2023/07/11 by Sascha Gaudlitz, Gaudlitz, Sascha · 1 citation
Economics, Econometrics and Finance · Mathematics · #60H15 #FOS: Mathematics #Financial Risk and Volatility Modeling #Primary: 62G05 #Probability (math.PR) #Secondary: 60H07 #Statistical Methods and Inference #Statistics Theory (math.ST) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2307.05457
openalex publication_date 2023/07/11 · openalex created_date 2023/07/13 · openalex updated_date 2026/08/01
This paper discusses the non-parametric estimation of a non-linear reaction term in a semi-linear parabolic stochastic partial differential equation (SPDE). The estimator's consistency is due to the spatial ergodicity of the SPDE while the time horizon remains fixed. The analysis of the estimation error requires the concentration of spatial averages of non-linear transformations of the SPDE. The method developed in this paper combines the Clark-Ocone formula from Malliavin calculus with the Markovianity of the SPDE and density estimates. The resulting variance bound utilises the averaging effect of the conditional expectation in the Clark-Ocone formula. The method is applied to two realistic asymptotic regimes. The focus is on a coupling between the diffusivity and the noise level, where both tend to zero. Secondly, the observation of a fixed SPDE on a growing spatial observation window is considered. Furthermore, the concentration of the occupation time around the occupation measure is proved.