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Journal of the Nigerian Mathematical Society

Caratheodory Solution Of Quantum Stochastic Differential Inclusions

M. O. Ogundiran, E. O. Ayoola

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

This work is concerned with the existence of solution of Quantum stochastic differential inclusions in the sense of Caratheodory. The multivalued stochastic process involved which is non-convex is Scorza-Dragoni lower semicontinuous (SD-l.s.c.) hence giving rise to a directionally continuous selection. The Quantum stochastic differential inclusion is driven by annihilation, creation and gauge operators.

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Lower Semicontinuous Quantum Stochastic Differential Inclusions

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Abstract. In this paper we established the existence of solutions of Lower semicontinuous quantum stochastic differential inclusions(QSDI). The existence of a continuous selection of a predefined integral operator was established. This selection which is an adapted stochastic process is a solution of the Lower semicontinuous quantum stochastic differential inclusions.

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ABSTRACT. We prove that the map that associates to the initial value the set of solutions to the Lipschitzian Quantum Stochastic Differential Inclusion (QSDI) admits a selection which is continuous from the locally convex space of stochastic processes to the space of adapted and weakly absolutely continuous solutions. As a corollary, the reachable set multifunction admits a continuous selection. In the framework of the Hudson-Parthasarathy formulation of quantum stochastic calculus, these results are achieved subject to some compactness conditions on the set of initial values and on some coefficients of the inclusion.

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Research Article Continuous Interpolation of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

E. O. Ayoola, John O. Adeyeye

Given any finite set of trajectories of a Lipschitzian quantum stochastic differential inclu-sion (QSDI), there exists a continuous selection from the complex-valued multifunction associated with the solution set of the inclusion, interpolating the matrix elements of the given trajectories. Furthermore, the difference of any two of such solutions is bounded in the seminorm of the locally convex space of solutions. Copyright © 2007 E. O. Ayoola and J. O. Adeyeye. This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.

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LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED KURZWEIL EQUATIONS

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Kurzweil or generalized differential equations associated with Lipschitzian quantum stochastic differential equations (QSDEs) are introduced and studied. This is accomplished within the framework of the Hudson-Parthasarathy formulations of quantum stochastic calculus. Results concerning the equivalence of these classes of equations satisfying the Caratheodory conditions are presented. It is further shown that the associated Kurzweil equation may be used to obtain a reasonably high accurate approximate solutions of the QSDEs. This generalize analogous results for classical initial value problems to the noncommutative quantum setting involving unbounded linear operators on a Hilbert space. Numerical examples are given.

Stochastic Analysis and Applications•Jun 30, 2001•10 citations
This work is concerned with the existence of solution of Quantum stochastic differential inclusions in the sense of Caratheodory. The multivalued stochastic process involved which is non-convex is Scorza-Dragoni lower semicontinuous (SD-l.s.c.) hence giving rise to a directionally continuous selection. The Quantum stochastic differential inclusion is driven by annihilation, creation and gauge operators.
We prove that the map that associates to the initial value the set of solutions to the Lipschitzian Quantum Stochastic Differential Inclusion (QSDI) admits a selection continuous from the locally convex space of stochastic processes to the adapted and weakly absolutely continuous space of solutions. As a corollary, we show that the reachable sets admit some continuous selections. In the framework of the Hudson - Parthasarathy formulations of quantum stochastic calculus, our results are achieved subject to some compactness conditions on the set of initial values and on some coefficients of the inclusion. The results here compliment similar results in our previous work in [3] where continuous selections defined on the set of the matrix elements of initial values were established. JONAMP Vol. 11 2007: pp. 71-82
ABSTRACT. We prove that the map that associates to the initial value the set of solutions to the Lipschitzian Quantum Stochastic Differential Inclusion (QSDI) admits a selection which is continuous from the locally convex space of stochastic processes to the space of adapted and weakly absolutely continuous solutions. As a corollary, the reachable set multifunction admits a continuous selection. In the framework of the Hudson-Parthasarathy formulation of quantum stochastic calculus, these results are achieved subject to some compactness conditions on the set of initial values and on some coefficients of the inclusion.
Given any finite set of trajectories of a Lipschitzian quantum stochastic differential inclu-sion (QSDI), there exists a continuous selection from the complex-valued multifunction associated with the solution set of the inclusion, interpolating the matrix elements of the given trajectories. Furthermore, the difference of any two of such solutions is bounded in the seminorm of the locally convex space of solutions. Copyright © 2007 E. O. Ayoola and J. O. Adeyeye. This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
Kurzweil or generalized differential equations associated with Lipschitzian quantum stochastic differential equations (QSDEs) are introduced and studied. This is accomplished within the framework of the Hudson-Parthasarathy formulations of quantum stochastic calculus. Results concerning the equivalence of these classes of equations satisfying the Caratheodory conditions are presented. It is further shown that the associated Kurzweil equation may be used to obtain a reasonably high accurate approximate solutions of the QSDEs. This generalize analogous results for classical initial value problems to the noncommutative quantum setting involving unbounded linear operators on a Hilbert space. Numerical examples are given.