IJPAM: Volume 32, No. 4 (2006)

DIFFUSION PROCESSES HAVING GENERALIZED
PARETO PROBABILITY DENSITY FUNCTIONS

Mario Lefebvre
Department of Mathematics and Industrial Engineering
Ecole Polytechnique
C.P. 6079, Succursale Centre-Ville, Montreal, Quebec, H3C 3A7, CANADA
e-mail: mlefebvre@polymtl.ca


Abstract.Let $X(t)$ be a one-dimensional time-inhomogeneous diffusion process with infinitesimal mean $m(x,t)$ and variance $v(x,t)$. The initial state $X(0)$ of the process is random. We consider the Kolmogorov forward equation satisfied by the probability density function $f(x,t)$ of $X(t)$. We find particular cases for which $f$ is the density function of a random variable having a generalized Pareto distribution (depending on $t$). We consider different possibilities for the infinitesimal parameters of the diffusion process, in particular the case when $m(x,t)$ is a constant and that when $v(x,t) \equiv v_0>0$. The solutions obtained correspond to the case when there is a reflecting boundary at the origin or at $-t$.

Received: October 5, 2006

AMS Subject Classification: 60G70

Key Words and Phrases: Kolmogorov forward equation, infinitesimal parameters, reflecting boundary

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 32
Issue: 4