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Potentials of a Markov process are expected suprema

Published online by Cambridge University Press:  01 March 2007

Hans Föllmer
Affiliation:
Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany; foellmer@math.hu-berlin.de; knispel@math.hu-berlin.de
Thomas Knispel
Affiliation:
Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany; foellmer@math.hu-berlin.de; knispel@math.hu-berlin.de
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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).

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2007

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References

Bank, P. and El Karoui, N., Stochastic Representation Theorem, A with Applications to Optimization and Obstacle Problems. Ann. Probab. 32 (2005) 10301067.
P. Bank and H. Föllmer, American Options, Multi-armed Bandits, and Optimal Consumption Plans: A Unifying View, in Paris-Princeton Lectures on Mathematical Finance 2002, Lect. Notes Math. 1814 (2003) 1–42.
C. Dellacherie and P. Meyer, Probabilités et potentiel. Chapitres XII–XVI: Théorie du potentiel associée à une résolvante, Théorie des processus de Markov, Hermann, Paris (1987).
N. El Karoui, Les aspects probabilistes du contrôle stochastique, in Ninth Saint Flour Probability Summer School-1979 (Saint Flour, 1979), Lect. Notes Math. 876 (1981) 73–238.
N. El Karoui, Max-Plus Decomposition of Supermartingale - Application to Portfolio Insurance, http://www.ima.umn.edu/talks/workshops/4-12-16.2004/el_karoui/IMA2004.pdf (2004).
N. El Karoui and H. Föllmer, A non-linear Riesz representation in probabilistic potential theory, in Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques 41 (2005) 269–283.
El Karoui, N. and Meziou, A., Constrained optimization with respect to stochastic dominance: Application to portfolio insurance. Math. Finance 16 (2006) 103117. CrossRef
T. Knispel, Eine nichtlineare Riesz-Darstellung bezüglich additiver Funktionale im potentialtheoretischen Kontext. Diploma Thesis, Humboldt University, Berlin (2004).
A.N. Shiryaev, Statistical Sequential Analysis. AMS, Providence, Transl. Math. Monographs 38 (1973).

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