 Tanaka's formula

In the stochastic calculus, Tanaka's formula states that

where "B""t" is the standard Brownian motion, sgn denotes the sign function

and "L""t" is its local time at 0 (the local time spent by "B" at 0 before time "t") given by the "L"2-limit

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), 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 formula_4, with formula_5 and formula_6; see local time for a formal explanation of the Itō term.