I think the notation and conceptualization is needlessly confusing. Some of it relates to old, often philosophical and even theological, debates on ontological status of infinitesimals.
The difference quantinent ended up as the Official Blessed Formulation of differential calculus, but it's very rarely used in practice, even though that's how calculus is used. And in practice calculus is still done using ad-hoc infinitesimal notations, but they are some weird thing with rules of their own which very few actually know (at least I don't).
Nonstandard calculus allows using infinitesimals in algebra with more or less the usual rules. Not sure if it's not more popular due to some fundamental technical or philosophical problems, or if it's just conservatism.
Stochastic calculus is quite bizarre indeed. Never understood e.g. the "proper" formulation of continuous time Kalman filters. Just limiting the timestep to zero seems to make sense and produces the right result with some massaging, but I've understood it's not really formally correct.
The difference quantinent ended up as the Official Blessed Formulation of differential calculus, but it's very rarely used in practice, even though that's how calculus is used. And in practice calculus is still done using ad-hoc infinitesimal notations, but they are some weird thing with rules of their own which very few actually know (at least I don't).
Nonstandard calculus allows using infinitesimals in algebra with more or less the usual rules. Not sure if it's not more popular due to some fundamental technical or philosophical problems, or if it's just conservatism.
Stochastic calculus is quite bizarre indeed. Never understood e.g. the "proper" formulation of continuous time Kalman filters. Just limiting the timestep to zero seems to make sense and produces the right result with some massaging, but I've understood it's not really formally correct.