The present paper contains some new conditions implying unboundedness of the above type set-valued stochastic integrals

doi:10.7151/dmdico.1173

BOUNDEDNESS OF SET-VALUED STOCHASTIC INTEGRALS

Micha l Kisielewicz

Faculty of Mathematics, Computer Science and Econometrics University of Zielona G´ora

Prof. Z. Szafrana 4a, 65–516 Zielona G´ora, Poland e-mail: M.Kisielewicz@wmie.uz.zgora.pl

Abstract

The paper deals with integrably boundedness of Itˆo 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 in- tegrably 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 Itˆo set-valued stochastic in- tegrals 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, Itˆo set-valued integral, set-valued stochas- tic 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 Itˆo 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 Itˆo set-valued stochastic integrals, defined as subsets of the spaces IL²(Ω, IRⁿ) and IL²(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 IRⁿ and X , because they are not decomposable subsets

of IL²(Ω, IRⁿ) and IL²(Ω, X ), respectively. J.Jung and J.H. Kim (see [4]) defined the Itˆo 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 Itˆo 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 = (Ft)_0≤t≤T satisfying the usual conditions and such that there are real stochasticaly independet IF-Brownian motions B^j = (B_t^j)0≤t≤T, j = 1, 2, . . . , m, defined on PIF. By k · k and || · ||

we denote the norms of the spaces IL²([0, T ] × Ω, ΣIF, IR^d×m) and IL²(Ω, F , IR^d), respectively, where Σ_IFdenotes the σ-algebra of all IF-nonanticipative subsets of [0, T ] × Ω. For a given set Λ ⊂ IL²([0, T ] × Ω, ΣIF, IR^d×m) by decΣIFΛ and decΣIFΛ 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 ⊂ IL²([0, T ] × Ω, ΣIF, IR^d×m) is said to be decomposable if for every u, v ∈ Λ and D ∈ ΣIF one has 1IDu + 1ID^∼v ∈ Λ, where D^∼ = ([0, T ] × Ω) \ D.

In a similar way for a given set Λ ⊂ IL²(Ω, F , IR^d) decomposable sets decFΛ and decFΛ are defined.

The Fr´echet-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 ∈ ΣIF : µ(C4D) = 0} for C ∈ ΣIF and λ([A], [B]) = µ(A4B) for A, B ∈ Σ_IF, A4B = (A\B)∪(B \A). It is a complete metric space homeomorphic with the set K_IF ⊂ IL²([0, T ] × Ω, Σ_IF, IR) defined by K_IF = {1I_D ∈ IL²([0, T ] × Ω, ΣIF, IR) : D ∈ ΣIF}. Indeed, for every A, B ∈ Σ_IF we have λ([A], [B]) = µ(A4B) =R R

A4BdP dt = ERT

0 1I_A4Bdt = ERT

0 |1I_A− 1I_B|²dt = k1IA− 1I_Bk². Let J_T^j denotes the isometry on the space IL²([0, T ] × Ω, Σ_IF, IR) defined by J_T^j(g) =RT

0 gtdB_t^j. It can be proved that J_T^j(KIF) is not an integrably bounded subset of the space IL²(Ω, F , IR) for every j = 1, 2, . . . , m (see [11], Corollary 3.14). It implies (see [11], Theorem 2.2) that decFJ_T^j(K_IF) is an unbounded subset of the space IL²(Ω, F , IR), where the closure is taken with respect to the norm topology of the space IL²(Ω, F , IR). Hence it follows that decFJ_T^j(K_IF) and decFJ_T^j(co K_IF) are unbounded subsets of this space, because they contain an unbounded subset J_T^j(K_IF). Therefore, co[decFJ_T^j(K_IF)] is unbounded, which by

virtue of ([9], Proposition 3.1) implies that decFJ_T^j(co KIF) is an unbounded subset of the space IL²(Ω, F , IR). Finally let us observe that for every α > 0 a set decFJ_T^j(α · K_IF) is an unbounded subset of IL²(Ω, F , IR^d), because decFJ_T^j(α · K_IF) = α · decFJ_T^j(KIF). Hence the following basic results of the paper follows.

Lemma 1. If α > 0 and h ∈ IL²([0, T ] × Ω, Σ_IF, IR) are such that h_t(ω) ≥ α for a.e. (t, ω) ∈ [0, T ] × Ω then the set decFJ_T^j(h · K_IF) is an unbounded subset of IL²(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 KIF = decΣIF[co {0, α}] and h · co KIF = decΣIF[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, decFJ_T^j(α · co KIF) ⊂ decFJ_T^j(h · co KIF), which implies decFJ_T^j(α · co KIF) ⊂ decFJ_T^j(h · co K_IF). But decFJ_T^j(α · K_IF) is an unbounded subset of IL²(Ω, F , IR^d). Therefore, a set co[decFJ_T^j(α · KIF)] is unbounded, which implies that decFJ_T^j(h · co KIF) is unbounded, because by ([8], Remark 3.6 of Chap. 2) we have decFJ_T^j(h·co K_IF) = co[decFJ_T^j(h · KIF)]. Finally, unboundedness of decFJ_T^j(h · co KIF) implies that decFJ_T^j(h · K_IF) is unbounded.

Lemma 2. Let C ∈ ΣIF be a set of positive measure µ = dt × P such that decFJ_T^j(1I_C·K_IF) is unbounded subset of IL²(Ω, F , IR). If α > 0 and h ∈ IL²([0, T ]×

Ω, ΣIF, IR) are such that 1ICh ≥ 1ICα and 1IC^∼h = 0 then decFJ_T^j(1ICh · KIF) is an unbounded subset of IL²(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_Cα} ⊂ co{0, 1I_Ch}. Therefore, 1ICα · co KIF = 1ICα · co[decΣ{0, 1}] = dec_Σ[co{0, 1ICα}] ⊂ decΣ[co{0, 1ICh}] = 1I_Ch · co K_IF. But dec J_T^j(1I_Cα · co K_IF) = α · J_T^j(1I_C· co K_IF) and J_T^j(1I_C· co K_IF) is unbounded. Then dec J_T^j(1ICh · co KIF) contains an unbounded subset J_T^j(1ICα · co K_IF). Thus decFJ_T^j(1I_Ch · co K_IF) is an unbounded subset of IL²(Ω, F , IR).

Hence, similarly as in the proof of Lemma 1, it follows that decFJ_T^j(1ICh · KIF) is unbounded.

Let us recall (see [4]) that for an m-dimensional IF-Brownian motion B = (B¹, . . . , B^m) defined on PIF and a given IF-nonanticipative set-valued process Φ = (Φt)_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 Rt

0ΦτBτ is defined on [0, t] ⊂ [0, T ] to be a set-valued random variable such that a set SFt(Rt

0 ΦτdBτ) of all Ft-measurable selectors of Rt

0ΦτdBτ covers with a closed decomposable hull decFJ_T^j(S_IF(Φ)) of the set Jt(S_IF(Φ)), where S_IF(Φ)) denotes the set of all square

integrable IF-nonanticipative selectors of Φ and Jt(f )(ω) =Rt

0fτ(ω)dBτ for every ω ∈ Ω and f ∈ SIF(Φ). A set-valued stochastic integral Rt

0ΦτdBτ is said to be integrably bounded if there exists a square integrably bounded random variable m : Ω → IR⁺ such that ρ(Rt

0Φ_τdB_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. Imme- diately from ([2], Theorem 3.2) it follows thatRt

0ΦτdBτ is integrably bounded if and only if SFt(R_t

0ΦτdBτ) is a bounded subset of the space IL²(Ω, F , IR^d), which by virtue of the above definition of R_t

0ΦτdBτ is equivalent to boundedness of the set decFJ_t(SIF(Φ)). But sup{E|u|² : u ∈ decFJ_t(SIF(Φ))} = sup{E|u|² : u ∈ decFJ_t(S_IF(Φ))}. Therefore, Rt

0Φ_τdB_τ is integrably bounded if and only if decFJ_t(S_IF(Φ)) is a bounded subset of the space IL²(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{g_tⁿ(ω) : n ≥ 1} for (t, ω) ∈ [0, T ] × Ω we have that S_IF(G) = dec_ΣIF{gⁿ : n ≥ 1} (see [8], Remark 3.6 of Chap. 2).

Hence, it follows that for every arbitrarily taken f, g ∈ {gⁿ : 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{gⁿ : n ≥ 1} = SIF(G). Then decFJ_T(SIF(Φ)) ⊂ decFJ_T(SIF(G)). Therefore, for the proof that sup{E|u|² : u ∈ decFJ_T(S_IF(G))} = ∞ it is enough only to verify that there are f, g ∈ {gⁿ: n ≥ 1} such that sup{E|u|² : u ∈ decFJ (S_IF(Φ))} = ∞.

2. Boundedness of decFJ_T(h · K_IF) for matrix-valued processes We shall consider here properties of the set decFJ_T(h · K_IF) with K_IF defined above and h ∈ IL²([0, T ] × Ω, ΣIF, IR^d×m), where JT(h · KIF) is defined by vector valued Itˆo integral of d × m-matrix processes with respect to an m-dimensional IF-Brownian motion B = (B¹, . . . , B^m). Let us recall that for h ∈ IL²([0, T ] × Ω, ΣIF, IR^d×m) and h^ij ∈ IL²([0, T ] × Ω, ΣIF, IR) such that ht(ω) = (h^ij_t (ω))d×m

for every (t, ω) ∈ [0, T ] × Ω, the norm khk is defined by khk² = ERT

0 |h_t|²dt, where |h_t|² = Pd

i=1

Pm

j=1|h^ij_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 ∈ IL²([0, T ] × Ω, Σ_IF, IR^d×m) a set decFJ_T(h · K_IF) is a bounded subset of IL²(Ω, F , IR^d) if and only if decFJ_T^j(h^i,j·K_IF) is a bounded subset of IL²(Ω, F , IR) for every i = 1, . . . , d and j = 1, . . . , m, where h^ij ∈ IL²([0, T ] ×

Ω, ΣIF, IR) are such that ht(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 J_T(1I_Dh) =

 m

X

j=1

J_T^j(1I_Dh^1,j), . . . ,

m

X

j=1

J_T^j(1I_Dh^d,j)

∗

,

where J_T^j(1IDh^i,j) = RT

0 1IDh^i,j_t dB^j_t for every D ∈ ΣIF, 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 ∈ decFJ_T(h · KIF)} = sup{E|u|² : u ∈ decF{J_T(1IDh) : D ∈ ΣIF}

= sup

N ≥1

sup

 d

X

i=1

E

N

X

k=1

1IAk

m

X

j=1

J_T^j(1IDkh^i,j)    

2

:(Ak)^N_k=1 ∈ Π(Ω, F ), (D_k)^N_k=1⊂ Σ_IF

 .

Hence it follows that if decFJ_T^j(h^i,j· K_IF) is a bounded subset of IL²(Ω, F , IR) for every i = 1, . . . , d and j = 1, . . . , m then there is a positive random variable ϕ ∈ IL²(Ω, F , IR) such that |J_T^j(1I_Dkh^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 ∈ decFJ_T(h · KIF)} ≤ m²d kϕk²< ∞.

Suppose a set decFJ_T(h · K_IF) is a bounded subset of IL²(Ω, F , IR^d) and there is I × J ⊂ {1, . . . , d} × {1, . . . , m}, such that a set decFJ_T^j(h^i,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 decFJ_T(h · K_IF) implies that P_m

j=1decFJ_T^j(h^i,jK_IF) is bounded for every i = 1, . . . , d, which implies thatPd

i=1

Pm

j=1decFJ_T^j(h^i,j·K_IF) is bounded. Indeed, suppose decFJ_T(h · KIF) is bounded and there is ¯i ∈ {1, . . . , d}

such that we have sup{E|u|²: u ∈ decF{Pm

j=1J_T^j(1I_Dh^¯i,j) : D ∈ Σ_IF} = ∞. But sup



E|u|²: u ∈ decF

m

X

j=1

J_T^j(1I_Dh^¯i,j):D ∈Σ_IF



= sup

N ≥1

sup

 E

N

X

k=1

1IAk

m

X

j=1

J_T^j(1IDkh^¯i,j)    

2

:(Ak)^N_k=1∈ Π(Ω, F ), (D_k)^N_k=1⊂ Σ_IF



≤ sup

N ≥1

sup

 d

X

i=1

E

N

X

k=1

1I_Ak    

m

X

j=1

J_T^j(1I_Dkh^i,j)    

2

:(A_k)^N_k=1∈ Π(Ω, F ), (D_k)^N_k=1⊂ Σ_IF



= sup{E|u|² : u ∈ decFJ_T(h · K_IF)}.

Then sup{E|u|²: u ∈ decFJ_T(h · K_IF)} = ∞. A contradiction. Thus decF{Pm j=1

J_T^j(1I_Dh^i,j) : D ∈ Σ_IF} = P_m

j=1decFJ_T^j(h^i,j · K_IF) is bounded for every i =

1, . . . , d, which implies that a set Pd i=1

Pm

j=1decFJ_T^j(h^i,j· K_IF) is bounded. Let us observe now that

(1)

X

(i,j)∈I×J

decFJ_T^j(h^i,j· K_IF)    

2

≤ 2    

X

(i,j)∈(I×J )^∼

decFJ_T^j(h^i,j· K_IF)    

2

+ 2    

d

X

i=1 m

X

j=1

decFJ_T^j(h^i,j· K_IF)    

2

,

where (I × J )^∼= {1, . . . , d} × {1, . . . , m} \ (I × J ) and |kΛk| = sup{ || u || : u ∈ Λ}

for Λ ⊂ IL²(Ω, F , IR). Indeed, for every u ∈ Pd i=1

Pm

j=1decFJ_T^j(h^i,j · K_IF) and every (i, j) ∈ {1, . . . , d} × {1, . . . , m} there is u_i,j ∈ dec_FJ_T^j(h^i,j· K_IF) such that u = Pd

i=1

Pm

j=1ui,j. But Pd i=1

Pm

j=1ui,j = P

(i,j)∈I×Jui,j +P

(i,j)∈(I×J )^∼ui,j. Therefore, P

(i,j)∈I×Jui,j = Pd i=1

Pm

j=1ui,j −P

(i,j)∈(I×J )^∼ui,j, which implies that

X

(i,j)∈I×J

u_i,j ∈

d

X

i=1 m

X

j=1

decFJ_T^j(h^i,j· K_IF) + (−1) · X

(i,j)∈(I×J )^∼

decFJ_T^j(h^i,j· K_IF).

Then for every v =P

(i,j)∈I×Jui,j ∈P

(i,j)∈(I×J )decFJ_T^j(h^i,j· K_IF) one has

|| v ||² ≤ 2 sup



|| u ||² : u ∈

d

X

i=1 m

X

j=1

decFJ_T^j(h^i,j· K_IF)



+ 2 sup



|| u ||² : u ∈ (−1) · X

(i,j)∈(I×J )^∼

decFJ_T^j(h^i,j· K_IF)

 .

But sup{ || u ||² : u ∈ (−1) ·P

(i,j)∈(I×J )^∼decFJ_T^j(h^i,j· K_IF)} = sup{ || u ||² : u ∈ P

(i,j)∈(I×J )^∼decFJ_T^j(h^i,j · K_IF)}. Therefore, from the above inequality, the in- equality (1) follows, which implies thatP

(i,j)∈I×JdecFJ_T^j(h^i,j· K_IF) is a bounded subset of the space IL²(Ω, F , IR). A contradiction. Then boundedness of a set decFJ_T(h·KIF) implies that decFJ_T^j(h^i,j·K_IF) is a bounded subset of IL²(Ω, F , IR) for every i = 1, . . . , d and j = 1, . . . , m.

Corollary 1. For every matrix-valued process h = (h^ij)_d×m ∈ IL²([0, T ] × Ω, ΣIF, IR^d×m) a set decFJ_T(h · KIF) is an unbounded subset of the space IL²(Ω, F , IR^d) if there exist (¯i, ¯j) ∈ {1, . . . , d} × {1, . . . , m} and a set C ∈ Σ_IF of positive measure µ = dt × P such that |(h^{¯i ¯}_t^j(ω)| > 0 for (t, ω) ∈ C, h^{¯i ¯}_t^j(ω) = 0 for (t, ω) ∈ C^∼ and decFJ_T^¯j(1ICh^{¯i ¯j} · K_IF) is an unbounded subset of the space IL²(Ω, F , IR), where C^∼= [0, T ] × Ω \ C.

Proof. Immediately from Lemma 3 it follows that a set decFJ_T(h · KIF) is un- bounded if there exists a pair (¯i, ¯j) ∈ {1, . . . , d} × {1, . . . , m} such that a set decFJ_T^¯j(h^{¯i ¯j} · K_IF) is an unbounded subset of the space IL²(Ω, F , IR). It is clear that a set decFJ_T^¯j(1I_Dh^{¯i ¯j}· K_IF) = {0} for every D ∈ Σ_IF such that |(h^{¯i ¯}_t^j(ω)| = 0 for a.e. (t, ω) ∈ D and decFJ_T^¯j(h^{¯i ¯j} · K_IF) = decFJ_T^¯j(1I_D^∼h^{¯i ¯j} · K_IF), where D^∼= [0, T ] × Ω \ D. Then by unboundedness of a set decFJ_T^¯j(h^{¯i ¯j}· K_IF) there is a set C ∈ ΣIFof positive measure µ = dt × P such that |(h^{¯i ¯}_t^j(ω)| > 0 for (t, ω) ∈ C, h^{¯i ¯}_t^j(ω) = 0 for a.e. (t, ω) ∈ C^∼ and decFJ_T^¯j(1I_Ch^{¯i ¯j}· K_IF) is an unbounded subset of the space IL²(Ω, F , IR).

3. Unboundedness of Itˆo set-valued stochastic integrals Let B = (B¹, . . . , B^m) be an m-dimensional IF-Brownian motion defined on P_IF and G = (Gt)0≤t≤T be an IF-nonanticipative square integrably bounded set- valued 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 IF- nonanticipative Castaing’s representation (gⁿ)^∞_n=1 such that there are α > 0 and f, g ∈ {gⁿ : n ≥ 1} such that there exist (¯i, ¯j) ∈ {1, . . . , d} × {1, . . . , m} and real-valued processes f^{¯i ¯j} and g^{¯i ¯j}, elements of matrix-valued processes f and g, respectively such that |f^¯_t^{i ¯j}(ω) − g^¯_t^{i ¯j}(ω)| ≥ α for a.e. (t, ω) ∈ [0, T ] × Ω then a set-valued stochastic integralR_T

0 GtdBt 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 IF- nonanticipative Castaing’s representation (gⁿ)^∞_n=1 such that there are α > 0 and f, g ∈ {gⁿ : n ≥ 1} such that there exist (¯i, ¯j) ∈ {1, . . . , d} × {1, . . . , m} and real-valued processes f^{¯i ¯j} and g^{¯i ¯j}, elements of matrix-valued processes f and g, respectively and such that |f_t^{¯i ¯j}(ω) − g^¯_t^{i ¯j}(ω)| ≥ α for a.e. (t, ω) ∈ [0, T ] × Ω. There are an IF-nonanticipative Castaing’s representation (egⁿ)^∞_n=1 of G and matrix- valued processes ef ,eg ∈ {egⁿ : n ≥ 1} possessing elements ef^{¯i ¯j} = ( ef^¯_t^{i ¯j})_0≤t≤T and eg^{¯i ¯j} = (eg^¯_t^{i ¯j})_0≤t≤T, respectively and such that ef_t^{¯i ¯j}(ω) −eg^{¯i ¯}_t^j(ω) ≥ α for a.e. (t, ω) ∈ [0, T ] × Ω.

Proof. Let a set-valued process G, a Castaing’s representation (gⁿ)^∞_n=1 of G, and f, g ∈ {gⁿ : n ≥ 1} be such as above. For simplicity assume that |f_t^{¯i ¯j}(ω) − g^{¯i ¯}_t^j(ω)| ≥ α is satisfied for every (t, ω) ∈ [0, T ] × Ω and let us denote processes f^{¯i ¯j} and g^{¯i ¯j} by φ, ψ, respectively. We have φt(ω) 6= ψt(ω) for every (t, ω) ∈

[0, T ] × Ω. Let A = {(t, ω) ∈ [0, T ] × Ω : φt(ω) > ψt(ω)}. If A = [0, T ] × Ω then f_t^{¯i ¯j}(ω) − g^¯_t^{i ¯j}(ω) = |f^¯_t^{i ¯j}(ω) − g^{¯i ¯}_t^j(ω)| ≥ α for every (t, ω) ∈ [0, T ] × Ω.

Suppose 0 < µ(A^∼) < T, where A^∼ = ([0, T ] × Ω) \ A, let eφ = 1I_Aφ + 1I_A^∼ψ and eψ = 1IAψ + 1IA^∼φ. It is clear that {φt(ω), ψt(ω)} = { eφt(ω), eψt(ω)} for every (t, ω) ∈ [0, T ] × Ω. Furthermore, for every (t, ω) ∈ A we have eφ_t(ω) − eψ_t(ω) = φ_t(ω) − ψ_t(ω) = |φ_t(ω) − ψ_t(ω)| ≥ α. Similarly for (t, ω) ∈ A^∼ we get eφ_t(ω) − ψet(ω) = ψt(ω)−φt(ω) = −(φt(ω)−ψt(ω)) = |φt(ω)−ψt(ω))| ≥ α. Taking ef^{¯i ¯j} = eφ and eg^¯i,¯j = eψ we obtain ef_t^{¯i ¯j}(ω) −eg^¯_t^{i ¯j}(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 (gⁿ)^∞_n=1 its elements f and g by new matrix-valued functions ef andeg obtained from f and g by changing in matrices f and g theirs elements f^{¯i ¯j} and g^{¯i ¯j} by ef^{¯i ¯j} and eg^{¯i ¯j}, respectively.

Lemma 5. For every f, g ∈ IL²([0, T ] × Ω, Σ_IF, IR^d×m) a set-valued stochas- tic integral RT

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 decFJ_T[(f − g) · KIF] is a bounded subset of the space IL²(Ω, F , IR^d).

Proof. Let us observe that a set-valued stochastic integral RT

0 F_tdB_t is square integrably bounded if and only if a set decFJ_T(S_IF(F )) is a bounded subset of the space IL²(Ω, F , IR^d). By ([8], Remark 3.6 of Chap. 2) a set S_IF(F ) is defined by SIF(F ) = decΣIF{f, g}. But dec_ΣIF{f, g} = {1I_D(f − g) + g : D ∈ ΣIF} = {1I_D(f − g) : D ∈ Σ_IF}+g = (f −g)·K_IF+g. Then decFJ_T(S_IF(F )) = decFJ_T[(f −g)·K_IF]+

J_T(g). Thus decFJ_T(S_IF(F )) is bounded if and only if decFJ_T[(f − g) · K_IF] is a bounded subset of the space IL²(Ω, F , IR). Boundedness of decFJ_T[(f − g) · KIF] is equivalent to boundedness of decFJ_T[(f − g) · K_IF]. Then a set-valued stochastic integralRT

0 F_tdB_tis square integrably bounded if and only if decFJ_T[(f − g) · K_IF] is a bounded subset of the space IL²(Ω, F , IR^d).

Now we prove the main result of the paper.

Theorem 6. Let G = (Gt)0≤t≤T be an IF-nonanticipative square integrably bounded set-valued stochastic process with values in the space Cl(IR^d×m) possess- ing an IF-nonanticipative Castaing’s representation (gⁿ)^∞_n=1 such that there are α > 0 and f, g ∈ {gⁿ: n ≥ 1} such that there are (¯i, ¯j) ∈ {1, . . . , d} × {1, . . . , m}

and real-valued processes f^{¯i ¯j}, g^{¯i ¯j}, elements of matrix-valued processes f and g, respectively and such that |f^¯_t^{i ¯j}(ω) − g^¯_t^{i ¯j}(ω)| ≥ α for a.e. (t, ω) ∈ [0, T ] × Ω. A set-valued stochastic integral RT

0 GtdBt is not integrably bounded.

Proof. Let G = (Gt)0≤t≤T and IF-nonanticipative Castaing’s representation (gⁿ)^∞_n=1 of G possess properties described above. By virtue of Lemma 4 there

is an IF-nonanticipative Castaing’s representation (egⁿ)^∞_n=1 of G containing pro- cesses ef ,eg ∈ {egⁿ : n ≥ 1} and such that there exist real-valued stochastic pro- cesses ef^{¯i ¯j} = ( ef^¯_t^{i ¯j})0≤t≤T, ge^{¯i ¯j} = (eg^{¯i ¯}_t^j)0≤t≤T, elements of matrix-valued prosesses f ,eeg, respectively and such that ef_t^{¯i ¯j}(ω) −ge^¯_t^{i ¯j}(ω) ≥ α for a.e. (t, ω) ∈ [0, T ]. By virtue of Lemma 1 it follows that decFJ^¯j[( ef^{¯i ¯j}−eg^{¯i ¯j}) · K_IF] is an unbounded sub- set of IL²(Ω, F , IR). Hence, by Lemma 3 it follows that decFJ_T[( ef −eg) · KIF] is an unbounded subset of IL²(Ω, F , IR), which by Lemma 5 implies that a set- valued stochastic integral RT

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 RT

0 FtdBt ⊂ RT

0 GtdBt a.s. Therefore, a set-valued stochastic integral RT 0 GtdBt

is not integrably bounded.

It is natural to expect that a set-valued stochastic integralRT

0 GtdBtis integrably bounded if and only if G possesses an IF-nonanticipative Castaing’s representa- tion containing only one element. Such result can be obtained if the following hypothesis would be satisfied.

Hypothesis B. For every set C ∈ ΣIF of positive measure µ = dt × P a set decFJ_T(1I_C· K_IF) is an unbounded subset of the space IL²(Ω, F , IR).

To obtain such result we begin with the following lemma.

Lemma 7. Let h ∈ IL²([0, T ] × Ω, Σ_IF, IR) be a non-negative process such that there exists a set C ∈ ΣIF of positive measure µ = dt × P such that ht(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 αε> 0 such that ht(ω) ≥ αε for (t, ω) ∈ Cε.

Proof. Let C ∈ Σ_IFbe a set of positive measure µ = dt × P such that ht(ω) > 0 for (t, ω) ∈ C and h_t(ω) = 0 for (t, ω) ∈ C^∼. We have C = {(t, ω) ∈ [0, T × Ω : ht(ω) > 0}. Let Cm = {(t, ω) ∈ C : ht(ω) ≥ m} for every m > 0. We have C =S

m>0Cm and Cm ⊂ C_n for m ≥ n. Put eC_k = Cm_k, where m_k = 1/k. We have C =S∞

k=1Ce_k and eC_k⊂ eC_k+1 for k ≥ 1. Therefore, µ(C) = lim_k→∞µ( eC_k).

Then for every ε ∈ (0, µ(C)) there is k_ε such that µ(C) − µ( eC_kε) < ε. Thus µ( eCkε) > µ(C) − ε > 0. Let Cε= eCkε and αε= 1/kε. By the definition of a set Ce_kε we get 1/kε≤ h_t(ω) for (t, ω) ∈ Cε. Then ht(ω) ≥ αε for (t, ω) ∈ Cε.

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 = (Gt)0≤t≤T with val- ues in the space Cl(IR^d×m), a set-valued stochastic integralRT

0 G_tdB_tis integrably bounded if and only if there is an IF-nonanticipative Castaing’s representation (gⁿ)^∞_n=1 of G such that kgⁿ− g^mk = 0 for every n, m ≥ 1.

Proof. If (gⁿ)^∞_n=1 is an IF-nonanticipative Castaing’s representation of G such that if kgⁿ − g^mk = 0 for every n, m ≥ 1 then a set-valued stochastic inte- gral RT

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 ∈ IL²([0, T ] × Ω, Σ_IF, IR^d×m) and therefore, RT

0 GtdBt=RT 0 gtdBt.

Suppose (gⁿ)^∞_n=1is an IF-nonanticipative Castaing’s representation of G such that there are f, g ∈ {gⁿ : n ≥ 1} such that kf − gk > 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 |ft(ω) − gt(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 (egⁿ)^∞_n=1 of G such that there are ef ,eg ∈ {egⁿ : n ≥ 1} having elements ef^i,j and eg^i,j such that ef_t^i,j(ω) −ge_t^i,j(ω) > 0 for a.e. (t, ω) ∈ C and ef_t^i,j(ω) −eg_t^i,j(ω) = 0 for a.e. (t, ω) ∈ C^∼. By virtue of Lemma 7 for ε > 0 there is ΣIF-measurable set Cε⊂ C of positive measure µ = dt × P and a real number α_ε> 0 such that 1I_Cεeh^i,j ≥ 1I_Cεα_ε, where eh^i,j = ef^i,j −eg^i,j. Let h^i,j = 1I_Cεeh^i,j + 1I_Cε^∼· 0, where C_ε^∼ = [0, T ] × Ω \ C_ε. We have 1I_Cεh^i,j ≥ 1I_Cεα_ε, 1I_Cε^∼h^i,j = 0 and decFJ_T^j(1I_Cε· K_IF) is an unbounded subset of the space IL²(Ω, F , IR). Therefore, by virtue of Lemma 2 a set decFJ_T^j(1ICεeh^i,j · K_IF) is an unbounded subset of IL²(Ω, F , IR).

But 1ICεeh^i,j ≤ 1I_Ceh^i,j. Therefore, [0, 1ICεeh^i,j] ⊂ [0, 1ICeh^i,j], which similarly as in the proof of Lemma 1, implies that decFJ_T^j(1ICεeh^i,j· K_IF) ⊂ decFJ_T^j(1ICeh^i,j· K_IF).

Therefore, decFJ_T^j(1ICeh^i,j·K_IF) is an unbounded subset of the space IL²(Ω, F , IR), which by Corollary 1 implies that decFJ_T^j[( ef −eg) · K_IF] is an unbounded subset of the space IL²(Ω, F , IR^d). Hence by Lemma 5 it folows that a set-valued stochastic integralRT

0 F_tdB_tof 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 RT

0 FtdBt ⊂RT 0 GtdBt

a.s. Therefore, a set-valued stochastic integralRT

0 GtdBtis not square integrably bounded.

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Received 14 November 2015