Download Math G4153 - Columbia Math

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.