Wiki source code of Stochastic analysis, fall 2014

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1 = Stochastic analysis, fall 2014 =
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3 === Lecturer ===
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5 [[Dario Gasbarra>>doc:mathstatHenkilokunta.Gasbarra, Dario]]
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7 === Scope ===
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9 10 sp.
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11 === Type ===
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13 Advanced studies.
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15 === **Prerequisites: ** ===
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17 Probability Theory (Todennäköisyysteoria 1, 2).
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24 ==== Lectures: ====
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26 Weeks 36-42 and 44-50, Tuesday 12-14 in room C124 and Thursday 10-12 in room B120, with two hours of exercise classes per week.
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28 === Language: ===
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30 The course will be given in english or finnish depending on the audience. The materials are written in english.
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32 === Contents ===
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34 The subject of this course is martingale theory and stochastic integration.
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36 0. Introduction: functions with bounded variations, Riemann-Stieltjes integral. Pathwise quadratic variation,  Ito-Föllmer pathwise integral and Ito formula. Brownian motion and its quadratic variation.
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38 I. Paul Levy's construction of Brownian motion. Gaussian processes. Kolmogorov extension theorem and construction of stochastic process on its canonical space. Kolmogorov continuity theorem.
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40 II. Martingales in discrete time: Conditional expectation, martingale transform, forward and backward martingale convergence theorems, uniformly integrable martingales, square integrable martingales, Doob maximal inequality. Change of measure and Radon-Nikodym derivative.
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42 III. Continuous martingales. Ito isometry. Ito integral and Ito formula. Burkholder Davis Gundy inequality. Local time, Ito-Tanaka formula.
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44 IV Change of measure: Girsanov formula, stochastic exponential, Gronwall lemma. Applications in stochastic filtering.
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46 V. Stochastic differential equations, strong and weak solutions. Applications: Probabilistic solution of partial differential equations. Kakutani's theorem, Feynman-Kac formula.
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48 VI. Ito-Clarck martingale representation. Application: option pricing in Black & Scholes market model.
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50 (% style="color: rgb(96,96,96);" %)[[Lecture Notes >>attach:stochastic_analysis_dario_2014.pdf]] (last updated  18.11.2014),(%%)  [[Exercises>>doc:mathstatKurssit.Syksy 2014.Stochastic analysis, fall 2014.Exercises Stochastic Analysis, Fall 2014.WebHome]] , [[home exam>>attach:stochastic_analysis_final_exam_2014.pdf]]
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52 Old webpages[[ spring 2008>>doc:mathstatKurssit.57456.Stokastinen analyysi, kevät 2008.WebHome]] , [[ fall 2011>>doc:mathstatKurssit.Syksy 2011.Stokastinen analyysi, syksy 2011.WebHome]] ,[[ spring 2013>>doc:mathstatKurssit.Kevät 2013.Stokastinen analyysi, kevät 2013.Stochastic analysis, spring 2013.WebHome]]
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54 === (% style="line-height: 1.5;" %)Exams (%%) ===
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56 You pass this course by solving exercises in the weekly tutorial sessions and writing and home exam.
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58 === Bibliography ===
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60 Karatzas and Shreve Brownian motion and stochastic calculus, Second edition, 1998 Springer.
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62 David Williams: Probability with Martingales (Cambridge Mathematical Textbooks).
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64 Mörters and Peres: Brownian motion, Cambridge 2010.
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67 Daniel Revuz ja Marc Yorin kirjaa "Continuous martingales and Brownian motion", 2nd edition Springer 2005
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71 Dieter Sondermann: Introduction to Stochastic Calculus for Finance: A New Didactic Approach, Springer 2007
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75 Timo Seppääinen: [[Basics of Stochastic Analysis>>url:http://www.math.wisc.edu/~~seppalai/courses/735/notes.pdf||shape="rect"]], Lecture Notes, University of Wisconsin-Madison.
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79 Richard Bass: Stochastic Processes, Cambridge 2011.
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82 === [[Registration>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57456||shape="rect"]] ===
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84
85 Did you forget to register? [[ What to do?>>doc:mathstatOpiskelu.Kysymys4]]
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87 === Tutorials: ===
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