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Tanaka's formula

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Not to be confused with Tanaka equation.

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

| B t | = ∫ 0 t sgn ⁡ ( B s ) d B s + L t {\displaystyle |B_{t}|=\int _{0}^{t}\operatorname {sgn}(B_{s})\,dB_{s}+L_{t}}

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

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

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

L t = lim ε ↓ 0 1 2 ε | { s ∈ [ 0 , t ] | B s ∈ ( − ε , + ε ) } | . {\displaystyle L_{t}=\lim _{\varepsilon \downarrow 0}{\frac {1}{2\varepsilon }}|\{s\in |B_{s}\in (-\varepsilon ,+\varepsilon )\}|.}

One can also extend the formula to semimartingales.

Properties

Tanaka's formula is the explicit Doob–Meyer decomposition of the submartingale |Bt| 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 | {\displaystyle f(x)=|x|} , with f ′ ( x ) = sgn ⁡ ( x ) {\displaystyle f'(x)=\operatorname {sgn}(x)} and f ″ ( x ) = 2 δ ( x ) {\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 C in x at x = 0, so we cannot apply Itō's formula directly. But if we approximate it near zero (i.e. in ) by parabolas

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

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

References

  1. Rogers, L.G.C. "I.14". Diffusions, Markov Processes and Martingales: Volume 1, Foundations. p. 30.
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