Hostname: page-component-848d4c4894-cjp7w Total loading time: 0 Render date: 2024-06-17T12:40:04.585Z Has data issue: false hasContentIssue false
  • English
  • Français

On the solution of Kac-type partial differential equations

Part of: Inference from stochastic processes Stochastic analysis Parabolic equations and systems

Published online by Cambridge University Press:  14 July 2016

Mátyás Arató
Get access Rights & Permissions [Opens in a new window]

Abstract

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

Keywords

AUTOREGRESSIONCAUCHY PROBLEMBROWNIAN MOTIONRADON–NIKODYM DERIVATIVESTOCHASTIC FUNCTIONALSDISTRIBUTIONSLAPLACE TRANSFORMPARTIAL DIFFERENTIAL EQUATIONS

MSC classification

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 62M10: Time series, auto-correlation, regression, etc. 35K99: None of the above, but in this section
Type
Part 6 Stochastic Processes
Information
Journal of Applied Probability , Volume 31 , Issue A , 1994 , pp. 311 - 324
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Arató, M. (1962) Estimation of the parameters of a stationary Gaussian Markov process. Dokl. Akad. Nauk SSSR 145, 13–14 (Soviet Math. Dokl. 3 (1962), 905-909).Google Scholar
[2] Arato, M. (1982) Linear Stochastic Systems with Constant Coefficients. Lecture Notes in Control and Information, Springer-Verlag, Berlin. (In Russian: Nauka, Moscow, 1989).Google Scholar
[3] Arato, M. and Benczúr, A. (1970) Distribution function of the damping parameter of stationary Gaussian processes. Studia Sci. Math. Hungar. 5, 445–456.Google Scholar
[4] Arato, M., Kolmogorov, A. N. and Sinai, Ja. G. (1962) Estimation of the parameters of a complex stationary Gaussian Markov process. Dokl. Akad. Nauk SSR 146, 747–750.Google Scholar
[5] Barra, J. R. (1971) Notions Fondamentales de Statistique Mathématique. Dunod, Paris.Google Scholar
[6] Friedman, A. (1975) Stochastic Differential Equations and Applications. Academic Press, New York.Google Scholar
[7] Gikhman, I. I. and Skorokhod, A. V. (1969) Introduction to the Theory of Random Processes. W. B. Saunders, Philadelphia, PA.Google Scholar
[8] Levy, P., Processus Stochastiques et Mouvement Brownien , 2ème édn. Gauthier-Villars, Paris.Google Scholar
[9] Liptser, R. and Shiryayev, A. N. (1977) Statistics of Random Processes. Springer-Verlag, New York.Google Scholar
[10] Kac, M. (1951) On some connections between probability theory and differential and integral equations. Proc. 2nd Berkeley Symp. Math. Statist. Prob. 189–215.Google Scholar
[11] Koncz, K. (1987) On the parameter estimation of diffusional type processes with constant coefficients. Anal. Math. 13, 75–91.Google Scholar
[12] Novikov, A. (1972) On the estimation of parameters of diffusion processes. Studia Sci. Math. Hungar. 7, 201–209.Google Scholar