Abstract
We study the precise relationship between the subordinate killed and killed subordinate processes in the case of an underlying Hunt process, and show that, under minimal conditions, the former is a subprocess of the latter obtained by killing at a terminal time. Moreover, we also show that the killed subordinate process can be obtained by resurrecting the subordinate killed one at most countably many times.
Highlights
Let X be a strong Markov process on a state space E
We study the precise relationship between the subordinate killed and killed subordinate processes in the case of an underlying Hunt process, and show that, under minimal conditions, the former is a subprocess of the latter obtained by killing at a terminal time
The first one is subordination of X via an independent subordinator T giving a Markov process Y = (Yt : t ≥ 0) on E defined by Yt = X(Tt)
Summary
Let X be a strong Markov process on a state space E. One can kill Y upon exiting D giving the process Y D, and subordinate XD by the same subordinator T giving the process that we will denote by ZD Both processes are Markov with the same state space D. It is an interesting problem to investigate the precise relationship between these two processes This question can be traced back to [4] in the case when X is a Brownian motion and T a stable subordinator. We go a step further and show that the process Y D can be recovered from ZD by resurrecting the latter at most countably many times This follows from our setting in which both ZD and Y D are described explicitly in terms of the underlying Hunt process X and the subordinator T. As an application, we give sufficient conditions for Y to be not on the boundary ∂D at the exit time from D
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