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arXiv:math/0602216

Quantum stochatic integrals and Doob-Meyer decomposition

Andrzej Łuczak

Published Feb 10, 2006• 1 citation
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Abstract

We show that for a quantum $L^p$-martingale $(X(t))$, $p>2$, there exists a Doob-Meyer decomposition of the submartingale $(|X(t)|^2)$. A noncommutative counterpart of a classical process continuous with probability one is introduced, and a quantum stochastic integral of such a process with respect to an $L^p$-martingale, $p>2$, is constructed. Using this construction, the uniqueness of the Doob-Meyer decomposition for a quantum martingale `continuous with probability one' is proved, and explicit forms of this decomposition and the quadratic variation process for such a martingale are obtained.

DOI
10.48550/arxiv.math/0602216

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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.
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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 classical stochastic differential equation given as dX(t,ω)=a(t,X) dt+b(t,X) dQ(t),X(t0)=X0,t∈[t0,T],dX(t, \omega) = a(t, X) \, dt + b(t, X) \, dQ(t), \quad X(t_0) = X_0, \quad t \in [t_0, T],dX(t,ω)=a(t,X)dt+b(t,X)dQ(t),X(t0​)=X0​,t∈[t0​,T], where the driving process Q(t) Q(t) Q(t) is a martingale, and a a a and b b b are adapted functionals has several numerical schemes developed for it, where each exhibits distinct stability and convergence properties depending on the nature of the driving process and the functional space of the solution. A generalization of this equation is the quantum stochastic differential equation introduced by Hudson and Parthasarathy, expressed as dX(t)=E(t,X(t)) dΛπ(t)+F(t,X(t)) dAf†(t)+G(t,X(t)) dAg(t)+H(t,X(t)) dt,dX(t) = E(t, X(t)) \, d\Lambda_\pi(t) + F(t, X(t)) \, dA_f^\dagger(t) + G(t, X(t)) \, dA_g(t) + H(t, X(t)) \, dt,dX(t)=E(t,X(t))dΛπ​(t)+F(t,X(t))dAf†​(t)+G(t,X(t))dAg​(t)+H(t,X(t))dt, where E,F,G, E, F, G, E,F,G, and H H H are stochastic operator-valued functions defined on a Hilbert space. In 1992, Ekhaquere reformulated this equation in an equivalent form involving the non-classical differential equation stated as; ddt⟨η,X(t)ξ⟩=P(t,X(t))⟨η,ξ⟩,X(t0)=X0,t∈[t0,T],\frac{d}{dt} \langle \eta, X(t) \xi \rangle = P(t, X(t)) \langle \eta, \xi \rangle, \quad X(t_0) = X_0, \quad t \in [t_0, T],dtd​⟨η,X(t)ξ⟩=P(t,X(t))⟨η,ξ⟩,X(t0​)=X0​,t∈[t0​,T], which represents an ordinary differential equation of non-classical type. The stochastic process X(t) X(t) X(t) acts as a densely defined linear operator on a tensor product of two Hilbert spaces, one of which is the Boson Fock space. The vectors η \eta η and ξ \xi ξ belong to a dense subset of this tensor product Hilbert space, and the mapping (η,ξ)↦P(t,X)(η,ξ)(\eta, \xi) \mapsto P(t, X)(\eta, \xi)(η,ξ)↦P(t,X)(η,ξ) defines a sesquilinear form for each fixed pair (t,X)(t, X)(t,X). We analyze the numerical approximation of the QSDE using its equivalent inner-product form. The numerical investigation employs three principal methods: the Lagrangian Quadrature Method, the Linear Multistep Scheme, and the Newton–Cotes Method. Each method is carefully designed to ensure the consistency, stability, and convergence of the approximate solution X(t)X(t)X(t) toward the exact stochastic process. These schemes are implemented using Python programming language, which provides a flexible and efficient computational environment for testing the proposed algorithms. Our computational results show that the developed schemes generalize Taylor-type approximations of Itô processes under weak convergence criteria. The proposed algorithms exhibit numerical stability and yield results that compare favorably with analytical benchmarks. Furthermore, the implementation avoids direct approximation of the driving quantum processes, thereby preserving the natural adaptation of the schemes to quantum dynamical models.