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 Bt 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 Lt 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 |Bt| 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 $$f(x)=|x|$$, 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 [&minus;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.