General solution to system of stochastic linear differential equations

Question

Assume we are given the system of linear stochastic differential equations $$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\sigma_j(t)\cdot dB_{j,t}$$ for $i = 1,\ldots,n$, where $B_{ij,t}$ and $B_{j,t}$ are brownian motions and $a_{ij},\sigma_{ij},b_j$ and $\sigma_j$ are real-values functions. Also of interest is the simpler case, where $\sigma_{ij}=0$ for all $(i,j)$, i.e., $$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + b_j(t)\cdot dt+\sigma_j(t)\cdot dB_{j,t}$$ If we consider just their deterministic equivalent $dx/dt = A(t)\cdot x +b(t)$, then there is a well-established formalism to solve it. Given these two systems of linear stochastic differential equations, if am wondering whether there is any known general solution that works for them? (If the first one, turns out to be too difficult, we might want to put our attention just to the last one.)

Many thanks for your help in advance!

Answer 1

Finding an explicit solution is probably not possible (but I could be wrong). However, the one-dimensional stochastic differential equation \begin{align*} dX_{t} = \bigl( a(t)X_{t} + b(t) \bigr) dt + (\alpha(t) X_{t} + \beta(t) \bigr) dB_{t} \end{align*} has an explicit (and complicated!) solution. See chapter 4.2 in Kloeden, P. E. and Platen, E.: Numerical Solution of Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, 1992.