I am working with stochastic differential equations (SDEs), which can be expressed as $$ dx = f(x, t) dt + G(x, t) dw $$ where $x \in \mathbb{R}^{n}$ is the state of the stochastic system, $t$ is time, and $w \in \mathbb{R}^{m}$ is a multidimensional Wiener process, $f(x, t) \in \mathbb{R}^{n}$ denotes the system unperturbed dynamics, and the matrix $G(x, t) \in \mathbb{R}^{n \times m}$ determines how the external disturbances affect the system.
I have some trouble understanding the physical dimensions of $G$ and $w$. I would expect the Wiener process to be non-dimensional, but my confusion arises from the fact that, according to Kloeden and Platen's book "Numerical Solution of SDEs", if the SDE is linear in the narrow sense (i.e., $f(x, t) = A(t) x + c(t)$ and $G(x, t) = G(t)$), the time evolution of the covariance matrix $P$ of the system state $x$ is described by the following ordinary differential equation (ODE) $$ \dot{P} = A(t) P + P A(t)^T + G(t) G(t)^T $$
It appears that, according to the SDE, the physical dimensions of $G$ must be the same as $dx / dw$, but, according to the differential equation of P, which has dimensions equal to $xx^T$, the physical dimensions of $GG^T$ must be the same as $xx^T / t$. The only way to obtain consistency is that the Wiener process dimensions are $\sqrt{t}$, but it does not make much sense to me. Can someone explain to me what am I doing wrong? And what are the dimensions of $w$ and $G$?