Abstract
We consider the decreasing and the increasing $r$-excessive functions $\varphi _r$ and $\psi _r$ that are associated with a one-dimensional conservative regular continuous strong Markov process $X$ with values in an interval with endpoints $\alpha < \beta $. We prove that the $r$-excessive local martingale $\bigl ( e^{-r (t \wedge T_\alpha )} \varphi _r (X_{t \wedge T_\alpha }) \bigr )$ $\bigl ($resp., $\bigl ( e^{-r (t \wedge T_\beta )} \psi _r (X_{t \wedge T_\beta }) \bigr ) \bigr )$ is a strict local martingale if the boundary point $\alpha $ (resp., $\beta $) is inaccessible and entrance, and a martingale otherwise.
Highlights
We consider a one-dimensional conservative regular continuous strong Markov process X = (Ω, F, Ft, Px, Xt; t ≥ 0, x ∈ I) with values in an interval I ⊆ [−∞, ∞] with endpoints α < β that is open, closed or semi-open
We consider the decreasing and the increasing r-excessive functions φr and ψr that are associated with a one-dimensional conservative regular continuous strong Markov process X with values in an interval with endpoints α < β
We prove that the r-excessive local martingale e−r(t∧Tα)φr(Xt∧Tα ) resp., e−r(t∧Tβ )ψr(Xt∧Tβ ) is a strict local martingale if the boundary point α is inaccessible and entrance, and a martingale otherwise
Summary
We prove that the r-excessive local martingale e−r(t∧Tα)φr(Xt∧Tα ) resp., e−r(t∧Tβ )ψr(Xt∧Tβ ) is a strict local martingale if the boundary point α (resp., β) is inaccessible and entrance, and a martingale otherwise. We prove that, if β is inaccessible, (i ) e−rtψr(Xt) is a Px-martingale for all x ∈ I if β is a natural boundary point, and (ii ) e−rtψr(Xt) is a strict Px-local martingale for all x ∈ I if β is an entrance boundary point, unless α is absorbing and x = α, in which case the process e−rtψr(Xt) under Pα is identically equal to 0.
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