Tanaka's formula

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In the stochastic calculus, Tanaka's formula 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


Tanaka's formula is the explicit Doob–Meyer decomposition of the submartingale |Bt| 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 , with and ; see local time for a formal explanation of the Itō term.

Outline of proof[edit]

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


  • Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1.  (Example 5.3.2)
  • Shiryaev, Albert N.; trans. N. Kruzhilin (1999). Essentials of stochastic finance: Facts, models, theory. Advanced Series on Statistical Science & Applied Probability No. 3. River Edge, NJ: World Scientific Publishing Co. Inc. ISBN 981-02-3605-0.