Tanaka's formula is a result in stochastic calculus that decomposes the absolute value of Brownian motion into two parts: a stochastic integral of the sign of the Brownian motion against itself, and a process called the local time of the Brownian motion at zero. This gives an explicit Doob-Meyer decomposition of the absolute value of Brownian motion, which is a submartingale, into a martingale part and a continuous, increasing part, the local time. The formula serves as an analogue of Ito's lemma for the absolute value function, which unlike the smooth functions Ito's lemma normally applies to, is not differentiable at zero, so Tanaka's formula shows how to make sense of the resulting stochastic calculus anyway. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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1. Tanaka's Formula (Wikipedia)
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In the stochastic calculus, Tanaka's formula for the Brownian motion states that | B t | = ∫ 0 t sgn ( B s ) d B s + L t {\displaystyle |B_{t}|=\int _{0}^{t}\operatorname {sgn}(B_{s})\,dB_{s}+L_{t}} where Bt is the standard Brownian motion, sgn denotes the sign function sgn ( x ) = { + 1 , x > 0 ; 0 , x = 0 − 1
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In the stochastic calculus, Tanaka's formula for the Brownian motion states that
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