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http://hdl.handle.net/11320/6551
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Pole DC | Wartość | Język |
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dc.contributor.author | Jaeger, Peter | - |
dc.date.accessioned | 2018-05-11T07:20:21Z | - |
dc.date.available | 2018-05-11T07:20:21Z | - |
dc.date.issued | 2017 | - |
dc.identifier.citation | Formalized Mathematics, Volume 25, Issue 4, Pages 261–268 | - |
dc.identifier.issn | 1426-2630 | - |
dc.identifier.uri | http://hdl.handle.net/11320/6551 | - |
dc.description.abstract | We start proceeding with the stopping time theory in discrete time with the help of the Mizar system [1], [4]. We prove, that the expression for two stopping times k1 and k2 not always implies a stopping time (k1 + k2) (see Theorem 6 in this paper). If you want to get a stopping time, you have to cut the function e.g. (k1 + k2) ⋂ T (see [2, p. 283 Remark 6.14]). Next we introduce the stopping time in continuous time. We are focused on the intervals [0, r] where r ∈ ℝ. We prove, that for I = [0, r] or I = [0,+∞[ the set {A ⋂ I : A ∈ Borel-Sets} is a σ-algebra of I (see Definition 6 in this paper, and more general given in [3, p.12 1.8e]). The interval I can be considered as a timeline from now to some point in the future. This set is necessary to define our next lemma. We prove the existence of the σ-algebra of the τ -past, where τ is a stopping time (see Definition 11 in this paper and [6, p.187, Definition 9.19]). If τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we can prove, that the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past (see Theorem 9 in this paper and [6, p.187 Lemma 9.21]). Suppose, that you want to use Lemma 9.21 with some events, that never occur, see as a comparison the paper [5] and the example for ST(1)={+∞} in the Summary. We don’t have the element +1 in our above-mentioned time intervals [0, r[ and [0,+1[. This is only possible if we construct a new σ-algebra on ℝ {−∞,+∞}. This construction is similar to the Borel-Sets and we call this σ-algebra extended Borel sets (see Definition 13 in this paper and [3, p. 21]). It can be proved, that {+∞} is an Element of extended Borel sets (see Theorem 21 in this paper). Now we use the interval [0,+∞] as a basis. We construct a σ-algebra on [0,+∞] similar to the book ([3, p. 12 18e]), see Definition 18 in this paper, and call it extended Borel subsets. We prove for stopping times with this given σ-algebra, that for τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we have the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past, see Theorem 25 in this paper. It is obvious, that {+∞} 2 extended Borel subsets. In general, Lemma 9.21 is important for the proof of the Optional Sampling Theorem, see 10.11 Proof of (i) in [6, p. 203]. | - |
dc.language.iso | en | - |
dc.publisher | DeGruyter Open | - |
dc.subject | stopping time | - |
dc.subject | stochastic process | - |
dc.title | Introduction to Stopping Time in Stochastic Finance Theory. Part II | - |
dc.type | Article | - |
dc.identifier.doi | 10.1515/forma-2017-0025 | - |
dc.description.Affiliation | Siegmund-Schacky-Str. 18a, 80993 Munich, Germany | - |
dc.description.references | Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261-279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi: 10.1007/978-3-319-20615-8_17. | - |
dc.description.references | Hans Föllmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004. | - |
dc.description.references | Hans-Otto Georgii. Stochastik, Einführung in die Wahrscheinlichkeitstheorie und Statistik. deGruyter, Berlin, 2nd edition, 2004. | - |
dc.description.references | Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007/s10817-015-9345-1. | - |
dc.description.references | Peter Jaeger. Introduction to stopping time in stochastic finance theory. Formalized Mathematics, 25(2):101-105, 2017. doi: 10.1515/forma-2017-0010. | - |
dc.description.references | Achim Klenke. Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin, Heidelberg, 2006. | - |
dc.description.references | Andrzej Nedzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401-407, 1990. | - |
dc.identifier.eissn | 1898-9934 | - |
dc.description.volume | 25 | - |
dc.description.issue | 4 | - |
dc.description.firstpage | 261 | - |
dc.description.lastpage | 268 | - |
dc.identifier.citation2 | Formalized Mathematics | - |
Występuje w kolekcji(ach): | Formalized Mathematics, 2017, Volume 25, Issue 4 |
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