Quasimartingales associated to Markov processes

Beznea, Lucian; Cîmpean, Iulian

doi:10.1090/tran/7214

Quasimartingales associated to Markov processes
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Trans. Amer. Math. Soc. 370 (2018), 7761-7787 Request permission

Abstract:

For a fixed right process $X$ we investigate those functions $u$ for which $u(X)$ is a quasimartingale. We prove that $u(X)$ is a quasimartingale if and only if $u$ is the difference of two finite excessive functions. In particular, we show that the quasimartingale nature of $u$ is preserved under killing, time change, or Bochner subordination. The study relies on an analytic reformulation of the quasimartingale property for $u(X)$ in terms of a certain variation of $u$ with respect to the transition function of the process. We provide sufficient conditions under which $u(X)$ is a quasimartingale, and finally, we extend to the case of semi-Dirichlet forms a semimartingale characterization of such functionals for symmetric Markov processes, due to Fukushima.

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