Suppose we have the following parabolic PDE in $X(s, t)$:
$\frac{\partial X}{\partial t} + sM_1 \frac{\partial X}{\partial s} + \frac{1}{2} s^2 M_2 \frac{\partial^2 X}{\partial s^2} + (M_3 - M_1)X = F(s, t)$.
This can be concisely written as $\frac{\partial X}{\partial t} + \mathbb{A}X = F(s, t)$. ($\mathbb{A}$ is a differential operator)
Some conditions are: $X(s, t), M_1, M_2, M_3, F(s, t)$ are $n \times n$ matrices, $F$ is a "known" function but we don't know how it looks like exactly. $M_1, M_2$ are diagonal matrices, and $M_3$ is a rate matrix, i.e., all its rows add up to zero, and all of its non-diagonal entries are non-negative.
My question is: what conditions on $F(s, t)$ must hold so that a solution to the above PDE exists? So far I know that $F(s, t)$ has at most polynomial growth w.r.t. $s$.
I'd also be grateful if someone could direct me to a reference where the above problem is dealt with
Thanks