Tanaka's formula

In the stochastic calculus, Tanaka's formula for the Brownian motion states that where Bt is the standard Brownian motion, sgn denotes the sign function and Lt is its local time at 0 (the local time spent by B at 0 before time t) given by the L2-limit One can also extend the formula to semimartingales.

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[1]), 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

; see local time for a formal explanation of the Itō term.

The function |x| is not C2 in x at x = 0, so we cannot apply Itō's formula directly.