Stochastic Analysis and Applications

LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED KURZWEIL EQUATIONS

E. O. Ayoola

Published Jun 30, 2001• 10 citations

Abstract

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.

DOI: 10.1081/sap-100002103

Related papers

On computational procedures for weak solutions of quantum stochastic differential equations

E. O. Ayoola

A continuous time Euler approximation scheme and a computational theorem for weak solutions of Lipschitzian quantum stochastic differential equations (QSDEs) are established. The work is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to the equivalent forms of the equations satisfying the Caratheodory conditions. Our results generalize analogous results concerning classical Ito stochastic differential equations and those of differential equations in a Banach space context to the noncommutative quantum setting involving unbounded linear operators on a Hilbert space. Numerical examples are given.

Stochastic Analysis and ApplicationsMar 21, 20021 citation

EXISTENCE, UNIQUENESS AND STABILITY OF A MILD SOLUTION OF LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS

Sheila Amina Bishop, MICHAEL CHIKODI AGARANA, O.O. Agboola +2 more

We introduce the concept of a mild solution of Lipschitzian quantum stochastic differential equations (QSDEs). Results on the existence, uniqueness and stability of a mild solution of QSDEs are established. This is accomplished within the framework of the Hudson-Parthasarathy formulation of quantum stochastic calculus. Here, the results on a mild solution are weaker compared with the ones in the literature.

Dec 1, 20141 citation

Viable Solutions of Lipschitzian Quantum Stochastic Differential Equations

Akinwumi Titilay, Ayoola Ezekiel

We establish the existence of viable solutions of quantum stochastic differential equations (QSDE) within the framework of Hudson and Parthasarathy formulation of quantum stochastic calculus.The main results are established for QSDE whose coefficients are Lipschitzian and quasicompact.This work extends the Nagumo viability results for classical differential equations on finite dimensional Euclidean spaces.Viable solutions of QSDE in the present formulation take values in some closed subsets of the infinite dimensional locally convex spaces of non-commutative stochastic processes.

Feb 27, 20210 citations

Existence and Uniqueness of Solution for the Kurzweil Equation Associated with Quantum Stochastic Differential Equation

Sheila Amina Bishop

We study the existence and uniqueness of solution of the Kurzweil equations ~ with the Non-Lipschitz quantum stochastic differential equation. This is accomplished within the framework of the Hudson and Parthasarathy formulations of quantum stochastic calculus and the Schwabic generalized ordinary differential equations.

Jan 1, 20110 citations

Converging multistep schemes for weak solutions of quantum stochastic differential equations

E. O. Ayoola

Multistep schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with the basic field operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to matrix element of solution being sufficiently differentiable. Results concerning convergence of explicit schemes of class A in the topology of the locally convex space of solution are presented.Numerical examples are given.

Stochastic Analysis and ApplicationsJan 1, 20008 citations

Discontinuous Quantum Stochastic Differential Equations and The Associated Kurzweil Equations

Sheila Amina Bishop

Quantum stochastic differential equations (QSDEs) of systems that exhibit discontinuity are introduced with the Kurzweil equations associated with this class of equations. The formulations are simple extensions of the methods applied by Schwabik [10] to ODEs to this present noncommutative quantum setting. Here the solutions of a QSDE are discontinuous functions of bounded variation that is they have the same properties as the Kurzweil equations associated with QSDEs introduced in [1].

Aug 1, 20121 citation