A Poisson representation of the positive sojourn time of Lévy processes
Helmut H. Pitters
TL;DR
The paper identifies a general Poisson representation for the positive occupation (sojourn) time $A_t$ of an arbitrary Lévy process, showing that at an independent exponential time $E^{(q)}$, the occupation time satisfies $A_{E^{(q)}}=_d \sum_{T\in\Pi} T$ where $\Pi$ is a Poisson process with intensity $e^{-qt} t^{-1} \mathbb{P}\{X_t>0\}$. It provides a fundamental Laplace-transform identity $G(q,\lambda)=\mathbb{E}[\int_0^\infty e^{-qt} e^{-\lambda A_t} dt] = q^{-1} \exp(-\int_0^\infty e^{-qt} t^{-1} \mathbb{P}\{X_t>0\}(1-e^{-\lambda t}) dt)$ and proves the equivalence between constant positivity, the arcsine law for $A_t/t$, and a gamma/arcsine decomposition for $A_{E^{(q)}}/E^{(q)}$. The work unifies classical arcsine results (Brownian motion, symmetric stable processes) within this Poisson framework and extends explicit results to the $(1/2)$-stable subordinator with drift, yielding exact density and transform formulas. The approach leverages sampling of occupation times, Bell polynomials, Spitzer's lemma, and Campbell’s theorem, linking occupation times to Poisson-Dirichlet structure and offering new, computable representations beyond existing moment-based characterizations. Overall, it broadens the understanding of occupation times for Lévy processes and connects probabilistic structure to explicit, actionable formulas with potential applications in finance and stochastic analysis.
Abstract
We study the distribution of the positive sojourn time $$ A_t:= \int_0^t \mathbf 1\{ X_s>0 \}ds $$ of an arbitrary Lévy 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 Π} T \end{align*} as the sum of points of a Poisson process $Π$ 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 $Π$ 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 Lévy process with constant positivity, we obtain an extension and new derivation of classical (generalised) arcsine laws going back to Lévy (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 Lévy 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.