Exact variations for stochastic heat equations with piecewise constant coefficients and application to parameter estimation

Estimation of the drift parameter for the fractional stochastic heat equation via power variation

“…and γ 0 = θ 1 1−α , we immediately obtain the convergence (24). Concerning the rate of convergence, we can write, with γ 0 a random point located between X n and γ…”

“…The study in [17] has been then extended in [2], by adding a time-varying volatility in the noise term and by using power variation techniques to estimate the parameter of the model. Other recent works on parameter estimates for discretely sampled SPDEs via power variations are [5,3,1,21] and [24].…”

Fractional stochastic heat equation with piecewise constant coefficients

“…An explicit expression of the fundamendal solution of equation (1.6) was given in [6,[22][23][24]. SPDEs with the operator L have been introduced and investigated in [25,26], with an additive noise W defined as a centered Gaussian field W = {W (t, C); t ∈ [0, T ], C ∈ B b (R)} with covariance E(W (t, C)W (s, D)) = (t ∧ s)λ(C ∩ D), where λ denotes the Lebesgue measure. Comparing to those two articles, the equation (1.2) contains more complicated noise, which is multiplicative and fractional.…”

On mild and weak solutions for stochastic heat equations with piecewise-constant conductivity

“…In Zili and Zougar (2018), the authors proved the existence of the mild solution to (1), they presented explicit expressions of its covariance and variance functions, and they analyzed some regularity properties of its sample paths. Then, Zili and Zougar (2019a) presented an estimation method of the parameters a 1 and a 2 appearing in (3) and, after that, in Zili and Zougar (2019b), they made a deep study of the spatial quadratic variations of the mild solution process.…”