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Lower Semicontinuous Quantum Stochastic Differential Inclusions

M. O. Ogundiran, E. O. Ayoola

Published Mar 6, 2013• 1 citation
OpenAlex

Abstract

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. 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.
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 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