Suppose we have a matrix-valued ODE: $$ i\tag{1} \frac{d}{dt}U(t) = HU(t) $$ where $U(t)$ and $H$ are complex square matrices. Given a time-independent vector $\psi$, we can convert (1) into a vector-valued differential equation: $$ i\frac{d}{dt}U(t)\psi = HU(t)\psi. $$ The above can be further rewritten as $$ i \frac{d}{dt}\psi(t) = H \psi(t). $$ So, now instead of a matrix-valued ODE, we have a simpler vector-valued ODE. Some might recognize that this is the Schrodinger equation with $\hbar=1$.
I would like to know if the same logic applies to a stochastic matrix-valued differential equation: $$ \tag{2} dX(t) = X(t)A(t)dt + X(t)\sum_j B_j dW_j, $$ where $X(t)$ is a matrix-valued stochastic process, $A(t)$ and $B_j$ are square complex matrices and $dW_j$ is a scalar increment of Wiener process.
Given a time-independent vector $\psi$, is it possible to bring (2) into a vector-valued stochastic differential equation?
\begin{align} \psi^*dX(t) &= \psi^*X(t)A(t)dt + \psi^*X(t)\sum_j B_j dW_j\\ d\psi(t)^* &= \psi(t)^*A(t)dt + \psi(t)^*\sum_j B_j dW_j.\\ \end{align}
Remark: $\psi^*$ denotes the conjugate transpose of $\psi$.
