Basic question about the stochastic integral $\int \limits_{0}^{t} X(s) \,dM(s)$ Suppose $(X_{t})_{t \geq 0}$ and $(M_{t})_{t \geq 0 }$ are stochastic processes, where the index is continuous and the probability space is $(\Omega, \Sigma, P)$.
We say for each fixed $\omega \in \Omega$ that the set $\{ X_{t}(\omega) \}_{t \geq 0 }$ is a path.
I'm trying to understand the stochastic integral $\int \limits_{0}^{t} X(s) \,dM(s)$.  Specifically, it seems this integral is in some sense being integrated with respect to the continuous index.  In other words, if I define $h(t, \omega) := X_{t}(\omega)$ and $g(t, \omega) := M_{t}(\omega)$, then it seems like the stochastic integral is really just asking for the Riemann-Stieltjes integral:
$\int \limits_{0}^{t} h(s, \omega) \,dg(s, \omega)$, where $\omega$ is a fixed value.  In other words, since $\omega$ is fixed, we can regard $h(t, \omega)$ as a function of $t$, i.e., $h(t)$, and similarly for $g(t, \omega)$.
Then the stochastic integral is just the Riemann-Stieltjes integral $\int \limits_{0}^{t} h(s) \,dg(s)$.
I don't think the explanation above is correct.  Can anyone tell me what I am wrong about?  Also, was I right that we are regarding $\omega$ as a fixed value when evaluating the integral?
 A: I will treat the case where  M is a continuous semimartingale. 
Unfortunately it is generally not the Riemann Stieltjes integral. You know that the Stieltjes measure of g, is only defined if g has finite variation. However, as you know, many stochastic processes does not have sample paths with finite variation, and therefore such an integral does not exist.
What is it then? In the special case M is a finite variation process, the stieltjes integral exists and you are correct. (Sokol Theorem 2.2.1)
In the general case, you may know, or should look up, the covariation process $[Z,W]$ between two continuous semimartingales Z and W. This is always a process of finite variation.
It then turns out, that there is a unique process(up to indistinguishability) $(L_t)_{t\geq 0}$ satisfying that:
 $$[L,N]=\int_0^t X_s d[M,N]_s.$$
For any third continuous semimartingale $N$
(note that the integral above is w.r.t. a finite variation process, and is thus already well-defined, as we just discussed).
We define L as the stochastic integral of X w.r.t. M. (Sokol Theorem 2.2.3)
Intuitive definition huh? Never the less it has properties that justifies such a definition. For example,  the Riemann sums converges in probability to L.
(See Sokol Theorem 2.3.2)
There are many books treating this subject, but many are not very detailed and hard to read. May i suggest the lecture notes that i have also referred to in the answer : "An introduction to stochastic
integration with respect to continuous
semimartingales"
By Sokol, Alexander.
