Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.
Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.
In Section 21 we have already seen how the classical stochastic processes with independent increments can be realised as suitable linear combinations of the creation, conservation and annihilation operators in the boson Fock space Γ s (ℋ) over a Hilbert space ℋ. This includes, in particular, the Brownian motion and Poisson process. Since a well-developed theory of stochastic integration with respect to these classical processes exists, it is natural to wonder whether it would be possible to introduce quantum probabilistic analogues of stochastic integrals with respect to processes defined by the creation, conservation and annihilation operators. To this end one needs the notions of filtration and adapted processes. This, in turn, depends on the notion of time. We adopt the view that time is an ℝ+-valued observable in a Hilbert space ℋ which, as a spectral measure, has no jumps. The absence of jumps is assumed more as a matter of mathematical convenience than a philosophical or conceptual necessity. We shall now examine how such a notion of time leads to a filtration and the definition of adapted processes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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].
Classical stochastic integration is based upon a probability space involving a filtration of sigma‐algebras. This construction lends itself to non‐commutative quantum analogues based for example, on a Hilbert space, a filtration of von Neumann algebras and gage. We recall a non‐commutative construction for the two parameter case, these being integrals in the plane, resulting in type one and type two stochastic integrals which are orthogonal, centred L2 — martingales, obeying isometry properties and develop the construction to obtain an Ito‐Clifford Wong‐Zakai martingale representation.
Martingales are fundamental stochastic process used to model the concept of fair game. They have a multitude of applications in the real world that include, random walks, Brownian motion, gamblers fortunes and survival analysis, Just as commutative integration theory may be realised as a special case of the more general non-commutative theory for integrals, so too, we find classical probability may be realised as a limiting, special case of quantum probability theory. In this thesis we are concerned with the development of multiparameter quantum stochastic integrals extending non-commutative constructions to the general n parameter case, these being multiparameter quantum stochastic integrals over the positive n - dimensional plane, employing martingales as integrator. The thesis extends previous analogues of type one, and type two stochastic integrals, for both Clifford and quasi free representations. As with one and two dimensional parameter sets, the stochastic integrals constructed form orthogonal, centred L2 - martingales, obeying isometry properties. We further explore analogues for weakly adapted processes, properties relating to the resulting quantum stochastic integrals, develop analogues to Fubini’s theorem, and explore applications for quantum stochastic integrals in a security setting.
On the other hand, based on the white noise theory [8, 22, 30] introduced by Hida, Kuo & Russek [23] studied anticipating (classical) stochastic integrals by applying the quantum decomposition of a Brownian motion.In this paper, we study some regularity properties of the quantum Hitsuda-Skorohod integrals as anticipating quantum stochastic integrals.Also, motivated by the results in [23], we discuss new types of anticipating quantum stochastic integrals in terms of pointwisely defined quantum white noise derivatives.2 Admissible Generalized Operators Admissible Rigging of Fock SpaceWe now review a construction of admissible rigging of Fock space which provides the basic structure of this paper.Let H=L^{2}(\mathbb{R}_{+}, dt) be the Hilbert space of complex valued Anticipating Quantum Stochastic Integrals for BasicQuantum Martingales
The basic integrator processes of quantum stochastic calculus, namely, creation, conservation, and annihilation, are introduced in the Hilbert space of square integrable Brownian functionals. Stochastic integrals with respect to these processes and a quantum Itô’s formula are described. As an application two examples of quantum stochastic differential equations are discussed. A continuous time version of Stinespring’s theorem on completely positive maps in C *-algebras is exploited to formulate the notion of a quantum Markov process and indicate how classical Markov processes are woven into the fabric of the Schodinger-Heisenberg dynamics of quantum theory.