Supriya Ghosh (Editor)

Tanaka's formula

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In the stochastic calculus, Tanaka's formula states that

Contents

| B t | = ∫ 0 t sgn ⁡ ( B s ) d B 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.

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

L t = lim ε ↓ 0 1 2 ε | { s ∈ [ 0 , t ] | B s ∈ ( − ε , + ε ) } | .

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), 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 δ ( 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

x 2 2 | ε | + | ε | 2 .

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

References

Tanaka's formula Wikipedia
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