The continuous-time Markov chain has an infinitesimal generating element Q.
For all $f \gt$ $0$,define
$$Z(t)=f(X_t)\exp\left(-\int_0^t\left(\frac{Qf}{f}\right)(X_s)ds\right).$$
Define $\tau_n$ as the nth jump time of the Markov chain and $\rho_n=\tau_n-\tau_{n-1},n=1,2,3……,\tau_0=0,$ satisfying $P(\tau_n\lt\infty)=1,P(\tau_n\to\infty)=1.$
Prove $E_iZ(t\land \tau_n)=f(i),n\ge1,t\in[0,+\infty)$.
My attempt:
$$E_iZ(t\land \tau_n)=\sum_{k=1}^nE_if(X_{t\land \tau_n})\exp\left(-\int_0^{t\land \tau_n}(\frac{Qf}{f})(X_s)ds\right)*I_{\tau_{k-1}\lt t\lt\tau_{k}}+E_if(X_{t\land \tau_n})\exp\left(-\int_0^{t\land \tau_n}(\frac{Qf}{f})(X_s)ds\right)*I_{ t\gt\tau_{n}},$$where $I_{\tau_{k-1}\lt t\lt\tau_{k}}$ is characteristic function of a set
$=\sum_{k=1}^nE_if(X_{t})\exp\left(-\int_0^{t}(\frac{Qf}{f}\right)(X_s)ds)*I_{\tau_{k-1}\lt t\lt\tau_{k}}$+$E_if(X_{\tau_n})\exp\left(-\int_0^{\tau_n}(\frac{Qf}{f})(X_s)ds\right)*I_{ t\gt\tau_{n}}$
$=I_1+I_2+……I_n+I_{n+1}$
I hope to decompose it to finish calculation. But,I can't finish it.I need some help!
