Let $X(t),t\in [a,b]$ be a stochastic process with $\mathbb E[X(t)]\equiv 0$ and uncorrelated increments, $f$ a continuously differentiable function.
With the above conditions, the following equality holds for a partition of $[a,b]$: $$ \int_a^b f(t)dX(t) = \lim_{n\rightarrow \infty} \sum_{k=0}^{n-1} f(t_i)(X(t_{i-1}) - X(t_i)) \text{ in }L^2 \quad (*)$$
If I'm not mistaken, an intuitive interpretation is that the function $f$ is randomly weighted by increments determined by the process $X(t)$. (pretty much like the Stieltjes integral just with a random second function)
Now two questions:
1) If I drop the above conditions, such that the equality (*) is no longer valid, is there still an intuitive picture about this kind of integration?
2) What would be a real-world example where this kind of integration is used to model? Or even better, what would be a class of models in the context of which this kind of integration naturally arises?
Added: With the conditions as in (*), what would then be the interpretation of integration by parts, i.e. changing from integrating with the process to integrating over the process?
$$\int_a^b f(t)dX(t) = f(b)X(b) - f(a)X(a) - \int_a^b X(t)f'(t)dt$$