Itô or Stratonovich calculus: which one is more relevant from the point of view of physics?

Question

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

Answer 1

Both are relevant, and "the misconception that Langevin equation is the universal stochastic differential equation for all kinds of noisy systems is responsible for the difficulties mentioned"* in your post.

Take the SDE from Thomas' answer, $$\frac{dy}{dt} = A(y) + C(y)L(t)$$ where $L(t)$ is the noise term. Suppose we can turn the noise off, so we'd only be looking at the isolated, deterministic, system, and suppose that we got: $$\frac{dy}{dt} = A(y)$$ It is thus obvious that the noise term in this case must be interpreted as per Stratonovich, because an Itô interpretation would change the dynamics of the isolated system.

Above we had assumed, as one does, that $\langle L(t)L(t') \rangle = D \delta(t-t')$, but rarely are actual physical processes made out of Dirac deltas. Now if they are not, then by Wong-Zakai, Stratonovich is the only possible interpretation here (i.e. if $L(t)$ is more general, and as it approaches a delta, Stratonovich is the integration form that arises).

Now by turning the noise on and off we have arrived at the conclusion that Stratonovich is the only way forward, but in the very beginning I said that both formulations are relevant. Indeed, what if the noise cannot be separated out?

The important distinction is to make between external and internal sources of noise. We dealt with the external, but what if the noise is internal, and is impossible to turn off like say in chemical reactions? It's not that we have a deterministic process with a noise term slapped on top, we have a stochastic process and one might be able to argue that the averages behave in a deterministic manner, but by no means can then one then simply just add a noise term back on top without more justification: "For internal noise one cannot just postulate a nonlinear Langevin equation or a Fokker-Planck equation and then hope to determine its coefficients from macroscopic data. The more fundamental approach of [the expansion of the master equation] is indispensable."

* Any quotes in my answer have been taken from the definitive book on the matter: Stochastic Processes in Physics and Chemistry by van Kampen, which I strongly suggest you consult for more detailed explanations of the issues and how one goes about dealing with them.

Answer 2

Both Ito and Stratonovich stochastic PDEs can be used to derive a Fokker-Planck equation. Indeed, for simple one-dimensional processes the Ito process $$ dx = a\, dt + b\, dW(t) $$ is equivalent to the Stratonovich process $$ dx =\left( a\,-\frac{1}{2}b\partial_x b\right) dt + b\, dW(t) $$ The answer is then that both are physically reasonable, for a given FP equation I can find both and Ito and a Stratonovich process that will realize it. These are physically equivalent.

Having said this, Physicists tend to think of Stratonovich as the more reasonable scheme, because the FP equation is the one that can be derived by taking moments and "naive" integration by parts.