A note on occupation times of stationary processes

DOI10.1214/ECP.V10-1138zbMATH Open1110.60029arXivmath/0408178OpenAlexW1967037245MaRDI QIDQ2433638

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Publication date: 3 November 2006

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Abstract: In this paper excursions of a stationary diffusion in stationary state are studied. In particular, we compute the joint distribution of the occupation times $I_t^{(+)}$ and $I_t^{(-)}$ above and below, respectively, the observed level at time $t$ during an excursion. We consider also the starting time $g_t$ and the ending time $d_t$ of the excursion (straddling $t$) and discuss their relations to the Levy measure of the inverse local time. It is seen that the pairs $(I_t^{(+)},I_t^{(-)})$ and $(t-g_t,d_t-t)$ are identically distributed. Moreover, conditionally on $I_t^{(+)}+I_t^{(-)}=v$, the variables $I_t^{(+)}$ and $I_t^{(-)}$ are uniformly distributed on $(0,v)$. Using the theory of the Palm measures, we derive an analoguous result for excursion bridges.

Full work available at URL: https://arxiv.org/abs/math/0408178

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