Tanaka's formula

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

${\displaystyle |B_{t}|=\int _{0}^{t}\operatorname {sgn}(B_{s})\,dB_{s}+L_{t}}$

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

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

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

${\displaystyle L_{t}=\lim _{\varepsilon \downarrow 0}{\frac {1}{2\varepsilon }}|\{s\in [0,t]|B_{s}\in (-\varepsilon ,+\varepsilon )\}|.}$

One can also extend the formula to semimartingales.