Set-Valued Functions of Bounded Generalized Variation and Set-Valued Young Integrals

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.


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. 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 or Itô-type integrals are useful tools in the investigation of properties of solution sets to differential or stochastic inclusions and set-valued equations [2,15,16,22]. Therefore, it is quite natural to introduce set-valued Young-type integrals. Motivated by this, the aim of this work is to introduce such set-valued integrals and to investigate their properties, especially these which seem to be useful in the Young set-valued inclusions theory. It is known that three of properties of Aumann set-valued integrals are crucial in the differential inclusions theory. Namely, they are the existence of a Castaing representation of the set of integrable selectors, decomposability of this set and valuation of a Hausdorff distance between set-valued integrals by the distance between integrated multifunctions (see, e.g., [20]).
Set-valued Young integrals considered in the paper deal with the class of set-valued functions having a bounded Riesz p-variation. 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 a space of set-valued functions of a finite Riesz p-variation. Section 3 deals with the properties of sets of appropriately regular selections of such set-valued functions. Here, we shall establish a new type of decomposability for sets of functions with a finite Riesz p-variation as well as their integral property. Finally, in Sect. 4, we introduce a set-valued Youngtype integral which is based on the sets of selections examined in Sect. 3. We shall investigate properties of this set-valued integral.

The Hausdorff metric H X in Comp (X ) is defined by
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., [23] for details. We use the notation Let T > 0 and β ∈ (0, 1]. For every function f : [0, T ] → X , we define By C β (X ), we denote the space of β-Hölder-continuous ( or shortly β-Hölder) functions with a finite norm It can be shown that C β , · β is a Banach space. Similarly, for a set-valued function F : [0, T ] → Comp (X ), let A set-valued function F is said to be β-Hölder if F β < ∞. By C β (Comp(X )), we denote the space of all such set-valued functions. The space of β-Hölder set-valued functions having compact and convex values will be denoted by C β (Conv(X )).
Let (E, d) be a metric space. For every 0 ≤ a < b ≤ T , by Π n = {t i } n i=0 , we denote a partition a = t 0 < t 2 < · · · < t n = b of the interval [a, b]. For every function f : [0, T ] → E and 1 ≤ p < ∞, we define its Young p-variation on [a, b] by the formula respectively, are Banach spaces. For X = R d and considered with the Euclidean norm, we will use the notation x instead of x R d .
We collect some properties of functions of bounded V p -variation in the following proposition.

Selections of Finite p-Variation Set-Valued Functions
Let T > 0 be given and let F : 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].
the set of selections of F with a bounded Riesz p-variation.
Let F ∈ C β Comp R d . Such set-valued functions need not admit any Hölder or even continuous selection, see, e.g., [10]. However, considering the smaller class BV p Comp R d ⊂ C β Comp R d , the following selection theorem holds true.
Let us note that the set S V p (F) need not be closed in the topology of point convergence even if F is bounded.

Example 1
The set S V p (F) need not be closed in the topology of point convergence even if F is bounded. To see this, let W be a Wiener process defined on some adequate probability space (Ω, F, P). Let W (·,ω) denote its trajectory connected with a fixed Therefore, we get by Proposition 1(a), be the V p -Castaing representation of F given in Proposition 2. Then, for every f ∈ S V p (F) and every > 0, there exist a finite measurable covering A 1 , . . . , A n of the interval [0, T ] and functions Moreover, for every f ∈ S V p (F) and every > 0, there exist n ≥ 1, a partition Proof Since S V p (F) ⊂ S L p (F) and the V p -Castaing representation of F is also an L p -Castaing representation of F introduced in [9], then the proof follows by Lemma 1.3 of [20].
We prove second inequality. Let f ∈ S V p (F) be arbitrary taken. There exists δ such that f (t) − f (s) < /3 and f m (t) − f m (s) < /3 for every |t − s| < δ (see Proposition 1). Let us take a partition Π n : 0 < δ < 2δ < · · · < nδ < T . Since Thus, Let us note that a similar approximation property with respect to V p -variation norm need not hold true. Now we introduce the notion of V p -decomposable selections of set-valued functions and investigate their properties. Let For a given set By dec L p (B), we denote a closed L p -decomposable hull of a set B. Similarly as in the case of convex and closed convex hulls, they are the smallest L p -decomposable and closed L p -decomposable sets containing the set B, respectively.
From this, it follows that the set S L p (F) consisting of all L p -selectors of a given measurable set-valued function F is always L p -decomposable and therefore, [20]). For other properties of L p -decomposable sets, see [19].
L 1 -decomposability of the set of L 1 -selectors of a given measurable set-valued function F is crucial for investigating properties of a set-valued Aumann integral of F defined by the formula Unfortunately, the set S V p (F) need not be L p -decomposable for any p ≥ 1, and therefore, if one defines a set-valued Young integral in the Aumann's sense, it is difficult to obtain its reasonable properties. This leads to the idea of a different type of decomposability called V p -decomposability.
It follows from Proposition 1(d) that a function f belongs to BV p (R d ) if and only if its strong derivative f belongs to . This property has been inspiring to the following definition.
For a given set B ⊂ BV p R d by dec V p (B), we denote a V p -decomposable hull of a set B, i.e., the smallest V p -decomposable set containing the set B.
Remark 1 Every function f = f 1 ⊕ a f 2 from Definition 2 can be represented by the formula We denote by R(t) the set .
Then, R is closed with respect to the norm · ∞ and V p -decomposable.
Proof If R is an integral, then for every t ∈ [0, T ] R(t) is a closed subset of R d by Theorem 8.6.7 of [5]. Let ( f n ) ∞ n=1 ⊂ R be a sequence convergent to some f with respect to the norm · ∞ . Since R is an integral, then f n (t) = x 0 + t 0 φ n (s) ds for some φ n ∈ S L p (Φ). But f n (0) = x 0 and therefore, f (0) = x 0 . Moreover, since Φ is p-integrably bounded by some function g ∈ L p ([0, T ]), then sup n V p ( f n ) ≤ g L p . It follows from Proposition 1(c) that V p ( f ) ≤ g L p . Therefore, f ∈ BV p (R d ) and f (t) = x 0 + t 0 f (s) ds. Since Φ is p-integrably bounded and has closed and bounded values, then the set S L p (Φ) is closed, bounded and convex in L p ([0, T ]). Therefore, it is weakly compact there. Thus, there exists a subsequence (φ n k ) of (φ n ) weakly convergent to some φ ∈ S L p (Φ). Let J : L p ([0, T ]) → C([0, T ]) be a linear operator defined by formula J (ψ) = x 0 + · 0 ψ(s) ds. Since J is norm-to-norm continuous, then it is also weak-to-weak continuous. Thus, f n k = x 0 + · 0 φ n k (s) ds tends weakly to . This implies f ∈ R, which proves the closedness of R.
Now let us take f 1 , f 2 ∈ R. There exist a set-valued function Φ and functions and closed with respect to the norm · ∞ . Then, R is an integral.
Proof Assuming that R ⊂ BV p (R d ), let f 1 , f 2 ∈ R and a ∈ [0, T ] be arbitrarily taken. If f = f 1 ⊕ a f 2 , then f ∈ R by the assumption of V p -decomposability. We define the set M by the formula Then, M is convex in L p ([0, T ]). It is bounded and closed in L p ([0, T ]) by Proposition 1(d). Since We will show that the set M is L p -decomposable in L p ([0, T ], β([0, T ]), λ), i.e., we will show that for every set A ∈ β([0, T ]) and any φ, ψ ∈ M, the function γ = 1I A · φ + 1I A ∼ · ψ belongs to the set M. β([0, T ]), as usual, denotes here the Borel σ algebra of subsets of the interval [0, T ], and λ is a Lebesgue measure.
We have shown that the set contains a ring generating β([0, T ]) and a monotone class From the monotone class theorem, we deduce that for every φ, ψ ∈ M and every set Q ∈ β([0, T ]) the set 1I Q φ + 1I Q ∼ ψ belongs to M. Therefore, M is L p -decomposable and by Theorem 3.1 of [20] there exists a measurable set-valued function Φ : Since S L p (Φ) = M is convex, then Φ has convex values by Theorem 1.5 from [20]. Moreover, Φ is p-integrably bounded by the boundedness of M. Therefore, R should be an integral.

Definition 4
Let X be a real normed linear space. Let A, B ∈ Conv(X ). The set C ∈ Conv(X ) is said to be the Hukuhara difference A ÷ B if A = B + C. Consider a setvalued mapping G : R 1 → Conv(X ). We say that G admits a Hukuhara differential at t 0 ∈ R 1 , if there exists a set D H G(t 0 ) ∈ Conv(X ) and such that the limits For a detailed discussion of the properties and applications of the Hukuhara differentiable multifunctions, we refer the reader to [23]. Now we are ready to prove the main decomposability results of the section.

then there exists a measurable and p-integrably bounded set-valued function Φ : [0, T ] → Comp(R d ) such that the set-valued function t → R(t) is Hukuhara differentiable for almost every t ∈ [0, T ] and D H R(t) = coΦ(t).
Proof Assume that a closed and bounded set R in It is also closed with respect to · ∞ . Therefore, it follows by Theorem 2 that R is an integral, i.e., there exists a measurable and a p-integrably bounded set-valued function Since From this, we deduce that the Hukuhara derivative D H (R(t)) exists for almost every t ∈ [0, T ] and D H R(t) = coΦ(t), see, e.g., [29].

Remark 2
If a set R ⊂ BV p (R d ) is an integral, then R(t) = x 0 + t 0 Φ(s) ds for every t ∈ [0, T ] and some measurable and p-integrably bounded set-valued function Φ. The reverse implication need not hold as the following example shows.

Theorem 4 Let F : [0, T ] → Conv(R d ) be a Hukuhara differentiable set-valued function, F ∈ BV p
is V p -decomposable and therefore, it is an integral.
and therefore, ( f ⊕ a g) ∈ R. We proved that R is V p -decomposable and it is an integral by Theorem 2.
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 }.
Let ξ μ := dμ/dp V , and let ∇ξ μ denote the gradient of ξ μ . By ω, we denote a Lebesgue measure on Σ. The function St μ : Conv R d → R d called a generalized Steiner center, and given by the formula for every μ ∈ M, has the following properties.
For A, B, C ∈ Conv R d and a, b ∈ R 1 where L μ = d max p∈Σ ξ μ ( p) + max p∈B(0,1) ξ μ ( p) (see e.g., [13]). Since the set C 1 d = {ξ ∈ C 1 (B(0, 1), R + ) : B ξ d p V = 1} is separable, then there exists a countable subset {ξ n } ⊂ C 1 d dense in C 1 d with respect to supremum norm. Let {μ n } be a sequence of measures from M with densities {ξ n }. It is known that every set C ⊂ Conv(R d ) has a representation where St μ (C) are generalized Steiner points of C given by formula (4), see also [13]. Therefore, by separability of C 1 d , we have Let F : [0, T ] → Conv(R d ) be a set-valued function. Then, for every t ∈ [0, T ] by [8] and we obtain Assume that F ∈ BV p (Conv(R d )) is Hukuhara differentiable, F(0) = x 0 , and consider again a set IS(F) defined by (3). This set is an integral by Theorem 4. We prove the following result.

F)(t)) ∈ D H (F)(t).
We have to show that f n ∈ BV p (R d ) and f n ∈ ) by equality (6). It follows from formula (5) that where L μ = d max p∈Σ ξ μ ( p) + max p∈B(0, 1) ξ μ ( p) . Therefore, for every 0 ≤ a < b <≤ T , Therefore, f n ∈ BV p (R d ). Now, we are able to apply Corollary 3.4(a) from [11] to deduce that f n satisfies t 0 f n (s) p ds < ∞. Since f n is a measurable selection of D H (F), then f n ∈ S L p (D H (F)). Therefore, f n ∈ IS(F) for every n = 1, 2, . . . .

Set-Valued Young Integrals
At the beginning of this section, we recall the notion of a Young integral in a singlevalued case introduced by Young in [30]. For details, see also [17]. Let f : [0, T ] → R d and g : [0, T ] → R d be given functions. For the partition Π m : 0 = t 0 < t 1 < · · · < t m = T of the interval [0, T ], we consider the Riemann sum of f with respect to g Then, the following version of Proposition 2.4 in [18] holds.
holds for every 0 ≤ s < t ≤ T , where the constant C(α, p) depends only on p and α.
Let us consider again a set IS(F) given in (3)

Definition 5
We define a set-valued Young integral of Hukuhara differentiable F ∈ BV p (Conv R n ) with respect to a function g ∈ C α R 1 , 1/ p + α > 1, by the formula for every 1/ p + α > 1 and g ∈ C α (R 1 ).
We have The following lemma was proved in [27].
Using this lemma, we are able to prove the following result.
It was proved in [20] that for measurable and p-integrably bounded set-valued functions F 1 , F 2 : [0, T ] → Comp(R d ), the equality holds. Therefore, a set-valued Aumann integral satisfies We will show that a set-valued Young integral is additive also.

Moreover, if the set F(0) is bounded in R d , then (IS)
· 0 F dg and (IS) t s F dg are bounded sets in C α (R d ) and in R d , respectively.

) takes on compact and convex values in
We show that the set IS(F) is bounded in the space (BV p (R d ), · V p ). Let f ∈ IS(F) be arbitrarily taken. By assumption, the set S L p (D H (F)) is bounded in L p norm by some constant M. Since f ∈ S L p (D H (F)), then (V p ( f )) 1/ p = ( T 0 f (s) p ds) 1/ p ≤ M. Moreover, f (t) − f (0) ≤ T 1−1/ p (V p ( f )) 1/ p for every t ∈ [0, T ] by Proposition 1(b). This implies Thus, we obtain the appropriate boundedness of both integrals.

Compliance with Ethical Standards
Conflict of interest The authors declare that they have no conflict of interest.
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