On exponentials of additive functionals of Markov processes

We give necessary and sufficient conditions in order that exponentials of additive functionals of Markov processes have finite expectations. Furthermore, we obtain sharp estimates for these expectations. More precisely, we investigate both the Stieltjes exponential and the ordinary exponential of right-continuous additive functionals of general right-continuous, time-inhomogenous Markov processes. The well-known Khasʹminskii Lemma (1959, Probab. Appl. 4, 309–318) follows as a corollary.