Invariant, super and quasi-martingale functions of a Markov process

2017/09/06 by Iulian Cîmpean, Cîmpean, Iulian, Lucian Beznea +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1709.01864

openalex publication_date 2017/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. Finally, using the co-excessive functions, we present a two-step approach to the existence of invariant probability measures.

Related

• Quasimartingales associated to Markov processes
• On characterisation of Markov processes via martingale problems
• Stochastic Calculus for Markov Processes Associated with Semi-Dirichlet Forms
• Comparison of Markov processes by the martingale comparison method
• Invariant measures of Lévy-type operators and their associated Markov processes