Question about Stochastic Integral – Mathematics Stack Exchange

Usually the stochastic integral is defined for processes indexed over $[0,infty).$

I wonder about the standard way to define the integral for processes indexed over $[0,T].$ That is, for a continuous local martingale $M = (M_t)_{t in [0,T]}$ and a progressive process (sufficiently integrable) $H = (H_t)_{t in [0,T]} $ I want to define
$$
int_0^cdot H_s dM_s, quad t in [0,T].
$$

I guess one could do two things:
1. Do the proof of the existence of the stochastic integral all over with processes defined on $[0,T]$
2. Reduce it to the general case. That is, define
$$
tilde{M}_t = begin{cases} M_t, t in [0,T] \
M_T, t in [T, infty) end{cases}
$$

Then define $tilde{H}$ in the same way. And then define
$$
int_0^t H_s dM_s = int_0^t tilde{H}_s d tilde{M}_s.
$$

Do both approaches work?