2020/04/30 by Randolf Altmeyer, Altmeyer, Randolf, Igor Cialenco +3 · 1 citation
Economics, Econometrics and Finance · Physics and Astronomy · #62G05 62F12 #62M05 #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Primary 60F05 #Probability (math.PR) #Secondary 60H15 #Statistics Theory (math.ST) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2004.14728
openalex publication_date 2020/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work contributes to the limited literature on estimating the diffusivity or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that the solution is measured locally in space and over a finite time interval, we show that the augmented maximum likelihood estimator introduced in Altmeyer, Reiss (2020) retains its asymptotic properties when used for semilinear SPDEs that satisfy some abstract, and verifiable, conditions. The proofs of asymptotic results are based on splitting the solution in linear and nonlinear parts and fine regularity properties in Lp-spaces. The obtained general results are applied to particular classes of equations, including stochastic reaction-diffusion equations. The stochastic Burgers equation, as an example with first order nonlinearity, is an interesting borderline case of the general results, and is treated by a Wiener chaos expansion. We conclude with numerical examples that validate the theoretical results.