A Poisson representation of the positive sojourn time of L\'evy processes

Abstract

We study the distribution of the positive sojourn time $$ A_t:= \int_0^t \mathbf 1\{ X_s>0 \}ds $$ of an arbitrary L\'evy process $X:= (X_t)_{t\geq 0}$. For an exponential random variable $E^{(q)}$ of rate $q>0$ independent of $X$ we show the representation in law \begin{align*} A_{E^{(q)}} =_d \sum_{T\in \Pi} T \end{align*} as the sum of points of a Poisson process $\Pi$ with intensity given explicitely in terms of the positivity $t\mapsto \mathbb P\{X_t>0\}$. This representation raises some fundamental questions, not least because $\Pi$ turns out to be intimately connected to the celebrated Poisson-Dirichlet distribution. Moreover, we characterise $A_t$ by working out its double Laplace transform, and thus complement a recent result in which the distribution of $A_t$ was characterised via its higher moments. As a Corollary of the Poisson representation, in the special cases where $X$ is Brownian motion, a symmetric stable process, a L\'evy process with constant positivity, we obtain an extension and new derivation of classical (generalised) arcsine laws going back to L\'evy (1939), Kac (1951), and Getoor and Sharpe (1994), respectively. Even in these cases the Poisson representation is new. As an application, if $X$ is the $(1/2)$-stable subordinator with drift, we obtain both the Laplace transform of $A_t$ and the density of its distribution. This is the second example of a L\'evy process whose occupation time distribution is known explicitely but is not generalized arcsine, the first example being Brownian motion with drift that was studied earlier in the context of option prizing.