Set-valued Stochastic Equations Research Articles

Stochastic inclusions and set-valued stochastic equations with mixed integrals in the plane

The paper refers to the study of properties of solutions to two-parameter stochastic inclusions and set-valued stochastic equations with set-valued mixed integrals driven by finite variation processes and martingales. We present new types of such inclusions and equations that generalize those studied earlier. Apart from existence results to such inclusions and equations also topological properties of their solutions are studied. Additionally some connections between their solutions are established. The results obtained in the paper present a set-valued counterpart dealing with this topic known both in deterministic and stochastic cases.

Properties of set-valued integrals and set-valued stochastic equations driven by two-parameter martingales

In the paper we study properties of Aumann's stochastic integral driven by a two-parameter increasing process and set-valued Itô's integral with respect to the two-parameter martingale. Both types of integrals are understood as set-valued processes. Next, the existence, uniqueness and convergence properties of solutions to set-valued stochastic integral equations with respect to such integrators are investigated. We present new types of such equations that generalize those studied earlier. The results obtained in the paper present a set-valued counterpart dealing with this topic known both in single-valued deterministic and stochastic cases.

Stochastic inclusions and set-valued stochastic equations driven by a two-parameter Wiener process

In the paper we study properties of solutions to stochastic differential inclusions and set-valued stochastic differential equations driven by a two-parameter Wiener process. We establish new connections between their solutions. We prove that attainable sets of solutions to such inclusions are subsets of values of multivalued solutions of associated set-valued stochastic equations. Next we show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set-valued stochastic equation. Additionally we establish other properties of such solutions. The results obtained in the paper extends results dealing with this topic known both in deterministic and stochastic cases.

On properties of set-valued integrals driven by martingales and set-valued stochastic equations

On properties of set-valued integrals driven by martingales and set-valued stochastic equations

Integrably bounded set-valued stochastic integrals

The paper is devoted to properties of Aumann and Itô set-valued stochastic integrals, defined as some set-valued random variables. In particular the problem of integrable boundedness of the generalized Itô set-valued stochastic integrals is considered. Unfortunately, Itô set-valued stochastic integrals, defined by E.J. Jung and J.H. Kim in the paper [5], are not in general integrably bounded (see [8,15]). Therefore, in the present paper we consider generalized Itô set-valued stochastic integrals (see [10,11]) defined for absolutely summable and countable subsets of the space IL2(IR+×Ω,ΣIF,IRd×m) of all square integrable IF-nonanticipative matrix-valued stochastic processes. Such integrals are integrably bounded and possess properties needed in the theory of set-valued stochastic equations.

Set-Valued Stochastic Equation with Set-Valued Square Integrable Martingale

In this paper, we shall introduce the stochastic integral of a stochastic process with respect to set-valued square integrable martingale. Then we shall give the Aumann integral measurable theorem, and give the set-valued stochastic Lebesgue integral and set-valued square integrable martingale integral equation. The existence and uniqueness of solution to set-valued stochastic integral equation are proved. The discussion will be useful in optimal control and mathematical finance in psychological factors.

On connections between stochastic differential inclusions and set-valued stochastic differential equations driven by semimartingales

In the paper we study properties of solutions to stochastic differential inclusions and set-valued stochastic differential equations with respect to semimartingale integrators. We present new connections between their solutions. In particular, we show that attainable sets of solutions to stochastic inclusions are subsets of values of multivalued solutions of certain set-valued stochastic equations. We also show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set-valued stochastic equation. The results obtained in the paper generalize results dealing with this topic known both in deterministic and stochastic cases.

Viable solutions of set-valued stochastic equation

The existence of viable solutions to set-valued Itô equation, i.e., solutions remaining at any time in a ilxed sublet of a slate space, is established for a functional inclusion with dissipative set-valued operators which need not he Lipsehitz continuous nor have to satisfy the Pardoux “monotone” property

Continuity properties of solutions of multivalued equations with white noise perturbation

In the paper, we consider a set‐valued stochastic equation with stochastic perturbation in a Banach space. We prove first the existence theorem and then study continuity properties of solutions.