A representation of excessive functions as expected suprema

• Autorzy: Föllmer, H. , Knispel, T.
• Języki publikacji: EN

Abstrakty

EN

For a nice Markov process such as Brownian motion on a domain in R^d, we prove a representation of excessive functions in terms of expected suprema. This is motivated by recent work of El Karoui [5] and El Karoui and Meziou [8] on the max-plus decomposition for supermartingales. Our results provide a singular analogue to the non-linear Riesz representation in El Karoui and Föllmer [6], and they extend the representation of potentials in Föllmer and Knispel [10] by clarifying the role of the boundary behavior and of the harmonic points of the given excessive function.

Słowa kluczowe

EN

Markov processes; excessive functions; expected suprema

• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2006
• Tom: Vol. 26, Fasc. 2
• Strony: 379--394
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Föllmer, H.

• Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany,

autor

Knispel, T.

• Humboldt-Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, 10099 Berlin, Germany,

Bibliografia

• [1] P. Bank and N. El Karoui, A stochastic representation theorem with applications to optimization and obstacle problems, Ann. Probab. 32 (1B) (2004), pp. 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, Lecture Notes in Math. No 1814, Springer, Berlin 2003, pp. 1-42.
• [3] P. Bank and F. Riedel, Optimal consumption choice with intertemporal substitution, Ann. Appl. Probab. 11 (3) (2001), pp. 750-788.
• [4] R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Pure Appl. Math., Academic Press, New York and London 1968.
• [5] N. El Karoui, Max-plus decomposition of supermartingale - Application to portfolio insurance, http://www.ima.umn.edu/talks/worksbps/4-12-l6.2004/el_karaui/IMA2004.pdf (2004).
• [6] N. El Karoui and H. Föllmer, A non-linear Riesz representation in probabilistic potential theory, Ann. Inst. H. Poincaré Probab. Statist. 41 (3) (2005), pp. 269-283.
• [7] N. El Karoui and I. Karatzas, Synchronisation and optimality for multi-armed bandit problems in continuous time, Comput. Appl. Math. 16 (2) (1997), pp. 117-152.
• [8] N. El Karoui and A. Meziou, Constrained optimization with respect to stochastic dominance - Application to portfolio insurance, Math. Finance 16 (1) (2006), pp. 103-117.
• [9] H. Föllmer, Feine Topologie am Martinrand eines Standardprozesses, Z. Wahrscheinlichkeitstheorie verw. Gebiete 12 (1969), pp. 127-144.
• [10] H. Föllmer and T. Knispel, Potentials of a Markov process are expected suprema, to appear in: ESAIM Probab. Stat. (Volume in honor of Nicole El Karoui).
• [11] H. Kaspi and A. Mandelbaum, Multi-armed bandits in discrete and continuous time, Ann. Appl. Probab. 8 (4) (1998), pp. 1270-1290.
• [12] K. Urbanik, Lectures on prediction theory, Lecture Notes in Math. No 44, Springer, Berlin-Heidelberg-New York 1967, 50 pp.

Uwagi

Dedicated to the memory of Kazimierz Urbanik.