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Exact variations for stochastic heat equations with piecewise constant coefficients and application to parameter estimation. (English) Zbl 1454.60100

Theory Probab. Math. Stat. 100, 77-106 (2020) and Teor. Jmovirn. Mat. Stat. 100, 75-101 (2019).
In this paper, the authors expand the quartic variations in time and the quadratic variations in space of the solution to a stochastic partial differential equation with piecewise constant coefficients. In particular, the authors obtain an explicit expansion of the temporal quartic variations in Section 4. Such expansion allows to deduce an estimation method for the parameters appearing in the given equation. In the last section, also an explicit expansion of the spatial quadratic variations is given that also allows to estimate the parameters appearing in the given equation.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
60G17 Sample path properties
35R05 PDEs with low regular coefficients and/or low regular data
60G60 Random fields
35K10 Second-order parabolic equations
33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
62F10 Point estimation
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