Exact variations for stochastic heat equations with piecewise constant coefficients and application to parameter estimation
Authors:
M. Zili and E. Zougar
Journal:
Theor. Probability and Math. Statist. 100 (2020), 77-106
MSC (2010):
Primary 60H15, 60G15, 60G17, 35R05, 60G17; Secondary 60G60, 35K10, 33B20, 62F10
DOI:
https://doi.org/10.1090/tpms/1099
Published electronically:
August 4, 2020
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We 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. Both expansions allow us to deduce an estimation method of the parameters appearing in the equation.
References
- 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.
- Zhen-Qing Chen and Mounir Zili, One-dimensional heat equation with discontinuous conductance, Sci. China Math. 58 (2015), no. 1, 97–108. MR 3296333, DOI https://doi.org/10.1007/s11425-014-4912-1
- Robert C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s, Electron. J. Probab. 4 (1999), no. 6, 29. MR 1684157, DOI https://doi.org/10.1214/EJP.v4-43
- Robert C. Dalang and Lluís Quer-Sardanyons, Stochastic integrals for spde’s: a comparison, Expo. Math. 29 (2011), no. 1, 67–109. MR 2785545, DOI https://doi.org/10.1016/j.exmath.2010.09.005
- Antoine Lejay, Monte Carlo methods for fissured porous media: a gridless approach, Monte Carlo Methods Appl. 10 (2004), no. 3-4, 385–392. MR 2105066, DOI https://doi.org/10.1515/mcma.2004.10.3-4.385
- S. Nicaise, Some results on spectral theory over networks, applied to nerve impulse transmission, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 532–541. MR 839024, DOI https://doi.org/10.1007/BFb0076584
- M. H. Protter and C. B. Morrey, Intermediate Calculus, Springer-Verlag, Berlin, Heidelberg, 1985.
- Jan Pospíšil and Roger 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. MR 2321899, DOI https://doi.org/10.1080/07362990701282849
- John J. Shynk, Probability, random variables, and random processes, John Wiley & Sons, Inc., Hoboken, NJ, 2013. Theory and signal processing applications. MR 3088510
- J. Swanson, Variations of the solution to a stochastic heat equation, Ann. Probab. 35 (2007), no. 6, 2122–2159.
- Ciprian A. Tudor, Analysis of variations for self-similar processes, Probability and its Applications (New York), Springer, Cham, 2013. A stochastic calculus approach. MR 3112799
- C. Vignat, A generalized Isserlis theorem for location mixtures of Gaussian random vectors, Statist. Probab. Lett. 82 (2012), no. 1, 67–71. MR 2863025, DOI https://doi.org/10.1016/j.spl.2011.09.008
- Mounir Zili, Développement asymptotique en temps petits de la solution d’une équation aux dérivées partielles de type parabolique, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 8, 1049–1052 (French, with English and French summaries). MR 1360571
- M. Zili, Construction d’une solution fondamentale d’une équation aux dérivées partielles à coefficients constants par morceaux, Bull. Sci. Math. 123 (1999), no. 2, 115–155 (French, with English and French summaries). MR 1679034, DOI https://doi.org/10.1016/S0007-4497%2899%2980017-7
- Mounir Zili and Eya Zougar, One-dimensional stochastic heat equation with discontinuous conductance, Appl. Anal. 98 (2019), no. 12, 2178–2191. MR 3988829, DOI https://doi.org/10.1080/00036811.2018.1451642
References
- 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.
- Z. Q. Chen and M. Zili, One-dimensional heat equation with discontinuous conductance, Science China Mathematics 58 (2015), no. 1, 97–108. MR 3296333
- 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. MR 1684157
- R. C. Dalang and L. Q. Sardanyons, Stochastic integrals for spde’s: a comparison, Expositiones Mathematicae 29 (2011), 67–109. MR 2785545
- A. Lejay, \text{Monte Carlo methods for fissured porous media: a gridless approach}, Monte Carlo Methods Appl. 10 (2004), 385–392. MR 2105066
- 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. MR 839024
- M. H. Protter and C. B. Morrey, Intermediate Calculus, Springer-Verlag, Berlin, Heidelberg, 1985.
- 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. MR 2321899
- J. J. Shynk, Probability, Random Variables and Random Processes, Wiley, Hoboken, 2013. MR 3088510
- J. Swanson, Variations of the solution to a stochastic heat equation, Ann. Probab. 35 (2007), no. 6, 2122–2159.
- C. A. Tudor, Analysis of Variations for Self-Similar Processes, Springer, 2013. MR 3112799
- C. Vignat, A generalized Isserlis theorem for location mixtures of Gaussian random vectors, Stat. Probab. Letters 82 (2012), no. 1, 67–71. MR 2863025
- M. Zili, Développement asymptotique en temps petits de la solution d’une équation aux dérivées partielles de type parabolique généralisée au sens des distributions-mesures, Note des Comptes Rendues de l’Académie des Sciences de Paris 321 (1995), 1049–1052. MR 1360571
- M. Zili, Construction d’une solution fondamentale d’une équation aux dérivées partielles à coefficients constants par morceaux, Bull. Sci. Math. 123 (1999), 115–155. MR 1679034
- Mounir Zili and Eya Zougar, One-dimensional stochastic heat equation with discontinuous conductance, Appl. Anal. 98 (2019), 2178–2191. MR 3988829
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60H15,
60G15,
60G17,
35R05,
60G17,
60G60,
35K10,
33B20,
62F10
Retrieve articles in all journals
with MSC (2010):
60H15,
60G15,
60G17,
35R05,
60G17,
60G60,
35K10,
33B20,
62F10
Additional Information
M. Zili
Affiliation:
Department of Mathematics, Faculty of sciences of Monastir, University of Monastir, Avenue de l’environnement, 5019 Monastir, Tunisia
Email:
mounir.zili@fsm.rnu.tn
E. Zougar
Affiliation:
Department of Mathematics, Faculty of sciences of Monastir, University of Monastir, Avenue de l’environnement, 5019 Monastir, Tunisia
Email:
zougareya@gmail.com
Keywords:
Quartic and quadratic variations,
stochastic partial differential equations,
discontinuity,
integration techniques,
special functions,
estimation of parameters
Received by editor(s):
September 13, 2018
Published electronically:
August 4, 2020
Article copyright:
© Copyright 2020
American Mathematical Society
