Local Time and Tanaka’s Formula

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:

$$L\left( {t,x} \right) = \mathop {\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{{2\varepsilon }}\int\limits_0^t {1\left( {x - \varepsilon ,x + \varepsilon } \right)} \left( {{B_s}} \right)ds{\text{ }} = \mathop {\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{{2\varepsilon }}\lambda \left\{ {s \in \left[ {0,t} \right]:{B_s} \in \left( {x - \varepsilon ,x + \varepsilon } \right)} \right\}{\text{ }}$$

(7.1)

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.