P. Lévy’s Theory of Brownian Local Time

  • Ioannis Karatzas
  • Steven E. Shreve
Part of the Graduate Texts in Mathematics book series (GTM, volume 113)


This chapter is an in-depth study of the Brownian local time first encountered in Section 3.6. Our approach to this subject is motivated by the desire to perform computations. This is manifested by the inclusion of the conditional Laplace transform formulas of D. Williams (Subsections 6.3.B, 6.4.C), the derivation of the joint density of Brownian motion, its local time at the origin and its occupation time of the positive half-line (Subsection 6.3.C), and the computation of the transition density for Brownian motion with two-valued drift (Section 6.5). This last computation arises in the problem of controlling the drift of a Brownian motion, within prescribed bounds, so as to keep the controlled process near the origin.


Brownian Motion Local Time Occupation Time Bessel Process Poisson Random Variable 
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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Ioannis Karatzas
    • 1
  • Steven E. Shreve
    • 2
  1. 1.Departments of Mathematics and StatisticsColumbia UniversityNew YorkUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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