Some properties of set-valued stochastic integrals of multiprocesses with finite Castaing representations

Michał Kisielewicz

Some properties of set-valued stochastic integrals of multiprocesses with finite Castaing representations

In honour of professor Julian Musielak, on the occasion of his 85th birthday.

Abstract. The paper contains new properties of set-valued stochastic integrals defi- ned as multifunctions with subtrajectory integrals equal to closed decomposable hulls of functional set – valued integrals defined in the author paper [8]. In particular, it is proved that such defined integrals for set – valued predictable square integrably bounded processes having finite Castaing representations are square integrably bo- unded. Up to now this property has not been proved. Unfortunately, in the general case the above boundedness problem is still open.

2001 Mathematics Subject Classification: 60H05, 28B20, 47H04.

Key words and phrases: Set-valued mappings, set-valued integrals, set-valued sto- chastic processes.

1. Introduction. The present paper deals with set-valued stochastic inte- grals defined as some special type of set-valued random variables with values in r-dimensional Euclidean space IR^r. The first paper dealing with set-valued sto- chastic integrals due to B. Bocsan ([1]). Unfortunately, the definition and some properties of such defined integrals are not quite correct. Quite different definitions of the set-valued stochastic integrals have been independently given by F. Hiai [2]

and M. Kisielewicz [8], [9]. The set-valued stochastic integrals are treated there as some subsets of the space IL²(Ω,X ) of all square integrable (equivalent classes of) random variables with values in a Hilbert space X . In what follows such defined set-valued stochastic integrals are called functional set-valued stochastic integrals.

Unfortunately, the functional stochastic integrals are not decomposable subsets of IL²(Ω,X ) and therefore, they do not define (see [4], Th.2.3.8) set-valued random variables with subtrajectory integrals equal to these integrals. By subtrajectory integrals of a measurable multifunction we mean a set of all its integrable selectors.

B.K. Kim and J.H. Kim ([7]) did not notice that and have defined a set – valued

stochastic integral as a random variable with its subrajectory integrals equal to a set – valued functional integral defined in [8]. J. Jung and J.H. Kim [6] have corrected the above definition by taking there the closed decomposable hull of a set – valued functional integral defined in [8] instead of this integral. Unfortunately, some properties of such defined integrals presented in [6] are not true and proofs of some theorems are not correct. Some properties of integrals defined in [6] are given in the author paper [10]. Unfortunately, the main property, integrable boundedness of such integrals for integrably bounded processes has not been decided even for the simplest multifunctions defined by closed decomposable hulls of two predictable square integrable stochastic processes. In the present paper we solve this problem for multifunctions having finite Castaing’s representations.

Let (X, ρ) be a metric space and denote by Cl(X) a space of all nonempty closed subsets of X. For every A, C ∈ Cl(X) let h(A, C) = sup{d(a, C) : a ∈ A}, where d(a, C) = inf{ρ(a, c) : c ∈ C}. The Hausdorff distance h(A, C) between A, C ∈ Cl(X) is defined by h(A, C) = max{ h(A, C), h(C, A)}. Convergence of a sequence (An)^∞_n=1⊂ Cl(X) to A ∈ Cl(X) in the Hausdorff metric will be denoted by An

→ A. It can be verified that if Ah ⁿ → A then A =^h T

n≥1

S

m≥nAm. Given a sequence (An)n≥1 ⊂ Cl(X) ∪ {∅} by Lim Aⁿ and Lim An we denote the Kuratowski limit inferior and superior, respectively of the sequence (An)n≥1 defined by Lim An ={x ∈ X : lim d(x, Aⁿ) = 0} and Lim Aⁿ ={x ∈ X : lim d(x, Aⁿ) = 0}. It can be verified (see [4]) that Lim Aⁿ={x ∈ X : x = lim xⁿ, xn∈ Aⁿ, n≥ 1}

and Lim An ={x ∈ X : x = lim xⁿk, xnk ∈ Aⁿk, n1 < n2 < ... < nk < ...}. We call a sequence (An)n≥1 convergent in the Kuratowski sense to A ∈ Cl(X) ∪ {∅}

if A = Lim An = Lim An. We denote this convergence by An

→ A. The limit AK

is denoted by Lim An and said to be Kuratowski’s limit of a sequence (An)_n≥1. Immediately from the above definions we get Lim An ⊂ Lim Aⁿ and Lim An = T

n≥1

S

m≥nAm. It can be verified (see [4]) that for every sequence (An)_n≥1 of Cl(X)∪ {∅} such that Aⁿ ⊂ Aⁿ⁺¹ for every n ≥ 1 the limit Lim Aⁿ exists and is equal to Lim An. It can be proved ([4], Prop.1.1.51) that if X = IR^r then a sequence (An)n≥1 converges in Kuratowski sense to A ∈ Cl(IR^r)∪ {∅} if and only if d(x, An)→ d(x, A) for every x ∈ X.

2. Set-valued stochastic processes. Throughout the paper we shall deal with a complete filtered probability space PF = (Ω,F, F, P ) with a filtration F = (F^t)_t≥0 satisfying the usual conditions. By an r – dimensional set-valued random variable we mean a closed valued F – measurable multifunction, i.e., a multifunction Z : Ω → Cl(IR^r) such that {ω ∈ Ω : Z(ω) ∩ C 6= ∅} ∈ F for every C ∈ Cl(IR^r). A family G = (Gt)t≥0 of set – valued random variables Gt: Ω→ Cl(IR^r) is said to be a set – valued stochastic process defined on P^IF. Si- milarly as in the theory of point valued stochastic processes a set – valued stochastic process can be defined as a multifunction G : IR⁺× Ω → Cl(IR^r) such that G(t, · ) is a set – valued random variable for every t ≥ 0. Such defined stochastic process is said to be IF – predictable if G is P(IF) – measurable, where P(IF) denotes the IF– predictable σ – algebra of subsets of IR⁺× Ω. Properties of set – valued random variables and set – valued stochastic processes follow immediately from properties

of measurable multifunctions. Therefore, we begin this section with recalling pro- perties of measurable multifunctions needed in the next part of the paper.

For a given separable Banach space X and σ – finite measure space (T, A, µ) a multifunction Z : T → Cl(X ) is measurable (see [4], Th.2.2.4) if and only if there exists a sequence (zn)^∞_n=1 of A-measurable selectors of Z such that Z(t) = cl{zⁿ(t) : n≥ 1} for every t ∈ T. Such sequence is said to be a Castaing represen- tation of Z. In what follows by S(Z) we shall denote subtrajectory integrals of Z, i.e., the set of all square µ – integrable selectors of Z. If S(Z) is nonempty then Z is said to be Aumann integrable. It can be verified (see [4], Prop. 3.29, Corollary 2.3.5) that if Z and G are Aumann integrable and S(Z) = S(G) then Z(t) = G(t) for µ – a.e. t ∈ T. It is clear that S(Z) is a closed subset of IL²(T, A, X ). It is also decomposable subset of this space, i.e., for every A ∈ A and u, v ∈ S(Z) one has 1Au +1_T\Av ∈ S(Z). If Z is Aumann integrable and (zⁿ)^∞_n=1 is a Castaing represetation of Z then (see [3], Lemma 1.3) for every z ∈ S(Z) and ε > 0 there exist the finite A – measurable partition (Ak)^N_k=1 of T and a family (znk)^N_k=1⊂ {zⁿ : n≥ 1} such that E|z −PN

k=11Akznk|²< ε. In what follows the family of all finite A – measurable partitions of T is denoted by Π(T, A).

We call Z square integrably bounded if there exists m ∈ IL²(T, A, IR⁺) such that kZ(t)k ≤ m(t) for µ – a.e. t ∈ T, where kAk = sup{|a| : a ∈ A} for A∈ Cl(X ). It can be proved (see [3]) that a multifunction Z is square integrably bounded if and only if S(Z) is a bounded subset of IL²(T, A, X ). For a given Λ ⊂ IL^p(T, A, X ) by decA(Λ) we denote the decomposable hull of Λ , i.e., the smallest decomposable subset of IL^p(T, A, X ) containing Λ. The closure of decA(Λ) in the norm topology of IL²(T, A, X ) is denoted by decA(Λ).

Immediately from the above properties of the set S(Z) it follows that if (zn)^∞_n=1⊂ S(Z) is a Castaing representation of Z then S(Z) = decA{zⁿ: n≥ 1}.

Indeed, it is clear that decA{zⁿ : n ≥ 1} ⊂ S(Z). On the other hand, for every z ∈ S(Z) and ε > 0 there exist a partition (A^k)^N_k=1 ∈ Π(T, A) and a family (znk)^N_k=1 ⊂ {zⁿ : n ≥ 1} such that E|z −PN

k=11Akznk| ≤ ε, which implies that z∈ decA{zⁿ: n≥ 1}. Thus S(Z) = decA{zⁿ: n≥ 1}.

Finally, let us note that (see [4], Th.2.3.8) that a nonempty closed set K ⊂ IL^p(T,A, IR^q) is decomposable if and only if there exists an A – measurable multifunction F : T → Cl(X ) such that K = S(F ). Summing up the above remarks we obtain the following result.

Proposition 2.1 Let (T, A, µ) be σ – finite measure space and Z : T → Cl(X ) an Aumann integrable multifunction. Then

(i) S(Z) is a closed decomposable subset of IL²(T, A, X ),

(ii) Z is square integrably bounded if and only if S(Z) is a nonempty bounded subset of IL²(T, A, X ).

(iii) there exists a sequence (zn)^∞_n=1 of A – measurable functions zⁿ :T → X such that zn(t) ∈ Z(t) and Z(t) = cl{zⁿ(t) : n≥ 1} for n ≥ 1 and t ∈ T. If (zn)^∞_n=1⊂ S(Z) then S(Z) = decA{zⁿ: n≥ 1},

(iv) if F and G are Aumann integrable A – measurable multifunctions such that S(F ) = S(G) then F (t) = G(t) for µ – a.e. t ∈ T

(v) for every nonempty set Λ ⊂ IL²(T, A, X ) there exists an A – measurable mul- tifunction F : T → Cl(X ) such that S(F ) = decA(Λ), where the closure is taken in the norm topology of IL²(T, A, X ).

Given a measurable function u : T × X → IR ∪ {−∞, +∞} and an A – measurable multifunction Z : T → Cl(X ) let ξ(t) = sup{u(t, x) : x ∈ Z(t)} for every t ∈ T. It can be proved ([3], Lemma 2.1) that if u(t, · ) is upper semicon- tinuous or lower semicontinuous for every t ∈ T then ξ is A – measurable. The following result can be proved.

Proposition 2.2 ([3], Th.2.2) . If Z is Aumann integrable, u : T × X → IR ∪ {−∞, +∞} is measurable, u(t, · ) is upper semicontinuous for every t ∈ T and the functional K(z) := R

Tu(·, z(·))dµ is defined for every z ∈ S(Z) and there exists at least one z0 ∈ S(Z)) such that K(z⁰) < ∞ , then sup{R

Tu(·, z(·))dµ : z ∈ S(Z)} =R

T[sup{u(t, x) : x ∈ Z(t)}]dµ.

Remark 2.3 A similar result is also true for the equality inf{R

Tu(·, z(·))dµ : z ∈ S(Z)} =R

T[inf{u(t, x) : x ∈ Z(t)}]dµ.

In what follows we shall consider a set – valued stochastic processes Φ = (Φt)_0≤t≤T defined by Φ : [0, T ] × Ω → Cl(IR^r) with r = m × d and such that Φ(t, · ) is for every t ∈ [0, T ] an r – dimensional set – valued random variable.

Recall, that if the multifunction Φ is P(IF) – measurable, where P(IF) denotes the IF- predictable σ – algebra on [0, T ] × Ω, then the set – valued process Φ is said to be IF – predictable. Similarly as above we can define square integrable boundedness of set-valued processes. If a set-valued process Φ = (Φt)0≤t≤T is square integrably bounded then ERT

0 kΦ^tk²dt <∞ , where kΦ^t(ω)k = sup{|x| : x ∈ Φ^t(ω)}. In what follows by SIF(Φ) we denote an IF – subtrajectory integrals of an IF – predictable set-valued stochastic process Φ : [0, T ]×Ω → Cl(IR^r),i.e., a set of all IF – predictable and square dt × P – integrable selectors of Φ. If S^IF(Φ)6= ∅ then Φ is said to be Itô integrable. If Φ is square integrably bounded then SIF(Φ) is a closed bounded decomposable subset of IL²([0, T ]× Ω, P(IF), IR^r). In what follows the last space will be denoted by IL²(T, IR^r). We shall consider the above space with r = m × d.

Let us observe that for a given above filtered probability space P^IF the σ– algebra P(IF) is generated (see[5], Th.1.2.2) by a family H = {A × {0}, A ∈ F⁰, (s, t]× C, 0 ≤ s < t < ∞, C ∈ F^s}, i.e., P(IF) = σ(H). Let us observe that for every D, E ∈ H one has D ∩ E ∈ H. In general it is not true for D ∪ E. For a fixed T > 0, a family (Dk)^N_k=1⊂ H is said to be a finite H – partition of [0, T ] × Ω if Di∩ D^j =∅ for i 6= j with i, j ∈ {1, ..., N} and [0, T ] × Ω =SN

k=1Dk. The family of all finite H - partitions of [0, T ]×Ω will be denoted by Π(T, H). It is clear that Π(T, H) 6= ∅. Furthermore, for every D ∈ H there exists an H - partition of

[0, T ]×Ω containing the set D. The family of all finite P(IF) – measurable partitions of [0, T ] × Ω will be denoted by Π(T, IF). It is clear that Π(T, H) ⊂ Π(T, IF). We shall also consider the family Π(Ω, FT)of all finite FT– measurable partitions of Ω.Given a number α ≥ 1 we shall consider subfamilies Πα(T,H) and Π^α(T, IF) of Π(T,H) and Π(T, IF), respectively defined by Π^α(T,H) = {(D^j)^M_j=1 ∈ Π(T, H) : M ≤ α} and Π^α(T, IF) = {(A^k)^N_k=1 ∈ Π(T, IF) : N ≤ α}. For a nonempty set Λ ⊂ IL²(T, IR^m×d) and α ≥ 1 by dec^IF(Λ), dec(IF,α)(Λ), and dec(H,α)(Λ), we denote sets of the form {PN

k=11Akg^k : (Ak)^N_k=1 ∈ Π(T, IF), (g^k)^N_k=1 ⊂ Λ}, {PN

k=11Akg^k : (Ak)^N_k=1∈ Π^α(T, IF), (g^k)^N_k=1⊂ Λ} and {PM

j=11Dkg^j: (Dk)^M_k=1∈ Πα(T,H), (g^k)^M_k=1 ⊂ Λ}, respectively. The closure of these sets with respect to the norm topology of IL²(T, IR^m×d) are denoted by decIF(Λ), dec(IF,α)(Λ), and dec_(H,α)(Λ), respectively. For simplicity, the decomposable hull and the closed decomposable hull of a set C ⊂ IL²(Ω,F, IR^d) are denoted by dec(C) and dec(C), respectively.

Proposition 2.4 If Λ = {g¹, ..., g^p} ⊂ IL²(T, IR^m×d) then decIF(Λ) = dec(IF,p)(Λ).

Proof It is clear that {Pp

k=11Akg^k : (Ak)^p_k=1 ∈ Π(T, IF)} ⊂ dec^IF(Λ). Let u ∈ dec^IF(Λ) and (Ak)^m_k=1 ∈ Π(T, IF) and (g^ik)^m_k=1 ⊂ Λ be such that u = Pm

k=11Akg^ik. If m < p then we can extend a partition (Ak)^m_k=1 to the form ( eAk)^p_k=1 by taking eAk = Ak for k = 1, ..., m and eAk=∅ for k = m + 1, ..., p and get u = Pp

k=11Aekg^k ∈ {Pp

k=11Akg^k : (Ak)^p_k=1 ∈ Π(T, IF)}. It is easy to verify that in the case m ≥ p we also have u ∈ {Pp

k=11Akg^k: (Ak)^p_k=1∈ Π(T, IF)}. Then decIF(Λ)⊂ {Pp

k=11Akg^k : (Ak)^p_k=1 ∈ Π(T, IF)}. Thus dec^IF(Λ) ={Pp

k=11Akg^k : (Ak)^p_k=1∈ Π(T, IF)}.

In a similar way we can verify that dec(IF,p)(Λ) ={Pp

k=11Akg^k: (Ak)^p_k=1∈ Π(T, IF)}. Indeed, for u ∈ dec^(IF,p)(Λ) there exist a partition (Ak)^m_k=1∈ Π(T, IF) and a family (g^ik)^m_k=1 ⊂ Λ with m ≤ p such that u = Pm

k=11Akg^ik. If m = p then u ∈ {Pp

k=11Akg^k : (Ak)^p_k=1 ∈ Π(T, IF)}. If m < p then similarly as above we can extend a partition (Ak)^m_k=1∈ Π(T, IF) to ( eAk)^p_k=1,defined above such that u = Pp

k=11Aekg^k ∈ {Pp

k=11Akg^k : (Ak)^p_k=1 ∈ Π(T, IF)}. Then, dec(IF,p)(Λ) ⊂ {Pp

k=11Akg^k : (Ak)^p_k=1 ∈ Π(T, IF)}. It is clear that {Pp

k=11Akg^k : (Ak)^p_k=1 ∈ Π(T, IF)} ⊂ dec^(IF,p)(Λ). Therefore, dec(IF,p)(Λ) = {Pp

k=11Akg^k : (Ak)^p_k=1 ∈

Π(T, IF)}. Thus dec^IF(Λ) = dec(IF,p)(Λ). _

In what follows we shall need the following result.

Proposition 2.5 For every nonempty set Λ ⊂ IL²(T, IR^m×d) and α ≥ 1 one has dec(H,α)(Λ) = dec(IF,α)(Λ).

Proof It is clear that dec(H,α)(Λ) ⊂ dec^(IF,α)(Λ). Let u ∈ dec(IF,α)(Λ) and (un)^∞_n=1 be a sequence of dec(IF,α)(Λ) such that kuⁿ− uk → 0 as n → ∞, where k · k is a norm of IL²(T, IR^m×d). For every un∈ dec^(IF,α)(Λ) there are a partition (Aⁿ_k)^N_k=1ⁿ ∈ Π^α(T, IF) and a family (g^k,n)^N_k=1ⁿ ⊂ Λ such that uⁿ =PNn

k=11Aⁿ_kg^k,n.

For every n ≥ 1 and k = 1, ..., Nn there exists a sequence (h^k,n,m)^∞_m=1 of sim- ple P(IF) – predictable processes (h^k,n,mt )0≤t≤T such that kh^k,n,m− g^k,nk → 0 as m → ∞. Then for every fixed n ≥ 1 and ε > 0 there exists m^ε > 0 such that kh^k,n,mε− g^k,nk ≤ ε/2α² for every k = 1, ..., Nn, which can be written in the form of the inclusion h^k,n,mε ∈ {g^k,n} + (ε/2α²)B, where B denotes the closed unit ball of IL²(T, IR^m×d). Hence, for every H – partition (Dkⁿ)^N_k=1ⁿ ∈ Π^α(T,H), we get

(2.1)

Nn

X

k=1

1Dⁿ_kh^k,n,mε ∈ (_Nn

X

k=1

1Dⁿ_kg^k,n )

+ε

2· B ⊂ dec(H,α)(Λ) +ε 2 · B for every n ≥ 1 and ε > 0. On the other hand we have

un−

Nn

X

k=1

1Dⁿ_kh^k,n,mε =

Nn

X

k=1

1Aⁿ_kg^k,n−

Nn

X

k=1

1Dⁿ_kh^k,n,mε =

Mn

X

k=1

1C_kⁿeg^k,n−

Mn

X

k=1

1C_kⁿeh^k,n,mε , where (Ckⁿ)^M_k=1ⁿ ∈ Π(T, IF), (eg^k,n)^M_k=1ⁿ ⊂ {g^k,n: k = 1, .., N} and (eh^k,n,mε)^M_k=1ⁿ ⊂ {h^k,n,mε : k = 1, ..., N} are such that

Mn

X

k=1

1C_kⁿeg^k,n=

Nn

X

k=1

1Aⁿ_kg^k,n and

Mn

X

k=1

1Cⁿ_keh^k,n,mε=

Nn

X

k=1

1Dⁿ_kh^k,n,mε

for every n ≥ 1 and ε > 0 with Mn≤ α² because the sets C_kⁿ are of the form Aⁿ_i ∩ D^j for every i, j = 1, ..., Nn . But

Mn

X

k=1

1C_kⁿeg^k,n−

Mn

X

k=1

1C_kⁿeh^k,n,mε =



EZ T 0

Mn

X

k=1

1Cⁿ_k

hegt^k,n− eh^k,n,mt ^ε

i

2

dt





1/2

≤

Mn

X

k=1

E Z T

0

eg^k,nt − eh^k,n,mt ^ε

2

dt

!1/2

=

Mn

X

k=1

keg^k,n− eh^k,n,mεk <

Mn· (ε/2α²)≤ ε/2

because keg^k,n− eh^k,n,mεk < ε/2α² for fixed n ≥ 1 and k = 1, ..., Mn. Then for fixed n ≥ 1 and ε > 0 one has kun−PNn

k=11Dⁿ_kh^k,n,mεk ≤ ε/2, which can be written in the form of the inclusion

(2.2) un∈

(_Nn X

k=1

1D_kⁿh^k,n,mε )

+ (ε/2)· B

for every n ≥ 1 and ε > 0. Immediately from (2.1) and (2.2) it follows un∈n

dec_(H,α)(Λ) + ε 2 · Bo

+ε

2· B = dec(H,α)(Λ) + ε· B

for every n ≥ 1 and ε > 0. Therefore, un ∈ \

ε>0

dec(H,α)(Λ) + ε· B

⊂ dec⁽H,α)(Λ)

for every n ≥ 1, which implies that u ∈ dec(H,α)(Λ) because kun− uk → 0 as

n→ ∞. Then dec^(IF,α)(Λ)⊂ dec⁽H,α)(Λ). 

Corollary 2.6 If Λ = {g¹, ..., g^p} ⊂ IL²(T, IR^m×d) then decIF(Λ) = dec_(H,p)(Λ).

Proof By Proposition 2.4 one has decIF(Λ) = dec(IF,p)(Λ). By virtue of Proposi- tion 2.5 we have dec(IF,p)(Λ) = dec(H,p)(Λ). Therefore, decIF(Λ) = dec(H,p)(Λ). _ Assume P^IF is such that there exists an m - dimensional IF - Brownian mo- tion B = (Bt)t≥0 defined on P^IF. In what follows we shall deal with the linear mapping J : IL²(T, IR^m×d)3 g →RT

0 gtdBt∈ IL²(Ω,F^T, IR^d). It is clear that J is continuous and for every closed sets Γ ⊂ IL²(T, IR^m×d) its image by the mapping J is a closed subset of IL²(Ω,F^T, IR^d). Hence and by the properties of continu- ous mappings it follows that J(U) = clIL{J(U)} for every set U ⊂ IL²(T, IR^m×d) where U denotes the closure of a set U in the norm topology of IL²(T, IR^m×d) and clIL{J(U)} is the closure of J(U) in the norm topology of IL²(Ω,F^T, IR^d). Indeed, we have J(U) ⊂ J(U). Therefore, cl^IL{J(U)} ⊂ J(U) because J(U) is a closed subset of IL²(Ω,F^T, IR^d). By properties of continuous mappings we have J(U )⊂ cl^IL{J(U)}. Then J(U) = cl^IL{J(U)}.

3. Set-valued stochastic integrals. Let G = (Gt)_0≤t≤T be an IF – predic- table square integrably bounded set – valued stochastic process G : [0, T ] × Ω → Cl(IR^m×d) and let (gⁿ)^∞_n=1 be a Castaing representation of G, i.e., let Gt(ω) = cl{gⁿt(ω) : n≥ 1} for every (t, ω) ∈ [0, T ]×Ω. Immediately from (iii) of Proposition 2.1, applied to the measure space ([0, T ]×Ω, P(IF), dt×P ), it follows that S^IF(G) = decIF{gⁿ : n ≥ 1}. In particular, if a Castaing representation of G is finite, i.e., if {gⁿ; n ≥ 1} = {g¹, ..., g^p} a set – valued process G will be denoted by G^p. If G is a set – valued process with an infinite Castaing representation (gⁿ)^∞_n=1 we can define a sequence (G^p)^∞_p=1 of set – valued processes (Gtp)0≤t≤T having finite Castaing representations of the form {g¹, ..., g^p}. It can be verified that Gt(ω) = Lim G^p_t(ω) for every (t, ω) ∈ [0, T ]×Ω. Indeed, it is clear that Lim G^pt(ω) exists for every (t, ω) ∈ [0, T ] × Ω because G^pt(ω) ⊂ G^p+1t (ω) ⊂ G^t(ω) for every (t, ω)∈ [0, T ]×Ω. Hence in particular, it follows that Lim G^pt(ω)⊂ G^t(ω) for every (t, ω)∈ [0, T ] × Ω. For fixed (t, ω) ∈ [0, T ] × Ω and every u ∈ G^t(ω) = cl{gⁿt(ω) : n ≥ 1} there exists a subsequence (g^nk)^∞_k=1 of (gⁿ)^∞_k=1 converging to u. Then u ∈ Lim G^pt(ω) = Lim G^p_t(ω). Therefore, Gt(ω) ⊂ Lim G^pt(ω), where Lim G^pt(ω) denotes Kuratowski’s limit of the sequence (G^pt(ω))^∞_p=1 of sbsets of IR^m×d.

Let B = (Bt)_t≥0 be an IF - Brownian motion. For the given above process G and a positive integer α ≥ 1 by RT

0 GtdBt and (α)RT

0 GtdBt we denote F^T- measurable multifunctions Ω 3 ω → (RT

0 GtdBt)(ω) ∈ Cl(IR^d) and

Ω 3 ω → ((α)RT

0 GtdBt)(ω) ∈ Cl(IR^d) such that ST{RT

0 GtdBt} = decJ[S^IF(G)]

and ST{(α)RT

0 GtdBt} = decJ[S^αIF(G)], where ST(F ) denotes the set of all FT- measurable selectors of a mutifunction F : Ω → Cl(IR^d) and S_IF^α(G) = dec(IF,α){gⁿ: n≥ 1}. The set – valued random variables RT

0 GtdBt and (α)RT

0 GtdBt are called set-valued stochastic integral and α – integral, respectively of G over the interval [0, T ] with respect to the IF - Brownian motion B. It is clear that (α)RT

0 GtdBt⊂ RT

0 GtdBt a.s. for every α ≥ 1. Immediately from Proposition 2.4 it follows that (p)RT

0 G^p_tdBt = RT

0 G^p_tdBt a.s. for every p ≥ 1. Indeed, by definitions of the stochastic integral and the (p)-integral of a set valued process G^p one has ST[(p)RT

0 G^p_tdBt] = dec(IF,p)(Λ) = decIF(Λ) = ST[RT

0 G^p_tdBt], where Λ = {g¹, ..., g^p}. Therefore, by (iv) of Proposition 2.1 we get (p)RT

0 G^p_tdBt=RT

0 G^p_tdBt. We shall show now that for the given above set – valued process G one has RT

0 GtdBt= Lim[(r)RT

0 GtdBt] a.s., where Lim Ar denotes Kuratowski’s limit of a sequence (Ar)^∞_r=1 of subsets of IR^d. We begin with the following result.

Proposition 3.1 If G = (Gt)_0≤t≤T is an IF – predictable square integrably bo- unded set – valued stochastic process G : [0, T ] × Ω → Cl(IR^m×d) then SIF(G) = Lim S_IF^r(G).

Proof Let (gⁿ)^∞_n=1 be a Castaing representation of G. For every r ≥ 1 one has SIF^r(G)⊂ SIF^r+1(G). Then Lim SIF^r(G) exists and is equal to clIF{S

r≥1S_IF^r(G)}, where the closure is taken in the norm topology of IL²(T, IR^m×d). On the other hand for every r ≥ 1 we have SIF^r(G)⊂ S^IF(G). Then Lim SIF^r(G) = clIF{S

r≥1S_IF^r(G)} ⊂ SIF(G) because SIF(G) is a closed subset of IL²(T, IR^m×d). Let u ∈ SIF(G) and let (um)^∞_m=1 be a sequence of decIF{gⁿ : n≥ 1} such that ku^m− u k → 0 as m → ∞, where k · k is a norm of IL²(T, IR^m×d). For every m ≥ 1 there exist a partition (A^m_k )^N_k=1^m ∈ Π(T, IF) and a family (g^nk(m))^N_k=1^m such that um =PNm

k=11A^m_kg^nk(m). For every m ≥ 1 there exists r^m≥ 1 such that N^m≤ r^m. Then um∈ SIF^r(G) for every m ≥ 1 and r ≥ r^m, which implies that um∈S

r≥1S_IF^r(G) for every m ≥ 1.

Thus u ∈ cl^IF{S

r≥1S^r_IF(G)} and therefore, S^IF(G)⊂ Lim SIF^r(G). 2 _

Theorem 3.2 If G = (Gt)0≤t≤T is an IF – predictable square integrably bounded set – valued stochastic process G : [0, T ] × Ω → Cl(IR^m×d) then

(3.1)

Z T 0

GtdBt= Lim

"

(r) Z T

0

GtdBt

# a.s.

Proof We shall show that J[SIF(G)] = Lim J[S_IF^r(G)]. It is clear that J[SIF^r(G)]⊂ J[SIF(G)] for every r ≥ 1, which by closedness of J[S^IF(G)], implies that Lim J[S_IF^r(G)]⊂ J[S^IF(G)] because Lim J[SIF^r(G)] = clIF{S

r≥1J[S_IF^r(G)]}, where the closure is taken with respect to the norm topology of IL²(Ω,F^T, IR^d). Let a∈ J[S^IF(G)] and u ∈ S^IF(G) be such that a = J(u). By virtue of Proposition 3.1

one has u ∈ Lim SIF^r(G), which implies that a ∈ J[Lim SIF^r(G)]. Then J[SIF(G)]⊂ J[Lim S_IF^r(G)]. We shall show now that J[Lim S_IF^r(G)] = Lim J[S_IF^r(G)]. It is clear that

Lim J[S_IF^r(G)] ⊂ J[Lim SIF^r(G)]. Let a ∈ J[Lim SIF^r(G)] and u ∈ Lim SIF^r(G) be such that a = J(u). Since u ∈ Lim SIF^r(G), there exists a sequence (ur)^∞_r=1 of IL²(T, IR^m×d) such that ku^r− uk → 0 as r → ∞ and u^r ∈ SIF^r(G) for every r≥ 1. Hence, by continuity of J, it follows that J(u^r)→ J(u) as r → ∞ in the norm topology of IL²(Ω,F^T, IR^d). We also have J(ur)∈ J[SIF^r(G)] for every r ≥ 1.

Thus J(u) ∈ LimJ[SIF^r(G)]⊂ LimJ[SIF^r(G)]. Therefore, a = J(u) ∈ LimJ[SIF^r(G)]

for every a ∈ J[S^IF(G)]. Then J[Lim SIF^r(G)]⊂ Lim J[SIF^r(G)]. Hence from the inc- lusion J[SIF(G)]⊂ J[Lim SIF^r(G)] it follows that J[SIF(G)]⊂ Lim J[SIF^r(G)]. There- fore, J[SIF(G)] = Lim J[S_IF^r(G)], which implies that dec{J[S^IF(G)]} = dec{Lim J[SIF^r(G)]}.

We shall show now that dec{Lim J[S^rIF(G)]} = Lim dec{J[SIF^r(G)]} and that ST{Lim[(r)RT

0 GtdBt]} = Lim S^T{(r)RT

0 GtdBt}. It is clear that Lim dec{J[SIF^r(G)]}

⊂ dec{Lim J[SIF^r(G)]} and Lim S^T{(r)RT

0 GtdBt} ⊂ S^T{Lim[(r)RT

0 GtdBt]}. For every a ∈ dec{Lim J[SIF^r(G)]} there exists a sequence (am)^∞_m=1 of dec{Lim J[SIF^r(G)]} = dec{J[Lim SIF^r(G)]} converging to a in the norm topology of IL²(Ω,F^T, IR^d).For every m ≥ 1 there are a partition (A^mk)^N_k=1^m and a family (u^m_k)^N_k=1^m ⊂ J[Lim SIF^r(G)] such that am = PNm

k=11A^m_ku^m_k. For every m ≥ 1 and k = 1, ..., Nm there is a sequence (v^k,mr )^∞_r=1 such that v^k,mr ∈ SIF^r(G) for every r ≥ 1 and J(vr^k,m) → u^mk in the norm topology of IL²(Ω,F^T, IR^d) as r → ∞.

Then PNm

k=11A^m_kJ(v_r^k,m) ∈ dec{J[SIF^r(G)]} for r ≥ 1 and PNm

k=11A^m_kJ(v_r^k,m) → PNm

k=11A^m_ku^m_k = am as r → ∞ for every m ≥ 1.

Therefore, a ∈ Lim[dec{J[SIF^r(G)]}] ⊂ Lim[dec{J[S^rIF(G)]}] for every m ≥ 1, which implies that a ∈ Lim[dec{J[SIF^r(G)]}]. Thus dec{Lim J[S^rIF(G)]} = Lim dec{J[SIF^r(G)]}.

Let a ∈ S^T{Lim[(r)RT

0 GtdBt]}. Then a ∈ Lim[(r)RT

0 GtdBt] a.s. The- refore, d[a, (r)RT

0 GtdBt] → 0 a.s. as r → ∞. Let us observe that the sequence {d[a, (r)RT

0 GtdBt]}^∞r=1 is integrably bounded by a function ϕ := d[a, (1)RT 0 GtdBt] because d[a, (r + 1)RT

0 GtdBt] ≤ d[a, (r)RT

0 GtdBt] for every r ≥ 1. Therefore, E{d[a, (r)RT

0 GtdBt]} → 0 as r → ∞. By virtue of Proposition 2.2 one has d a, ST

"

(r) Z T

0

GtdBt

#!

=

inf (

|a − u|²: u∈ S^T

"

(r) Z T

0

GtdBt

#)

= E

"

d a, (r) Z T

0

GtdBt

!#

for every r ≥ 1. Then d(a, S^T[(r)RT

0 GtdBt])→ 0 as r → ∞. Therefore, a ∈ Lim ST[(r)RT

0 GtdBt] ⊂ Lim ST[(r)RT

0 GtdBt], which implies that a∈ Lim S^T[(r)RT

0 GtdBt].