AN INTRODUCTION TO BROWNIAN MOTION AND STOCHASTIC CALCULUS

MATH 531

A rigorous introduction to probability theory at an advanced undergraduate level. Only a minimal amount of measure theory is used, in particular, the theory of Lebesgue integrals is not needed. It is aimed at math majors and Master's degree students, or students in other fields who will need probability in their future careers. Gives an introduction to the basics (Kolmogorov axioms, conditional probability and independence, random variables, expectation) and discusses some classical results with proofs (DeMoivre-Laplace limit theorems, the study of simple random walk on the one dimensional lattice, applications of generating functions).

MATH 735

Foundations of continuous time stochastic processes, semimartingales and the semimartingale integral, Ito's formula, stochastic differential equations, stochastic equations for Markov processes, application in finance, filtering, and control. The course relies on measure theoretic probability theory that can be reviewed at the beginning of the semester.

FINANCE 830

Introduces continuous-time financial models essential for the advanced analysis of derivative securities. Discuss the fundamental mathematical concepts and tools from continuous-time stochastic processes including Brownian motion, Poisson processes, stochastic calculus, and change of measure. This provides a framework for analyzing derivative securities including their pricing, hedging, and risk management. In particular, covers the Black-Scholes and stochastic volatility models for equity options; basic term-structure modeling for interest rate derivatives; and reduced-form credit-risk models. Emphasis is put on applications and economic interpretation rather than mathematical rigor.

MATH/STAT 734

Possible topics include martingales, weak convergence of measures, introduction to Brownian motion.

MATH 717

Introduction to computational methods that use stochastic algorithms and/or methods that are applied to random or stochastic mathematical problems. The main emphasis will be placed on learning practical tools, while some aspects of theoretical foundations will also be covered (e.g., basic error analysis for numerical solution of stochastic differential equations (SDEs), and basic convergence of Monte Carlo methods). Topics include Monte Carlo methods, Bayesian inference and Bayesian sampling, simulation of Markov chains, numerical analysis for SDEs, data assimilation / state estimation, stochastic optimization methods and random sketching. Applications to science, engineering, finance, data science, and other practical problems also included.