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