LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED KURZWEIL EQUATIONS

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.

Keywords:

• QSDE
• Fock spaces
• Exponential vectors
• Kurzweil equations and integrals
• Noncommutative stochastic processes
• AMS Subject Classification (1991): 60H10, 60H20, 65L05, 81S25

ACKNOWLEDGMENTS

I am grateful to Professor G.O.S.Ekhaguere for useful discussions and suggestions in the course of this work. This paper is a generalization of similar results reported in my thesis [2] to the case of QSDES that satisfy the Caratheodory conditions. I also thank The National Mathematical Centre, Abuja, Nigeria, for numerous supports and hospitality at the centre on several occassions.