Potentials of a Markov process are expected suprema

Hans Föllmer; Thomas Knispel

ESAIM: Probability and Statistics (2007)

  • Volume: 11, page 89-101
  • ISSN: 1292-8100

Abstract

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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).

How to cite

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Föllmer, Hans, and Knispel, Thomas. "Potentials of a Markov process are expected suprema." ESAIM: Probability and Statistics 11 (2007): 89-101. <http://eudml.org/doc/250094>.

@article{Föllmer2007,
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). },
author = {Föllmer, Hans, Knispel, Thomas},
journal = {ESAIM: Probability and Statistics},
keywords = {Markov processes; potentials; optimal stopping; max-plus decomposition.; max-plus decomposition},
language = {eng},
month = {3},
pages = {89-101},
publisher = {EDP Sciences},
title = {Potentials of a Markov process are expected suprema},
url = {http://eudml.org/doc/250094},
volume = {11},
year = {2007},
}

TY - JOUR
AU - Föllmer, Hans
AU - Knispel, Thomas
TI - Potentials of a Markov process are expected suprema
JO - ESAIM: Probability and Statistics
DA - 2007/3//
PB - EDP Sciences
VL - 11
SP - 89
EP - 101
AB - 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).
LA - eng
KW - Markov processes; potentials; optimal stopping; max-plus decomposition.; max-plus decomposition
UR - http://eudml.org/doc/250094
ER -

References

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  1. P. Bank and N. El Karoui, A Stochastic Representation Theorem with Applications to Optimization and Obstacle Problems. Ann. Probab.32 (2005) 1030–1067.  
  2. 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.  
  3. 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).  
  4. 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.  
  5. N. El Karoui, Max-Plus Decomposition of Supermartingale - Application to Portfolio Insurance, (2004).  URIhttp://www.ima.umn.edu/talks/workshops/4-12-16.2004/el_karoui/IMA2004.pdf
  6. 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 Statistiques41 (2005) 269–283.  
  7. N. El Karoui and A. Meziou, Constrained optimization with respect to stochastic dominance: Application to portfolio insurance. Math. Finance16 (2006) 103–117.  
  8. T. Knispel, Eine nichtlineare Riesz-Darstellung bezüglich additiver Funktionale im potentialtheoretischen Kontext. Diploma Thesis, Humboldt University, Berlin (2004).  
  9. A.N. Shiryaev, Statistical Sequential Analysis. AMS, Providence, Transl. Math. Monographs38 (1973).  

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