Tanaka's formula
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In the stochastic calculus, Tanaka's formula states that
where Bt is the standard brownian motion and Lt(0) is its local time at zero. It is the explicit Doob-Meyer decomposition of the submartingale |Bt| into the martingale part (the integral on the right-hand side), and a non-decreasing predictable process (local time).
[edit] Outline of proof
The function |x| is not smooth, so we cannot apply Ito's formula directly. But if we approximate it near zero (i.e. in [-ε; ε]) by parabolas x2/ε and use Ito's formula we can then take the limit as ε → 0, leading to Tanaka's formula.
[edit] References
A. N. Shiryaev Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific, 1999, ISBN 978-9810236052
