Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 30K / b / Solution

The process is adapted to its natural filtration. At each fixed time it has finite support and is therefore integrable. Conditional on $X_{n-1}=0$,Conditional on $X_{n-1}\ne 0$,The Markov property now givesso $(X_n)$ is a martingale.