stochastic integration with respect to quadratic variation

I have been studying stochastic integral with respect to Brownian motion. At some point my professor generalized our approach such that we are able to integrate with respect to general Martingales. That means, if we have a simple function $H_s(w)=\sum_{i=0}^n \eta_i(w) I([t_i,t_{i+1}])(s)$ where $I$ is an indicator function we said that the Integral w.r.t M to be $J(H)=\sum_{i=0}^n \eta_i (M_{t_i+1}-M_{t_i})$ where M is a continuous $L^2$ martingale. with absolutely continuous quadratic variation process $(<M>_t)_{t\geq 0}$. I can't make something out of this, in how far is the quadratic variation a process? Normally the quadratic variation is defined for instants $t_0,\dots,t_n$, right? Any help would be apprechiated.

Cheers

• It sounds like you're missing some bits of theory in order to understand stochastic integration theory properly. You might consider taking a look at "Stochastic Differential Equations" by B. Øksendal. Otherwise, take a look at the somewhat more difficult "Brownian motion and Stochastic Calculus" by I. Karatzas and S. Shreve. You'll find a good deal of answers there. – Alexander Sokol Jul 28 '14 at 20:27