Tanaka's formula

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

$|B_t|=\underset{0}{\overset{t}{\int }}\mathrm{sgn}(B_s)d{B}_s+L_t$

where B_t is the standard Brownian motion, sgn denotes the sign function

$\mathrm{sgn}(x)=\{\begin{array}{ll}+1,& x>0;\\ 0,& x=0\\ -1,& x<0.\end{array}$

and L_t is its local time at 0 (the local time spent by B at 0 before time t) given by the L²-limit

$L_t=\underset{\epsilon \downarrow 0}{\lim }\frac{1}{2\epsilon }|\{s\in [0,t]|B_s\in (-\epsilon ,+\epsilon )\}|.$

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 $f^{\prime }(x)=\mathrm{sgn}(x)$ 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 C² in x at x = 0, so we cannot apply Itō's formula directly. But if we approximate it near zero (i.e. in [−ε, ε]) by parabolas

$\frac{x^2}{2|\epsilon |}+\frac{|\epsilon |}{2}.$

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

References

• 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. https://archive.org/details/essentialsofstoc0000shir.