printer friendly pageMATH 230A: Theory of Probability (STATS 310A)
Mathematical tools: asymptotics, metric spaces; measure and integration; Lp spaces; some Hilbert spaces theory. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Weak convergence and characteristic functions; central limit theorems; local limit theorems; Poisson convergence. Prerequisites: 116,
MATH 171.
Terms: Aut
| Units: 2-4
Instructors:
Montanari, A. (PI)
MATH 230B: Theory of Probability (STATS 310B)
Stopping times, 0-1 laws, Kolmogorov consistency theorem. Uniform integrability. Radon-Nikodym theorem, branching processes, conditional expectation, discrete time martingales. Exchangeability. Large deviations. Laws of the iterated logarithm. Birkhoff's and Kingman's ergodic theorems. Recurrence, entropy. Prerequisite: 310A or
MATH 230A.
Terms: Win
| Units: 2-4
Instructors:
Siegmund, D. (PI)
MATH 230C: Theory of Probability (STATS 310C)
Infinitely divisible laws. Continuous time martingales, random walks and Brownian motion. Invariance principle. Markov and strong Markov property. Processes with stationary independent increments. Prerequisite: 310B or
MATH 230B.
Terms: Spr
| Units: 2-4
Instructors:
Dembo, A. (PI)
MATH 232: Topics in Probability: Malliavin Calculus, Fractional Brownian Motion and Applications
Malliavin calculus: derivative and divergence operators, Skorohod integral. Fractional Brownian motion: relavance for financial mathematics, Ito and Tanaka formula, driving force for the heat equation. Ito formula for irregular Gaussian processes and other applications of Malliavin calculus. May be repeated for credit. Prerequisites:
MATH 236,
STATS 310C or equivalent.
Last offered: Winter 2009
| Repeatable
for credit
MATH 234: Large Deviations
Combinatorial estimates and the method of types. Large deviation probabilities for partial sums and for empirical distributions, Cramer's and Sanov's theorems and their Markov extensions. Applications in statistics, information theory, and statistical mechanics. Prerequisite:
MATH 230A or
STATS 310.
Terms: Aut
| Units: 3
Instructors:
Dembo, A. (PI)
MATH 236: Introduction to Stochastic Differential Equations
Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Prerequisite: 136 or equivalent and differential equations.
Terms: Win
| Units: 3
Instructors:
Papanicolaou, G. (PI)
MATH 238: Mathematical Finance (STATS 250)
Stochastic models of financial markets. Forward and futures contracts. European options and equivalent martingale measures. Hedging strategies and management of risk. Term structure models and interest rate derivatives. Optimal stopping and American options. Corequisites:
MATH 236 and 227 or equivalent.
Terms: Win
| Units: 3
Instructors:
Papanicolaou, G. (PI)
MATH 239: Computation and Simulation in Finance
Monte Carlo, finite difference, tree, and transform methods for the numerical solution of partial differential equations in finance. Emphasis is on derivative security pricing. Prerequisite: 238 or equivalent.
Terms: Spr
| Units: 3
Instructors:
Toussaint, A. (PI)
MATH 240: Topics in Financial Mathematics: Fixed Income Models
Introduction to continuous time models for arbitrage-free pricing of interest rate derivatives. Bonds, yields, and the construction of yield curves. Caps, floors, swaps, swaptions, and bond options. Short rate models. Yield curve models. Forward measures. Forward and futures. LIBOR and swap market models. Prerequisite:
MATH 238.
Terms: Spr
| Units: 3
Instructors:
Toussaint, A. (PI)
MATH 248: Algebraic Number Theory
Harmonic analysis on number fields; Tate's Thesis; locally compact groups; adeles; Hecke Grossencharakters. Applications may include Sato-Tate for CM elliptic curves, nonvanishing theorems, and extensions to larger groups. Prerequisite: basic knowledge of number theory and p-adic fields. May be repeated for credit.
Terms: Aut
| Units: 3
| Repeatable
for credit
Instructors:
Trotabas, D. (PI)
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