I have a stochastic differential equation with multiplicative noise $\alpha(t)$
\begin{equation} \dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X} \end{equation}
where $T$ is the transpose and the stochastic variable $\alpha(t)$ is a complex-variable Ornstein-Uhlenbeck process. Physically, I have a set of variables $\textbf{X}$ driven by colored noise.
Important here is that $X$ and $B$ can be either column vectors or matrices, both representations are the same. If the problem would be 1-D, then I wouldn't have any problem.
My plan is to use UCNA method to solve it (http://www.physik.uni-augsburg.de/theo1/hanggi/Papers/79.pdf). However, the key step is to transform my stochastic differential equation with multiplicative noise into a stochastic differential equation with additive noise. And here is where the problem starts:
$\textbf{B}$ is singular, so I cannot invert them and isolate the variable $\alpha(t)$.
Even if I could invert one of them $\textbf{B}^{-1}$, I would still get $\textbf{B}^{-1}\textbf{B}^T$, which it doesn't have to be the identity.
So my questions are: what can be done considering I have a singular matrix? Is there a way to solve for both $\textbf{B}$ and $\textbf{B}^T$? Mathematically, I have heard one can use the Moore–Penrose inverse, however, this does not allow me to isolate $\alpha(t)$.
My goal is to find the steady-state (the long-term) value of the average of , i.e. $\overline{(t\rightarrow \infty)}$.
Note: If the OU process is slow enough (static limit), I know how to solve it: I can assume that ()=, then everything is time-independent, I can solve the exponential and then average over the OU distribution. I want now the quasi-slow limit, the OU process is slow but still time-dependent