Mariusz Michta

On set-valued stochastic integrals and fuzzy stochastic equations

In this paper we extend the notion of set-valued and fuzzy stochastic integrals to semimartingale integrators. We present their main properties and finally we establish the existence of solutions of a fuzzy integral stochastic equation driven by a Brownian motion. The approach is based on the existence of solutions of an appropriately formulated martingale problem for a system of stochastic inclusions and the theorem of Negoita and Ralescu.

STOCHASTIC INCLUSIONS WITH MULTIVALUED INTEGRATORS

Set-valued semimartingales are introduced, as an extension of single-valued ones. For such multivalued processes, say X, we define a set-valued stochastic integral and study its selection properties. Finally, a stochastic inclusion, with mixed type of integrals: is considered.

On Weak Solutions to Stochastic Differential Inclusions Driven by Semimartingales

Abstract In this paper we consider weak solutions to stochastic inclusions driven by a general semimartingale. We prove the existence of weak solutions and equivalence with the existence of solutions to the martingale problem formulated to such inclusion. Using this we then analyze compactness property of solutions set. Presenting results extend some of those being known for stochastic differential inclusions of Itôs type.

The Method of Upper and Lower Solutions of Stochastic Differential Equations and Applications

Abstract The purpose of this article is to consider a stochastic integral equation driven by semimartingale with discontinuous and increasing drift part. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered topological space. Finally we present some applications in finance.

Set-Valued Stochastic Integrals and Equations with Respect to Two-Parameter Martingales

This article is concerned with notions of set-valued stochastic integrals driven by two-parameter martingales and increasing processes. We investigate their main properties and we consider next multivalued stochastic integral equations in the plane. We establish the existence and uniqueness of solutions to such equations as well as their additional properties.

Second Order Stochastic Inclusion

Abstract The purpose of the paper is to consider some stochastic control problems as a particular case of a more general theory, the stochastic inclusions theory. We discuss the existence of weak solutions to a stochastic inclusion of second order, driven by two general semimartingales. Finally we present some examples.

Properties of set-valued stochastic differential equations

The paper is devoted to properties of set-valued stochastic differential equations. The main result of the paper deals with existence and uniqueness of solutions. Furthermore, a connection between solutions of stochastic differential inclusions and solutions of set-valued stochastic differential equations are given. The result of the paper extends a lot of particular results dealing with such type equations.

Fuzzy Stochastic Integral Equations Driven by Martingales

Exploiting the properties of set-valued stochastic trajectory integrals we consider a notion of fuzzy stochastic Lebesgue–Stieltjes trajectory integral and a notion of fuzzy stochastic trajectory integral with respect to martingale. Then we use these integrals in a formulation of fuzzy stochastic integral equations. We investigate the existence and uniqueness of solution to such the equations.

The Upper and Lower Solutions Method for Stochastic Inclusions with Discontinuous Multivalued Mappings

Abstract In this article, we consider a stochastic integral inclusion driven by semimartingale with discontinuous multivalued right hand side. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered sets. The presented studies extend some recent results both for deterministic differential inclusions and stochastic differential equations for increasing operators.

Convex selections of multifunctions and their applications

The notion of upper separated set-valued functions which forms a necessary and sufficient condition for the existence of convex selections for convex-valued multifunctions is introduced. The results obtained in this article lead to a new class of the multifunctions admitting continuous selections and therefore they are applicable to the existence of solutions to differential and stochastic inclusions. †Dedicated to N.U. Ahmed on the occassion of his 70th birthday.