Parameter Estimation for Stochastic Partial Differential Equations Driven by an Additive Multi-Order Fractional Brownian Motion

We investigate the parameter estimation problem for a diagonalizable stochastic evolution equation driven by an additive noise that is white in space and fractional in time. The fractional component in the noise is described by the so-called multi-order fractional Brownian motion \(\displaystyle W^{\varvec{H}}\) ${\displaystyle W^{\mathit{H}}}$ with Hurst sequence \(\displaystyle \varvec{H}=\left( H_1,H_2,H_3,\cdots \right)\) ${\displaystyle \mathit{H}=\left(H_1,,,H_2,,,H_3,,,\cdots \right)}$ , introduced in El Omari (2021). By using the spectral approach, we study the maximum likelihood estimator (MLE) as the number of Fourier modes becomes sufficiently large. A necessary and sufficient conditions for consistency and asymptotic normality are presented in terms of the eigenvalues of the operators in the equation.