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

  • P. J. Fitzsimmons
  • S. C. Port
Part of the Progress in Probability book series (PRPR, volume 18)


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 → Nt(a) and aR 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.


Lebesgue Measure Local Time Occupation Time Regular Case Compound Poisson Process 
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Copyright information

© Birkhäuser Boston 1990

Authors and Affiliations

  • P. J. Fitzsimmons
    • 1
  • S. C. Port
    • 2
  1. 1.Department of Mathematics, C-012University of California, San DiegoLa JollaUSA
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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