I want to solve a 2 equation system of SDEs:
$dF_{1t} = (\mu_1 + \rho_{11}F_{1t})dt + dW_{1t}\\ dF_{2t} = (\mu_2 + \rho_{21}F_{1t} + \rho_{22}F_{2t})dt + dW_{2t} $
with $F_{10} = F_{20} = 0$. Here $W_{1t}$ and $W_{2t}$ are 2 independent Brownian Motions. I solved the first equation as a regular Ornstein Uhlenbeck process to obtain: \begin{align*} F_{1t} = (e^{\rho_{11}t} - 1)\frac{\mu_1}{\rho_{11}} + e^{\rho_{11}t} \int_0^t e^{-\rho_{11}s}dW_{1s} \end{align*} I now substitute this solution into the second equation to obtain the following SDE: \begin{align*} dF_{2t} &= \bigg(\mu_2 + (e^{\rho_{11}t}-1)\frac{\mu_1 \rho_{21}}{\rho_{11}} + \rho_{21}e^{\rho_{11}t}\int_0^t e^{-\rho_{11}s}dW_{1s} + \rho_{22}F_{2t}\bigg)dt + dW_{2t} \end{align*} My question is: How can I solve equations of the second form, which basically have an Ito integral inside the drift? I need a strong solution.
The system of SDEs can be written in matrix vector form as $$\tag{1} d\boldsymbol{F}_t=(\boldsymbol{\mu}+\boldsymbol{A}\boldsymbol{F})\,dt+d\boldsymbol{W}_t $$ where $$ \boldsymbol{A}=\left(\begin{matrix}\rho_{11} & \rho_{12}\\\rho_{21} &\rho_{22}\end{matrix}\right)\,. $$ The solution to (1) is $$ \boldsymbol{F}_t=e^{\boldsymbol{A}t}\Big(\boldsymbol{F}_0+\int_0^t e^{-\boldsymbol{A}s}\boldsymbol{\mu}\,ds+\int_0^t e^{-\boldsymbol{A}s}\,dW_s\Big)\,. $$ This can easily be checked using Ito's formula. See also Wikipedia.