MATH 605, Fall 2019.
Topics in Probability (39788R)
Numerical Methods in Stochastic Analysis
Class meetings: MW 4:30-5:50pm, KAP 245
Information on this and linked pages changes frequently.
Instructor: Sergey Lototsky.
Office: KAP 248D.
Phone: 213 740 2389
E-mail: lototsky usc edu.
Office Hours: MWF 11-11:50am in KAP 248D
Walk-ins and appointments at other time are welcome.
Course objective To learn the main problems, tools, and algorithms related to Monte Carlo and similar methods.
Course work Class participation, homework assignments, final presentation.
Official grading scheme 20% class participation, 40% homework assignments, 40% final presentation.
Main reference Stochastic Simulation: Algorithms and Analysis by S. Asmussen and P. W. Glynn, published by Springer in the series “Stochastic Modeling and Applied Probability”, vol. 57, 2007.
Class notes (my version, frequently updated)
I expect you to (1) do the problems, (2) think about the questions, (3) have a one-sentence description of the each of the key ideas and “other points”, (4) remember some of the “random bits and pieces”.
Our Progress
August 26 Generation of Uniform(0,1) random variables; discrete distributions from Uniform(0,1).
August 28 Randomness and related ideas.
September 2 Labor Day, no class.
September 4 Generation of continuous random variables; infinite divisibility, stability, and related notions.
September 9 Variance reduction.
September 11 “Random” bits and pieces related to generation of random objects.
September 16 Markov chains, regeneration identity, and the Propp-Wilson algorithm.
September 18 Rare events and importance sampling.
September 23 Siegmund’s algorithm.
September 25 Standard Brownian motion.
September 30 SODEs: Existence and uniqueness of solution
October 2 SODEs: Euler-Maryama and Milstein schemes
October 7 Stochastic Approximation: Robbins-Monro et al.
October 9 Stochastic Approximation: estimation with heavy-tailed noise
October 14 MCMC: Metropolis-Hastings
October 16 MCMC: Langevin and beyond
October 21 Orthogonal polynomials: an overview
October 23 Polynomial Chaos: Gaussian bi-linear case
October 28 Polynomial Chaos: Non-linear case
October 30 Orthogonal polynomials: Levy-Sheffer and Levy-Meixner systems
Novermber 4 and 6 Select topics in asymptotic analysis
Student presentations begin.
November 11 John on extensions and analysis of the Metropolis-Hastings algorithm
November 13 Maria on statistical approach to inverse problems for partial differntial equations
November 18 Austin on optimal control approach to the Kelly criterion
November 20 Apoorva on importance sampling and the random Lorenz system
November 25 Ujan on non-centeral limit theorems for martingales
November 27 Thanksgiving Break, no class
December 2 Anna on generating random logic circuits on a quantum computer
December 4 Lernik on non-parametric inference via optimiziation in the space of measures
Supplemental material
About Buffon’s needle and some further reading on Lazzarini’s experiment and on a higer-dimensional application in biology
The review of the “Stochastic Simulation” book from MathSciNet
USC Math Department Homepage