Abstract
In the paper we study the existence of solutions of the random differential inclusion urn:x-wiley:20903332:media:ista802365:ista802365-math-0001 where G is a given set‐valued mapping value in the space Kn of all nonempty, compact and convex subsets of the space ℝn, and μ is some probability measure on the Borel σ‐algebra in ℝn. Under certain restrictions imposed on F and μ, we obtain weak solutions of problem (I), where the initial condition requires that the solution of (I) has a given distribution at time t = 0.
Highlights
(z) x0 #, where G is a given set-valued mapping value in the space Kn of all nonempty, n, compact and convex subsets of the space and # is some probability measure on the Borel a-algebra in Rn
Let us notice that the set-valued stochastic process X can be though as a random element X: ft--CI
Xo d where the initial condition requires that the set-valued solution process X (Xt) E I has a given distribution # at the time t =0
Summary
For S- Rn and Kn- Kc(Rn), we denote by CI -C(I,Kn) the space of all H-continuous setvalued mappings. Let us notice that the set-valued stochastic process X can be though as a random element X: ft--CI It follows immediately from [3] and from the fact that the topology of the uniform convergence and the compact-open topology in CI are the same. A) is a measurable multifunction for every A Kn. Let us consider the multivalued random differential equation: DHX F(t, Xt) P.1, t e [0, T]-a.e. Xo d where the initial condition requires that the set-valued solution process X (Xt) E I has a given distribution # at the time t =0. By a weak solution of (II) we understand a system (,,P(Xt)tE I) where (Xt)te I is a set-valued process on some probability space (,,P)such that (II) isomer. G 0 be such that An- CV for n- 1, 2,
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