# Potentials of a Markov process are expected suprema

ESAIM: Probability and Statistics (2007)

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

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topFöllmer, 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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- 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
- 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.
- N. El Karoui and A. Meziou, Constrained optimization with respect to stochastic dominance: Application to portfolio insurance. Math. Finance16 (2006) 103–117. Zbl1128.91022
- 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. Monographs38 (1973).

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