Itō’s Insight
Consider the stochastic process
where is standard Brownian motion (or the Wiener process) and
is a twice-differentiable function.
Ito’s lemma states that
The first term is recognizable from the chain rule in classical calculus, but why the second term? If is truly infinitesimal, it doesn’t even seem possible that
.
To understand Ito’s lemma intuitively, think of as a little stochastic variable, specifying
‘s change during the next
.
This models Brownian motion (or the Wiener process) completely. Now should be a little stochastic variable too, modeling the stochastic process
.
The picture at the start considers an example where
, thereby suppressing the “intuitive” or classical term in Ito’s lemma. The reason why
in that picture is the reason why that “non-intuitive” term is needed.
Loosely speaking, wherever has curvature,
will diffuse around that curvature sufficiently to influence the expected result on the order of
. (The specific example in the picture makes this trivial, as
.)
Why doesn’t dominate away
(i.e.
)? Because
does not come from
as happens with classical differentials. The strong law of large numbers implies that a stochastic differential’s expected value pushes its integral on a faster order than its deviation does.
Hey super cool content here!
Is there any way from someone that has knowledge of calculus 3 and mathematical statistics 2 or less to understand that? I would love to share Ito’s lemma with my Financial Technology club.
Thanks!
Hello,
why is the derivative f’ in the considered example of f(B_t) = (B_t)^2 equal to zero?
In that sentence we are considering only the (example) case where B_t = 0.
I will clarify the text; thank you for pointing this out.