Question about Stochastic Integral

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 $$\int_0^t H_s dM_s = \int_0^t \tilde{H}_s d \tilde{M}_s, \quad t \in [0,T].$$

Do both approaches work?

• Yeah, should work fine. – saz Nov 14 '18 at 18:48
• Thanks. Regarding the second option: I would also like to get an existence and uniqueness theorem like in the $[0,\infty)$ case. What is the definition of a local martingale for $[0,T]$ ? – White Nov 14 '18 at 18:53