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Professor Ioannis Karatzas
G4151-G6105: ANALYSIS AND PROBABILITY I
COURSE SYLLABUS
I. Measure Theory
i. Construction of the integral, limits and integration
ii. Lp - spaces of functions
iii. Construction of measures, Lebesgue-Stieltjes and product measures
iv. Examples: ergodicity, Liouville measure, Hausdorff measure
II. Elements of Probability
i. The coin-tossing or random walk model
ii. Independent events and independent random variables
iii. The Strong Law of Large Numbers
iv. Notions of convergence of random variables
v. The Central Limit, Cramer and Iterated Logarithm Theorems
III. Elements of Fourier Analysis
i. Fourier transforms of measures, Fourier-Lévy Inversion Formula
ii. Convergence of distributions and characteristic functions
iii. Proof of the Central Limit Theorem
iv. Fourier transforms on Euclidean spaces
v. Fourier series, the Poisson summation formula
vi. Spectral decompositions of the Laplacian
vii. The Heat equation and heat kernel, the Wave equation and D’Alembert’s formula
IV. Brownian Motion
i. Brownian motion as a Gaussian process
ii. Brownian motion as scaling limit of random walks
iii. Brownian motion as random Fourier series
iv. Brownian motion and the heat equation
v. Elementary properties of Brownian paths
Professor Ioannis Karatzas
G4153-G6106: PROBABILITY II
TENTATIVE 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
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