ECE 586: Lecture Schedule

The schedule will be updated and revised as the course progresses. Reading from the lecture notes will be indicated on the left.

Warning: the lecture notes will be updated and revised throughout the semester!

Jan 17 - 19 [🎥 video]: Introduction
history, motivation through problems in physics, control, mathematical finance, optimization, and machine learning
Jan 24 - Jan 31 Chapter 2: Brownian motion and diffusion processes
definition, construction, and basic properties;
transition densities;
connection to second-order linear differential operators;
forward Kolmogorov (Fokker-Planck) and backward Kolmogorov equations;
the martingale problem (Stroock-Varadhan)
Feb 2 - Feb 16 Chapter 3: Stochastic integrals and differential equations
basic construction;
the Itô isometry;
the (local) martingale property;
stochastic differentials and Itô’s rule;
SDEs with explicit solutions;
the chain rule and the Stratonovich integral;
SDEs as models of physical and engineering systems with stochastic disturbances;
the Wong-Zakai theorem
Feb 21: Change of measure (Cameron-Martin-Girsanov theory)
motivation via importance sampling;
absolute continuity and change of drift;
martingale proof of Girsanov’s theorem
Feb 23: No class: CSL Student Conference
Feb 28: The Feynman-Kac formula
statement, intuition, and proof sketch;
links between diffusion processes and solutions of partial differential equations;
connection to change of measure, importance sampling and stochastic simulation

Mar 2 - Mar 9: Stochastic control
filtering, both linear (Kalman-Bucy) and nonlinear;
introduction to controlled diffusions and optimal control;
the Hamilton-Jacobi-Bellman equation and martingales;
the LQG problem and the separation principle;
optimal control via Girsanov change of measure (following Beneš);
the Schrödinger bridge problem, its control formulation, and solution via Fleming’s logarithmic transformation
Mar 14 - 16: No class: spring break
Mar 21 - Apr 6: Optimization
gradient flows;
the Langevin dynamics and variants;
simulated annealing in continuous time;
stochastic neural nets;
variational free energy minimization, importance sampling, and interpretation through optimal control
Apr 11 - Apr 27 (no class on Apr 13): Machine learning
stochastic gradient Langevin dynamics;
sampling and score-based diffusion models;
duality between inference and sampling via time reversal;
the variational formulation and construction of the solution via optimal control;
connections to Schrödinger bridges and stochastic thermodynamics
May 2: Review and Q&A