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2 = Stochastic analysys, spring 2013 ([[Suomeksi>>doc:mathstatKurssit.Kevät 2013.Stokastinen analyysi, kevät 2013.WebHome]]) =
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4 === Teacher ===
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6 [[Dario Gasbarra>>url:https://wiki.helsinki.fi/display/mathstatHenkilokunta/Gasbarra%2C+Dario||shape="rect"]]
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8 === Credits ===
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10 10 cr.
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12 === Type ===
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14 Advanced course
15
16 === Prerequisites ===
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18 Probabllity Theory
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20 === Language: ===
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22 this course will be given in english or finnish depending on the audience.
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24 The teaching material is in english.
25
26 === Contents ===
27
28 The subject of this course is martingale theory and stochastic integration.
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30 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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32 I. Kolmogorov extension theorem and construction of stochastic process on its canonical space. Kolmogorov continuity theorem. Paul Levy's construction of Brownian motion.
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34 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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36 III. Continuous martingales. Ito isometry. Ito integral and Ito formula. Burkholder Davis Gundy inequality. Local time, Ito-Tanaka formula.
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38 IV Change of measure: Girsanov formula, stochastic exponential, Gronwal lemma. Applications in stochastic filtering.
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40 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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42 VI. Ito-Clarck martingale representation. Application: option pricing in Black & Scholes market model.
43
44 ==== [[Study materials>>doc:mathstatKurssit.Kevät 2013.Stokastinen analyysi, kevät 2013.Course materials Stochastic Analysis Spring 2013.WebHome]] ====
45
46 Old webpages
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48 [[doc:mathstatKurssit.57456.Stokastinen analyysi, kevät 2008.WebHome]]
49 [[doc:mathstatKurssit.Syksy 2011.Stokastinen analyysi, syksy 2011.WebHome]]
50
51 === Schedule ===
52
53 During weeks 3-9 and 11-18 tu 12-14 B322, we 10-12 B120, and 2 hours of tutorials. First lecture on tuesday 15.1.
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55 Easter holyday 28.3.-3.4.
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57 === Exams ===
58
59 You pass this course by solving exercises in the weekly tutorial sessions and writing and home exam.
60
61 === Literature ===
62
63 Karatzas and Shreve Brownian motion and stochastic calculus, Second edition, 1998 Springer.
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65 David Williams: Probability with Martingales (Cambridge Mathematical Textbooks).
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67 Mörters and Peres: Brownian motion, Cambridge 2010.
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69 (((
70 Daniel Revuz ja Marc Yorin kirjaa "Continuous martingales and Brownian motion", 2nd edition Springer 2005
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73 (((
74 Dieter Sondermann: Introduction to Stochastic Calculus for Finance: A New Didactic Approach, Springer 2007
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77 (((
78 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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81 (((
82 Richard Bass: Stochastic Processes, Cambridge 2011.
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85 (((
86 (% class="title" %)
87 ==== [[Register>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57456||shape="rect"]] ====
88 )))
89
90 Forgot to register ? [[What to do>>doc:mathstatOpiskelu.Kysymys4]].
91
92 === Tutorials ===
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94 |=(((
95 Group
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97 Day
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99 Time
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101 Class
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103 Lecturer
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106 1.
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108 thu
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110 10-12
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112 B322
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114 Dario Gasbarra
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