Selection Properties and Set-Valued Young Integrals of Set-Valued Functions
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Results in Mathematics
Selection Properties and Set-Valued Young Integrals of Set-Valued Functions Mariusz Michta and Jerzy Motyl Abstract. The paper deals with some selection properties of set-valued functions and different types of set-valued integrals of a Young type. Such integrals are considered for classes of H¨ older continuous or with bounded Young p-variation set-valued functions. Two different cases are considered, namely set-valued functions with convex values and without convexity assumptions. The integrals contain as a particular case setvalued 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. Mathematics Subject Classification. 26A33, 26A16, 26A45, 28B20, 47H04. Keywords. H¨ older-continuity, set-valued function, set-valued Young and Riesz p-variation, set-valued Young integral, selection, generalized Steiner center.
1. Introduction Since the pioneering work of Aumann [6], the notion of set-valued integrals for multivalued functions has attracted the interest of many authors both from 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. [1,4,13,22,26,37]. Later, the notion of the integral for set-valued functions has been extended to a stochastic case where set-valued Itˆ o and Stratonovich integrals have been studied and applied to stochastic differential inclusions and set-valued stochastic differential equations, see e.g., 0123456789().: V,-vol
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M. Michta and J. Motyl
Results Math
[23–25,33–35]. 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., [16,21,27,32]. On the other hand, in a single-valued case, one can consider stochastic integration with respect to non-semimartingale integrators such as the Mandelbrot fractional Brownian motion which has H¨ older continuous sample paths ([31]). Such integrals can be understood in the sense of Young [38]. This kind of integrals have been developed and widely used in the theory of differential equations by many authors, see e.g., [8,12,18,19,28,30]. Furthermore, 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. A similar idea was used for a stochastic inclusion with a fractional Brownian motion in [29]. Both the Aumann and
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