I have the following definition for an
Ito process:
For $a(\omega, t), b(\omega, t)$ real valued, adapted stochastic processes that respectively satisfy the conditions $$ P(\int_0^t \vert a(\omega, s)\vert ds < \infty ) = 1 \quad \text{and} \quad P(\int_0^t b(\omega, s)^2 ds < \infty ) = 1 $$
and Ito-process has the form $$ X_t = x_0 + \int_0^t a(\omega, s) ds + \int_0^t b(\omega, s) dB_s. $$
Now for another adapted stochastic process $f(\omega, t)$ that satisfies
$$
P(\int_0^t \vert f(\omega, s) a(\omega, s)\vert ds < \infty ) = 1
\quad \text{and} \quad
P(\int_0^t (f(\omega, s)b(\omega, s))^2 ds < \infty ) = 1
$$
the stochastic integral of $f$ with respect to $X$ is defined as $$ \int_0^t f(\omega, s) dX_s := \int_0^t f(\omega, s)a(\omega, s) ds + \int_0^t f(\omega, s)b(\omega, s) dB_s $$
For the Ito integral I have seen the interpretation of say the cumulative gains or losses of a gaming/investing strategy $f(\omega,t)$ w.r.t. an underlying asset price (the Brownian motion). It makes sense to me to extend the original definition to more complicated asset price models (here an Ito-process), but I don't see how the above defined stochastic integral is
a) consistent with the formal definition of the Ito integral - i.e. whether this integral can also be derived as a limit of sums of this form $$ \lim_{\vert \mathcal{P} \vert \to 0} \sum_{t_i \in \mathcal{P}} f(t_i) (X_{t_i} - X_{t_{i-1}}) $$ b) consistent with the interpretation given above
So essentially I'm asking: why does this definition make sense?
Expanding on my comment here.
It does indeed follow from linearity of the stochastic integral. One may also note, that the Itô integral wrt some function/process does not necessarily have to yield an integral wrt a Brownian motion. Your process $X$ does have a drift term, which is a Riemann integral, so naturally such Riemann integrals are likely to appear when integrating wrt $X$.
Heuristically speaking, we can write out what $dX_{s}$ is and obtain $$ \int_{0}^{t}f(s)dX_{s} = \int_{0}^{t}f(s)\left( a(s)ds+b(s)dB_{s} \right) $$ Depending on the regularity of your functions $f,a,b$, this stochastic integral may also be obtained by taking limits of Riemann sums as you suggest. However, the Itô integral is not usually defined as the limit of Riemann sums. This type of limit is usually only valid in probability and for nice integrands $f$ (for example left-continuous integrands).
Instead, one may define the Itô integral more abstractly as the unique adapted process satisfying certain "nice" properties, such as linearity and local martingality. This allows for a large class of integrators and integrands, but is naturally less intuitive to think of.
In order to make the above heuristics more formal, I suppose you could start by showing that the Itô integral behaves that way for simple integrands, i.e. where $f$ is of the form $$ f(\omega , t) = f_{0}(\omega)\mathbb{1}_{\lbrace 0\rbrace}(t) + \sum_{i=1}^{n}f_{i}(\omega)\mathbb{1}_{(t_{i},t_{i+1}]}(t) $$ For such functions/processes we know exactly how the Itô integral should be defined. Then we may take appropriate limits of these simple functions and extend the class of integrable $f$'s to more interesting functions. In the end, the largest class of integrands will be the class of predictable processes.