Starting from the following differential equation \begin{equation} dS=aSdt+\sqrt{S}dW \tag{1} \end{equation} with $dW$ denoting Wiener increment, if one splits the above SDE $(1)$ into stochastic part and deterministic part as follows $$ dS_1=\sqrt{S}dW\tag{2}$$$$dS_2=aS dt\tag{3} $$ in which sense am I given that "the numerical solution to $(2)$ is used as the initial condition in $(3)$, with the solution to $(3)$ providing the approximation to the true solution at each timestep"?
I used to consider a typical initial condition differential equation as follows for example $$dS=aSdt + \sqrt{S}dW\\S(0)=0.43$$ How can the above example typical scheme be applied to the above mentioned system? That is, let us pretend that I am able to find a numerical solution to $(2)$. Then? What I do with such a solution so as to solve $(3)$, or, better, in which sense does solution to $(2)$ represent the initial condition to $(3)$?
I suggest away, but honestly, I did not use it.
rewrite it as $$\begin{equation} dS_1=0dt+\sqrt{S}dW\\ dS_2=asdt+0dW\\ s(0)=0.43,W(0)=0 \end{equation}$$ and turn it to matrix form $$d\begin{bmatrix}S_1 \\S_2 \end{bmatrix}=\begin{bmatrix}0 & \sqrt s \\as & 0 \end{bmatrix}\begin{bmatrix}dW \\ds \end{bmatrix}$$ hope it will helpful