We compute the joint density of Brownian motion, its local time at the origin, and its occupation time of $\lbrack 0, \infty)$. Two derivations of the main result are offered; one is computational, whereas the other uses some of the deep properties of Brownian local time. We use the result to compute the transition probabilities of the optimal process in a stochastic control problem.
Publié le : 1984-08-14
Classification:
Brownian motion,
local time,
occupation time,
Feynman-Kac formula,
Girsanov theorem,
Tanaka formula,
bang-bang stochastic control,
60J65,
93E20,
60G17
@article{1176993230,
author = {Karatzas, Ioannis and Shreve, Steven E.},
title = {Trivariate Density of Brownian Motion, Its Local and Occupation Times, with Application to Stochastic Control},
journal = {Ann. Probab.},
volume = {12},
number = {4},
year = {1984},
pages = { 819-828},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993230}
}
Karatzas, Ioannis; Shreve, Steven E. Trivariate Density of Brownian Motion, Its Local and Occupation Times, with Application to Stochastic Control. Ann. Probab., Tome 12 (1984) no. 4, pp. 819-828. http://gdmltest.u-ga.fr/item/1176993230/