Langevin equation provides an example of a physical model which involves a differential equation with a stochastic term. Now, I wonder, how should one treat this?
When I studied stochastic processes, I learned about Itô and Stratonovich calculus*. Back then, I just saw these things as technical tools devised to define the stochastic integral in a meningful way. Yes, depending on which one you use, you would obtain some results or some others, but this didn't bother me, since I was looking at these objects as mere mathematical structures.
But, when faced to the Langevin equation, I wonder, how does one make a meaningful interpretation of it? Both Itô and Stratonovich calculus seem like reasonable alternatives, but I don't find any strong argument to prefer one over the other. The point is, there must be a way to decide which type of stochastic calculus we should use! Two different definitions of the stochastic integral will lead to contradictory physics law, which is troublesome.
I believe that the Fokker-Planck equation can be derived from the Langevin equation via Itô's formula, but I don't think there's any way to achieve this using Stratonovich's calculus... But of course, this would be a poor ad hoc argument in favor of Itô's calculus, so this is clearly unsatisfactory.
suming up: how would you proceed to find out which version of the stochastic integral is more convenient to deal with stochastic equations in physics? Is there any physical assumption which makes Itô's calculus more reasonable?
- A note: Of course, there is a whole parametric family of possible "interpolating" calculus between both Itô and Stratonovich, but I believe these two are the only ones that are relevant: one of them gives rise to the martingale property, and the other one preserves the classical integration-by-parts rule