Set-Valued Stochastic Integrals and Applications by Michał Kisielewicz

Author:Michał Kisielewicz
Language: eng
Format: epub
ISBN: 9783030403294
Publisher: Springer International Publishing

(ii) ,

(iii)if is bounded decomposable, then is a convex weakly compact subset of ,

(iv)if is closed decomposable and integrably bounded, then , for every 0 ≤ s < t < ∞,

(v)if is decomposable, then is integrably bounded if and only if is bounded. □

Differently to Aumann stochastic functional integrals, Aumann stochastic integrals are defined as set-valued mappings with values in the space . We begin with the definition of Aumann stochastic integral defined, for a given nonempty set and fixed 0 ≤ s < t < ∞, by setting for every ω ∈ Ω. Hence in particular, it follows that if is a countable set, then is a set-valued random variable. Indeed, in such case there is a sequence such that . Therefore, . Hence, by Theorem 2.​2.​3 of Chapter 2, it follows that is -measurable. In particular, if , where S(F) is a set of all measurable selectors of a measurable p-integrably bounded multifunction of , a set-valued stochastic integral is denoted by and said to be the Aumann stochastic integral of F over the interval [s, t]. If a multifunction is -non-anticipative p-integrably bounded, then the Aumann stochastic integral is denoted by and defined by setting for every ω ∈ Ω. The second type of Aumann stochastic integrals of multifunction F can be defined immediately by the Aumann integral depending on a random parameter. More precisely, for a given above multifunction F the set-valued mapping is denoted by or by and called the Aumann stochastic integral depending on a random parameter. From this definition it follows that for every ω ∈ Ω, where S(⋅, ω) denotes subtrajectory integrals of the set-valued mapping for every ω ∈ Ω. But, for every v ∈ S(F) one has v(⋅, ω) ∈ S(⋅, ω). Then for every ω ∈ Ω, which implies that for every ω ∈ Ω. Thus, for every ω ∈ Ω. Similarly, the inclusion for every ω ∈ Ω, for an -non-anticipative p-integrably bounded multifunction , can be obtained. From properties of Aumann integrals it follows that the Aumann stochastic integral is a set-valued random variable. It is a consequence of the following theorem.

Theorem 4.2.2

If is a measurable ( -non-anticipative) integrably bounded set-valued stochastic process, then the set-valued mapping is -measurable ( -measurable) for every t ≥ 0.

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