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?