Tanaka's formula

In the stochastic calculus, Tanaka's formula states that

${\displaystyle |B_{t}|=\int _{0}^{t}\operatorname {sgn}(B_{s})\,dB_{s}+L_{t}}$

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

${\displaystyle \operatorname {sgn}(x)={\begin{cases}+1,&x>0;\\0,&x=0\\-1,&x<0.\end{cases}}}$

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

${\displaystyle L_{t}=\lim _{\varepsilon \downarrow 0}{\frac {1}{2\varepsilon }}|\{s\in [0,t]|B_{s}\in (-\varepsilon ,+\varepsilon )\}|.}$

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), 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 ${\displaystyle f(x)=|x|}$, with ${\displaystyle f'(x)=\operatorname {sgn}(x)}$ and ${\displaystyle f''(x)=2\delta (x)}$; 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

${\displaystyle {\frac {x^{2}}{2|\varepsilon |}}+{\frac {|\varepsilon |}{2}}.}$

And using 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.