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Generalized Ito Formula and Change of Time

  • K. L. Chung
  • R. J. Williams
Part of the Progress in Probability and Statistics book series (PRPR, volume 4)

Abstract

In this chapter we shall first obtain a generalized Itô formula for convex functions of Brownian motion. Next we shall prove a result which shows that Brownian motion is truly the canonical example of a continuous local martingale. Namely, if M is a continuous local martingale with quadratic variation [M] t increasing to infinity as t → ∞, then there is a random change of time τ t such that {M τt ,t ∈ ℝ+} is a Brownian motion. An application of this result shows that there is a time change τ t such that {B τt , t ∈ ℝ+} is equivalent in law to |B|.

Keywords

Brownian Motion Convex Function Compact Support Quadratic Variation Local Martingale 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • K. L. Chung
    • 1
  • R. J. Williams
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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