197 0894-9840/03/0100-0197/0 © 2003 Plenum Publishing Corporation Journal of Theoretical Probability, Vol. 16, No. 1, January 2003 (© 2003) On Non-Continuous Dirichlet Processes François Coquet, 1 Jean Mémin, 1, 2 and Leszek Slomin ´ski 3 1 IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France. E-mail: Francois.Coquet@univ-rennes1.fr 2 To whom correspondence should be addressed. E-mail: memin@univ-rennes1.fr 3 Faculty of Mathematics and Computer Science, N. Copernicus University, ul. Chopina, 87-100 Torun, Poland. E-mail: leszeks@mat.uni.torun.pl Received June 18, 2001; revised October 18, 2001 We introduce here some Ito ˆ calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C 1 transformation. KEY WORDS: Non-continuous Dirichlet process; stochastic integral; Ito ˆ formula. 0. INTRODUCTION Since the pioneering paper (7) by Föllmer, Dirichlet processes viewed as natural generalisations of semimartingales have taken increasing interest, be it for their own sake or for the specific processes (e.g., fractional brow- nian motion) they can give a new light on. Up to now, people have mostly concentrated on continuous Dirichlet processes. Although the definitions vary a little bit from an author to another one, continuous Dirichlet processes appear to be the natural class to introduce when dealing with C 1 transformations of continuous semi- martingales. More generally, C 1 transformations of continuous Dirichlet processes are again Dirichlet processes (see, for instance, Refs. 1, 3, or 13). On another hand, it is to some extent possible to define stochastic integrals with respect to Dirichlet processes, allowing for some of the usual tools of stochastic calculus and differential equations.