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.

Properties

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

Outline of proof

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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 [−εε]) by parabolas

 

and use Itō's formula, we can then take the limit as ε → 0, leading to Tanaka's formula.

References

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  1. ^ Rogers, L.G.C. "I.14". Diffusions, Markov Processes and Martingales: Volume 1, Foundations. p. 30.