Abstract
In this chapter B denotes a Brownian motion in IR. For each x ∈ IR we shall obtain a decomposition, known as Tanaka’s formula, of the positive submartingale |B − x| as the sum of another Brownian motion \(hat B\) and a continuous increasing process L( · , x). The latter is called the local time of B at x, a fundamental notion invented by P. Lévy (see [54]). It may be expressed as follows:
where λ is the Lebesgue measure. Thus it measures the amount of time the Brownian motion spends in the neighborhood of x. It is well known that {t ∈ I R + : B t = x} is a perfect closed set of Lebesgue measure zero. The existence of a nonvanishing L defined in (7.1) is therefore far from obvious. In fact, the limit in (7.1) exists both in L2 and a.s., as we shall see. Moreover, L(t, x) may be defined to be a jointly continuous function of (t, x). This was first proved by H. F. Trotter, but our approach follows that of Stroock and Varadhan [73, p. 117]. The local time plays an important role in many refined developments of the theory of Brownian motion. One application, given at the end of Section 7.3, is a derivation of the exponential distribution of the local time accumulated up until the hitting time of a fixed level. Other applications of local time and Tanaka’s formula are discussed in the next two chapters.
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© 1990 Birkhäuser Boston
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Chung, K.L., Williams, R.J. (1990). Local Time and Tanaka’s Formula. In: Introduction to Stochastic Integration. Probability and Its Applications. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4480-6_7
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DOI: https://doi.org/10.1007/978-1-4612-4480-6_7
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8837-4
Online ISBN: 978-1-4612-4480-6
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