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**