The Malliavin Calculus for Pure Jump Processes and Applications to Local Time

Abstract

A Malliavin calculus is developed whose scope includes point processes, pure jump Markov processes, and purely discontinuous martingales. An integration by parts formula for functionals of Poisson point processes is proved. This is used to develop a criterion for pure jump Markov processes to have a density in $L^p$. The integration by parts formula is then used to give conditions for a purely discontinuous martingale to have a jointly continuous local time $L^x_t$ that is an occupation time density with respect to Lebesgue measure.