I've read some chapters of several textbooks about stochastic processes and SDEs. I've faced that we use a special stochastic process in order to define $"noise"$ term. It appears that the $\bf{only}$ suitable stochastic process is white noise process denoted by $\xi(t)$.
Moving forward, when it comes to the integral of $\int^t_0 \xi(s)ds$, because of the time derivative of Brownian motion equals to white noise process, which is $\frac{dB(t)}{dt}=\xi(t)$, we can write $\int^t_0 \xi(s)ds=\int^t_0 dB(s)$. So here are my questions:
1-) Why do we use white noise process? What makes it important? Can't we use any other process?
2-) Is there any other process whose time derivative equals to white noise process or Brownian motion is the only one?
I would also be appreciated if you can recommend me any beginner level source. Thank you so much!
It is the paradigm process because it is tied to many paradigm objects that show up in physics eg.
There are other types of noise see https://en.wikipedia.org/wiki/Colors_of_noise and "Coloured Noise" in the time continuous SDE setting
To be clear Brownian motion is not differentiable and so it doesn't not have a time derivative and so it doesn't make sense to talk about its deritivate. A more formal connection with white noise is described in Le-Gall "Brownian Motion, Martingales, and Stochastic Calculus".
see formal descriptions here What is "white noise" and how is it related to the Brownian motion? too that describe their unique connection.
If we use the distributional definition of white noise $\xi$
$$(\xi,\phi)=-\int B_{t}\phi'(t)dt$$
then we can get uniqueness. See What is the "distributional derivative" of a Brownian motion?.