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Set-valued random differential equations in Banach space

Mariusz Michta — 1995

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We consider the problem of the existence of solutions of the random set-valued equation: (I) D H X t = F ( t , X t ) P . 1 , t ∈ [0,T] -a.e.; X₀ = U p.1 where F and U are given random set-valued mappings with values in the space K c ( E ) , of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.

Weak solutions of stochastic differential inclusions and their compactness

Mariusz Michta — 2009

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, we consider weak solutions to stochastic inclusions driven by a semimartingale and a martingale problem formulated for such inclusions. Using this we analyze compactness of the set of solutions. The paper extends some earlier results known for stochastic differential inclusions driven by a diffusion process.

Optimal solutions to stochastic differential inclusions

Mariusz Michta — 2002

Applicationes Mathematicae

A martingale problem approach is used first to analyze compactness and continuous dependence of the solution set to stochastic differential inclusions of Ito type with convex integrands on the initial distributions. Next the problem of existence of optimal weak solutions to such inclusions and their dependence on the initial distributions is investigated.

On risk reserve under distribution constraints

Mariusz Michta — 2000

Discussiones Mathematicae Probability and Statistics

The purpose of this work is a study of the following insurance reserve model: R ( t ) = η + 0 t p ( s , R ( s ) ) d s + 0 t σ ( s , R ( s ) ) d W s - Z ( t ) , t ∈ [0,T], P(η ≥ c) ≥ 1-ϵ, ϵ ≥ 0. Under viability-type assumptions on a pair (p,σ) the estimation γ with the property: i n f 0 t T P R ( t ) c γ is considered.

Stochastic fuzzy differential equations with an application

Marek T. MalinowskiMariusz Michta — 2011

Kybernetika

In this paper we present the existence and uniqueness of solutions to the stochastic fuzzy differential equations driven by Brownian motion. The continuous dependence on initial condition and stability properties are also established. As an example of application we use some stochastic fuzzy differential equation in a model of population dynamics.

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