Seminar on Stochastic Processes, 1989 pp 59-73 | Cite as

# 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.

## Keywords

Lebesgue Measure Local Time Occupation Time Regular Case Compound Poisson Process## Preview

Unable to display preview. Download preview PDF.

## References

- 1.V. BALLY. Approximation theorems for the local time of a Markov process.
*Studii si Cercetări Matematice***38**(1986) 139–147. Bucuresti.MathSciNetMATHGoogle Scholar - 2.V. BALLY and L. STOICA. A class of Markov processes which admit local times.
*Ann. Probab.***15**(1987) 241–262.MathSciNetMATHCrossRefGoogle Scholar - 3.R. M. BLUMENTHAL and R. K. GETOOR. Local times for Markov processes.
*Z. Wahrsch. verw. Gebiete***3**(1964) 50–74.MathSciNetMATHCrossRefGoogle Scholar - 4.J. BRETAGNOLLE. Resultats de Kesten sur les processus a acroissements indépendants.
*Sem. Prob.*V.*Lecture Notes in Math*.**191**21–36, Springer, Berlin, 1971.Google Scholar - 5.B. E. FRISTEDT and S. J. TAYLOR. Constructions of local time for a Markov process.
*Z. Wahrsch. verw. Gebiete***62**(1983) 73–112.MathSciNetMATHCrossRefGoogle Scholar - 6.R. K. GETOOR. Another limit theorem for local time.
*Z. Wahrsch. verw. Gebiete***34**(1976) 1–10.MathSciNetMATHCrossRefGoogle Scholar - 7.R. J. GRIEGO. Local time as a derivative of occupation times.
*Ill. J. Math.***11**(1967) 53–64.MathSciNetGoogle Scholar - 8.
- 9.S.C. PORT and C. J. STONE. Infinitely divisible processes and their potential theory I.
*Ann. Inst. Fourier***21**(1971) 157–275.MathSciNetMATHCrossRefGoogle Scholar - 10.S. C. PORT. Stable processes with drift on the line.
*Trans. Amer. Math. Soc.*(to appear)Google Scholar