Stopping of functionals with discontinuity at the boundary of an open set
We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set O . The stopping horizon is either random, equal to the first exit from the set O , or fixed (finite or infinite). The...
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| Published in | Stochastic processes and their applications Vol. 121; no. 10; pp. 2361 - 2392 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier B.V
01.10.2011
Elsevier |
| Series | Stochastic Processes and their Applications |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0304-4149 1879-209X 1879-209X |
| DOI | 10.1016/j.spa.2011.05.013 |
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| Abstract | We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set
O
. The stopping horizon is either random, equal to the first exit from the set
O
, or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of
O
. Using a generalization of the penalty method, we derive a numerical algorithm for approximation of the value function for general Feller–Markov processes and show existence of optimal or
ε
-optimal stopping times.
► Optimal stopping of Feller–Markov processes with discontinuous functionals. ► Properties of the value function and existence of optimal stopping times. ► A generalization of the penalty method. ► Numerical algorithm for approximation of the value function and optimal stopping times. |
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| AbstractList | We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set . The stopping horizon is either random, equal to the first exit from the set , or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of . Using a generalization of the penalty method, we derive a numerical algorithm for approximation of the value function for general Feller-Markov processes and show existence of optimal or [epsilon]-optimal stopping times. We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set O . The stopping horizon is either random, equal to the first exit from the set O , or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of O . Using a generalization of the penalty method, we derive a numerical algorithm for approximation of the value function for general Feller–Markov processes and show existence of optimal or ε -optimal stopping times. ► Optimal stopping of Feller–Markov processes with discontinuous functionals. ► Properties of the value function and existence of optimal stopping times. ► A generalization of the penalty method. ► Numerical algorithm for approximation of the value function and optimal stopping times. |
| Author | Stettner, Łukasz Palczewski, Jan |
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| Keywords | Optimal stopping Discontinuous functional Feller–Markov process Penalty method Markov process Discontinuity Stopping time Feller-Markov process Continuous function Numerical method Discontinuous functional: Penalty method Stochastic process Algorithm Numerical approximation Functional Value function Payoff function |
| Language | English |
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| References | Mertens (br000080) 1973; 26 Palczewski, Stettner (br000090) 2010; 48 El Karoui (br000040) 1981; vol. 876 Menaldi (br000075) 1980; 18 Karatzas, Shreve (br000055) 1991 Dynkin (br000030) 1965; vol. 1 Zabczyk (br000110) 1984; vol. II Ł. Stettner, J. Zabczyk, Optimal Stopping for Feller Markov Processes, Preprint No. 284, IMPAN, Warsaw, 1983. M. Robin, Controle impulsionnel des processus de Markov (Thesis), University of Paris IX, 1978. Fakeev (br000050) 1971; 16 P.A. Meyer, Markov Processes, Research and Training School, Indian Statistical Institute (translation by MN Sastry), 1969. Blumenthal, Getoor (br000025) 1968 Bassan, Ceci (br000005) 2002; 72 Lamberton (br000065) 2009; 119 El Karoui, Lepeltier, Marchal (br000045) 1982; vol. 42 Bassan, Ceci (br000010) 2002; 74 Fleming, Soner (br000035) 1993 Bismut, Skalli (br000020) 1977; 39 Kushner, Dupuis (br000060) 1992 Stettner (br000105) 2011; 49 Bensoussan, Lions (br000015) 1978 Matskyavichyus (br000070) 1973; 13 Stettner (10.1016/j.spa.2011.05.013_br000105) 2011; 49 Fleming (10.1016/j.spa.2011.05.013_br000035) 1993 10.1016/j.spa.2011.05.013_br000100 Bensoussan (10.1016/j.spa.2011.05.013_br000015) 1978 Bassan (10.1016/j.spa.2011.05.013_br000005) 2002; 72 Fakeev (10.1016/j.spa.2011.05.013_br000050) 1971; 16 Mertens (10.1016/j.spa.2011.05.013_br000080) 1973; 26 Blumenthal (10.1016/j.spa.2011.05.013_br000025) 1968 El Karoui (10.1016/j.spa.2011.05.013_br000045) 1982; vol. 42 Kushner (10.1016/j.spa.2011.05.013_br000060) 1992 Matskyavichyus (10.1016/j.spa.2011.05.013_br000070) 1973; 13 Bismut (10.1016/j.spa.2011.05.013_br000020) 1977; 39 Menaldi (10.1016/j.spa.2011.05.013_br000075) 1980; 18 El Karoui (10.1016/j.spa.2011.05.013_br000040) 1981; vol. 876 10.1016/j.spa.2011.05.013_br000085 Palczewski (10.1016/j.spa.2011.05.013_br000090) 2010; 48 Bassan (10.1016/j.spa.2011.05.013_br000010) 2002; 74 10.1016/j.spa.2011.05.013_br000095 Karatzas (10.1016/j.spa.2011.05.013_br000055) 1991 Zabczyk (10.1016/j.spa.2011.05.013_br000110) 1984; vol. II Lamberton (10.1016/j.spa.2011.05.013_br000065) 2009; 119 Dynkin (10.1016/j.spa.2011.05.013_br000030) 1965; vol. 1 |
| References_xml | – volume: vol. 1 year: 1965 ident: br000030 publication-title: Markov Processes – volume: 48 start-page: 4874 year: 2010 end-page: 4909 ident: br000090 article-title: Finite horizon optimal stopping of time-discontinuous functionals with applications to impulse control with delay publication-title: SIAM Journal on Control and Optimization – volume: vol. 876 start-page: 73 year: 1981 end-page: 238 ident: br000040 article-title: Les aspects probabilistes du contrôle stochastique, Ninth Saint Flour Probability Summer School–1979 publication-title: Lecture Notes in Mathematics – volume: 18 start-page: 697 year: 1980 end-page: 721 ident: br000075 article-title: On the optimal stopping time problem for degenerate diffusions publication-title: SIAM Journal on Control and Optimization – year: 1978 ident: br000015 article-title: Applications des Inéquations Variationnelles en contrôle Stochastique – year: 1991 ident: br000055 article-title: Brownian Motion and Stochastic Calculus – volume: vol. 42 start-page: 106 year: 1982 end-page: 112 ident: br000045 article-title: Optimal stopping of controlled Markov processes publication-title: Advances in Filtering and Optimal Stochastic Control – reference: Ł. 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Meyer, Markov Processes, Research and Training School, Indian Statistical Institute (translation by MN Sastry), 1969. – volume: 72 start-page: 55 year: 2002 end-page: 77 ident: br000005 article-title: Optimal stopping with discontinuous reward: Regularity of the value function and viscosity solution publication-title: Stochastics and Stochastics Reports – volume: 74 start-page: 633 year: 2002 end-page: 649 ident: br000010 article-title: Regularity of the value function and viscosity solutions in optimal stopping problems for general Markov processes publication-title: Stochastics and Stochastics Reports – year: 1993 ident: br000035 article-title: Controlled Markov Processes and Viscosity Solutions – volume: 16 start-page: 708 year: 1971 end-page: 710 ident: br000050 article-title: On the question of the optimal stopping of a Markov process publication-title: Teoriya Veroyatnostei i ee Primeneniya – volume: 26 start-page: 119 year: 1973 end-page: 139 ident: br000080 article-title: Strongly supermedian functions and optimal stopping publication-title: Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete – volume: vol. 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| SubjectTerms | Discontinuous functional Exact sciences and technology Feller–Markov process Markov processes Mathematics Optimal stopping Optimal stopping Feller-Markov process Discontinuous functional Penalty method Penalty method Probability and statistics Probability theory and stochastic processes Sciences and techniques of general use Sequential methods Statistics Stochastic analysis Stochastic processes |
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| Title | Stopping of functionals with discontinuity at the boundary of an open set |
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