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.
Publié le : 1986-04-14
Classification:
Malliavin calculus,
point processes,
pure jump Markov processes,
stochastic differential equations,
local times,
martingales,
60G44,
60J35,
60J55,
60G57
@article{1176992528,
author = {Bass, R. F. and Cranston, M.},
title = {The Malliavin Calculus for Pure Jump Processes and Applications to Local Time},
journal = {Ann. Probab.},
volume = {14},
number = {4},
year = {1986},
pages = { 490-532},
language = {en},
url = {http://dml.mathdoc.fr/item/1176992528}
}
Bass, R. F.; Cranston, M. The Malliavin Calculus for Pure Jump Processes and Applications to Local Time. Ann. Probab., Tome 14 (1986) no. 4, pp. 490-532. http://gdmltest.u-ga.fr/item/1176992528/