Zili, M.; Zougar, E. 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. Reviewer: Udhayakumar Ramalingam (Vellore) Cited in 5 Documents 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 Keywords:quartic and quadratic variations; stochastic partial differential equations; special functions; estimation of parameters PDFBibTeX XMLCite \textit{M. Zili} and \textit{E. Zougar}, Theory Probab. Math. Stat. 100, 77--106 (2020; Zbl 1454.60100) Full Text: DOI References: [1] Can R. Cantrell and C. Cosner, Diffusion models for population dynamics incorporating individual behavior at boundaries: Applications to refuge design, Theor. Population Biology 55 (1999), 198-207. · Zbl 0958.92028 [2] CZ Z. Q. Chen and M. Zili, One-dimensional heat equation with discontinuous conductance, Science China Mathematics 58 (2015), no. 1, 97-108. · Zbl 1322.60129 [3] Dalang R. C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E.’s, Electronic J. Probab. 4 (1999), 1-29. · Zbl 0922.60056 [4] DalQuer R. C. Dalang and L. Q. Sardanyons, Stochastic integrals for spde’s: a comparison, Expositiones Mathematicae 29 (2011), 67-109. · Zbl 1234.60064 [5] Lej A. Lejay, Monte Carlo methods for fissured porous media: a gridless approach, Monte Carlo Methods Appl. 10 (2004), 385-392. · Zbl 1109.76378 [6] Nic S. Nicas, Some results on spectral theory over networks, applied to nerve impulse transmission, Orthogonal Polynomials and Applications (Bar-le-Duc, 1984), Lect. Notes Math., vol. 1171, Springer, pp. 532-541. [7] Protter M. H. Protter and C. B. Morrey, Intermediate Calculus, Springer-Verlag, Berlin, Heidelberg, 1985. · Zbl 0555.26002 [8] PsTrib J. Pospisil and R. Tribe, Parameter estimates and exact variations for stochastic heat equations driven by space-time white noise, Stoch. Anal. Appl. 25 (2007), no. 3, 593-611. · Zbl 1118.60030 [9] JS J. J. Shynk, Probability, Random Variables and Random Processes, Wiley, Hoboken, 2013. [10] Swanson J. Swanson, Variations of the solution to a stochastic heat equation, Ann. Probab. 35 (2007), no. 6, 2122-2159. · Zbl 1135.60041 [11] Tud1 C. A. Tudor, Analysis of Variations for Self-Similar Processes, Springer, 2013. [12] V C. Vignat, A generalized Isserlis theorem for location mixtures of Gaussian random vectors, Stat. Probab. Letters 82 (2012), no. 1, 67-71. · Zbl 1241.60011 [13] Zi M. Zili, D\'eveloppement asymptotique en temps petits de la solution d’une \'equation aux d\'eriv\'ees partielles de type parabolique g\'en\'eralis\'ee au sens des distributions-mesures, Note des Comptes Rendues de l’Acad\'emie des Sciences de Paris 321 (1995), 1049-1052. [14] Zil M. Zili, Construction d’une solution fondamentale d’une \'equation aux d\'eriv\'ees partielles \`a coefficients constants par morceaux, Bull. Sci. Math. 123 (1999), 115-155. · Zbl 0919.35006 [15] ZZMounir Zili and Eya Zougar, One-dimensional stochastic heat equation with discontinuous conductance, Appl. Anal. 98 (2019), 2178-2191. · Zbl 1427.60138 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.