{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T21:57:59Z","timestamp":1649109479513},"reference-count":9,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2017,7,1]],"date-time":"2017-07-01T00:00:00Z","timestamp":1498867200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,7,1]]},"abstract":"Summary<\/jats:title>\n We start with the definition of stopping time according to [4], p.283. We prove, that different definitions for stopping time can coincide. We give examples of stopping time using constant-functions or functions defined with the operator max or min (defined in [6], pp.37\u201338). Finally we give an example with some given filtration. Stopping time is very important for stochastic finance. A stopping time is the moment, where a certain event occurs ([7], p.372) and can be used together with stochastic processes ([4], p.283). Look at the following example: we install a function ST: {1,2,3,4} \u2192 {0, 1, 2} \u222a {+\u221e}, we define:<\/jats:p>\n a. ST(1)=1, ST(2)=1, ST(3)=2, ST(4)=2.<\/jats:p>\n b. The set {0,1,2} consists of time points: 0=now,1=tomorrow,2=the day after tomorrow.<\/jats:p>\n We can prove:<\/jats:p>\n c. {w, where w is Element of \u03a9: ST.w=0}=\u2205 & {w, where w is Element of \u03a9: ST.w=1}={1,2} & {w, where w is Element of \u03a9: ST.w=2}={3,4} and<\/jats:p>\n ST is a stopping time.<\/jats:p>\n We use a function Filt as Filtration of {0,1,2}, \u03a3 where Filt(0)=\u03a9\n now<\/jats:sub>\n <\/jats:italic>, Filt(1)=\u03a9\n fut<\/jats:sub>\n <\/jats:italic>\n 1<\/jats:sub> and Filt(2)=\u03a9\n fut<\/jats:sub>\n <\/jats:italic>\n 2<\/jats:sub>. From a., b. and c. we know that:<\/jats:p>\n d. {w, where w is Element of \u03a9: ST.w=0} in \u03a9\n now<\/jats:sub>\n <\/jats:italic> and<\/jats:p>\n {w, where w is Element of \u03a9: ST.w=1} in \u03a9\n fut<\/jats:sub>\n <\/jats:italic>\n 1<\/jats:sub> and<\/jats:p>\n {w, where w is Element of \u03a9: ST.w=2} in \u03a9\n fut<\/jats:sub>\n <\/jats:italic>\n 2<\/jats:sub>.<\/jats:p>\n The sets in d. are events, which occur at the time points 0(=now), 1(=tomorrow) or 2(=the day after tomorrow), see also [7], p.371. Suppose we have ST(1)=+\u221e, then this means that for 1 the corresponding event never occurs.<\/jats:p>\n As an interpretation for our installed functions consider the given adapted stochastic process in the article [5].<\/jats:p>\n ST(1)=1 means, that the given element 1 in {1,2,3,4} is stopped in 1 (=tomorrow). That tells us, that we have to look at the value f<\/jats:italic>\n 2<\/jats:sub>(1) which is equal to 80. The same argumentation can be applied for the element 2 in {1,2,3,4}.<\/jats:p>\n ST(3)=2 means, that the given element 3 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f<\/jats:italic>\n 3<\/jats:sub>(3) which is equal to 100.<\/jats:p>\n ST(4)=2 means, that the given element 4 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f<\/jats:italic>\n 3<\/jats:sub>(4) which is equal to 120.<\/jats:p>\n In the real world, these functions can be used for questions like: when does the share price exceed a certain limit? (see [7], p.372).<\/jats:p>","DOI":"10.1515\/forma-2017-0010","type":"journal-article","created":{"date-parts":[[2017,9,25]],"date-time":"2017-09-25T10:00:55Z","timestamp":1506333655000},"page":"101-105","source":"Crossref","is-referenced-by-count":1,"title":["Introduction to Stopping Time in Stochastic Finance Theory"],"prefix":"10.1515","volume":"25","author":[{"given":"Peter","family":"Jaeger","sequence":"first","affiliation":[{"name":"Siegmund-Schacky-Str. 18a, 80993 Munich , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,9,23]]},"reference":[{"key":"2021040704590216631_j_forma-2017-0010_ref_001_w2aab3b7b2b1b6b1ab1ab1Aa","unstructured":"[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41\u201346, 1990."},{"key":"2021040704590216631_j_forma-2017-0010_ref_002_w2aab3b7b2b1b6b1ab1ab2Aa","unstructured":"[2] Czes\u0142aw Byli\u0144ski. Functions and their basic properties. Formalized Mathematics, 1(1): 55\u201365, 1990."},{"key":"2021040704590216631_j_forma-2017-0010_ref_003_w2aab3b7b2b1b6b1ab1ab3Aa","unstructured":"[3] Czes\u0142aw Byli\u0144ski. Functions from a set to a set. Formalized Mathematics, 1(1):153\u2013164, 1990."},{"key":"2021040704590216631_j_forma-2017-0010_ref_004_w2aab3b7b2b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Hans F\u00f6llmer and Alexander Schied. Stochastic Finance: An Introduction in Discrete Time, volume 27 of Studies in Mathematics. de Gruyter, Berlin, 2nd edition, 2004.","DOI":"10.1515\/9783110212075"},{"key":"2021040704590216631_j_forma-2017-0010_ref_005_w2aab3b7b2b1b6b1ab1ab5Aa","unstructured":"[5] Peter Jaeger. Modelling real world using stochastic processes and filtration. 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Formalized Mathematics, 1(2): 329\u2013334, 1990."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/25\/2\/article-p101.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2017-0010","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,7]],"date-time":"2021-04-07T13:59:25Z","timestamp":1617803965000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.sciendo.com\/article\/10.1515\/forma-2017-0010"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,7,1]]},"references-count":9,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,9,23]]},"published-print":{"date-parts":[[2017,7,1]]}},"alternative-id":["10.1515\/forma-2017-0010"],"URL":"https:\/\/doi.org\/10.1515\/forma-2017-0010","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2017,7,1]]}}}