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

November, 2001
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Jim Pitman and Marc Yor

Basic relations between the distributions of hitting, occupation, and inverse local times of a one-dimensional diffusion process $X$, first discussed by \Ito-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 $T$.

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