zbMATH — the first resource for mathematics

Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches. (English) Zbl 1024.60032
Summary: Basic relations between the distributions of hitting, occupation and inverse local times of a one-dimensional diffusion process \(X\), first discussed by K. ItĂ´ 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 \(T\).

60J55 Local time and additive functionals
Full Text: DOI