Past exam of the stochastic analysis and partial differential equations course of the University of Oxford / 2025 / 1 / a

Let $Z=(Z_n{)}_{n\ge 0}$ be a Markov chain with transition kernel $K$, and let $N$ be an independent Poisson process of constant rate $r>0$. A Pseudo-Poisson process is $X_t=Z_{N_t}$. On the Banach space $B_b(E)$ of bounded Borel measurable functions, with the supremum norm, writeIts infinitesimal generator, transition semigroup, and resolvent operator are respectivelyand, for $\lambda >0$,The transition semigroup averages $K^n$ against the Poisson distribution of the number of updates. Since $K$ is a Markov operator, $\Vert K\Vert \le 1$; both series converge in operator norm. Updates with $Z_{n+1}=Z_n$ are allowed and do not count as changes of the observed state. The degenerate rate $r=0$ gives $L=0$, $P_t=I$, and $R_{\lambda }=I/\lambda$.