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s -\sqrt{2}\tau } dy \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s-\sqrt{2}\tau } d\tilde{y} |y - \tilde{y}|^{2H-2} \\ &\, \cdot \left( \left| s - \frac{\tau+\lambda}{\sqrt{2}} \right| + \left| y - \frac{-\tau+\lambda}{\sqrt{2}} \right| \right)^{H} \cdot \left( \left| s - \frac{\tau+\lambda}{\sqrt{2}} \right| + \left| \tilde{y} - \frac{-\tau+\lambda}{\sqrt{2}} \right| \right)^{H}\\ =&\, \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s -\sqrt{2}\tau } dy \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s -\sqrt{2}\tau } d\tilde{y} |y - \tilde{y}|^{2H-2} \cdot \left( s +y-\sqrt{2} \lambda \right)^{H} \left( s+\tilde{y}-\sqrt{2} \lambda \right)^{H} \\ \lesssim & \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \left (2s-\sqrt{2}(\tau+\lambda)\right)^{2H} \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s -\sqrt{2}\tau } ds \int_{\frac{\lambda-\tau}{\sqrt 2}}^{s -\sqrt{2}\tau } d\tilde{y} |y - \tilde{y}|^{2H-2} \\ = &\, \int_{\frac{\tau+\lambda}{\sqrt 2} }^ \frac{\tau+\lambda+\varepsilon}{\sqrt 2}ds \left ( s- \frac{\tau+\lambda}{\sqrt{2}}\right)^{4H} \\ =&\, \frac{1}{(1+4H) 2^{\frac{1}{2}+2H}} \varepsilon^{1+4H}, \end{align}\] where ?? is used in last second step.
Using the Hölder inequality, we have \[\begin{align} \mathbb{E}\left[|I_2|^p\right]^{\frac{2}{p}} \lesssim &\, \mathbb{E}\left[ \left| u(P_2)- u(P_1)\right|^{2p}\right]^{\frac{1}{p}} \cdot \mathbb{E}\left[ \left|\iint_{D_2} W(ds,dy)\right|^{2p}\right]^{\frac{1}{p}}. \end{align}\] By the Lipschitz continuity of \(F\), ?? , and 20 , we have \[\begin{align} \mathbb{E}\left[ \left| u(P_2)- u(P_1)\right|^{2p}\right]^{\frac{1}{p}} \lesssim \varepsilon^{2H}. \end{align}\] By ?? , we have \[\begin{align} \mathbb{E}\left[ \left|\iint_{D_2} W(ds,dy)\right|^{2}\right]=&\, \int_{\frac{\tau+\lambda+\varepsilon}{\sqrt 2}}^{ \frac{\tau+\lambda+2\varepsilon}{\sqrt 2}}ds\int_{ s-\sqrt{2}(\tau+\varepsilon)}^{ -s+\sqrt 2(\lambda+ \varepsilon)}dy \int_{ s-\sqrt{2}(\tau+\varepsilon)}^{ -s+\sqrt 2(\lambda+ \varepsilon)}d\tilde{y} |y- \tilde{y}|^{2H-2}\\ \lesssim &\, \varepsilon^{1+2H}. \end{align}\]
Finally, for \(I_3\), applying the BDG inequality, the Lipschitz continuity of \(F\), 2 , ?? , and ?? , we have \[\begin{align} \mathbb{E}\left[\left|I_3\right|^p\right]^{\frac{2}{p}} \lesssim &\, \int_{\frac{\tau+\lambda+\varepsilon}{\sqrt 2}}^{\frac{\tau+\lambda+2\varepsilon}{\sqrt 2}}ds \int_{s-\sqrt{2}(\tau+\varepsilon)}^{-s+\sqrt 2(\lambda+ \varepsilon)} dy \int_{s-\sqrt{2}(\tau+\varepsilon)}^{-s+\sqrt 2(\lambda+ \varepsilon) } d\tilde{y} |y - \tilde{y}|^{2H-2}\\ & \;\;\;\cdot \left\| F(u(s,y)) - F(u(P_2)) \right\|_{L^p(\Omega)} \left\| F(u(s,\tilde{y})) - F(u(P_2)) \right\|_{L^p(\Omega)} \\ \lesssim &\, \int_{\frac{\tau+\lambda+\varepsilon}{\sqrt 2}}^{\frac{\tau+\lambda+2\varepsilon}{\sqrt 2}}ds \int_{s-\sqrt{2}(\tau+\varepsilon)}^{-s+\sqrt 2(\lambda+ \varepsilon)} dy \int_{s-\sqrt{2}(\tau+\varepsilon)}^{-s+\sqrt 2(\lambda+ \varepsilon) } d\tilde{y} |y - \tilde{y}|^{2H-2}\\ &\, \cdot \left( s+y-\sqrt{2}\lambda \right)^{H} \cdot \left( s+\tilde{y}-\sqrt{2}\lambda \right)^{H}\\ \lesssim &\, \varepsilon^2 \int_{\frac{\tau+\lambda+\varepsilon}{\sqrt 2}}^{\frac{\tau+\lambda+2\varepsilon}{\sqrt 2}}ds \int_{s-\sqrt{2}(\tau+\varepsilon)}^{-s+\sqrt 2(\lambda+ \varepsilon)} dy \int_{s-\sqrt{2}(\tau+\varepsilon)}^{-s+\sqrt 2(\lambda+ \varepsilon) } d\tilde{y} |y - \tilde{y}|^{2H-2}\\ \lesssim&\, \varepsilon^2 \int_{\frac{\tau+\lambda+\varepsilon}{\sqrt 2}}^{\frac{\tau+\lambda+2\varepsilon}{\sqrt 2}}ds \left(\frac{\tau+\lambda+2\varepsilon}{\sqrt 2}-s\right)^{2H}\\ \lesssim&\, \varepsilon^{1+4H}. \end{align}\]
The proof is complete. ◻
4 Quadratic variations and parameter estimation↩︎
The development of statistical inference for stochastic partial differential equations is largely motivated by the need to calibrate mathematical models using empirical data. In many physical systems, including wave propagation in heterogeneous media, the stochastic wave equation serves as a fundamental framework. Accurate estimation of key parameters, especially the diffusion constant or noise intensity, is essential for enhancing the predictive power of these models. Although parameter estimation under additive noise has been extensively studied (see, e.g., [22]–[25]; we also refer to the monograph [26] for a comprehensive overview), the multiplicative noise case remains less explored despite its greater practical relevance and theoretical challenges.
4.1 Parameter estimates↩︎
In this section, we address the problem of estimating the diffusion parameter in the following stochastic wave equation: \[\label{SWE32para} \begin{cases}\displaystyle{\frac{\partial^2}{\partial t^2} u_{\theta }(t, x) =\frac{\partial^2}{\partial x^2} u_{\theta }(t, x) + \theta F(u_{\theta}(t,x))\dot{W}(t, x), \quad t\ge 0, \, x \in \mathbb{R},}\\ \displaystyle{u_{\theta }(0, x) = 0, \quad \frac{\partial}{\partial t} u_{\theta}(0, x) = 0}, \end{cases}\tag{21}\] where \(\dot{W}\) is a space-time noise that is white in time and colored in space, and \(F\) is a Lipschitz continuous function, as stated in 1 .
We investigate the estimation of the parameter \(\theta\) from discrete observations of the solution \(u_{\theta}\) over a space-time grid. Our analysis provides a partial resolution to the open problem posed in [22]. Unlike the approach in [27], which relies on discrete temporal observations at a fixed spatial location, the proposed methodology fully exploits the spatiotemporal structure of the data.
For every \(\tau>0\) and \(\lambda\ge -\tau\), let \[\begin{align} \label{eq32v32theta} v_{\theta}(\tau, \lambda):= u_{\theta}\left(\frac{\tau+\lambda}{\sqrt{2}}, \frac{-\tau+\lambda}{\sqrt{2}}\right). \end{align}\tag{22}\] For every \(N\ge1\), let \[\begin{align} \label{akyhtesz} Q_{N}(v_{\theta}):= N^{2H-1}\sum_{i=0}^{N-1}\sum_{j=0}^{N-1}\Delta_{i,j}(v_{\theta})^2, \end{align}\tag{23}\] where \[\Delta_{i,j}(v_{\theta}) := v_{\theta}\left(\tau_{i+1}, \lambda_{j+1} \right)-v_{\theta}\left(\tau_{i+1}, \lambda_{j}\right)-v_{\theta}\left(\tau_{i}, \lambda_{j+1} \right) +v_{\theta}\left(\tau_{i}, \lambda_{j} \right),\] and \(\tau_i:=\frac{i}{N}, \lambda_j:=\frac{j}{N}\).
We postpone the proof of Proposition 3 to Subsection 4.2. Using Proposition 3, we know that \[\begin{align} \label{eq32variation32converge32theta} Q_{N}(v_{\theta})\longrightarrow \theta^2 2^{H-\frac{5}{2}} \int_0^1\int_0^1 F^2\big(v_{\theta}(\tau, \lambda) \big) d\tau d\lambda, \end{align}\tag{24}\] in \(L^{1}(\Omega)\), as \(N\rightarrow\infty\). This convergence allows us to construct a consistent estimator of the noise parameter \(\theta\) based on the observations \[\left\{u_{\theta}\left(\frac{\frac{i}{N}+\frac{j}{N} }{\sqrt 2}, \frac{-\frac{i}{N}+\frac{j}{N} }{\sqrt 2} \right);\, i, j = 0,1, \cdots, N\right\}.\]
Given the limit in 24 , a natural approach to estimating \(\theta\) is to use the approximation \[\begin{align} \label{eq32appr1} \theta \approx \sqrt{\frac{Q_{N}(v_{\theta}) }{2^{H-\frac{5}{2}} \int_0^1\int_0^1 F^2\big(v_{\theta}(\tau, \lambda) \big) d\tau d\lambda}}. \end{align}\tag{25}\] To evaluate the denominator, we discretize the double integral via a Riemann sum: \[\int_0^1\int_0^1 F^2\big(v(\tau, \lambda) \big) d\tau d\lambda\approx N^{-2} \sum_{i=0}^{N-1} \sum_{j=0}^{N-1} F^2\big(v(\tau_i, \lambda_j) \big).\] This leads to the following estimator for the diffusion parameter \(\theta\): for every \(N\ge1\), \[\begin{align} \label{eq32appr2} \hat{\theta}_N := \sqrt{\frac{Q_{N}(v_{\theta})}{ 2^{H-\frac{5}{2}} N^{-2} \sum_{i=0}^{N-1} \sum_{j=0}^{N-1} F^2\big(v_{\theta}(\tau_i, \lambda_j) \big)}}. \end{align}\tag{26}\]
From Proposition 3, we directly obtain the following consistency result.
Corollary 1. Assume that \(H\in \left[\frac{1}{2}, 1\right)\). The estimator \(\hat{\theta}_N\) defined in 26 is weakly consistent, i.e., \[\hat{\theta}_N \stackrel{\mathbb{P}}{\longrightarrow} \theta, \;\;\;\;\text{as } N\rightarrow\infty.\]
Proof. By Proposition 3, \(Q_{N}(v_{\theta})\) converges in probability to \(\theta^2 \int_0^1\int_0^1 F^2\big(v_{\theta}(\tau, \lambda)\big)d\tau d\lambda\). It remains to show that \[I:= \int_0^1\int_0^1 F^2\big(v_{\theta}(\tau, \lambda)\big)d\tau d\lambda -\frac{1}{N^2} \sum_{i=0}^{N-1} \sum_{j=0}^{N-1} F^2\big(v_{\theta}(\tau_i, \lambda_j) \big) \stackrel{\mathbb{P}}{\longrightarrow} 0.\] Note that \[\begin{align} I= & \, \sum_{i=0}^{N-1} \sum_{j=0}^{N-1} \int_{\tau_i}^{\tau_{i+1}}\int_{\tau_j}^{\tau_{j+1}} \left[ F^2\big(v_{\theta}(\tau, \lambda)\big)-F^2\big(v_{\theta}(\tau_i, \lambda_j) \big)\right]d\tau d\lambda. \end{align}\] For each \(\tau, \lambda\in [\tau_i, \tau_{i+1}) \times [\tau_j, \tau_{j+1})\), the Lipschitz continuity of \(F\) and the Hölder continuity of \(v_{\theta}\) imply \[\begin{align} & \mathbb{E}\left[ \left|F^2\big(v_{\theta}(\tau, \lambda)\big)-F^2\big(v_{\theta}(\tau_i, \lambda_j) \big)\right| \right]\\ \le &\, \left\{ \mathbb{E}\left[ \left|F\big(v_{\theta}(\tau, \lambda)\big)+F\big(v_{\theta}(\tau_i, \lambda_j) \big)\right|^2 \right]\right\}^{1/2} \left\{ \mathbb{E}\left[ \left|F\big(v_{\theta}(\tau, \lambda)\big)-F\big(v_{\theta}(\tau_i, \lambda_j) \big)\right|^2\right]\right\}^{1/2} \\ \lesssim &\, N^{-H}. \end{align}\] Therefore, we have \[\mathbb{E}\left[I \right] \lesssim N^{-H},\] which implies convergence in probability. ◻
4.2 Quadratic variations↩︎
For every \(N\ge1\) and \(0\le i\le N\), define the grid points \(\tau_i:=\frac{i}{N}, \lambda_j:=\frac{j}{N}\), and the second-order increment \[\Delta_{i,j}(V) := V\left(\tau_{i+1}, \lambda_{j+1} \right)-V\left(\tau_{i+1}, \lambda_{j}\right)-V\left(\tau_{i}, \lambda_{j+1} \right) +V\left(\tau_{i}, \lambda_{j} \right).\] The corresponding quadratic variation statistic is defined as \[\begin{align} \label{eq32V32V} Q_{N}(V):= N^{2H-1}\sum_{i=0}^{N-1}\sum_{j=0}^{N-1}\Delta_{i,j}(V)^2. \end{align}\tag{27}\]
We first study the asymptotic behavior of the sequence \(\{Q_{N}(V)\}_{ N\ge 1}\) as \(N\rightarrow\infty\).
Lemma 3. Assume that \(H\in \left[\frac{1}{2}, 1\right)\). Then, there exists a constant \(c>0\) satisfying that \[\label{2f-10} \mathbb{E}\left[ \left| Q_{N}(V) -2^{H-\frac{5}{2}}\right|^2 \right] \lesssim N^{-1}.\qquad{(9)}\]
Proof. From equation ?? , the random variable \(N^{H-\frac{1}{2}} \Delta_{i,j}(V)\) follows a Gaussian distribution with mean zero and variance \(2^{H-\frac{5}{2}}N^{-2}\). Using standard moment bounds for Gaussian random variables, we obtain that for all \(i, j\geq 1\) and \(p\in \mathbb{N}\), \[\label{2f-2} \mathbb{E}\left[ \left(N^{2H-1} \left( \Delta_{i,j}(V) \right)^{2} -2^{H-\frac{5}{2}}N^{-2}\right) ^{2p}\right]\lesssim N^{-4p}.\tag{28}\]
Due to the temporal independence of the noise \(\{W(t, x)\}_{t\ge0, x\in\mathbb{R}}\), which is white in time and colored in space, the increments \(\Delta_{i,j}(V)\) and \(\Delta_{i',j'}(V)\) are independent whenever \(|i-i'|\ge2\). Applying the Cauchy-Schwarz inequality and the moment estimate 28 , we derive \[\begin{align} & \mathbb{E}\left[ \left(\sum_{i=0}^{N-1}\sum_{j=0}^{N-1} \left( N^{2H-1} \left(\Delta_{i,j}(V) \right)^{2} -2^{H-\frac{5}{2}}N^{-2}\right) \right)^{2}\right] \\ \le &\, 4 \sum_{i=0}^{N-1} \mathbb{E} \left[ \left( \sum_{j=0}^{N-1} \left( N^{2H-1} \left(\Delta_{i,j}(V) \right)^{2} -2^{H-\frac{5}{2}}N^{-2}\right) \right)^{2}\right] \\ \le & \, 4 N \sum_{i=0}^{N-1} \sum_{j=0}^{N-1} \mathbb{E} \left[ \left( N^{2H-1} \left(\Delta_{i,j}(V) \right)^{2} -2^{H-\frac{5}{2}}N^{-2} \right)^{2}\right]\\ \lesssim &\, N^{-1}. \end{align}\]
The proof is complete. ◻
Next, we prove Proposition 3 by applying the approximation of the increment \(\Delta_{i,j}(v)\) by \(F(v(\tau_i, \lambda_j))\Delta_{i,j}(V)\), as established in Theorem 1.
Proof of Proposition 3. We decompose the target difference into three parts: \[\begin{align} &Q_{N}(v)- \int_0^1\int_0^1 F^2\big(v(\tau, \lambda) \big) d\tau d\lambda\\ =&\, N^{2H-1}\sum_{i=0}^{N-1}\sum_{j=0}^{N-1}\left\{\Delta_{i,j}(v)^2 - F^2(v(\tau_i, \lambda_j)) \Delta_{i,j}(V)^2 \right\}\\ & +\sum_{i=0}^{N-1}\sum_{j=0}^{N-1} F^2(v(\tau_i, \lambda_j)) \left\{N^{2H-1} \Delta_{i,j}(V)^2 -2^{H-\frac{5}{2}}N^{-2} \right\}\\ &+2^{H-\frac{5}{2}} \left[\sum_{i=0}^{N-1}\sum_{j=0}^{N-1} F^2(v(\tau_i, \lambda_j)) N^{-2} - \int_0^1\int_0^1 F^2\big(v(\tau, \lambda) \big) d\tau d\lambda\right]\\ =:&\, I_1+I_2+I_3. \end{align}\]
Step 1: Estimation of \(I_1\). By the Cauchy-Schwarz inequality, \[\begin{align} &\mathbb{E} \left[\left|\Delta_{i,j}(v)^2 - F^2(v(\tau_i, \lambda_j)) \Delta_{i,j}(V)^2\right| \right]\\ \le &\, \left( \mathbb{E} \left[\left|\Delta_{i,j}(v)- F (v(\tau_i, \lambda_j)) \Delta_{i,j}(V)\right|^2 \right]\right)^{\frac{1}{2}} \left( \mathbb{E} \left[\left|\Delta_{i,j}(v)+ F (v(\tau_i, \lambda_j)) \Delta_{i,j}(V)\right|^2 \right]\right)^{\frac{1}{2}} . \end{align}\] From Theorem 1, we obtain the bound for the first factor: \[\begin{align} \left( \mathbb{E} \left[\left|\Delta_{i,j}(v)- F (v(\tau_i, \lambda_j)) \Delta_{i,j}(V)\right|^2 \right]\right)^{\frac{1}{2}} \lesssim \, N^{-\frac{1}{2}-2H}. \end{align}\] For the second factor, applying Minkowski’s inequality, Hölder’s inequality, ?? , and Theorem 1 again, we have \[\begin{align} & \left( \mathbb{E} \left[\left|\Delta_{i,j}(v)+ F (v(\tau_i, \lambda_j)) \Delta_{i,j}(V)\right|^2 \right]\right)^{\frac{1}{2}}\\ \le &\, \left( \mathbb{E} \left[\left|\Delta_{i,j}(v) \right|^2 \right]\right)^{\frac{1}{2}}+\left( \mathbb{E} \left[\left| F (v(\tau_i, \lambda_j)) \Delta_{i,j}(V)\right|^2 \right]\right)^{\frac{1}{2}} \\ \lesssim & \, N^{-\frac{1}{2}-H}. \end{align}\] Combining the two bounds above and summing over \(i, j=0, \cdots, N-1\), we conclude that \[\begin{align} \mathbb{E} [I_1] \lesssim N^{-H}. \end{align}\]
Step 2: Estimation of \(I_2\). Let \(\mathcal{F}_t\) be the \(\sigma\)-algebra defined in 15 . Consider any pairs \((i, j), (i', j')\in \mathbb{N}^2\) such that \(i+j\ge i'+j'+2\). By 6 and the temporal independence of \(\{W(t, x)\}_{t\ge0, x\in\mathbb{R}}\), the random variables \(v(\tau_{i'}, \lambda_{j'}), v(\tau_i, \lambda_j)\), and \(\Delta_{i',j'}(V)\) are \(\mathcal{F}_{\sqrt{2}(\tau_i+\lambda_j)}\)-measurable, while \(\Delta_{i,j}(V)\) is independent of \(\mathcal{F}_{\sqrt{2}(\tau_i+\lambda_j)}\). Taking the conditional expectation with respect to \(\mathcal{F}_{\sqrt{2}(\tau_i+\lambda_j)}\), we obtain \[\begin{align} & \mathbb{E}\left[ F^2(v(\tau_i, \lambda_j) ) F^2(v(\tau_{i'}, \lambda_{j'})) \left( N^{2H-1} \Delta_{i,j}(V)^2 -2^{H-\frac{5}{2}}N^{-2}\right) \left( N^{2H-1} \Delta_{i',j'}(V)^2 -2^{H-\frac{5}{2}}N^{-2}\right) \right]\\ =&\, \mathbb{E}\Bigg[ F^2(v(\tau_i, \lambda_j) ) F^2(v(\tau_{i'}, \lambda_{j'})) \left( N^{2H-1} \Delta_{i',j'}(V)^2 - 2^{H-\frac{5}{2}}N^{-2}\right) \\ & \;\;\;\;\;\; \;\;\;\;\; \cdot \mathbb{E}\left[ \left( N^{2H-1} \Delta_{i,j}(V)^2 -2^{H-\frac{5}{2}}N^{-2}\right)|\mathcal{F}_{\sqrt{2}(\tau_i+\lambda_j)}\right] \Bigg] \\ =& \, 0, \end{align}\] where the last equality follows from ?? .
For pairs with \((i, j), (i', j')\in \mathbb{N}^2\) with \(|(i+j)-( i'+j')|<2\), we apply the Cauchy-Schwarz inequality and the moment estimate 28 to get \[\begin{align} & \mathbb{E}\left[ F^2(v(\tau_i, \lambda_j) F^2(v(\tau_{i'}, \lambda_{j'}) \left( N^{2H-1} \Delta_{i,j}(V)^2 - 2^{H-\frac{5}{2}}N^{-2}\right) \left( N^{2H-1} \Delta_{i',j'}(V)^2 -2^{H-\frac{5}{2}}N^{-2}\right) \right]\\ \lesssim & \, \left( \mathbb{E}\left[ F^8(v(\tau_i, \lambda_j) \right]\right)^{1/4 } \left( \mathbb{E}\left[ F^8(v(\tau_{i'}, \lambda_{j'}) \right]\right)^{1/4 } \\ & \cdot \left( \mathbb{E}\left[ \left( N^{2H-1} \left( \Delta_{i,j}(V) \right)^{2} -2^{H-\frac{5}{2}}N^{-2}\right) ^{4}\right] \right)^{1/4 } \left( \mathbb{E}\left[ \left( N^{2H-1} \left( \Delta_{i',j'}(V) \right)^{2} -2^{H-\frac{5}{2}}N^{-2}\right) ^{4}\right] \right)^{1/4 } \\ \lesssim &\, N^{-4}. \end{align}\] Summing over all such pairs yields \[\begin{align} \mathbb{E} [|I_2|]\le c N^{-1}. \end{align}\]
Step 3: Estimation of \(I_3\). Using the Hölder continuity of \(v\) from 19 , the Lipschitz continuity of \(F\), and the Cauchy-Schwarz inequality, we obtain \[\begin{align} \mathbb{E}[|I_3|] \le & \, 2^{H-\frac{5}{2}} \sum_{i=0}^{N-1}\sum_{j=0}^{N-1} \int_{\tau_i}^{\tau_{i+1}} \int_{\tau_j}^{\tau_{j+1}} \mathbb{E} \left[| F^2(v(\tau_i, \lambda_j)) - F^2(v(\tau, \lambda))| \right] d\tau d\lambda\\ \lesssim & \, \sum_{i=0}^{N-1}\sum_{j=0}^{N-1} \int_{\tau_i}^{\tau_{i+1}} \int_{\tau_j}^{\tau_{j+1}} \left( \mathbb{E} \left[| F(v(\tau_i, \lambda_j)) - F(v(\tau, \lambda))| ^2 \right] \right)^{\frac{1}{2}} d\tau d\lambda\\ \lesssim &\, N^{-H}. \end{align}\] The proof is complete. ◻
4.3 Numerical experiments↩︎
Let us consider the equation 21 with \(F(u)=1+\sin(u)\) and \(\theta=2\). We simulate the paths of the solution 21 , using the R procedure fieldsim which is available via [28]. The following numerical results are consistent with the theoretical results from Corollary 1.
4.3.1 Space-time white noise↩︎
Take the parameter \(H=0.5\). Using Corollary 1, we compute the estimator \(\hat{\theta}_N\) defined in 26 , and consequently their sample mean and sample standard deviation. In Figure ¿fig:fig:left1?, we present one realization of the estimator \(\hat{\theta}_N\) for \(N=20, \cdots, 60\), and we display the absolute and relative errors in Figure [fig:right1].
4.3.2 Time-white and space-colored noise↩︎
Take the parameter \(H=0.55\). Using Corollary 1, we compute the estimator \(\hat{\theta}_N\) defined in 26 , and consequently their sample mean and sample standard deviation. In Figure 3, we present one realization of the estimator \(\hat{\theta}_N\) for \(N=30, \cdots, 60\), and we display the absolute and relative errors in Figure [fig:right2].
The research of G. Liu is supported by the NSFC (No. 11801196). The research of R. Wang is partially supported by the NSF of Hubei Province (No. 2024AFB683) and the Wuhan University Social Science Digital Innovation Research Team Project (No. WDSZTD2024B05).
G. Liu and R. Wang designed the inference methodology, implemented the method, conducted the simulation studies and data analyses, and drafted the manuscript. All authors reviewed the manuscript.
No potential conflict of interest was reported by the authors.