Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals
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Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals Mariusz Michta1 · Jerzy Motyl1 Received: 8 June 2020 / Revised: 21 October 2020 / Accepted: 11 November 2020 © The Author(s) 2020
Abstract The paper deals with some properties of set-valued functions having bounded Riesz p-variation. Set-valued integrals of Young type for such multifunctions are introduced. Selection results and properties of such set-valued integrals are discussed. These integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties are crucial for the investigation of solutions to stochastic differential inclusions driven by a fractional Brownian motion. Keywords Hölder continuity · Set-valued function · Set-valued Riesz p-variation · Set-valued Young integral · Selection · Generalized Steiner center Mathematics Subject Classification (2020) Primary 26A33; Secondary 26A16 · 26A45 · 28B20 · 47H04
1 Introduction Since the pioneering work of Aumann in 1965 [6], the notion of set-valued integrals for multivalued functions has attracted the interest of many authors from both theoretical and practical points of view. In particular, the theory has been developed extensively, among others, with applications to optimal control theory, mathematical economics, theory of differential inclusions and set-valued differential equations, see, e.g., [1,3,4,21,23,29]. Later, the notion of the integral for set-valued functions has been extended to a stochastic case, where set-valued Itô integrals have been studied.
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Jerzy Motyl [email protected] Mariusz Michta [email protected]
1
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Gora, Poland
123
Journal of Theoretical Probability
Moreover, concepts of set-valued integrals, both deterministic and stochastic, were used to define the notion of fuzzy integrals applied in the theory of fuzzy differential equations, e.g., [14,24]. On the other hand, in a single-valued case, one can consider integration with respect to integrators such as fractional Brownian motion which has Hölder continuous sample paths. In some cases, such integrals can be understood in the sense of Young [30]. Controlled differential equations driven by Young integrals have been studied by Lejay in [25]. A more advanced approach to controlled differential equations is based on the rough path integration theory initiated by T. Lions [26] and further examined in [12,17]. Control and optimal control problems inspired the intensive expansion of differential and stochastic set-valued inclusions theory. Thus, it seems reasonable to investigate also differential inclusions driven by a fractional Brownian motion and Young-type integrals also. Recently, in [7] the authors considered a Young-type differential inclusion, where solutions were understood as Young integrals of appropriately regular selections of multivalued right-hand side. Set-valued Aumann
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