Introduction
I have a multidimensional, inhomogeneous SDE system:
$$ d\underline{X}(t) = \left[\underline{\underline{A}}(t) \cdot \underline{X}(t) + \underline{a}(t) \right] \cdot dt + \sum_{m = 1}^{M} \left[\underline{\underline{B_m}}(t) \cdot \underline{X}(t) + \underline{b_m}(t) \right] \cdot dW_m(t) , \quad (1) $$
where we know by the Ludwig Arnold book (https://epubs.siam.org/doi/10.1137/1018036?mobileUi=0) that the inhomogeneous solution is the following:
$$ \underline{X}(t) = \underline{\underline{\Phi}}(t) \cdot \left[ \underline{X}_0 + \int_0^t \underline{\underline{\Phi}}^{-1}(s) \cdot \left( \underline{a}(s) - \sum_{m = 1}^{M} \underline{\underline{B_m}}(s) \cdot \underline{b_m}(s) \right) ds + \sum_{m = 1}^{M} \int_0^t \underline{\underline{\Phi}}^{-1}(s) \cdot \underline{b_m}(s) dW_m(s) \right] , \quad (2) $$
where $\underline{\underline{\Phi}}(t)$ function has to be calculated by
$$ d\underline{\underline{\Phi}}(t) = \underline{\underline{A}}(t) \cdot \underline{\underline{\Phi}}(t) \cdot dt + \sum_{m = 1}^{M} \underline{\underline{B_m}}(t) \cdot \underline{\underline{\Phi}}(t) \cdot dW_m(t) , \quad (3) $$
equation and it has no explicit general form.
Question
If $\underline{\underline{A}}(t)$ and $\underline{\underline{B}}(t)$ has special properties (commutation) the I think equation 3 can be calculated in the following form:
$$ \underline{\underline{\Phi}}(t) = \underline{\underline{1}} \cdot e^{\int_0^t \underline{\underline{A}}(s) - \frac{1}{2} \sum_{m = 1}^{M} \underline{\underline{B_m}}^2(s) ds + \sum_{m = 1}^{M} \int_0^t \underline{\underline{B_m}}(s) dW_m(s)} , \quad (4) $$
If equation (4) is true, why is equation (1) not solveable? Im I correct if I calculate $\underline{\underline{\Phi}}(t)$ function with equation (4) using numerical integration? Can $\underline{X}(t)$ be solved stronger than simulation if I use the equations (2) and (4) but make the integrations numerically if I must?
Thank you very much for the answers!