On Occupation Times of One-Dimensional Diffusions

Paavo Salminen () and David Stenlund ()
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Paavo Salminen: Åbo Akademi University
David Stenlund: Åbo Akademi University

Journal of Theoretical Probability, 2021, vol. 34, issue 2, 975-1011

Abstract: Abstract In this paper, we study the moment generating function and the moments of occupation time functionals of one-dimensional diffusions. Assuming, specifically, that the process lives on $${{\,\mathrm{{\mathbb {R}}}\,}}$$ R and starts at 0, we apply Kac’s moment formula and the strong Markov property to derive an expression for the moment generating function in terms of the Green kernel of the underlying diffusion. Moreover, the approach allows us to derive a recursive equation for the Laplace transforms of the moments of the occupation time on $${{\,\mathrm{{\mathbb {R}}}\,}}_+$$ R + . If the diffusion has a scaling property, the recursive equation simplifies to an equation for the moments of the occupation time up to time 1. As examples of diffusions with scaling property, we study in detail skew two-sided Bessel processes and, as a special case, skew Brownian motion. It is seen that for these processes our approach leads to simple explicit formulas. The recursive equation for a sticky Brownian motion is also discussed.

Keywords: Additive functional; Arcsine law; Green kernel; Oscillating Brownian motion; Brownian spider; Lamperti distribution; 60J60; 60J55; 05A10; 60J65 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-020-00993-3

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