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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. 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
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