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ESAIM: P&S, February 2007, Vol. 11, pp. 89-101
DOI: 10.1051/ps:2007008
Potentials of a Markov process are expected suprema
Hans Föllmer and Thomas KnispelHumboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany; foellmer@math.hu-berlin.de; knispel@math.hu-berlin.de
(Invited paper accepted September 2005. Published online 1 March 2007.)
Abstract
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 El Karoui and
Meziou (2006) on the max-plus decomposition of supermartingales, and they provide a singular analogue to the non-linear Riesz representation in El Karoui and Föllmer (2005).
Mathematics Subject Classification. 31C05, 60J25, 60J45.
Key words: Markov processes, potentials, optimal stopping, max-plus decomposition.
© EDP Sciences, SMAI 2007