Tanaka's formula

In the stochastic calculus, Tanaka's formula states that

:$|B\_t|\; =\; int\_0^t\; sgn(B\_s)\; dB\_s\; +\; L\_t$

where "B"_"t" is the standard Brownian motion, sgn denotes the sign function

:

and "L"_"t" is its "local time" at 0 given by the "L"²-limit

:$L\_\{t\}\; =\; lim\_\{varepsilon\; downarrow\; 0\}\; frac1\{2\; varepsilon\}\; |\; \{\; s\; in\; [0,\; t]\; |\; B\_\{s\}\; in\; (-\; varepsilon,\; +\; varepsilon)\; \}\; |.$

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 non-decreasing predictable process (local time).

Outline of proof

The function |"x"| is not "C"² in "x" at "x" = 0, so we cannot apply Ito's formula directly. But if we approximate it near zero (i.e. in [−"ε", "ε"] ) by parabolas :$frac\{x^2\}\{2|epsilon+frac\{2\}$And use Ito's formula we can then take the limit as "ε" → 0, leading to Tanaka's formula.

References

* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
id = ISBN 3-540-04758-1 (Example 5.3.2)
* cite book
last = Shiryaev
first = Albert N.
authorlink= Albert Shiryaev
title = Essentials of stochastic finance: Facts, models, theory
series = Advanced Series on Statistical Science & Applied Probability No. 3
coauthors = trans. N. Kruzhilin
publisher = World Scientific Publishing Co. Inc.
location = River Edge, NJ
year = 1999
id = ISBN 981-02-3605-0