So I am learning stochastic calculus and I have seen this relationship be used many times: $$ \int_0^t dX_s = X_t-X_0 $$ where $X_t$ is some stochastic process.
It looks like some sort of application of the fundamental theorem of calculus, but is it? And does this hold for any stochastic process?
The definition of an ItΕ stochastic integral $ \int_0^t H_s \; dX_s$ (with respect to a semimartingale $X_t$) is the limit of $\sum_{j=1}^n H_{t_{j-1}} (X_{t_j} - X_{t_{j-1}})$ for partitions $0 = t_0 \le t_1 \le \ldots \le t_n= t$ of $[0,t]$ with mesh size going to $0$. For $H = 1$ the sum telescopes to $X_t - X_0$. So in this case the identity follows directly from the definition.
Similarly for the Stratonovich integral, and essentially any reasonable kind of stochastic integral that is defined using a limit of "Riemann sums".
