I am reading an economics paper in which the authors write at one point (page 11, section D):
We restrict attention to policies under which inflation is locally deterministic
$$ -\frac{\mathrm{d}\pi_{t}}{\pi_{t}} = \iota_{t}\mathrm{d}t.$$
What exactly does locally deterministic mean here? Though I understand it’s an ODE and there are concepts like deterministic or stochastic ODEs (the latter when it contains an additive shock component, for instance), I don’t get what locally deterministic is and why they have to assume it. Will also appreciate any suggestions for suitable references on this and related topics, preferably targeted for those in economics.
My educated guess is this (to confirm it, I would probably need to invest quite some time to dig into that paper):
The authors know that the dynamics they investigate is stochastic, but they assume that on short time scales (“locally”), it can be treated like a deterministic dynamics governed by the given equation. So, they do not claim that the equation leads to some special dynamics that has a miraculous property of being locally deterministic. Instead, the equation (which is deterministic) is a local description of the dynamics.
If I understand the paper correctly from my quick glance, $ι_t$ is the inflation rate. This may be subject to stochastic fluctuations on large time scales, but this is not relevant for modelling processes that happen on a scale of days. On such a time scale, it is probably a valid assumption that this rate is constant – and thus the dynamics is deterministic.