Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches. (English) Zbl 1024.60032

Pitman, Jim; Yor, Marc

Bernoulli 9, No. 1, 1-24 (2003).

Summary: Basic relations between the distributions of hitting, occupation and inverse local times of a one-dimensional diffusion process \(X\), first discussed by K. Itô and H. P. McKean jun. [“Diffusion processes and their sample paths” (1965; Zbl 0127.09503)], are reviewed from the perspectives of martingale calculus and excursion theory. These relations, and the technique of conditioning on \(L^y_T\), the local time of \(X\) at level \(y\) before a suitable random time \(T\), yield formulae for the joint Laplace transform of \(L^y_T\) and the times spent by \(X\) above and below level \(y\) up to time \(T\).

MSC:

60J55

Local time and additive functionals

Keywords:

arcsine law; Feynman-Kac formula; last exit decomposition