On the solution of Kac-type partial differential equations

1994 ◽  
Vol 31 (A) ◽  
pp. 311-324
Author(s):  
Mátyás Arató
Keyword(s):  
Stochastic Process ◽  
Cauchy Problem ◽  
Brownian Motion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Explicit Form ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
The Cauchy Problem ◽  
Constant Coefficients

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].

On the solution of Kac-type partial differential equations

1994 ◽  
Vol 31 (A) ◽  
pp. 311-324
Author(s):  
Mátyás Arató
Keyword(s):  
Stochastic Process ◽  
Cauchy Problem ◽  
Brownian Motion ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Explicit Form ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
The Cauchy Problem ◽  
Constant Coefficients

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].

scholarly journals On the Entropy of Fractionally Integrated Gauss–Markov Processes

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2031
Author(s):  
Mario Abundo ◽  
Enrica Pirozzi
Keyword(s):  
Dynamical System ◽  
Stochastic Process ◽  
Brownian Motion ◽  
Markov Process ◽  
Markov Processes ◽  
Numerical Approximation ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Ornstein Uhlenbeck Process ◽  
Application Fields

This paper is devoted to the estimation of the entropy of the dynamical system {Xα(t),t≥0}, where the stochastic process Xα(t) consists of the fractional Riemann–Liouville integral of order α∈(0,1) of a Gauss–Markov process. The study is based on a specific algorithm suitably devised in order to perform the simulation of sample paths of such processes and to evaluate the numerical approximation of the entropy. We focus on fractionally integrated Brownian motion and Ornstein–Uhlenbeck process due their main rule in the theory and application fields. Their entropy is specifically estimated by computing its approximation (ApEn). We investigate the relation between the value of α and the complexity degree; we show that the entropy of Xα(t) is a decreasing function of α∈(0,1).

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