## Bernoulli

### Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches

#### Abstract

Basic relations between the distributions of hitting, occupation and inverse local times of a one-dimensional diffusion process $X$, first discussed by It\^o and McKean, are reviewed from the perspectives of martingale calculus and excursion theory. These relations, and the technique of conditioning on $L_T^y$, the local time of $X$ at level $y$ before a suitable random time $T$, yield formulae for the joint Laplace transform of $L_T^y$ and the times spent by $X$ above and below level $y$ up to time $T$.

#### Article information

Source
Bernoulli, Volume 9, Number 1 (2003), 1-24.

Dates
First available in Project Euclid: 6 November 2003

https://projecteuclid.org/euclid.bj/1068129008

Digital Object Identifier
doi:10.3150/bj/1068129008

Mathematical Reviews number (MathSciNet)
MR1963670

Zentralblatt MATH identifier
1024.60032

#### Citation

Pitman, Jim; Yor, Marc. Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches. Bernoulli 9 (2003), no. 1, 1--24. doi:10.3150/bj/1068129008. https://projecteuclid.org/euclid.bj/1068129008