The Annals of Probability

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

R. F. Bass and M. Cranston

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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.

Article information

Ann. Probab., Volume 14, Number 2 (1986), 490-532.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G44: Martingales with continuous parameter
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J55: Local time and additive functionals 60G57: Random measures

Malliavin calculus point processes pure jump Markov processes stochastic differential equations local times martingales


Bass, R. F.; Cranston, M. The Malliavin Calculus for Pure Jump Processes and Applications to Local Time. Ann. Probab. 14 (1986), no. 2, 490--532. doi:10.1214/aop/1176992528.

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