Tanaka's formula explained

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

|B _{t|} = \int ^{t} _{0} sgn (B _{s)} dB _{s} + L _{t}

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

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

L _{t} = \lim _{\varepsilon} (1 { 2)/(\varepsilon}) | \{ s \in [ 0 , t] | 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 |B_t| into the martingale part (the integral on the right-hand side, which is a Brownian motion^[1]), 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 C² 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'',&nbsp;''ε'']) by parabolas

(x ^{2})/(+ 2 |\varepsilon|) (|\varepsilon|)/(2) .

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

References

• Book: Øksendal , Bernt K. . Bernt Øksendal

. Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications . Sixth. Springer. Berlin . 2003 . 3-540-04758-1. (Example 5.3.2)

• Book: Shiryaev , Albert N. . Albert Shiryaev

. Albert Shiryaev . Essentials of stochastic finance: Facts, models, theory . Advanced Series on Statistical Science & Applied Probability No. 3 . trans. N. Kruzhilin . World Scientific Publishing Co. Inc. . River Edge, NJ . 1999 . 981-02-3605-0 . registration .

Notes and References

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