Let $(X_n,P_x)$ be a canonical time-homogeneous Markov chain with measurable state space $(S, \mathcal{B})$ and generator $\mathcal{L}$. Let $w : S → \mathbb{R}$ be a non-negative measurable function.
For what functions $v$ is the process $M_n=e^{-\sum_{k=0}^{n-1}w(X_i)}v(X_n)$ a martingale?
Any ideas on how to approach this?
I tried to write $E(M_{n+1}|X_1,\dots,X_n)=E(e^{-W(X_n)})E(e^{-\sum_{k=0}^{n-1}w(X_i)}v(X_{n+1})|X_1,\dots,X_n)$. I don't know if $v(X_n)$ is a Markov process so I am not even sure how to use the Markov property. Also I know $M_t^f=f(X_t)-\sum_{s=0}^{t-1} (L_sf)(X_s)$ is martingale for any measurable function but I am not sure if I can apply this.