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Set-valued random differential equations in Banach space

100%

Michta M.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1995

tom 15

nr 2

191-200

We consider the problem of the existence of solutions of the random set-valued equation: (I) $D_HX_t = F(t,X_t)P.1$, 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.

Optimal solutions to stochastic differential inclusions

100%

Michta M.

Applicationes Mathematicae

2002

tom 29

nr 4

387-398

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.

Note on the selection properties of set-valued semimartingales

100%

Michta M.

nr 2

161-169

Set-valued semimartingales are introduced as an extension of the notion of single-valued semimartingales. For such multivalued processes their semimartingale selection properties are investigated.

Weak solutions of stochastic differential inclusions and their compactness

100%

Michta M.

nr 1

91-106

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.