Tanaka's formula

In the , 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 L²-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 ${\displaystyle f(x)=|x|}$, with ${\displaystyle f'(x)=\operatorname {sgn}(x)}$ and ${\displaystyle f''(x)=2\delta (x)}$; see local time for a formal explanation of the Itō term.

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