How to show that the transition probabilities are continuous.

Question

I have a continuous time Markov chain $X_t$. It satisfies these four assumptions:

1. It has a finite number, $N$ of states.

2. For arbitrary $j,k,s$ and $t$ with $s<t$, assume that $X_s=j$ and consider the probability $X_t=k$. This probability does not change if information about the behaviour of the process during the interval $[0,s)$ is added to the knowledge that $X_s=j$.

3. $\lim\limits_{t \downarrow s}P_{jk}(s,t)=\delta_{jk}$ for all $j,k, s \ge 0$where $\delta_{jk}$ is the Kronecker delta.

4. For all $j,k, j\ne k$ and all $t \ge 0$:

$\mu_{jk}(t)=\lim\limits_{\Delta t \downarrow 0}\frac{P_{jk}(t,t+\Delta t)}{\Delta t}$,

exits and is continuous in $t$.

Then the author says:

By these for assumptions, it may be shown that for each $s\ge 0$ each $P_{jk}(s,\cdot)$ is continuous for all $t\ge s$.

But how is this shown? I am able to show right-continuity, but not left-continuity. I show right-continuity like this:

$P_{jk}(s,t+\Delta t)=\sum\limits_{\nu} P_{j\nu}(s,t)P_{\nu k}(t,t+\Delta t)$. By taking the limits on both sides and using assumption $3$ the result follows.

But to show left-continuity I start like this:

$P_{jk}(s,t)=\sum\limits_{\nu} P_{j\nu}(s,t-\Delta t)P_{\nu k}(t-\Delta t,t)$.

And here I get stuck, because what can we say about the limits $\lim\limits_{\Delta t \rightarrow 0}P_{\nu k}(t-\Delta t,t)$?

Answer 1

I think you would use the continuity of $\mu_{jk}$ here to show $\lim_{\Delta t \rightarrow 0} P_{jk}(t - \Delta t,t) = \lim_{\Delta t \rightarrow 0} P_{jk}(t,t+ \Delta t)$. We have

\begin{align*} &\left| \frac{P_{jk}(t - \Delta t,t) - P_{jk}(t,t+ \Delta t)}{\Delta t} \right| = \left| \frac{P_{jk}(t - \Delta t,t)}{\Delta t} - \mu_{jk}(t - \Delta t) + \mu_{jk}(t - \Delta t) - \mu_{jk}(t + \Delta t) + \mu_{jk}(t+\Delta t) - \frac{ P_{jk}(t,t+ \Delta t)}{\Delta t} \right| \\ &\le \left| \frac{P_{jk}(t - \Delta t,t)}{\Delta t} - \mu_{jk}(t - \Delta t)\right| + \left|\mu_{jk}(t - \Delta t) - \mu_{jk}(t )\right| + \left|\mu_{jk}(t) - \frac{ P_{jk}(t,t+ \Delta t)}{\Delta t} \right| \end{align*}

and taking the limit in the first and third term gives $0$ from the definition of $\mu_{jk}$, and $0$ in the middle term from the continuity of $\mu_{jk}$.