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Columbia University in the City of New York | DEPARTMENT OF MATHEMATICS DEPARTMENT OF STATISTICS New York, N.Y. 10027 508 Mathematics Building 618 Mathematics Building 2990 Broadway Spring Semester 2004 Professor Ioannis Karatzas G4153-G6106: PROBABILITY & RANDOM PROCESSES II COURSE SYLLABUS I. Rare Events i. Cramér's Theorem ii. Introduction to the Theory of Large Deviations iii. The Shannon-Breiman-McMillan Theorem II. Conditional Distributions and Expectations i. Absolute continuity and singularity of measures ii. Radon-Nikodým theorem. Conditional distributions iii. Conditional expectations as least-square projections iv. Notion of conditional independence v. Introduction to Markov Chains. Harmonic functions III. Martingales i. Definitions, basic properties, examples, transforms ii. Optional sampling and upcrossings theorems, convergence iii. Burkholder-Gundy and Azuma inequalities iv. Doob decomposition, square-integrable martingales v. Strong laws of large numbers and central limit theorems IV. Applications i. Optimal stopping ii. Branching processes and their limiting behavior. Urn schemes iii. Stochastic approximation. Probabilistic analysis of algorithms V. Stochastic Integrals and Stochastic Differential Equations i. Detailed study of Brownian motion ii. Martingales in continuous time iii. Doob-Meyer decomposition, stopping times iv. Integration with respect to continuous martingales, Itô's rule v. Girsanov's theorem and its applications vi. Introduction to stochastic differential equations. Diffusion processes VI. Elements of Potential Theory i. The Dirichlet problem. Poisson integral formula ii. Solution in terms of Brownian motion iii. Detailed study of the heat equation; Cauchy and boundary-value problems iv. Feynman-Kac theorems, applications Recommended Texts: D. WILLIAMS : “Probability with Martingales” Cambridge Univ. Press (1990). K.L. CHUNG : “A Course in Probability Theory”. 2nd Edition, Academic Press (1974). I. KARATZAS & S.E. SHREVE : “Brownian Motion and Stochastic Calculus”. Springer Verlag (1991). DETAILED SYLLABUS . Lecture #1: Tuesday, 20 January Conditional expectations and probabilities. Properties. Law of Total Probability. Examples. Properties of conditional expectations, least-squares prediction. . Lecture #2: Thursday, 22 January Notions of pressure, entropy and their properties. Notion and relevance of Large Deviations. Statement and discussion of Cramer’s theorem. . Lecture #3: Tuesday, 27 January Proof of Cramer’s theorem. The Portmanteau theorem of weak convergence. The multi-dimensional Cramer theorem. Example: the multinomial distribution, Shannon-Breiman-McMillan theorem. . Lecture #4: Thursday, 29 January Relative Entropy, its decrease by conditioning. Shannon entropy, its increase by convolution; Fisher information, and its decrease by convolution. Cramer-Rao inequality. . Lecture #5: Tuesday, 3 February Approach to the Central Limit Theorem via entropy-production. Convergence properties of subadditive sequences of real numbers. Results of Linnik-BrownBarron; de Bruijn and Stam inequalities. . Lecture #6: Thursday, 5 February Notions and elementary properties of Martingales and Stopping Times. Doob decomposition. Martingale transforms. Elementary Optional Sampling theorem. . Lecture #7: Tuesday, 10 February Statement and proof of the general Optional Sampling Theorem. Relevance of last elements and of uniform integrability. Statement of the (Super)Martingale Convergence Theorem. Nonnegative Supermartingales. . Lecture #8: Thursday, 12 February Martingale Systems Theorems; the Fundamental Theorem of Asset-Pricing. . Lecture #9: Tuesday, 17 February Doob’s Upcrossings inequality; proof of the convergence theorem. Submartingale and Maximal inequalities. . Lecture #10: Thursday, 19 February Optional Sampling (continuation). Wald identities. Applications to the simple random walk. Statements of the Marzinkiewicz-Zygmund, Burkholder-Gundy and Davis inequalities. Applications. . Lecture #11: Tuesday, 24 February Convergence theorems for uniformly integrable submartingales, Levy’s martingale Convergence theorem. Kolmogorov’s 0-1 law and Strong Law of Large Numbers, via martingales. . Lecture #12: Thursday, 26 February Square-integrable martingales; bracket sequence. Convergence properties of series of Independent random variables. Cesaro and Kronecker lemmata. . Lecture #13: Tuesday, 2 March Strong Law of Large Numbers for Martingales. Three-Series Theorem. Kolmogorov’s proof of the classical strong law of large numbers for IID random variables. . Lecture #14: Thursday, 4 March Applications of Martingales to the theory of Optimal Stopping. Dynamic Programming Algorithm. Examples. . Lecture #15: Tuesday, 9 March Applications of Martingales to Stochastic Approximation. The Robbins-Monro Algorithm and its Properties. The Robbins-Siegmund theorem on convergence of “almost-supermartingales”. . Lecture #16: Thursday, 11 March The Markov and strong Markov properties. Harmonic and sub-harmonic functions, relations with the tail sigma-algebra, the Hewitt-Savage 0-1 law. . Lecture #17: Tuesday, 23 March Continuous-parameter stochastic processes: filtrations and their properties, adaptivity and progressive measurability, stopping and optional times, first hitting times. . Lecture #18: Thursday, 25 March Continuous-time martingales: optional sampling theorem, regularity of the paths (right-continuity, exhaustion of jumps by a countable sequence of stopping times). Doob decomposition in discrete time; equivalent formulations of predictability. . Assignment: Read pages 1-21 in their entirety. Do the following (not to be turned in): Exercises 3.18, 3.30 Problems 2.2, 2.6, 2.7, 2.11, 2.21-24, 3.16, 3.28, 3.29 . Lecture #19: Tuesday, 30 March Doob-Meyer decomposition for continuous-parameter submartingales. Notion and continuity of the natural increasing process. Square-integrable martingales and their bracket-processes; orthogonality, continuity, quadratic variation, localization. . Lecture #20: Thursday, 1 April Mid-Term Examination. . Lecture #21: Tuesday, 6 April Local martingales and their properties: Doob decomposition, quadratic variation process, relation to the maximal process. Statement of the Burkholder-Gundy inequalities in this context, regularly-varying fujnctions. Statement of the representation of a continuous local martingale as time-changed Brownian motion. . Assignment: Read pages 21-38 in their entirety. Do the following (not to be turned in): Problems 4.15, 5.11, 5.12, 5.19, 5.24-26. . Lecture #22: Thursday, 8 April Integration with respect to continuous local martingales. Properties of the Ito integral, the new change-of-variable formula. Characterization of the stochastic integral. . Lecture #23: Tuesday, 13 April Applications: exponential (local) martingales, the Levy characterization of Brownian motion, representation of continuous martingales in terms of Brownian motion. . Lecture #24: Thursday, 15 April The Girsanov theorem and its applications; Brownian motion with drift. Good-λ inequalities, proof of the Burkholder-Gundy inequalities. . Lecture #25: Tuesday, 20 April The Kunita-Watanabe decomposition; the martingale representation property of the Brownian filtration. Notion and elementary properties of Brownian local time. . Lecture #26: Thursday, 22 April Proof of the Law of the Iterated Logarithm for normal variables and for Brownian Motion, using exponential martingales. Strong Markov property for Brownian Motion, the reflection principle, applications. . Assignment: Read pages 47-55 as well as Sections 2.6, 2.8 and 2.9 in their entirety. Do the following (not to be turned in): Exercises 9.17, 9.21 Problems 8.2, 8.8, 8.14, 8.18, 9.11 . Lecture #27: Tuesday, 27 April Harmonic functions and the Dirichlet problem. Poisson integral formulas. . Lecture #28: Thursday, 29 April Space-time harmonic functions, the Feynman-Kac formula. First-passage-time problems for Brownian motion, the arc-sine law.