On the interpretation of Stratonovich calculus

The Itô–Stratonovich dilemma is revisited from the perspective of the interpretation of Stratonovich calculus using shot noise. Over the long time scales of the displacement of an observable, the principal issue is how to deal with finite/ zero autocorrelation of the stochastic noise. The former (non-zero) noise autocorrelation structure preserves the normal chain rule using a mid-point selection scheme, which is the basis Stratonovich calculus, whereas the instantaneous autocorrelation structure of Itôʼs approach does not. By considering the finite decay of the noise correlations on time scales very short relative to the overall displacement times of the observable, we suggest a generalization of the integral Taylor expansion criterion of Wong and Zakai (1965 Ann. Math. Stat. 36 1560–4) for the validity of the Stratonovich approach.