Although my resarch focuses on PDEs (optimal transport, these days), I am currently trying to learn stochastic calculus and integration. I am just beginning in this topics, but I was wondering: why do we mainly integrate with respect to martingales? Typically $$ Y_t=\int_0^s X_s \,dH_s $$ seems to require that $H_s$ be a martingale, as far as I can tell after skimming through a few lecture notes and textbooks.

I'm aware that one can extend stochastic integration to semi-martingales (i.e. processes that are the sum of a martingale plus a "reasonable" process locally of bounded variations $H_s=\tilde H_s+B_s$): is this just a slight technical variation on the concept of martingale (which I should then think of as the sole and typical object w.r.t. which one can integrate), or are semi-martingales really very general processes and should I believe that one can essentially integrate w.r.t to (almost) ANY process?

I'm not interested in highly technical details (such as "if $H_s$ is not a martingale then this precise term is not square-integrable in the discrete construction of the stochastic integral", or whatever), I'm more focused on the big picture and I'd like a "philosophical/heuristic" explanation if possible. For example, I believe that one can think of martingales as processes that "do not see the future", which makes a whole lot of sense to me at least when it comes to what randomness should mean in real life. So what I'm wondering is: does it make sense to integrate w.r.t. processes that might be able to predict the future (some kind of non-locality in time, but why not after all?), or does that not make sense for some deep reason? If not, what is the said deep reason?

PS: feel free to migrate to MSE, although I do think that this is a fair research-level question.

adapted(or perhapsprogressively measurable). Martingales have the much stronger property of being a "fair game", whereas semimartingales allow for some "drift" which can bias the game one way or another. There are certainly theories of stochastic calculus that extend beyond semimartingales and even beyond adapted processes; see for instance the Skorokhod anticipating stochastic integral. $\endgroup$integrator. This issue is relevant instead for theintegrandand whether it is adapted to the integrator, which asks whether theintegrandcan see the future of theintegrator. I guess my comment came out looking misleading that way. But I agree that to focus on adaptedness for this question would be barking up the wrong tree. $\endgroup$7more comments