BOUNDEDNESS OF SET-VALUED STOCHASTIC INTEGRALS

Discussiones Mathematicae Differential Inclusions, Control and Optimization 35 (2015) 197 207 doi:10.7151/dmdico.1173 BOUNDEDNESS OF SET-VALUED STOCHASTIC INTEGRALS Micha l Kisielewicz Faculty of Mathematics, Computer Science and Econometrics University of Zielona Góra Prof. Z. Szafrana 4a, 65 516 Zielona Góra, Poland e-mail: M.Kisielewicz@wmie.uz.zgora.pl Abstract The paper deals with integrably boundedness of Itô set-valued stochastic integrals defined by E.J. Jung and J.H. Kim in the paper [4], where has not been proved that this integral is integrably bounded. The problem of integrably boundedness of the above set-valued stochastic integrals has been considered in the paper [7] and the monograph [8], but the problem has not been solved there. The first positive results dealing with this problem due to M. Michta, who showed (see [11]) that there are bounded set-valued IF-nonanticipative mappings having unbounded Itô set-valued stochastic integrals defined by E.J. Jung and J.H. Kim. The present paper contains some new conditions implying unboundedness of the above type set-valued stochastic integrals. Keywords: set-valued mapping, Itô set-valued integral, set-valued stochastic process, integrably boundedness of set-valued integral. 2010 Mathematics Subject Classification: 60H05, 28B20, 47H04. 1. Introduction The paper is devoted to the integrably boundedness problem of Itô set-valued stochastic integrals defined by E.J. Jung and J.H. Kim in the paper [4] as some set-valued random variables. The first Itô set-valued stochastic integrals, defined as subsets of the spaces IL 2 (Ω, IR n ) and IL 2 (Ω, X ), have been considered by F. Hiai and M. Kisielewicz (see [1, 5, 6]), where X is a Hilbert space. Unfortunately, such defined integrals do not admit their representations by set-valued random variables with values in IR n and X, because they are not decomposable subsets

198 M. Kisielewicz of IL 2 (Ω, IR n ) and IL 2 (Ω, X ), respectively. J.Jung and J.H. Kim (see [4]) defined the Itô set-valued stochastic integral as a set-valued random variable determined by a closed decomposable hull of the set-valued stochastic functional integral defined in [5]. Unfortunately, the proof of the result dealing with integrably boundedness of such integrals, presented in the paper [4] is not correct. Later on, integrably boundedness of such defined set-valued stochastic integrals has been considered in the paper [7] and the monograph [9]. However the problem has not been solved there. The first positive results dealing with this problem due to M. Michta, who showed in [11], that there are bounded set-valued IF-nonanticipative mappings having unbounded Itô set-valued stochastic integrals defined by E.J. Jung and J.H. Kim. The present paper contains some new conditions implying unboudednes of the above type set-valued stochastic integrals. In what follows we shall assume that we have given a filtered probability space P IF = (Ω, F, IF, P ) with a filtration IF = (F t ) 0 t T satisfying the usual conditions and such that there are real stochasticaly independet IF-Brownian motions B j = (B j t ) 0 t T, j = 1, 2,..., m, defined on P IF. By and we denote the norms of the spaces IL 2 ([0, T ] Ω, Σ IF, IR d m ) and IL 2 (Ω, F, IR d ), respectively, where Σ IF denotes the σ-algebra of all IF-nonanticipative subsets of [0, T ] Ω. For a given set Λ IL 2 ([0, T ] Ω, Σ IF, IR d m ) by dec ΣIF Λ and dec ΣIF Λ we denote a decomposable and a closed decomposable hull of Λ, respectively, i.e., the smallest decomposble and closed decomposable set containig Λ. Let us recall that a set Λ IL 2 ([0, T ] Ω, Σ IF, IR d m ) is said to be decomposable if for every u, v Λ and D Σ IF one has 1I D u + 1I D v Λ, where D = ([0, T ] Ω) \ D. In a similar way for a given set Λ IL 2 (Ω, F, IR d ) decomposable sets dec F Λ and dec F Λ are defined. The Fréchet-Nikodym metric space (S IF, λ) corresponding to a measure space ([0, T ] Ω, Σ IF, µ), with µ = dt P, is defined by S IF = {[C] : C Σ IF }, where [C] = {D Σ IF : µ(c D) = 0} for C Σ IF and λ([a], [B]) = µ(a B) for A, B Σ IF, A B = (A\B) (B\A). It is a complete metric space homeomorphic with the set K IF IL 2 ([0, T ] Ω, Σ IF, IR) defined by K IF = {1I D IL 2 ([0, T ] Ω, Σ IF, IR) : D Σ IF }. Indeed, for every A, B Σ IF we have λ([a], [B]) = µ(a B) = A B dp dt = E T 0 1I A Bdt = E T 0 1I A 1I B 2 dt = 1I A 1I B 2. Let J j T denotes the isometry on the space IL2 ([0, T ] Ω, Σ IF, IR) defined by J j T (g) = T 0 g tdb j t. It can be proved that J j T (K IF) is not an integrably bounded subset of the space IL 2 (Ω, F, IR) for every j = 1, 2,..., m (see [11], Corollary 3.14). It implies (see [11], Theorem 2.2) that dec F J j T (K IF) is an unbounded subset of the space IL 2 (Ω, F, IR), where the closure is taken with respect to the norm topology of the space IL 2 (Ω, F, IR). Hence it follows that dec F J j T (K IF) and dec F J j T (co K IF) are unbounded subsets of this space, because they contain an unbounded subset J j T (K IF). Therefore, co[dec F J j T (K IF)] is unbounded, which by

Boundedness of set-valued stochastic integrals 199 virtue of ([9], Proposition 3.1) implies that dec F J j T (co K IF) is an unbounded subset of the space IL 2 (Ω, F, IR). Finally let us observe that for every α > 0 a set dec F J j T (α K IF) is an unbounded subset of IL 2 (Ω, F, IR d ), because dec F J j T (α K IF ) = α dec F J j T (K IF). Hence the following basic results of the paper follows. Lemma 1. If α > 0 and h IL 2 ([0, T ] Ω, Σ IF, IR) are such that h t (ω) α for a.e. (t, ω) [0, T ] Ω then the set dec F J j T (h K IF) is an unbounded subset of IL 2 (Ω, F, IR) for every j = 1, 2,..., m. Proof. Let j = 1, 2,..., m be fixed and let us observe that K IF = dec ΣIF {0, 1}. Hence it follows that co K IF = co[dec ΣIF {0, 1}]= dec ΣIF [co {0,1}]= dec ΣIF [co {0,1}]. Therefore, α co K IF = dec ΣIF [co {0, α}] and h co K IF = dec ΣIF [co {0, h}]. But co {0, α} = [0, α] [0, h t (ω)] = co {0, h t (ω)} for a.e. (t, ω) [0, T ] Ω. Then α co K IF = dec ΣIF [co {0, α}] dec ΣIF [co {0, h}] = h cok IF. Therefore, dec F J j T (α co K IF ) dec F J j T (h co K IF), which implies dec F J j T (α co K IF) dec F J j T (h co K IF ). But dec F J j T (α K IF) is an unbounded subset of IL 2 (Ω, F, IR d ). Therefore, a set co[dec F J j T (α K IF)] is unbounded, which implies that dec F J j T (h co K IF) is unbounded, because by ([8], Remark 3.6 of Chap. 2) we have dec F J j T (h co K IF) = co[dec F J j T (h K IF)]. Finally, unboundedness of dec F J j T (h co K IF) implies that dec F J j T (h K IF) is unbounded. Lemma 2. Let C Σ IF be a set of positive measure µ = dt P such that dec F J j T (1I C K IF ) is unbounded subset of IL 2 (Ω, F, IR). If α > 0 and h IL 2 ([0, T ] Ω, Σ IF, IR) are such that 1I C h 1I C α and 1I C h = 0 then dec F J j T (1I Ch K IF ) is an unbounded subset of IL 2 (Ω, F, IR), where C = ([0, T ] Ω) \ C. Proof. Similarly as in the proof of Lemma 1 we get co{0, α} = [0, α] [0, h t (ω)] = co{0, h t (ω)} for every (t, ω) C. Then co{0, 1I C α} co{0, 1I C h}. Therefore, 1I C α co K IF = 1I C α co[dec Σ {0, 1}] = dec Σ [co{0, 1I C α}] dec Σ [co{0, 1I C h}] = 1I C h co K IF. But dec J j T (1I Cα co K IF ) = α J j T (1I C co K IF ) and J j T (1I C co K IF ) is unbounded. Then dec J j T (1I Ch co K IF ) contains an unbounded subset J j T (1I Cα co K IF ). Thus dec F J j T (1I Ch co K IF ) is an unbounded subset of IL 2 (Ω, F, IR). Hence, similarly as in the proof of Lemma 1, it follows that dec F J j T (1I Ch K IF ) is unbounded. Let us recall (see [4]) that for an m-dimensional IF-Brownian motion B = (B 1,..., B m ) defined on P IF and a given IF-nonanticipative set-valued process Φ = (Φ t ) 0 t T defined on P IF with values in the space Cl(IR d m ) of all nonempty closed subsets of IR d m, a set-valued stochastic integral t 0 Φ τ B τ is defined on [0, t] [0, T ] to be a set-valued random variable such that a set S Ft ( t 0 Φ τ db τ ) of all F t -measurable selectors of t 0 Φ τ db τ covers with a closed decomposable hull dec F J j T (S IF(Φ)) of the set J t (S IF (Φ)), where S IF (Φ)) denotes the set of all square

200 M. Kisielewicz integrable IF-nonanticipative selectors of Φ and J t (f)(ω) = t 0 f τ (ω)db τ for every ω Ω and f S IF (Φ). A set-valued stochastic integral t 0 Φ τ db τ is said to be integrably bounded if there exists a square integrably bounded random variable m : Ω IR + such that ρ( t 0 Φ τ db τ, {0}) m a.s., where ρ is the Hausdorff metric defined on the space Cl(IR d ) of all nonempty closed subsets of IR d. Immediately from ([2], Theorem 3.2) it follows that t 0 Φ τ db τ is integrably bounded if and only if S Ft ( t 0 Φ τ db τ ) is a bounded subset of the space IL 2 (Ω, F, IR d ), which by virtue of the above definition of t 0 Φ τ db τ is equivalent to boundedness of the set dec F J t (S IF (Φ)). But sup{e u 2 : u dec F J t (S IF (Φ))} = sup{e u 2 : u dec F J t (S IF (Φ))}. Therefore, t 0 Φ τ db τ is integrably bounded if and only if dec F J t (S IF (Φ)) is a bounded subset of the space IL 2 (Ω, F, IR d ). The idea of the proof of the main result of the paper is based on the following properties of measurable and integrably bounded multifunctions. For a given square integrably bounded IF-nonanticipative set-valued multifunction G : [0, T ] Ω Cl(IR d m ) such that G t (ω) = cl{gt n (ω) : n 1} for (t, ω) [0, T ] Ω we have that S IF (G) = dec ΣIF {g n : n 1} (see [8], Remark 3.6 of Chap. 2). Hence, it follows that for every arbitrarily taken f, g {g n : n 1} and a multifunction Φ : [0, T ] Ω Cl(IR d m ) defined by Φ t (ω) = {f t (ω), g t (ω)} for (t, ω) [0, T ] Ω, we have S IF (Φ) = dec ΣIF {f, g} dec ΣIF {g n : n 1} = S IF (G). Then dec F J T (S IF (Φ)) dec F J T (S IF (G)). Therefore, for the proof that sup{e u 2 : u dec F J T (S IF (G))} = it is enough only to verify that there are f, g {g n : n 1} such that sup{e u 2 : u dec F J (S IF (Φ))} =. 2. Boundedness of dec F J T (h K IF ) for matrix-valued processes We shall consider here properties of the set dec F J T (h K IF ) with K IF defined above and h IL 2 ([0, T ] Ω, Σ IF, IR d m ), where J T (h K IF ) is defined by vector valued Itô integral of d m-matrix processes with respect to an m-dimensional IF-Brownian motion B = (B 1,..., B m ). Let us recall that for h IL 2 ([0, T ] Ω, Σ IF, IR d m ) and h ij IL 2 ([0, T ] Ω, Σ IF, IR) such that h t (ω) = (h ij t (ω)) d m for every (t, ω) [0, T ] Ω, the norm h is defined by h 2 = E T 0 h t 2 dt, where h t 2 = d m i=1 j=1 hij t 2. By Π(Ω, F) we denote the family of all finite F-measurable partitions of Ω. We begin with the following lemma. Lemma 3. For every h IL 2 ([0, T ] Ω, Σ IF, IR d m ) a set dec F J T (h K IF ) is a bounded subset of IL 2 (Ω, F, IR d ) if and only if dec F J j T (hi,j K IF ) is a bounded subset of IL 2 (Ω, F, IR) for every i = 1,..., d and j = 1,..., m, where h ij IL 2 ([0, T ]

Boundedness of set-valued stochastic integrals 201 Ω, Σ IF, IR) are such that h t (ω) = (h ij t (ω)) d m for every (t, ω) [0, T ] Ω and i = 1,..., d and j = 1,..., m. Proof. Let us observe that for every D Σ IF one has ( m m J T (1I D h) = J j T (1I Dh 1,j ),..., J j T (1I Dh )) d,j, j=1 where J j T (1I Dh i,j ) = T 0 1I Dh i,j t db j t for every D Σ IF, i = 1,..., d and j = 1,..., m, and u denotes the transpose of a matrix u IR 1 d. Therefore, we have sup{e u 2 : u dec F J T (h K IF )} = sup{e u 2 : u dec F {J T (1I D h) : D Σ IF } { d N m = sup sup E 1I Ak J j T (1I D k h i,j 2 } ) :(A k ) N k=1 Π(Ω, F), (D k) N k=1 Σ IF. N 1 i=1 k=1 j=1 Hence it follows that if dec F J j T (hi,j K IF ) is a bounded subset of IL 2 (Ω, F, IR) for every i = 1,..., d and j = 1,..., m then there is a positive random variable ϕ IL 2 (Ω, F, IR) such that J j T (1I D k h i,j ) ϕ a.s. for (i, j) {1,..., d} {1,..., m}, N 1, every k = 1,..., N and every family (D k ) N k=1 Σ IF. Thus sup{e u 2 : u dec F J T (h K IF )} m 2 d ϕ 2 <. Suppose a set dec F J T (h K IF ) is a bounded subset of IL 2 (Ω, F, IR d ) and there is I J {1,..., d} {1,..., m}, such that a set dec F J j T (hi,j K IF ) is unbounded for every (i, j) I J and is bounded for (i, j) {1,..., d} {1,..., m}\(i J). But boundedness of dec F J T (h K IF ) implies that m j=1 dec FJ j T (hi,j K IF ) is bounded for every i = 1,..., d, which implies that d m i=1 j=1 dec FJ j T (hi,j K IF ) is bounded. Indeed, suppose dec F J T (h K IF ) is bounded and there is ī {1,..., d} such that we have sup{e u 2 : u dec F { m j=1 J j T (1I Dhī,j ) : D Σ IF } =. But { { m }} sup E u 2 :u dec F J j T (1I Dhī,j ):D Σ IF i=1 k=1 j=1 { N m = sup sup E 1I Ak J j T (1I 2 } D hī,j k ) :(A k ) N k=1 Π(Ω, F), (D k) N k=1 Σ IF N 1 k=1 j=1 { d N m sup sup E 1I Ak J j T (1I D k h i,j 2 } ) :(A k ) N k=1 Π(Ω, F), (D k) N k=1 Σ IF N 1 j=1 = sup{e u 2 : u dec F J T (h K IF )}. Then sup{e u 2 : u dec F J T (h K IF )} =. A contradiction. Thus dec F { m j=1 J j T (1I Dh i,j ) : D Σ IF } = m j=1 dec FJ j T (hi,j K IF ) is bounded for every i = j=1

202 M. Kisielewicz 1,..., d, which implies that a set d m i=1 j=1 dec FJ j T (hi,j K IF ) is bounded. Let us observe now that (1) dec F J j 2 T (hi,j K IF ) (i,j) I J 2 dec F J j T (hi,j K IF ) (i,j) (I J) 2 + 2 d m dec F J j 2 T (hi,j K IF ), i=1 j=1 where (I J) = {1,..., d} {1,..., m} \ (I J) and Λ = sup{ u : u Λ} for Λ IL 2 (Ω, F, IR). Indeed, for every u d m i=1 j=1 dec FJ j T (hi,j K IF ) and every (i, j) {1,..., d} {1,..., m} there is u i,j dec F J j T (hi,j K IF ) such that m j=1 u i,j. But d i=1 m j=1 u i,j = (i,j) I J u i,j + (i,j) (I J) u i,j. u = d i=1 Therefore, (i,j) I J u i,j that (i,j) I J u i,j d i=1 j=1 = d i=1 m j=1 u i,j (i,j) (I J) u i,j, which implies m dec F J j T (hi,j K IF ) + ( 1) dec F J j T (hi,j K IF ). (i,j) (I J) Then for every v = (i,j) I J u i,j (i,j) (I J) dec FJ j T (hi,j K IF ) one has { v 2 2 sup u 2 : u d m i=1 j=1 { + 2 sup u 2 : u ( 1) } dec F J j T (hi,j K IF ) } dec F J j T (hi,j K IF ). (i,j) (I J) But sup{ u 2 : u ( 1) (i,j) (I J) dec FJ j T (hi,j K IF )} = sup{ u 2 : u (i,j) (I J) dec FJ j T (hi,j K IF )}. Therefore, from the above inequality, the inequality (1) follows, which implies that (i,j) I J dec FJ j T (hi,j K IF ) is a bounded subset of the space IL 2 (Ω, F, IR). A contradiction. Then boundedness of a set dec F J T (h K IF ) implies that dec F J j T (hi,j K IF ) is a bounded subset of IL 2 (Ω, F, IR) for every i = 1,..., d and j = 1,..., m. Corollary 1. For every matrix-valued process h = (h ij ) d m IL 2 ([0, T ] Ω, Σ IF, IR d m ) a set dec F J T (h K IF ) is an unbounded subset of the space IL 2 (Ω, F, IR d ) if there exist (ī, ) {1,..., d} {1,..., m} and a set C Σ IF of positive measure µ = dt P such that (hī t (ω) > 0 for (t, ω) C, hī t (ω) = 0 for (t, ω) C and dec F J T (1I Chī K IF ) is an unbounded subset of the space IL 2 (Ω, F, IR), where C = [0, T ] Ω \ C.

Boundedness of set-valued stochastic integrals 203 Proof. Immediately from Lemma 3 it follows that a set dec F J T (h K IF ) is unbounded if there exists a pair (ī, ) {1,..., d} {1,..., m} such that a set dec F J T (hī K IF ) is an unbounded subset of the space IL 2 (Ω, F, IR). It is clear that a set dec F J T (1I Dhī K IF ) = {0} for every D Σ IF such that (hī t (ω) = 0 for a.e. (t, ω) D and dec F J T (hī K IF ) = dec F J T (1I D hī K IF ), where D = [0, T ] Ω\D. Then by unboundedness of a set dec F J T (hī K IF ) there is a set C Σ IF of positive measure µ = dt P such that (hī t (ω) > 0 for (t, ω) C, hī t (ω) = 0 for a.e. (t, ω) C and dec F J T (1I Chī K IF ) is an unbounded subset of the space IL 2 (Ω, F, IR). 3. Unboundedness of Itô set-valued stochastic integrals Let B = (B 1,..., B m ) be an m-dimensional IF-Brownian motion defined on P IF and G = (G t ) 0 t T be an IF-nonanticipative square integrably bounded setvalued stochastic process with values in the space Cl(IR d m ) of all nonempty closed subsets of the space IR d m. We will show that if G possesses an IFnonanticipative Castaing s representation (g n ) n=1 such that there are α > 0 and f, g {g n : n 1} such that there exist (ī, ) {1,..., d} {1,..., m} and real-valued processes f ī and gī, elements of matrix-valued processes f and g, respectively such that f ī t (ω) gī t (ω) α for a.e. (t, ω) [0, T ] Ω then a set-valued stochastic integral T 0 G tdb t is not integrably bounded. We begin with the following lemmas. Lemma 4. Let G = (G t ) 0 t T be an IF-nonanticipative square integrably bounded set-valued stochastic process with values in the space Cl(IR d m ) possessing an IFnonanticipative Castaing s representation (g n ) n=1 such that there are α > 0 and f, g {g n : n 1} such that there exist (ī, ) {1,..., d} {1,..., m} and real-valued processes f ī and gī, elements of matrix-valued processes f and g, respectively and such that f ī t (ω) gī t (ω) α for a.e. (t, ω) [0, T ] Ω. There are an IF-nonanticipative Castaing s representation ( g n ) n=1 of G and matrixvalued processes f, g { g n : n 1} possessing elements f ī = ( f ī t ) 0 t T and gī = ( gī t ) 0 t T, respectively and such that f ī t (ω) gī t (ω) α for a.e. (t, ω) [0, T ] Ω. Proof. Let a set-valued process G, a Castaing s representation (g n ) n=1 of G, and f, g {g n : n 1} be such as above. For simplicity assume that f ī t (ω) (ω) α is satisfied for every (t, ω) [0, T ] Ω and let us denote processes gī t f ī and gī by φ, ψ, respectively. We have φ t (ω) ψ t (ω) for every (t, ω)

204 M. Kisielewicz [0, T ] Ω. Let A = {(t, ω) [0, T ] Ω : φ t (ω) > ψ t (ω)}. If A = [0, T ] Ω then (ω) gī t (ω) = f ī t (ω) gī t (ω) α for every (t, ω) [0, T ] Ω. f ī t Suppose 0 < µ(a ) < T, where A = ([0, T ] Ω) \ A, let φ = 1I A φ + 1I A ψ and ψ = 1I A ψ + 1I A φ. It is clear that {φ t (ω), ψ t (ω)} = { φ t (ω), ψ t (ω)} for every (t, ω) [0, T ] Ω. Furthermore, for every (t, ω) A we have φ t (ω) ψ t (ω) = φ t (ω) ψ t (ω) = φ t (ω) ψ t (ω) α. Similarly for (t, ω) A we get φ t (ω) ψ t (ω) = ψ t (ω) φ t (ω) = (φ t (ω) ψ t (ω)) = φ t (ω) ψ t (ω)) α. Taking f ī = φ and gī, = ψ we obtain f ī t (ω) gī t (ω) α for every (t, ω) [0, T ] Ω. To get a required new Castaing s representation of G we can change in the given above Castaing s representation (g n ) n=1 its elements f and g by new matrix-valued functions f and g obtained from f and g by changing in matrices f and g theirs elements f ī and gī by f ī and gī, respectively. Lemma 5. For every f, g IL 2 ([0, T ] Ω, Σ IF, IR d m ) a set-valued stochastic integral T 0 F tdb t of a multiprocess F defined by F t (ω) = {f t (ω), g t (ω)} for (t, ω) [0, T ] Ω, is square integrably bounded if and only if dec F J T [(f g) K IF ] is a bounded subset of the space IL 2 (Ω, F, IR d ). Proof. Let us observe that a set-valued stochastic integral T 0 F tdb t is square integrably bounded if and only if a set dec F J T (S IF (F )) is a bounded subset of the space IL 2 (Ω, F, IR d ). By ([8], Remark 3.6 of Chap. 2) a set S IF (F ) is defined by S IF (F ) = dec ΣIF {f, g}. But dec ΣIF {f, g} = {1I D (f g) + g : D Σ IF } = {1I D (f g) : D Σ IF }+g = (f g) K IF +g. Then dec F J T (S IF (F )) = dec F J T [(f g) K IF ]+ J T (g). Thus dec F J T (S IF (F )) is bounded if and only if dec F J T [(f g) K IF ] is a bounded subset of the space IL 2 (Ω, F, IR). Boundedness of dec F J T [(f g) K IF ] is equivalent to boundedness of dec F J T [(f g) K IF ]. Then a set-valued stochastic integral T 0 F tdb t is square integrably bounded if and only if dec F J T [(f g) K IF ] is a bounded subset of the space IL 2 (Ω, F, IR d ). Now we prove the main result of the paper. Theorem 6. Let G = (G t ) 0 t T be an IF-nonanticipative square integrably bounded set-valued stochastic process with values in the space Cl(IR d m ) possessing an IF-nonanticipative Castaing s representation (g n ) n=1 such that there are α > 0 and f, g {g n : n 1} such that there are (ī, ) {1,..., d} {1,..., m} and real-valued processes f ī, gī, elements of matrix-valued processes f and g, respectively and such that f ī t (ω) gī t (ω) α for a.e. (t, ω) [0, T ] Ω. A set-valued stochastic integral T 0 G tdb t is not integrably bounded. Proof. Let G = (G t ) 0 t T and IF-nonanticipative Castaing s representation (g n ) n=1 of G possess properties described above. By virtue of Lemma 4 there

Boundedness of set-valued stochastic integrals 205 is an IF-nonanticipative Castaing s representation ( g n ) n=1 of G containing processes f, g { g n : n 1} and such that there exist real-valued stochastic processes f ī = ( f ī t ) 0 t T, gī = ( gī t ) 0 t T, elements of matrix-valued prosesses f, g, respectively and such that f ī t (ω) gī t (ω) α for a.e. (t, ω) [0, T ]. By virtue of Lemma 1 it follows that dec F J [( f ī gī ) K IF ] is an unbounded subset of IL 2 (Ω, F, IR). Hence, by Lemma 3 it follows that dec F J T [( f g) K IF ] is an unbounded subset of IL 2 (Ω, F, IR), which by Lemma 5 implies that a setvalued stochastic integral T 0 F tdb t of a multiprocess F defined by F t (ω) = {f t (ω), g t (ω)} for every (t, ω) [0, T ] Ω is not integrably bounded. But T 0 F tdb t T 0 G tdb t a.s. Therefore, a set-valued stochastic integral T 0 G tdb t is not integrably bounded. It is natural to expect that a set-valued stochastic integral T 0 G tdb t is integrably bounded if and only if G possesses an IF-nonanticipative Castaing s representation containing only one element. Such result can be obtained if the following hypothesis would be satisfied. Hypothesis B. For every set C Σ IF of positive measure µ = dt P a set dec F J T (1I C K IF ) is an unbounded subset of the space IL 2 (Ω, F, IR). To obtain such result we begin with the following lemma. Lemma 7. Let h IL 2 ([0, T ] Ω, Σ IF, IR) be a non-negative process such that there exists a set C Σ IF of positive measure µ = dt P such that h t (ω) > 0 for (t, ω) C and h t (ω) = 0 for (t, ω) C, where C = ([0, T ] Ω) \ C. For every ε (0, µ(c)) there is an Σ IF -measurable set C ε C of positive measure µ and a real number α ε > 0 such that h t (ω) α ε for (t, ω) C ε. Proof. Let C Σ IF be a set of positive measure µ = dt P such that h t (ω) > 0 for (t, ω) C and h t (ω) = 0 for (t, ω) C. We have C = {(t, ω) [0, T Ω : h t (ω) > 0}. Let C m = {(t, ω) C : h t (ω) m} for every m > 0. We have C = m>0 C m and C m C n for m n. Put C k = C mk, where m k = 1/k. We have C = C k=1 k and C k C k+1 for k 1. Therefore, µ(c) = lim k µ( C k ). Then for every ε (0, µ(c)) there is k ε such that µ(c) µ( C kε ) < ε. Thus µ( C kε ) > µ(c) ε > 0. Let C ε = C kε and α ε = 1/k ε. By the definition of a set C kε we get 1/k ε h t (ω) for (t, ω) C ε. Then h t (ω) α ε for (t, ω) C ε. We can prove now the following result. Theorem 8. If the Hypothesis B is satisfied then for every square integrably bounded IF-nonanticipative set-valued stochastic process G = (G t ) 0 t T with values in the space Cl(IR d m ), a set-valued stochastic integral T 0 G tdb t is integrably bounded if and only if there is an IF-nonanticipative Castaing s representation (g n ) n=1 of G such that gn g m = 0 for every n, m 1.

206 M. Kisielewicz Proof. If (g n ) n=1 is an IF-nonanticipative Castaing s representation of G such that if g n g m = 0 for every n, m 1 then a set-valued stochastic integral T 0 G tdb t is integrably bounded because in such a case we have G t (ω) = cl{g t (ω)} for (t, ω) [0, T ] Ω with g IL 2 ([0, T ] Ω, Σ IF, IR d m ) and therefore, T 0 G tdb t = T 0 g tdb t. Suppose (g n ) n=1 is an IF-nonanticipative Castaing s representation of G such that there are f, g {g n : n 1} such that f g > 0 and let C Σ IF be a set of positive measure µ = t P such that f t (ω) g t (ω) > 0 for a.e. (t, ω) C and f t (ω) g t (ω) = 0 for a.e. (t, ω) C. Similarly as in the proof of Lemma 4 we can select an IF-nonanticipative Castaing s representation ( g n ) n=1 of G such that there are f, g { g n : n 1} having elements f i,j and g i,j i,j such that f t (ω) g i,j i,j t (ω) > 0 for a.e. (t, ω) C and f t (ω) g i,j t (ω) = 0 for a.e. (t, ω) C. By virtue of Lemma 7 for ε > 0 there is Σ IF -measurable set C ε C of positive measure µ = dt P and a real number α ε > 0 such that 1I Cε hi,j 1I Cε α ε, where h i,j = f i,j g i,j. Let h i,j = 1I Cε hi,j + 1I C ε 0, where Cε = [0, T ] Ω \ C ε. We have 1I Cε h i,j 1I Cε α ε, 1I C ε h i,j = 0 and dec F J j T (1I C ε K IF ) is an unbounded subset of the space IL 2 (Ω, F, IR). Therefore, by virtue of Lemma 2 a set dec F J j T (1I C ε hi,j K IF ) is an unbounded subset of IL 2 (Ω, F, IR). But 1I Cε hi,j 1I C hi,j. Therefore, [0, 1I Cε hi,j ] [0, 1I C hi,j ], which similarly as in the proof of Lemma 1, implies that dec F J j T (1I C ε hi,j K IF ) dec F J j T (1I C h i,j K IF ). Therefore, dec F J j T (1I C h i,j K IF ) is an unbounded subset of the space IL 2 (Ω, F, IR), which by Corollary 1 implies that dec F J j T [( f g) K IF ] is an unbounded subset of the space IL 2 (Ω, F, IR d ). Hence by Lemma 5 it folows that a set-valued stochastic integral T 0 F tdb t of a set-valued process F defined by F t (ω) = {f t (ω), g t (ω)} for (t, ω) [0, T ] Ω is not square integrably bounded. But T 0 F tdb t T 0 G tdb t a.s. Therefore, a set-valued stochastic integral T 0 G tdb t is not square integrably bounded. References [1] F. Hiai, Multivalued stochastic integrals and stochastic inclusions, Division of Applied Mathematics, Research Institute of Applied Electricity, Sapporo 060 Japan (not published). [2] F. Hiai and H. Umegaki, Integrals, conditional expectations, and martingales of multivalued functions, J. Multivariate Anal. 7 (1977), 149 182. doi:10.1016/0047-259x(77)90037-9 [3] Sh. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis I (Kluwer Academic Publishers, Dordrecht, London, 1997). doi:10.1007/978-1-4615-6359-4 [4] E.J. Jung and J. H. Kim, On the set-valued stochastic integrals, Stoch. Anal. Appl. 21 (2) (2003), 401 418. doi:10.1081/sap-120019292

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