Selection Properties and Set-Valued Young Integrals of Set-Valued Functions

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 Hölder 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 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.


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 Itô and Stratonovich integrals have been studied and applied to stochastic differential inclusions and set-valued stochastic differential equations, see e.g., [23][24][25][33][34][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 Hölder 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 set-valued Itô integrals play a crucial role in the studies of properties of solution sets, e.g., [2,3,17,22,23,37]. Therefore, it is quite natural to introduce set-valued Young type integrals. Motivated by this, the aim of this work is to introduce some types of set-valued Young integrals and to investigate their properties, especially these which seem to be useful in the Young set-valued inclusions theory. Such set-valued integrals are considered for classes of Hölder continuous setvalued functions, set-valued functions having a bounded Young p-variation as well as for set-valued functions with a bounded Riesz p-variation. In the paper, we consider two different cases, namely set-valued functions with convex values and without convexity assumptions. The properties of the two integrals and selection theorems presented in the paper are significantly different. Such integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion. Therefore, in our opinion, their properties are crucial not only for the existence of solutions to stochastic differential inclusions and set-valued stochastic differential equations driven by a fractional Brownian motion but also for useful properties of their solution sets.
The paper is organized as follows. In Sect. 2, we define spaces of setvalued functions being Hölder continuous and spaces of set-valued functions of a finite Young or Riesz p-variation. Several basic relations between such spaces are given. Section 3 deals with the properties of sets of appropriately regular selections of such set-valued functions, both in convex and nonconvex cases. Finally, in Sect. 4, we introduce set-valued Young type integrals which are based on the sets of selections examined in Sect. 3. We shall give a number of properties of such set-valued integrals. Some of them are natural multivalued generalizations of those known from a single-valued Young integral theory.
where B + C := {b + c : b ∈ B, c ∈ C} denotes the Minkowski sum of B and C. Moreover, for B, C, D ∈ Conv (X) the equality holds, see e.g., [20] for details.
We use the notation Similarly, for a set-valued function F : we denote the space of all such set-valued functions. The space of β-Hölder setvalued functions having also convex values will be denoted by C β (Conv (X)).
Introducing metrics , one can prove the following result.
we denote a partition a = t 0 < t 2 < .. and Let (X, · ) be a Banach space and let F : [0, T ] → Comp(X) be a given set-valued function. In this case we can set (E, d) = (Comp(X), H X ) and define quantities V ar p (F ) and V p (F ) being a Young p-variation and Riesz p-variation for F , respectively.
By BV ar p (Comp (X)) (resp. BV p (Comp (X)) we denote the space of all set-valued functions from [0, T ] into Comp (X) having finite Young p-variations (resp., finite Riesz p -variations).
We set Similarly, as in a single valued case, every F ∈ C β (Comp (X)) satisfies F ∈ BV ar p (Comp (X)) with p = 1/β, and and [9]). The sequence (f n ) is called an L p -Castaing representation for F . For other properties of measurable set-valued functions and their measurable selections (see e.g. [5]).

Selections of Nonconvex Valued Set-Valued Functions
the sets of selections of F with a bounded Young p-variation and bounded Riesz p-variation, respectively.
First problem is connected with the nonemptiness of sets S V arp (F ) and S Vp (F ).
Note, that in the case p = 1 we have BV 1 (Comp(R d )) = BV ar 1 (Comp(R d )). When p > 1 it seems that the question whether any compact valued set-valued function of finite Young p-variation admits a selection with  [10]). Therefore, we consider the more regular class C β Comp R d .
Let F ∈ C β Comp R d . Such set-valued functions need not admit any Hölder or even continuous selection (see Example 1 in Sect. 6 of Chap. I in [4] and Proposition 8.2 in [10]). However, the following selection theorem holds true (Theorem 1.2 in [7]).
Hence, by the theorem above, we have the following result used in Sect. 5.

Corollary 1.
Let F ∈ C β Comp R d and p > 1/β. Then the sets S V arp (F ) and S β V arp (F ) are nonempty and closed subsets of BV ar p (R d ) . The latter one is also bounded.
In order to obtain more regular selections we put our attention on a class BV p (Comp(R d )). For such class of set-valued functions we obtain not only nonemptiness of the set S Vp (F ) but we get some additional properties of this set, which seem to be useful in the investigation of differental-integral inclusions with set-valued Young integrals.
for some 1 ≤ p < ∞ then there exist a function φ ∈ BV p (R d ) and a sequence of equi-Lipschitzean functions (g n ) ∞ n=1 with Lipschitz constants L n ≤ 1 such that taking f n : n=1 is a V p -Castaing representation for F . From the above proposition it is easy to deduce the following result.
Proof. The set S Vp (F ) is nonempty by Proposition 2. Let (f n ) ⊂ S Vp (F ) be a sequence convergent to some function f with respect to norm · Vp . Then f (t) ∈ F (t) for every t ∈ [0, T ]. Since V p (f + g) 1/p ≤ V p (f ) 1/p + V p (g) 1/p , then the real sequence (V p (f n )) ∞ n=1 is Cauchy and therefore, bounded in R 1 . Then there exists a constant M > 0 such that V p (f n ) ≤ M for every n ≥ 1. Theorem 1(e). Then, using Theorem 1(f), we get V p (f ) ≤ M and therefore, f ∈ S Vp (F ).
It is an open question if the analogues of Theorem 3 and Proposition 3 hold true for the larger class BV ar p (Comp(R d )).
The approximation property of the set S Vp (F ) is placed in the "Appendix" as Theorem 7, because it is not used in proofs of main results.

Selections of Convex Valued Set-Valued Functions
From a previous section we know that every set-valued function F : [0, T ] → Comp(R d ) which has a finite Riesz p-variation admits a selection f ∈ BV p (R d ), Proposition 2. However, if F ∈ C β Comp R d , then it need not admit any β-Hölder selection. It has only selections with a finite Young p-variation, see Theorem 2. It seems that the question whether any compact valued set-valued function of finite Young p -variation admits a selection with a finite Young pvariation is still open (see. e.g., Remark 8.1 in [10]). The property of convexity of values of a given set-valued function is helpful in the problem.
For F ∈ C β Conv R d and G ∈ BV ar p (Conv R d ) we consider the set S β (F ) of all β-Hölder selections of F , i.e., the set As before, the set S V arp (G) denotes the set of all selections of G having a finite Young p-variation.
It is clear that for every p = 1/β it holds S β (F ) ⊂ S V arp (F ). Thus in order to show nonemptiness of the sets S β (F ) for F ∈ C β Conv R d and S V arp (G) for G ∈ BV ar p (Conv R d ) we cannot apply Theorem 2. However, since both set-valued functons F and G have convex values, it is possible to use a generalized Steiner center (see e.g., [14]). Namely, let C ∈ Conv R d and let σ (·, C) : R d → R 1 , σ (p, C) = sup y∈C < p, y > be a support function of C. Let Σ denote the unit sphere in R d and let V denote a Lebesgue measure of a closed unit ball B(0, 1) in R d , i.e., V = π d/2 /Γ (1 + d 2 ) with Γ being the Euler function. Let p V be a normalized Lebesgue measure on B(0, 1), i.e., dp V = dp/V . Let M = { μ : μ is a probability measure on B(0, 1) having the C 1 − density dμ/dp V with respect to measure p V }.

Proposition 4. Let G ∈ BV ar p (Conv R d ). Then the set S V arp (G) is closed and convex in BV ar
Proof. The closedness and convexity of S V arp (G) in BV ar p (R d ) are evident. Let {g μn } ∞ n=1 be a V ar p -Castaing representation for G 1 for some {μ n } ∞ n=1 ⊂ M. Then, for every n ≥ 1, g μn ∈ S V arp (G 1 ) = S V arp (G 2 ) and therefore, g μn (t) ∈ G 2 (t) for every t ∈ [0, T ] and n ≥ 1. Since G 2 takes on closed values in R d , then we have G 1 (t) = {g μn (t)} ∞ n=1 ⊂ G 2 (t) for every t ∈ [0, T ]. The opposite inclusion can be proved in a similar way.
Let F ∈ C β Conv R d . Using a similar argumentation we deduce that for every μ ∈ M, the function f μ (t) := St μ (F (t)) belongs to S β (F ) and again by (1) we get Similarly as above we obtain the following result: In general, we cannot expect the boundedness of sets S β (F ) in the space C β R d or sets S V arp (F ) in BV ar p (R d ), even for a constant set-valued function F . for 0≤ t < 1/2 nt − n/2 for 1/2 ≤ t < 1/2 + 1/n 1 for 1/2 + 1/n ≤ t ≤ 1 .

Set-Valued Young Integrals
At the beginning of this section we recall the notion of a Young integral in a single valued case introduced by Young [38]. For details see the excelent monography of Friz and Victoir [18].
Then the following version of Proposition 2.4 in [19] holds.
holds for every 0 ≤ s < t ≤ T , where the constant C(α, p) depends only on p and α.
Then the integral T 0 fdg exists in the sense of Riemann and for every ρ ∈ (1 − α, β). Moreover, the following version of the inequality holds for every 0 ≤ t 1 < t 2 ≤ T , where C(α, β) depends only on α and β.

Nonconvex Case
Let g ∈ C α (R 1 ) and let F ∈ C β (Comp(R d )) where α, β ∈ (0, 1] and α + β > 1. By Corollary 1, for any fixed p > 1/β the sets S V arp (F ) and S β V arp (F ) are nonempty and closed in BV ar p (R d ). If 1/p + α > 1, then we define a setvalued Young integral by the formula ). Repeating the proof of Theorem 1.2 in [7] it follows, that for every p > 1/β and g ∈ C α (R l ) there exists f ∈ BV ar p (L(R l , R d )) being a selection of F . Therefore, the definition of a set-valued Young integral given in formula (3) can be extended to the above case.

Proposition 7.
For every 0 ≤ s < t ≤ T , we have Proof. Let f ∈ S V arp (F ) and let 0 ≤ s < t ≤ T . By (2) we have Then, taking the supremum over f ∈ S V arp (F ), we get Hence we get estimations (4). Similarly, by H V arp we mean a Hausdorff metric in the space Cl(BV ar p (R d )).
Proof. Let f 1 ∈ S V arp (F 1 ), f 1 ∈ S V arp (F 2 ). Then by (2) we have Hence we obtain In a similar way we get the same estimation forH R d t s F 2 dg, t s F 1 dg R d . This completes the proof.
In a single valued case, this theorem expresses continuity of the Young integration operator BV ar holds.
Proof. For any 0 ≤ s < t ≤ T , then applying Proposition 7, we get and it is sufficient to apply Theorem 5 and Proposition 7 to get the desired estimation.
In general, the value of H V arp (S V arp (F 1 ), S V arp (F 2 )) in the estimation (5) may be infinite. It is finite, if the definition of the set-valued Young integral given by formula (3 ) is based on the set In such a case, we have the following result.
holds for every 0 ≤ s < t ≤ T .

Convex Case
Let F : [0, T ] → Conv R d . Let α, β ∈ (0, 1] with α + β > 1. If S β (F ) = ∅ we define another type of a set-valued Young integral of F over the interval [0, t] , which is restricted to β-Hölder selections of F , i.e., In the whole Section 5.2 we consider integrals (6). (6) can be extended to the case g ∈ C α (R l ) while F ∈ C β (Conv(L(R l , R d ))) by a slight modification. The space of linear bounded
and it is also a selection, i.eS μ (A) ∈ A for every A ∈ Conv(L(R l , R d )). Moreover, we have for every A, B ∈ Conv(L(R l , R d )). HenceS μ is a Lipschitz selection. Finally, for F ∈ C β (Conv(L(R l , R d ))) one can define a function f μ : [0, T ] → L(R l , R d ) as f μ (t) :=S μ (F (t)).
By the properties above it follows that f μ ∈ C β ((L(R l , R d ))) and thus S β (F ) :={f ∈ C β ((L(R l , R d ))) : f (t) ∈ F (t), t ∈ [0, T ]} is nonempty. Therefore, the set-valued Young integral can be properly defined in this generalized case. Since S β (F ) ⊂ S V arp (F ), where p = 1/β, then the integral defined by formula (6) is a subset of the integral (3). If F ∈ C β Conv R d , then the set-valued Young integral is a nonempty, closed and convex subset of R d for any t ∈ [0, T ] by Proposition 5. Similarly as Proposition 7, one can prove the following.
Let C R d denote the space of continuous functions on [0, T ] with values in R d . We denote a Hausdorff distance in the space Cl C R d by H ∞ .

Theorem 6.
Let g ∈ C α R 1 . Then, for every ρ ∈ (1 − α, β), there exists a positive constant C(ρ) such that for every θ ∈ (0, 1], t ∈ [0, T ] and every holds. Remark 3. Let us note that the estimation in Theorem 6 does not follow by Theorem 5. Indeed, the estimation on he right hand side in Theorem 6 concerns the Hausdorff distance between sets S β (F 1 ) and S β (F 2 ) for which S β (F 1 ) ⊂ S V arp (F 1 ) and S β (F 2 ) ⊂ S V arp (F 2 ) where p = 1/β. The distances H ∞ (S β (F 1 ), S β (F 2 )) and H ∞ (S V arp (F 1 ), S V arp (F 2 )) are not comparable in general. From this, it follows that the Hausdorff distance between set-valued integrals (6) is not comparable with the Hausdorff distance between integrals (3). Therefore, we cannot say which of results of Theorem 5 or Theorem 6 is more useful. Continuity of a set-valued Young integral can be applicable among other to investigation the existence and properties of solutions of differential inclusions. Applicability of continuity results presented above is strictly connected with the choice of a space to be more appropriate in the investigation of