For the following Stochastic Differential Equation: \begin{align*} dX_1&=k_aX_1 dt + \sigma_1X_1 dW \\ dX_2&=(k_bX_2- k_aX_1)dt + (\sigma_2X_2 - \sigma_1X_1)dW \\ X_1(0&)=C, \quad X_2(0)=D. \end{align*}
I used the transformation $Y= \left[\begin{array}{l}1&0\\1&1\end{array}\right] X $ , where I define $X=[X_1,X_2]^T$.
The transformed problem becomes: \begin{align*} dY_1&=k_aY_1 dt + \sigma_1Y_1 dW \\ dY_2&=k_b(Y_2-Y_1)dt + \sigma_2(Y_2-Y_1) dW \\ Y_1(0&)=C, \quad Y_2(0)=C+D, \end{align*} by Ito's formula.
Solving the transformed problem we get: \begin{align*} &Y_1=Ce^{\sigma_1W(t)+\left(K_a -\frac{\sigma_1^2}{2}\right)t} ,&Y_2=(C+D)e^{\sigma_2W(t)+\left(K_b -\frac{\sigma_2^2}{2}\right)t} \end{align*}
Transforming back we get: \begin{align*} &X_1=Y_1, &X_2=Y_2 - Y_1.\end{align*}
This doesn't look like it should be correct, although I haven't gained an intuition for this yet.
I attempted to evaluate if this solution was correct by comparing this solution to one produced by an SDE solver on both MATLAB and Julia but cannot draw conclusive results.
Question: Where does the solution go wrong? What are some alternative methods to check a SDE has been solved correctly?