Set-valued functions of bounded generalized variation and set-valued Young integrals

The paper deals with some properties of set-valued functions having a bounded Riesz p-variation. Set-valued integrals of a Young type for such multifunctions are introduced. Selection results and properties of such setvalued 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.

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], [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 A. 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], [16]. 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 toolls in the investigation of properties of solution sets to differential or stochastic inclusions and setvalued equations [2], [15], [19], [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 setvalued 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 Section 2, we define a space of setvalued 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 Section 4, we introduce a set-valued Young type integral which is based on the sets 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 : 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 setvalued function F : [0, T ] → Comp (X) let 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)).
For every function f : [0, T ] → E and 1 ≤ p < ∞ we define its Young p-variation on [a, b] by the formula p and a Riesz 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.
Proposition 1 ([10], [11]). Let f : [0, T ] → E. Then, for every 1 ≤ p < ∞, the following conditions hold: Let (X, · ) be a Banach space and let Π m : 0 = t 0 < t 1 < ... < t m = T be a partition of the interval [0, T ]. Given a set-valued function F : Then by a Riesz p-variation on [0, T ] we mean the quantity By BV p (Comp (X)) we denote the space of all set-valued functions from [0, T ] into Comp (X) having finite Riesz p-variation.

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.
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 : Let us note, that the set S Vp (F ) need not be closed in the topology of point convergence even if F is bounded.

Example 1
The set S Vp (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 fixedω ∈ Ω. Then M = sup t∈[0,T ] |W (t,ω)| < ∞, because of continuity of trajectories of a Wiener process. Let F : [0, T ] → Comp(R 1 ) be a set-valued function defined by formula denote a sequence of normal partitions 0 = t 1 < t 2 < ... < t n = T of the interval [0, T ] and let W n (·,ω) denote regularizations of W (·,ω) defined by the formula below .
be the V p -Castaing representation of F given in Proposition 2. Then, for every f ∈ S Vp (F ) and every ǫ > 0, there exist a finite measuarable covering A 1 , ..., A n of the interval [0, T ] and functions f k1 , ..., 1I Aj · f kj L p < ǫ.
Moreover, for every f ∈ S Vp (F ) and every ǫ > 0, there exist n ≥ 1, a partition Proof Since S Vp (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 Vp (F ) be arbitrary taken. There exists δ such that f (t) − f (s) < ǫ/3 and f m (t) − f m (s) < ǫ/3 for every |t − s| < δ (see Proposition 1). Let us take a partition Π n : Thus, Let us note, that a similar approximation property with respect to V pvariation norm need not hold true. Now we introduce the notion of V p -decomposable selections of set-valued functions and investigate their properties.
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 [18].
L 1 -decomposability of the set of L 1 -selectors of a given measurable setvalued function F is crucial for investigating properties of a set-valued Aumann integral of F defined by the formula Unfortunately, the set S Vp (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
. This property has been inspiring to the following definition.
belongs to the set Λ. For a given set B ⊂ BV p R d by dec Vp (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 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, Since Φ is pintegrably 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 convex and closed with respect to the norm · ∞ . Then R is an integral.
Proof Assume 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 take a partition Π n : 0 = t 0 < t 1 < ... < t 2n < t 2n+1 = T and the set A of the form A = n i=0 [t 2i , t 2i+1 ). Since R is V p -decomposable it is easy to see that taking any f 1 , ..., f 2n+1 ∈ R a function f given by the formula ..n and f ′ 2i+1 = ψ for i = 0, 1, ...n, we have Then M is a ring generating a σ-algebra β([0, T ]). We will show that M is a monotone class also. To this end, assume that ( ∼ form a decreasing family, then a sequence It was shown in the first part of the proof that γ k (s) ∈ M , because of x 0 + γ k (s)ds ∈ R. We show that γ ∈ M , i.e., that f = x 0 + γ(s)ds ∈ R. We know that f k = x 0 + γ k (s)ds ∈ R. We have However, γ k (s) → γ(s) a.e. and the sequence γ k (s) − γ(s) admits a pintegrable majorant 2|φ(s)| + 2|ψ(s)|. Therefore, f k − f ∞ → 0. Since R is closed by the assumption, then f ∈ R and therefore, γ ∈ M . 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 setvalued function Φ : x 0 + φ(s)ds} ∈ R. It means that R = x 0 + Φ(s)ds. 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 set-valued 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. Proof Assume that a closed and bounded set R in BV p (R d ) is V p -decomposable. 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 Φ :

Theorem 3 If a closed and bounded set
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].
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.
is V p -decomposable and therefore, it is an integral.
Proof Really, let f, g ∈ R. Then f, g ∈ S Vp (F ). Therefore, , then (f ⊕ a g) ∈ S Vp (F ), 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 }.
Since the set C 1 d = {ξ ∈ C 1 (B(0, 1), R + ) : B ξdp 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

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.

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 L.S. Young in [30]. For details see also [16]. 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 [17] 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 f dg exists in the sense of Riemann and for every ρ ∈ (1 − α, β). Moreover, the following version of the inequality (8) holds for every 0 ≤ t 1 < t 2 ≤ T , where C(α, β) depends only on α and β.
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 (ConvR 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 folllowing lemma was proved in [27].

holds.
Using this Lemma we are able to prove the following result.
We obtain by Lemma 1, for any Hence,

The same estimation holds for
Therefore, .
Suppose that f ∈ IS(F ). Then, it is expressed as the integral, i.e., f (·) = · 0 φ(s)ds for some φ ∈ S L p (D H (F )) and we have From the other side, we get by the formula (2), the equalities and taking in the mind the begining of the proof, we get Let φ 1 ∈ S L p (D H (F 1 )). Then, by Theorem 2.2 from [20], we have Thus In a similar way we get and finally Hence, by the formula (10) together with (11), we obtain inequality (9). Then, Proof Let us note that for every n ≥ 1 we have Since β > ρ and θ ∈ (0, 1] is arbitrarily taken, we obtain formula (12).
It was proved in [20] that for measurable and p-integrably bounded setvalued 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.
be Hukuhara differentiable with p-integrably bounded Hukuhara derivatives, 1 < p < ∞. Let g ∈ C α (R 1 ), where 1/p + α > 1. Then 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.
We show that the set IS(F ) is bounded in the space (BV p (R d ), · Vp ). 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 = ( Thus we obtain the appropriate boundedness of both integrals.

Disclosure of potential conflicts of interest
Conflict of Interest: The authors declare that they have no conflict of interest.