November 27, 2023

Stochastic calculus is a fundamental mathematical framework that describes the evolution of noisy dynamic systems. In the classical world, stochastic differential equations are used for modeling non-Lagrangian systems and have applications ranging from physical system modeling to finance. In the quantum world, quantum stochastic differential equations are a fundamental tool in the theory of (noisy) open quantum systems that cannot be modeled with a simple Schrödinger/Heisenberg equation.

This course will develop the rigorous theory of classical and quantum stochastic calculus. Both classical and quantum theory will be developed together, and the emphasis will be on viewing the quantum theory as a generalization of well-understood concepts in classical probability theory.

Topics will include:

  1. Classical and quantum probability spaces, random variables, stochastic processes, filrations, adapted and predictable stochastic processes. For quantum probability spaces, we will also develop the theory in infinite-dimensional but separable Hilbert spaces.
  2. Classical Wiener processes and Wiener integrals, Ito integral, stochastic differential equations (existence of solutions, uniqueness, and computational aspects), and connections to Langevin and Fokker Planck equations.
  3. Fock spaces, quantum wiener process, and wiener integrals, quantum Ito integrals, and stochastic differential equations. Derivation of the Lindblad master equation and its dilation, GKSL theorem.

Course reading:

  1. Parthasarthy and Hudson, Introduction to quantum Stochastic Calculus.
  2. Ramon Von Handel, Stochastic calculus, Filtering and Stochastic control (Chapters 1 – 5ti.


The focus on this course will be on mathematical rigor and not just practical calculations. While this course will require basic concepts from analysis, measure-theoretic probability, and some basic quantum theory, all the required definitions and ideas will be introduced in class. However, exposure to graduate-level probability/stochastic processes courses and theoretical quantum physics (even if not done very mathematically rigorously) would be beneficial for connecting this class to practical applications.


Tuesday/Thursday, 15:00 – 17:00, ECE 031, credits: 4.

The evaluation will be entirely on class participation, scribing, and in-class exercises.

View course details in MyPlan: E E 559