[944] Doubts Itô Formula

[Gautam Iyer]

On Wed, Nov 15, 2017 at 11:28:14AM -0500, Lucas Duarte Bahia wrote:

> Is C1,2 a necessary and sufficient condition, or just a sufficient
> condition? What I mean is if f is not C1,2 is there any case where I
> can still apply Itô?

There are many more general versions of Itô's formula. The simplest one
to state would be if f is C^{1,1} everywhere, and is C^{1,2} except at
finitely many points, then Itô's formula holds.

For instance the function f(x) = (x^+)^2 is not C^2, but Itô's formula
still works. (However the function f(x) = x^+ is not of this class and
Itô's formula certainly fails for this.)

There are an even weaker versions of Itô's formula as well. If you're
interested, Krylov has a version of this in his paper about L^p theory
of stochastic partial differential equations. But he's from the old
Russian school that believes if it was hard to write, it should be hard
to read. So I don't recommend trying to dig it out... :)

GI

-- 
'Lactomangulation' -- Manhandling the 'open here' spout on a milk
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