Suppose that $T$ is an exponential random variable. Show that the process ${X_T (t)}:=1_{[T,\infty)}(t), t\geq0,$ with state space $S=\{0,1\}$ is a Pure Jump Markov Process.
Let $T$ ~ $Expo (q)$, where $q>0$. Then, we can write the sample path decomposition: $${X_T (t)}= 1_{[T,\infty)}(t)=\left\{ \begin{array}{lcc} 1 & t\geq T \\ \\ 0 &\mbox{ $t<T$} \end{array} \right.$$
The process is a jump process since the $t \in [0,\infty),$ the state space is $S=\{0,1\}$ (which is clearly countable), and the sample path is right continuous.
Now, here is where I get confused. How do I show that this process is a PJMP? This is what I have tried:
Consider $$\mathbb{P}({X_T (t+s)=0|{X_T (t)}=0)}$$ $$= \mathbb{P}({1_{[T,\infty)} (t+s)=0|{1_{[T,\infty)} (t)}=0)}$$ $$=\mathbb{P}(T>t+s|\ T>t) = \mathbb{P}(T>s)$$ $$\mathbb{P}({X_T (s)=0|{X_T (0)}=0)}$$
which I have used the memoryless property of the exponential distribution. But I am sure that the above haven't actually proved the Markovian property for a pure jump process.