Coupling between two CTMCs

Question

Suppose I have two random processes $X(t)$ and $Y(t)$ starting at time $t=0$ and $X(0)=Y(0)=0$. The processes obey the following transition rates:

$$ X(t):\begin{cases} 0\to 1,\text{at rate } A\\ 1\to 0,\text{at rate } B \end{cases}$$

$$ Y(t):\begin{cases} 0\to 1,\text{at rate } C\\ 1\to 0,\text{at rate } D \end{cases}$$

If $A\ge C$ and $B \le D$, then can we argue that $P(X(t)=1)\ge P(Y(t)=1), \ \forall\ t$? I understand some form of coupling argument must be used here, but given my unfamiliarity with this area, I do not know how to write a formal proof of this. Any help will be appreciated.

Answer 1

Coupling, indeed:

• When $(X_t,Y_t)=(0,0)$, move independently to $(1,1)$ at rate $C$ and to $(1,0)$ at rate $A-C$
• When $(X_t,Y_t)=(1,1)$, move independently to $(0,0)$ at rate $B$ and to $(1,0)$ at rate $D-B$
• When $(X_t,Y_t)=(1,0)$, move independently to $(0,0)$ at rate $B$ and to $(1,1)$ at rate $C$

Then the process $(X_t)$ has the desired transition rates, the process $(Y_t)$ has the desired transition rates, and, starting from any state $(X_0,Y_0)$ different from $(0,1)$, one never visits the state $(0,1)$, that is, $Y_t\leqslant X_t$ almost surely for every $t$.

In particular, $[Y_t=1]\subseteq[X_t=1]$ hence $P[Y_t=1]\leqslant P[X_t=1]$.