Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Communications on Stochastic Analysis

The early years of quantum stochastic calculus

R. L. Hudson

Published Mar 1, 2012• 5 citations
Publisher

Abstract

The origins and early history of quantum stochastic calculus are surveyed, with emphasis on the collaboration between K R Parthasarathy and the author.
DOI
10.31390/cosa.6.1.12

Related papers

Contributions of K. R. Parthasarathy (KRP) to Mathematics of Quantum theory - Perturbations of Operators and Quantum Stochastic Calculus

Kalyan B. Sinha

Indian Journal of Pure and Applied Mathematics•Jul 29, 2024•0 citations

Quantum Stochastic Calculus and Some Of Its Applications

Andreas Boukas

Abstract:-We describe the main features of the Hudson-Parthasarathy quantum stochastic calculus and some of its applications to systems control and, recently, to quantum economics.

•Jan 1, 2006•2 citations

A brief survey of quantum stochastic calculus

Kalyanapuram Rangachari Parthasarathy

Stochastic Processes and their Applications•Dec 1, 1985•0 citations

Generalized Quantum Stochastic Processes on Fock Space

Nobuaki Obata

As is highlighted in the excellent books by Meyer [21] and by Parthasarathy [26] quantum stochastic calculus on (Boson) Fock space has developed into a new field of mathematics keeping a profound contact with physical applications. Since Hudson and Parthasarathy [12] first formulated quantum stochastic integrals of Ito type in 1984 a crucial role has been played by three basic quantum stochastic processes:

Publications of the Research Institute for Mathematical Sciences•Aug 31, 1995•24 citations

Quantum Stochastic Calculus

Mou-Hsiung Chang

This chapter begins with introductions of symmetric Fock space Γ(ℍ) and symmetric Guichardet space Φ(ℍ) of a generic complex Hilbert space ℍ. It is shown that these 2 spaces are actually isomorphic and can and will be used interchangeably throughout the chapter. In particular, the simple and yet useful integral-summation formula (see Theorem (3.2.1)) for the Guichardet space is often employed to establish the essential results in quantum stochastic calculus. It has been shown that the symmetric Fock space (and hence the symmetric Guichadet space) provides a plausible mathematical tool for modeling phenomena in quantum optics or quantum electrodynamics. Many results in the quantum physics and quantum probability literature are actually established based on this concrete model space. When the generic complex Hilbert space ℍ = L 2 (ℝ + ; K) (where K is another complex Hilbert space), the class of exponential vectors along with 3 different types of quantum noise processes, namely, the creation, annihilation, and neutral (or guage) processes, can be introduced. It is shown that the subspace generated by the class of exponential vectors is dense in the Fock space. Therefore, it is convenient to verify the properties that hold for Fock space by verifying the same hold for the class of exponential vectors. Parallel to those of Itô integrals with respect to classical Brownian motion and/or Poisson process, the concepts of a quantum stochastic integral of a operator-valued process (as a member of the Fock space) with respect to each of the above mentioned quantum noise processes are constructed. The quantum stochastic calculus, parallel to those of classical Itô calculus, is developed. The theory of quantum stochastic calculus on the symmetric Fock space due originally to Hudson and Parthasarathy [HP84a] has been the topic of investigation by many researchers in the last 20 years. Much of he material presented in this chapter can be found in the books by Meyer [Mey95], Parthasarathy [Par92] and Attal [Att03].

Cambridge University Press eBooks•Feb 9, 2015•0 citations

The functional Ito formula in quantum stochastic calculus

G. O. S. Ekhaguere

Using an Op-*-algebraic approach, noncommutative analogs of the Ito formula of classical stochastic calculus within the framework of the Hudson–Parthasarathy formulation of Boson quantum stochastic calculus are proven.

Journal of Mathematical Physics•Dec 1, 1990•4 citations
This chapter begins with introductions of symmetric Fock space Γ(ℍ) and symmetric Guichardet space Φ(ℍ) of a generic complex Hilbert space ℍ. It is shown that these 2 spaces are actually isomorphic and can and will be used interchangeably throughout the chapter. In particular, the simple and yet useful integral-summation formula (see Theorem (3.2.1)) for the Guichardet space is often employed to establish the essential results in quantum stochastic calculus. It has been shown that the symmetric Fock space (and hence the symmetric Guichadet space) provides a plausible mathematical tool for modeling phenomena in quantum optics or quantum electrodynamics. Many results in the quantum physics and quantum probability literature are actually established based on this concrete model space. When the generic complex Hilbert space ℍ = L 2 (ℝ + ; K) (where K is another complex Hilbert space), the class of exponential vectors along with 3 different types of quantum noise processes, namely, the creation, annihilation, and neutral (or guage) processes, can be introduced. It is shown that the subspace generated by the class of exponential vectors is dense in the Fock space. Therefore, it is convenient to verify the properties that hold for Fock space by verifying the same hold for the class of exponential vectors. Parallel to those of Itô integrals with respect to classical Brownian motion and/or Poisson process, the concepts of a quantum stochastic integral of a operator-valued process (as a member of the Fock space) with respect to each of the above mentioned quantum noise processes are constructed. The quantum stochastic calculus, parallel to those of classical Itô calculus, is developed. The theory of quantum stochastic calculus on the symmetric Fock space due originally to Hudson and Parthasarathy [HP84a] has been the topic of investigation by many researchers in the last 20 years. Much of he material presented in this chapter can be found in the books by Meyer [Mey95], Parthasarathy [Par92] and Attal [Att03].