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

Sheila Amina Bishop, MICHAEL CHIKODI AGARANA, O.O. Agboola, Jimevwo Godwin Oghonyon, T. A. Anake

Published Dec 1, 2014• 1 citation
OpenAlex

Abstract

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.

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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.
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.
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.
We introduce the concept of a mild solution for the right Hudson-Parthasarathy quantum stochastic differential equation, prove existence and uniqueness results, and show the correspondence between our definition and similar ideas in the theory of classical stochastic differential equations. The conditions that a process must satisfy in order for it to be a mild solution are shown to be strictly weaker than those for it to be a strong solution by exhibiting a class of coefficient matrices for which a mild unitary solution can be found, but for which no strong solution exists.
We establish further results concerning the existence, uniqueness and stability of weak solutions of quantum stochastic differential equations (QSDEs). Our results are achieved by considering a more general Lipschit condition on the coefficients than our previous considerations in [1]. We exhibit a class of Lipschitzian QSDEs in the formulation of this paper, whose coefficients are only continuous on the locally convex space of the weak solution.Journal of the Nigerian Association of Mathematical Physics Vol. 8 2004: pp. 5-8
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.