Advances in Difference Equations
Volume 2011 (2011), Article ID 803508, 15 pages
doi:10.1155/2011/803508
Research Article

A Criterium for the Strict Positivity of the Density of the Law of a Poisson Process

Rémi Léandre

Institut de Mathématiques, Université de Bourgogne, 21000 Dijon, France

Received 26 August 2010; Revised 23 December 2010; Accepted 9 January 2011

Academic Editor: Dumitru Baleanu

Copyright © 2011 Rémi Léandre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We translate in semigroup theory our result (Léandre, 1990) giving a necessary condition so that the law of a Markov process with jumps could have a strictly positive density. This result express, that we have to jump in a finite number of jumps in a “submersive" way from the starting point 𝑥 to the end point 𝑦 if the density of the jump process 𝑝 1 ( 𝑥 , 𝑧 ) is strictly positive in ( 𝑥 , 𝑦 ) . We use the Malliavin Calculus of Bismut type of (Léandre, (2008;2010)) translated in semi-group theory as a tool, and the interpretation in semi-group theory of some classical results of the stochastic analysis for Poisson process as, for instance, the formula giving the law of a compound Poisson process.