Continuous-time Markov processes and discontinuous transition probabilities

Question

Are there any continuous-time Markov processes in which transition probabilities are discontinuous functions over time?
Discontinuity does not enable any deep mathematical analysis of the process (such as transition rates).Is this the reason why we do not study such processes or is there anything fundamentally wrong with transition probabilities being discontinuous?

Answer 1

Here's a simple, if somewhat artificial, example of a Markov process $X=(X_t)$ with state space $\Bbb R$ and discontinuous (in time) transition probabilities. Started at $x<0$, $X$ moves deterministically to the right until time $-x$, when it jumps to $+1$ and moves thereafter as a three-dimensional Bessel process (the radial part of a three-dimensional Brownian motion). Started at $x\ge 0$, $X$ evolves as a three-dimensional Bessel process. Let $P_tf(x):=\Bbb E^x[f(X_t)]$ for $t\ge 0$, $x\in \Bbb R$, and $f:\Bbb R\to\Bbb R$ bounded and continuous. (The superscript $x$ on the expectation indicates the starting point $x=X_0$.) Observe that for $x<0$, $$ \lim_{t\uparrow -x}P_tf(x) = \lim_{t\uparrow -x}f(x+t)=f(0), $$ while $$ P_{-x}f(x)=f(1). $$ Thus $t\mapsto P_tf(x)$ will (for $x<0$) be discontinuous at $t=-x$ unless $f(0)=f(1)$.

This process $X$ is a strong Markov process, but it isn't a Feller process.