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Zbl 1184.60030
Föllmer, Hans; Knispel, Thomas
Potentials of a Markov process are expected suprema. (English)
[J] ESAIM, Probab. Stat. 11, 89-101 (2007). ISSN 1292-8100; ISSN 1262-3318/e

Summary: Expected suprema of a function $f$ observed along the paths of a nice Markov process define an excessive function, and in fact a potential if $f$ vanishes at the boundary. Conversely, we show under mild regularity conditions that any potential admits a representation in terms of expected suprema. Moreover, we identify the maximal and the minimal representing function in terms of probabilistic potential theory. Our results are motivated by the work of {\it N. El Karoui} and {\it A. Meziou} [Math. Finance 16, No.~1, 103--117 (2006; Zbl 1128.91022)] on the max-plus decomposition of supermartingales, and they provide a singular analogue to the non-linear Riesz representation in {\it N. El Karoui} and {\it H. Föllmer} [Ann. Inst. Henri Poincaré, Probab. Stat. 41, No.~3, 269--283 (2005; Zbl 1078.60058)].

MSC 2000:: *60J45 Probabilistic potential theory
31C05 Generalizations of harmonic (etc.) functions
60J25 Markov processes with continuous parameter

Keywords: Markov processes; potentials; optimal stopping; max-plus decomposition

Citations: Zbl 1128.91022; Zbl 1078.60058

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