Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE models can be turned into stochastic models via adding a noise term in the dynamics, which changes the qualitative features of the model. However, it is not clear that the stochastic model is “more accurate” than the deterministic one.

I am interested in when an SDE model would be more appropriate than a deterministic one. As I am interested in the use of SDE in particular, I wish to restrict to continuous time examples.

One instance I am aware of is mathematical finance - most things in finance need to be stochastic to be of interest, indeed a deterministic model of stock prices or buy orders does not make too much sense, except in rare cases. What other phenomena/naturally occurring processes require the use of SDE over ODE? I am open to examples from machine learning/AI as well.