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Graduate Mathematical Physics · 8 Topics
Quantum Stochastic Calculus
The noncommutative extension of Itô calculus, built on Boson Fock space and the Hudson–Parthasarathy quantum Itô formula — with applications from open quantum systems to quantum models of financial markets.
1
From Classical to Quantum Probability
Why commuting random variables on a sample space are not enough, and how observables on a Hilbert space generalize classical probability
2
Bosonic Fock Space
The exponential (symmetric) Fock space Γ(L²) that serves as the noise reservoir for quantum stochastic processes
3
Creation, Annihilation & Conservation Operators
The three fundamental noise operators on Fock space and the canonical commutation relations that drive quantum noise
4
Quantum Stochastic Integrals
Defining adapted integrals against creation, annihilation, and conservation processes on exponential vectors
5
The Quantum Itô Formula
The Hudson–Parthasarathy noncommutative product rule and Itô correction table for quantum stochastic differentials
6
Quantum Stochastic Differential Equations
Unitary quantum stochastic differential equations and how they dilate the dynamics of an open quantum system
7
Quantum Langevin Equations & Open Systems
Connecting Hudson–Parthasarathy dilations to quantum Langevin equations and Lindblad master equations for dissipative systems
8
Applications to Mathematical Finance
The Accardi–Boukas quantum Black–Scholes framework, modeling markets as quantum observables on Fock space
Full lessons and exercises for this course are coming soon.
References & Further Reading
K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus
Monographs in Mathematics, Vol. 85, Birkhäuser (1992) — the standard graduate reference for Fock space quantum stochastic calculus, from Itô’s correction formulae to the Hudson–Parthasarathy quantum Itô formula.
R. L. Hudson & K. R. Parthasarathy, Quantum Itô’s formula and stochastic evolutions
Communications in Mathematical Physics, 93 (1984) — the original paper constructing quantum stochastic calculus on Boson Fock space.
L. Accardi & A. Boukas, The Quantum Black–Scholes Equation
Applies Hudson–Parthasarathy quantum stochastic calculus to derivative pricing, modeling the market as a quantum observable on Fock space (arXiv:0706.1300).
Quantum Logic and Probability Theory
Stanford Encyclopedia of Philosophy — an accessible entry point into the noncommutative probability framework that quantum stochastic calculus builds on.