{"id":1079,"date":"2023-06-25T19:35:40","date_gmt":"2023-06-26T02:35:40","guid":{"rendered":"https:\/\/dornsife.usc.edu\/sergey-lototsky\/?page_id=1079"},"modified":"2023-06-25T19:35:41","modified_gmt":"2023-06-26T02:35:41","slug":"math-606-summer-2020","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/sergey-lototsky\/math-606-summer-2020\/","title":{"rendered":"Math 606: Topics in Stochastic Processes, Summer 2020"},"content":{"rendered":"\n\n \n \n\n\n\n\n\n\n\n
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MATH 606, Summer 2020.
\nTopics in Stochastic Processes (054–39482R)
\nUnderstanding the Bichteler-Dellacherie Theorem
\n Class meetings: MW, 9:30am-12:30pm, On line. <\/span><\/h1>\n

 <\/p>\n

Information on this and related pages changes frequently.<\/h2>\n

Instructor: <\/strong> Sergey Lototsky. <\/a>
\n Office:<\/strong> KAP 248D.
\n Phone:<\/strong> 213 740 2389
\n E-mail:<\/strong> lototsky usc edu.<\/h2>\n

URL:\u00a0https:\/\/dornsife.usc.edu\/sergey-lototsky\/<\/strong><\/p>\n

Office Hours:<\/strong> MW 12:30-1:30pm, on line [May 20-July 6]
\nAppointments at other time are welcome.<\/p>\n

Course objective <\/strong>To understand why a consistent stochastic integration is only possible with respect to semi-martingales<\/p>\n

Course work: <\/strong>Class participation, homework assignments, final presentation.<\/p>\n

Official grading scheme: <\/strong>20% class participation, 40% homework assignments, 40% final presentation.<\/p>\n

Main reference: <\/strong> Stochastic Integration and Differential Equations, by Philip Protter; Any Edition, Springer, from 1990 on, 300 pp.<\/strong><\/p>\n

Other references <\/strong><\/p>\n

    \n
  1. Karatzas and Shreve, Browinian Motion and Stochastic Calculus, Springer, from 1988 on, 500 pp.<\/strong><\/li>\n
  2. Liptser and Shiryaev, Theory of Martingales, Kluwer\/Springer, from 1986 on, 800 pp.<\/strong><\/li>\n
  3. An alternative proof of the theorem<\/a> [pre-print; published as Beiglb\u00f6ck, M. and Siorpaes, P. Riemann-integration and a new proof of the Bichteler-Dellacherie theorem.<\/span> Stochastic Process. Appl.<\/em> 124 (2014), no. 3, pages 1226\u20131235.]<\/li>\n<\/ol>\n

    Closely related fun reads<\/strong><\/p>\n

      \n
    1. Burk, A Garden of Integrals, MAA, 2007.<\/li>\n
    2. Gelbaum and Olmsted, Counterexamples in Analysis, from 1964 on [e.g. Dover, 2003].<\/li>\n
    3. Stoyanov, Counterexamples in Probability, Wiley, from 1987 on.<\/li>\n
    4. Szekely, Paradoxes in Probability Theory and Mathematical Statistics, Kluwer, from 1986 on.<\/li>\n<\/ol>\n

      Highlights of the course, including homework problems, will be updated on a regular basis and appear at the top of the content section of Blackboard<\/p>\n

      Lecture notes: at least two versions will be uploaded to the content section of Blackboard after every lecture.<\/p>\n

      The plan <\/strong>
      \nIntroduction and outline; Deterministic integration; Summary of Functional Analysis; Review of Basic Probability; Stochastic Analysis I: sigma-algebras and stopping times; Stochastic Analysis II: different types of processes; Stochastic Analysis III: martingales and related concepts; Properties of the Stochastic Integral, Cameron-Martin-Girsanov-Meyer; The Proof; Some examples and counterexamples.<\/p>\n

      Our Progress<\/strong><\/p>\n

      May 20. Understanding the task: stochastic integration as a bounded linear operator.<\/p>\n

      May 25. Memorial Day: no class.
      \nMay 27. An overview of integration: from Cauchy and Riemann to Henstock-Kuzweil and Ito.<\/p>\n

      June 1. Deterministic integration: importance of bounded variation.
      \nJune 3. Functional analysis and probability: topology vs sigma-algebra.<\/p>\n

      June 8. Functional analysis: the main theorems.
      \nJune 10. Stochastic analysis: the foundations.<\/p>\n

      June 15. Martingales and the idea of the proof.
      \nJune 17. Quasimartingales and the proof.<\/p>\n

      June 22. What is behind: Theorems of Cameron-Martin-Girsanov-Meyer, Doob-Meyer, and Rao.
      \nJune 24. Semimartingales and stochastic integrals: general properties<\/p>\n

      June 9. Martingales: inequalities, convergence, and change of measure.
      \nJuly 1. A summary: examples and counterexamples.<\/p>\n

      July 6. A problem-solving session.<\/p>\n\n\n\n<\/div>\n\n\n <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":282,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-1079","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"\nMath 606: Topics in Stochastic Processes, Summer 2020 - Sergey Lototsky<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/math-606-summer-2020\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Math 606: Topics in Stochastic Processes, Summer 2020 - Sergey Lototsky\" \/>\n<meta property=\"og:url\" content=\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/math-606-summer-2020\/\" \/>\n<meta property=\"og:site_name\" content=\"Sergey Lototsky\" \/>\n<meta property=\"article:modified_time\" content=\"2023-06-26T02:35:41+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/math-606-summer-2020\/\",\"url\":\"https:\/\/dornsife.usc.edu\/sergey-lototsky\/math-606-summer-2020\/\",\"name\":\"Math 606: Topics in Stochastic Processes, Summer 2020 - 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