Quantum stochastic differential inclusions of hypermaximal monotone type

Abstract

In continuation of our study of the existence of solutions of quantum stochastic differential inclusions, we first introduce and develop some aspects of the theory of maximal [resp. hypermaximal] monotone multifunctions, including the description of a number of properties of their resolvents and Yosida approximations, in the present noncommutative setting. Then, it is proved that, under a certain continuity assumption, a quantum stochastic differential inclusion of hypermaximal monotone type has a unique adapted solution which is obtained as the limit of the unique adapted solutions of a one-parameter family of Lipschitzian quantum stochastic differential equations. As examples, we show that a large class of quantum stochastic differential inclusions which satisfy the assumptions and conclusion of our main result arises as perturbations of certain quantum stochastic differential equations by some multivalued stochastic processes.