ON THE FUNDAMENTAL HYPOTHESES OF HUNT PROCESSES

Let (Ω, ℱ, P) be a probability space and X = {X_t, t > 0} a homogeneous Markov process adapted to the Borel subfields {ℱ_t} of ℱ. The process X is supposed to be measurable and it takes values in (E, ε), where E is a locally compact space with countable base and ε its Borel field. The transition function P_t(x, A), t > 0, x ε E, A ∈ ε, is Borelian in (t, x). The field family {ℱ_t} is augmented by all P-null sets but not necessarily right continuous. A stopping time relative to {ℱ_t+} will be called « optional » here. The augmented Borel field generated by (…) will be denoted by σ(…). C_K denotes the class of functions continuous on E and having compact supports. « Almost surely » (« a.s. ») refers to P and is often omitted in an obvious context. Any statement below regarding X_T is automatically understood to be under the proviso « a.s. on {T < ∞} » since X_∞ is not defined. « The path » is a circumlocution for « almost all paths ». The symbol « ↑ » means « increasing » but not necessarily strictly. Other terminology, notation and conventions follow generally those in [1] and [5]…