Introduction to Stochastic Differential Equations with Applications to Modelling in Biology and Finance

9The issue of the Itô and Stratonovich calculi

9.1 Controversy

For the Black–Scholes model, we have shown in Chapter that, using Itô or Stratonovich calculus in the resolution of stochastic differential equations (SDEs) (i.e. using Itô or Stratonovich integrals in the integral form of such SDEs) usually leads to different results and the associated models (they are indeed different models) usually have qualitatively different behaviours. This is just an example of a much more general feature of SDEs, with the exception of SDEs having functions not depending on (i.e. ) since, as mentioned in Section  6.6, in this exceptional case the two calculi coincide.

In the example, when applied to population growth in a randomly varying environment (in which case the model is known as the stochastic Malthusian model), the two calculi give different behaviours with respect to population extinction. This also happens for other more realistic models of population growth, like the stochastic logistic model. There are situations where the Stratonovich calculus model predicts non‐extinction of the population (and even the existence of a stationary density, which corresponds to a stochastic equilibrium), ...