Tanaka's formula

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

[math]\displaystyle{ |B_t| = \int_0^t \sgn(B_s)\, dB_s + L_t }[/math]

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

[math]\displaystyle{ \sgn (x) = \begin{cases} +1, & x \gt 0; \\0,& x=0 \\-1, & x \lt 0. \end{cases} }[/math]

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

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

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), 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 [math]\displaystyle{ f(x)=|x| }[/math], with [math]\displaystyle{ f'(x) = \sgn(x) }[/math] and [math]\displaystyle{ f''(x) = 2\delta(x) }[/math]; 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

[math]\displaystyle{ \frac{x^2}{2|\varepsilon|}+\frac{|\varepsilon|}{2}. }[/math]

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

References

• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1.  (Example 5.3.2)
• Shiryaev, Albert N.; trans. N. Kruzhilin (1999). Essentials of stochastic finance: Facts, models, theory. Advanced Series on Statistical Science & Applied Probability No. 3. River Edge, NJ: World Scientific Publishing Co. Inc.. ISBN 981-02-3605-0. https://archive.org/details/essentialsofstoc0000shir.