# 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?