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|.
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© 1983 Springer Science+Business Media New York
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Chung, K.L., Williams, R.J. (1983). Generalized Ito Formula and Change of Time. In: Introduction to Stochastic Integration. Progress in Probability and Statistics, vol 4. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4757-9174-7_9
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DOI: https://doi.org/10.1007/978-1-4757-9174-7_9
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-3117-8
Online ISBN: 978-1-4757-9174-7
eBook Packages: Springer Book Archive