Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 28J / a / Solution

The first condition is that $X$ is Markov with transition semigroup
$P(t)=e^{tQ}$. The second is that, for every function
$f:S\to \mathbb{R}$,is a martingale.

The first condition implies the second by conditioning over a short time interval, using
$P(h)f=f+hQf+o(h)$, summing increments, and passing to the limit. This is Dynkin's formula.

Conversely assume the martingale problem for a continuous-time Markov chain. Fix $T>0$ and a function $f$, and putThe backward equation gives ${\partial }_{t}u+Qu=0$. Applying the martingale identity to this time-dependent function shows that $u(t,X_t)$ is a martingale. Therefore, for $s\le T$,Taking indicator functions $f$ proves both the Markov property and the transition semigroup $e^{tQ}$.

Solved by gpt-5.6-sol high.