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.