SDE Modeling: Ito vs. Stratonovich

Question

In my SDE class last semester there were some hints that sometimes an SDE model makes more sense in the Ito sense, and sometimes in the Stratonovich sense. This was explained very briefly and vaguely. The idea was that SDEs containing white noise are really the limit of models which contain noise with a positive correlation time scale $\tau$, as $\tau \to 0^+$. Yet the model may also have some other limit as well.

For instance, we may be interested in the motion of particles of small mass $m$ under the influence of noise with small $\tau$. My professor hinted that in one regime, the limit is an Ito SDE, while in another it is a Stratonovich SDE. As I recall we get the Ito SDE in particular when the correlation time scale is much smaller than the inertial time scale. Or more mathematically, by taking the limit in $\tau$ before the limit in $m$.

Can anyone clarify this, or point to references which do so?

Note that I am not asking about the theoretical advantages of Ito vs. Stratonovich integrals, I have already understood that part. I am asking about modeling advantages.

Answer 1

This paper:

http://link.springer.com/article/10.1007%2Fs10955-004-2273-9

quantifies the notions I was discussing in my question.

Answer 2

Typically, a nice feature of Ito SDE's is that they are not forward looking. As a result, mathematical finance relies on Ito calculus, not Stratonovich calculus.

Answer 3

I am late, but this article is very good at highlighting the philosophical difference between Ito and Stratonovich calculus. The section "does it matter?" discusses modeling implications

Itô integral is more popular in mathematics and finance, where the interpretation as the limit of a discrete game is somewhat appealing, and (more importantly) the martingale property is convenient. Stratonovich’s rule is more popular in physics, where the limit of smooth noise argument is more compelling.