So, I have two continuous time Markov chains, say $(X_t)_{t\geq 0}$ and $(Y_t)_{t\geq 0}$ and I know by an earlier result based on Kolmogorov's equation $$ \partial_t \mathbb{E}[X_t] \leq a\cdot \mathbb{E}[X_t] - b\cdot \mathbb{E}[X_t^2] $$ and
$$ \partial_t \mathbb{E}[Y_t] = a\cdot \mathbb{E}[Y_t] - b\cdot \mathbb{E}[Y_t^2] $$, where $a, b>0$ and $\mathbb{E}[Y_0] = \mathbb{E}[X_0]$. I would like apply ideas like super solutions for ODEs to that problem, but the second moments are in the way, because for $\mathbb{E}[Y_t]$ to be a super solution of $\mathbb{E}[X_t]$ I need $\mathbb{E}[X_t^2]$ to be larger than $\mathbb{E}[Y_t^2]$. This is a priori not obvious to me. Does anyone have an idea how to approach this?
EDIT 1: Maybe to clarify, what I wish to apply to this problem. Consider $f,g$ maps from the set of $\mathbb{N}$ valued random variables to the real numbers and $(X_t)_t$, $(Y_t)_t$ continuous time Markov chains s.t. $$ \partial_t \mathbb{E}[X_t] = f(X_t),\qquad \partial_t \mathbb{E}[Y_t] = g(Y_t)$$ and for all $t> 0$ it holds $f(Y_t)\leq g(Y_t)$ as well as $\mathbb{E}[Y_0] = \mathbb{E}[X_0]$. Then, $\mathbb{E}[X_t]\leq \mathbb{E}[Y_t]$ for all $t\geq 0$. I am quite certain that this will not hold in this generality for any $f$ and $g$ but maybe someone has an idea where I could find a result on this or has a solution.