Let $X=\{X_t: t \geq 0\}$ and $Y=\{Y_t : t \geq 0\}$ be two stochastic processes and consider the following system of SDE's: \begin{equation*} \begin{cases} dX_t = a X_t \, dt + Y_t \, dW_t & ,X_0 = x\\ dY_t = a Y_t \, dt - X_t \, dW_t & ,Y_0 = y \end{cases} \end{equation*} where $W=\{W_t:t \geq 0\}$ is the standard Brownian motion.
Question: Determine $\mathbb{E}[X_t]$
However the question only asks for the expcted value (which I don't know how to compute), I would like to know what's the explicit solution of the system.
Integrating the first equation on both sides from $0$ to $t$, we get $$X_t-x = a\int_0^t X_s \, ds + \int_0^t Y_s \, dW_s$$ and taking the expected value on both sides we get $$\mathbb{E}[X_t]= x + a \int_0^t \mathbb{E}[X_s] ds$$ because the expected value of the Ito's integral is zero.
Then, if $f(t)=\mathbb{E}[X_t]$, we need to solve \begin{equation*} \begin{cases} f'(t) = f(t) \\ f(0) = x \end{cases} \end{equation*} And we know that the solution of the previous differential equation is given by $$f(t)= \mathbb{E}[X_t]=xe^{at}$$