Tanaka's formula

In the stochastic calculus, Tanaka's formula for the Brownian motion states that

: $|B_t| = \int_0^t \sgn(B_s)\, dB_s + L_t$

where B_t is the standard Brownian motion, sgn denotes the sign function

: $\sgn (x) = \begin{cases} +1, & x > 0; \\0,& x=0 \\-1, & x < 0. \end{cases}$

and L_t is its local time at 0 (the local time spent by B at 0 before time t) given by the L²-limit

: $L_{t} = \lim_{\varepsilon \downarrow 0} \frac1{2 \varepsilon} | \{ s \in [0, t] | B_{s} \in (- \varepsilon, + \varepsilon) \} |.$

One can also extend the formula to semimartingales.

Properties

Tanaka's formula is the explicit Doob–Meyer decomposition of the submartingale |B_t| into the martingale part (the integral on the right-hand side, which is a Brownian motion{{Cite book|last=Rogers|first=L.G.C.|title=Diffusions, Markov Processes and Martingales: Volume 1, Foundations|pages=30|chapter=I.14}}), and a continuous increasing process (local time). It can also be seen as the analogue of Itō's lemma for the (nonsmooth) absolute value function $f(x)=|x|$ , with $f'(x) = \sgn(x)$ and $f''(x) = 2\delta(x)$ ; see local time for a formal explanation of the Itō term.

Outline of proof

The function |x| is not C² in x at x = 0, so we cannot apply Itō's formula directly. But if we approximate it near zero (i.e. in [−ε, ε]) by parabolas

: $\frac{x^2}{2|\varepsilon|}+\frac$

\varepsilon

{2}.

and use Itō's formula, we can then take the limit as ε → 0, leading to Tanaka's formula.

References

{{cite book

| last = Øksendal

| first = Bernt K.

| authorlink = Bernt Øksendal

| title = Stochastic Differential Equations: An Introduction with Applications

| edition = Sixth

| publisher=Springer

| location = Berlin

| year = 2003

| isbn = 3-540-04758-1

}} (Example 5.3.2)

{{cite book |last = Shiryaev

|first = Albert N.

|authorlink = Albert Shiryaev

|title = Essentials of stochastic finance: Facts, models, theory

|series = Advanced Series on Statistical Science & Applied Probability No. 3

|author2 = trans. N. Kruzhilin

|publisher = World Scientific Publishing Co. Inc.

|location = River Edge, NJ

|year = 1999

|isbn = 981-02-3605-0

|url-access = registration

|url = https://archive.org/details/essentialsofstoc0000shir

}}

Category:Equations

Category:Martingale theory

Category:Theorems in probability theory

Category:Stochastic calculus