I am solving a set of stochastic differential equations and I need some feedback about if what I am doing is correct.
Given a vector $\boldsymbol{C}(t)=(C_+(t),C_-(t))^T$, we can writte a set of differential equations as $\frac{d}{dt}\boldsymbol{C}(t)=\boldsymbol{A}(t)\boldsymbol{C}(t)$, where $\boldsymbol{A}(t)$ is a 2x2 matrix with stochastic variables. Specifically,
\begin{equation} \boldsymbol{A}(t)= \begin{pmatrix} -\gamma/2 -i \alpha(t)\alpha^*(t) & \alpha^*(t)\dot{\alpha}(t) \\ \alpha(t)\dot{\alpha}^*(t) & -\gamma/2 +i \alpha(t)\alpha^*(t) \end{pmatrix} \end{equation}
where $\gamma>0$ and $\alpha(t)$ is a Ornstein-Uhlenbeck process described by $\frac{d}{dt} \alpha(t)=-(\kappa/2) \alpha(t)+\sqrt{\kappa n_{th}} \eta(t)$, such that $\overline{\eta(t)}=0$ and $\overline{\eta(t)\eta(t')}=\delta(t-t')$. Therefore, one can find that $\overline{\alpha^*(t)\alpha(t)}=n_{th}$.
I am interested in the average of $\overline{|C_{\pm}(t)|^2}=\overline{C_{\pm}(t)C^*_{\pm}(t)}$, or if it can not be done exactly, at least some flavour of it.
I would like to know if my attempted solution is correct or not, such because I am doing some risky assumptions. First, we formally integrate the set of equations
\begin{equation} \boldsymbol{C}(t)=e^{\int_0^t {\rm d}\tau\boldsymbol{A}(\tau)}\boldsymbol{C}(0) \end{equation}
Then, the quantity I am interested in can be found by the matrix elements of
\begin{equation} \boldsymbol{C}(t)\boldsymbol{C}^\dagger(t)=e^{\int_0^t {\rm d}\tau\boldsymbol{A}(\tau)}\boldsymbol{C}(0)\boldsymbol{C}^\dagger(0) e^{\int_0^t {\rm d}\tau\boldsymbol{A}^\dagger(\tau)} \end{equation}
While the matrix in the middle is just some coefficient, it is surrounded by stochastic integrals. So I just take the average. And this is where I make my first approximation which I think it's wrong, but I do not know better. I assume that
\begin{equation} \overline{e^{\int_0^t {\rm d}\tau\boldsymbol{A}(\tau)}\boldsymbol{C}(0)\boldsymbol{C}^\dagger(0) e^{\int_0^t {\rm d}\tau\boldsymbol{A}^\dagger(\tau)}}\simeq \overline{e^{\int_0^t {\rm d}\tau\boldsymbol{A}(\tau)}}\boldsymbol{C}(0)\boldsymbol{C}^\dagger(0)\overline{e^{\int_0^t {\rm d}\tau\boldsymbol{A}^\dagger(\tau)}} \end{equation}
Finally, I again do the following approximation
\begin{equation} \overline{e^{\int_0^t {\rm d}\tau\boldsymbol{A}(\tau)}}\simeq e^{\int_0^t {\rm d}\tau\overline{\boldsymbol{A}(\tau)}} \end{equation}
And this is something I can evaluate: \begin{equation} \overline{\boldsymbol{A}(t)}= \begin{pmatrix} -\gamma/2 -i n_{th} & -\kappa/2 n_{th} \\ -\kappa/2 n_{th} & -\gamma/2 +i n_{th} \end{pmatrix} \end{equation}
where I have used $\overline{\alpha^*(t)\eta(t)}=0$ because in an Itô stochastic equation, the variable and the noise are uncorrelated.
After this, I can solve the equations for $|C_{\pm}(t)|^2$. It is, however, highly unlike that this is the correct result, as there are plenty of (unjustified) approximations.
I would appreciate it if somebody would tell me if my approximations are justified, and if not, what would be the correct approach?
PD: In case somebody wonders where that set of stochastic differential equations, they appear when one tries to solve the adiabatic theorem (https://en.wikipedia.org/wiki/Adiabatic_theorem) where one considers the time-dependent Hamiltonian to be stochastic time-dependent.