Context:
I am currently working with Kalman-like filters. As a result I deal with linear Stochastic Differential Equations (SDEs) with constant coefficients such as: $$ dx(t) = Ax(t)dt + Bdw(t) $$ where $A,B$ are known constant matrices of appropriate dimension and $w(t)$ is an $n$-dimensional Wiener process. Also, the initial condition $x(0)$ is a gaussian random variable with known mean and covariance.
I got asked by a "peer": what is the probability space you are working on? which took me for surprise at that moment. Due to the nature of the SDE, when dealing with it, it suffices to keep track of the mean and covariance of the solution of the differential equation in order to perform computations. Hence, many books and papers do not seem to specify what the probability space they are using. Some works even start by saying "Let $(\Omega, \mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a filtered probability space..." and then proceed to write the SDE, without mentioning the filtered probability space again, or even define the Wiener Process over it as if the filtered probability space was arbitrary.
The default answer I have stored in my brain is the filtered probability space ensured to exist due to Kolmogorov extension theorem for the Wiener process, which may be equivalent to $\Omega$ being the set of continuous functions, $\mathcal{F}_t$ the natural filtration of the Wiener process and $\mathbb{P}$ the Wiener measure which coincides with finite-dimensional distributions at discrete time points. However, I may be mistaken.
Also, this probability space is for the Wiener process, and not for the solution of the SDE, so I think there is something missing here.
Question:
What should be the default/standard answer to the question "What probability space are you working on?" when dealing with linear SDEs in the context I described before?