Local Times, Occupation Times, and the Lebesgue Measure of the Range of a Levy Process

Abstract

Let X = (X _t:t ≥ 0) be a Lévy process on the line for which singletons are non-polar. We assume that X does not have the form ${\tilde X_t} + bt$, where $\tilde X$ is a compound Poisson process. Let N _t(a) denote the occupation time, up to time t, of (0, a] if a > 0, and the negative of the occupation time, up to time t, of (a, 0] if a ≤ 0. Let R _t(a) denote the Lebesgue measure of the partial range $\left \{X_s:0\leq s\leq t \right \}$ intersected with (0,a] if a > 0 (and the negative of the measure of this range intersected with (a, 0] if a ≤ 0). Our purpose in this paper is to investigate the L ²-differentiability of a → N_t(a) and a → R _t(a). As it turns out, the derivatives of N _t(a) coincide with certain “local times,” even when singletons are semipolar for X. These local times also arise as limits of upcrossing and downcrossing processes. In the following discussion we consider the cases “0 regular for {0}” and “0 irregular for {0}” separately.

Research supported in part by NSF grant DMS 8419377.