Itô vs. Stratonovich SDE

For the multiplicative SDE dX = f(X)dt + g(X)dW, the Itô and Stratonovich conventions differ by a drift correction term g'(X)g(X)/2. This leads to different stationary distributions — physically observable!

Itô paths
Stratonovich paths
Theoretical distributions

SDE Parameters

Multiplicative SDE model:
dX = a·X·dt + σ·X·dW

Itô drift: a·x
Stratonovich drift: (a + σ²/2)·x
(Itô-Stratonovich correction: +σ²x/2)

Stationary distributions:
Itô: p(x) ∝ x^(2a/σ²-1) · e^(-2a/σ²)
Stratonovich: p(x) ∝ x^(2a/σ²-2)

The two conventions give physically distinct predictions — only experiments select the correct one. For thermal noise: Stratonovich; for white-noise limits: Itô.