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1 = Stochastic analysis, spring 2010[[ (suomeksi)>>doc:mathstatKurssit.Home.Kevät 2010.Stokastinen analyysi, kevät 2010.WebHome]] =
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3 === Lecturer ===
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5 [[Dario Gasbarra>>doc:mathstatHenkilokunta.Gasbarra, Dario]]
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7 === Credits ===
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9 10 op.
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11 === Type ===
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13 Advanced course.
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15 === Prerequisites: Probability theory. ===
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17 === Language: finnish or/and english, according to the audience. ===
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19 === Schedule ===
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21 During weeks 3-9 and 11-18, mo 10-12, tu 12-14 C124, plus 2 hours/week of exercise class. The first lecture on monday 18.1. Easter holiday on 1.-7.4.
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23 === Exams. ===
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25 There will be a mid-term exam and a final exam.
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27 === [[Lecture notes,>>attach:mathstatKurssit.Stokastinen analyysi, kevät 2010.WebHome@stochastic_analysis_dario.pdf]] [[exercises>>doc:mathstatKurssit.Home.Kevät 2010.Stokastinen analyysi, kevät 2010.exercises.WebHome]] ===
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29 === Content: ===
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31 We will study the trheory of continuous martingales and stochastic integration.
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33 I. Discrete time martingales. Conditional expectation, filtration and stopping times. Uniformly integrable martingales, square integrable martingales, martingale convergence theorem. Doob's maximal inequality.
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35 II. Stochastic prosesses in continuous time. Kolmogorov's consitency theorem, construction and properties of Brownian motion. Poisson and Levy processes. Markov processes. Strong Markov property.
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37 III. Ito calculus: functions of finite variation and Stieltjes integrals. Quadratic variation. Ito isometry for Brownian motion and Ito integral. Localization, Burkholder Davis Gundy inequality, Föllmer's pathwise inegral, Ito formula, Local time, Ito-Tanaka formula.
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39 IV Change of measure: Girsanov theorem, stochastic exponential, Gronwall lemma. Application to stochastic filtering theory.
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41 V Stochastic differential equations, weak and strong solutions, martingale-problem. Application: Probabilistic solutions to partial differential equations: Kakutani's theorem, Feynman-Kac formula.
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43 VI. Ito-Clarck martingale representation theorem. Applications: option pricing in Black & Scholes financial market model.
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45 === Literature: ===
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47 For the discrete-time martingale theory we follow David Williams' book: Probability with Martingales (Cambridge Mathematical Textbooks).
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49 Karatzas, Shreve: Brownian motion and stochastic calculus, Springer 1998.
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51 Revuz, Yor: Continuous martingales and Brownian motion, Springer 2005.
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53 === [[Registration>>url:https://oodi-www.it.helsinki.fi/hy/opintjakstied.jsp?html=1&Tunniste=57456||shape="rect"]] ===
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55 === Tutorials ===
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