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This paper deals with the drift estimation in linear stochastic evolutionequations (with emphasis on linear SPDEs) with additive fractional noise (withHurst index ranging from 0 to 1) via least-squares procedure. Since theleast-squares estimator contains stochastic integrals of divergence type, weaddress the problem of its pathwise (and robust to observation errors)evaluation by comparison with the pathwise integral of Stratonovich type andusing its chain-rule property. The resulting pathwise LSE is then definedimplicitly as a solution to a non-linear equation. We study its numericalproperties (existence and uniqueness of the solution) as well as statisticalproperties (strong consistency and the speed of its convergence). Theasymptotic properties are obtained assuming fixed time horizon and increasingnumber of the observed Fourier modes (space asymptotics). We also conjecturethe asymptotic normality of the pathwise LSE.

Pathwise least-squares estimator for linear SPDEs with additive fractional noise