Why do we mainly integrate with respect to martingales?

Question

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.

Answer 1

I am aware that this is a really old post, but I feel it is a very important question. My personal favourite answer to this question lies in the theory of rough paths. I will give both a technical answer and a heuristic/qualitative answer, as my view of maths is that both should be present and linked as tightly as possible.

The main problem when trying to define a stochastic integral in the usual Riemann Stiltjes way is the fact that the squared increments $(M_{t+s} - M_{s})^2$ of Brownian motion, and martingales in general - are not of order $o(t)$. In other words, the paths of the integrator are too rough.

Recalling how Riemann Stiltjes integration works, the key observation is that the above increments of second order and higher are of order $o(t)$, and thus vanish in the limit as the mesh size goes to $0$. This is not the case for martingales.

However, this problem only occurs on a pathwise level. The orthogonality of martingale increments leads to the miraculous fact that the conditional expectations of all the problematic second order terms vanish, so long as the integrand is adapted (this is why adaptedness is such a crucial assumption). This leads to the well definedness of the Riemann-Stiltjes limits in $L^2$ sense, and this is none other than the Ito integral. This also explains why typically Ito integrals are defined as a limit $L^2$ or in probability instead of almost sure limits.

Now, the observation of rough paths theory is that we do not need the full martingale property! We simply need the conditional expectations of the higher increments to be of order $o(t)$ in the limit. In proving this theorem, we gain a finer understanding of why martingale integration works to begin with, and how the Ito theory may be naturally extended.

All of this is made precise in the so called Stochastic Sewing Lemma in rough path theory. I highly recommend the exposition in Hairer’s A Course in Rough Paths, where the Stochastic Sewing Lemma is found as Proposition 4.19.

On a more philosophical level, what the above is saying is this - adapted processes cannot see into the future. Martingales are fair in the future given the present. These two aspects together give rise to an “averaging effect” that allows the stochastic integral to be well defined despite the roughness of paths. I like to think of it as the integrand being “constant” from the point of view of the future martingale increments. If the integrand were not adapted, it could conspire with the martingale paths to ruin the fairness, and hence the averaging property.