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On Unique Solution of Quantum Stochastic Differential Inclusions

Sheila Amina Bishop, MICHAEL CHIKODI AGARANA, Hilary Izuchukwu Okagbue, Jimevwo Godwin Oghonyon

Published Oct 25, 2017• 0 citations
OpenAlex

Abstract

We investigate the existence of solutions of quantum stochastic differential inclusion (QSDI) with some uniqueness properties as a variant of the results in the literature. We impose some weaker conditions on the coefficients and show that under these conditions, a unique solution can be obtained provided the functions are measurable such that their integral is finite.

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We investigate the existence of solutions of quantum stochastic differential inclusion (QSDI) with some uniqueness properties as a variant of the results in the literature. We impose some weaker conditions on the coefficients and show that under these conditions, a unique solution can be obtained provided the functions are measurable such that their integral is finite.
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.
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