Tanaka's Formula

In the stochastic calculus, Tanaka's formula for the Brownian motion states that

:$, B_t, = \int_0^t \sgn(B_s)\, dB_s + L_t$

where ''B''_''t'' is the standard Brownian motion, sgn denotes the sign function

:$\sgn (x) = \begin +1, & x > 0; \\0,& x=0 \\-1, & x < 0. \end$

and ''L''_''t'' is its local time at 0 (the local time spent by ''B'' at 0 before time ''t'') given by the ''L''²-limit

:$L_ = \lim_ \frac1 , \ , .$

One can also extend the formula to semimartingales.

Properties

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, which is a Brownian motion), 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 $f(x)=, x,$, with $f'(x) = \sgn(x)$ and $f''(x) = 2\delta(x)$; see local time for a formal explanation of the Itō term.

Outline of proof

The function , ''x'', is not ''C''² in ''x'' at ''x'' = 0, so we cannot apply Itō's formula directly. But if we approximate it near zero (i.e. in minus;''ε'', ''ε'' by parabolas

:$\frac+\frac.$

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

References

* (Example 5.3.2)
* {{cite book , last = Shiryaev
, first = Albert N.
, authorlink = Albert Shiryaev
, title = Essentials of stochastic finance: Facts, models, theory
, series = Advanced Series on Statistical Science & Applied Probability No. 3
, author2 = trans. N. Kruzhilin
, publisher = World Scientific Publishing Co. Inc.
, location = River Edge, NJ
, year = 1999
, isbn = 981-02-3605-0
, url-access = registration
, url = https://archive.org/details/essentialsofstoc0000shir

Equations
Martingale theory
Probability theorems
Stochastic calculus