Michta (see [11]) that in the general case set-valued stochastic integrals defined by E.J

doi:10.7151/dmdico.1155

PROPERTIES OF GENERALIZED 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 e-mail: M.Kisielewicz@wmie.uz.zgora.pl

Abstract

The paper is devoted to properties of generalized set-valued stochastic integrals defined in [10]. These integrals generalize set-valued stochastic in- tegrals defined by E.J. Jung and J.H. Kim in the paper [4]. Up to now we were not able to construct any example of set-valued stochastic processes, different on a singleton, having integrably bounded set-valued integrals de- fined in [4]. It was shown by M. Michta (see [11]) that in the general case set-valued stochastic integrals defined by E.J. Jung and J.H. Kim, are not in- tegrably bounded. Generalized set-valued stochastic integrals, considered in the paper, are in some non-trivial cases square integrably bounded and can be applied in the theory of stochastic differential equations with set-valued solutions.

Keywords:set-valued mappings, set-valued integrals, set-valued stochastic processes.

2010 Mathematics Subject Classification:60H05, 28B20, 47H04.

1. Introduction

The paper is devoted to properties of generalized set-valued stochastic inte- grals, defined in the author paper [10] for a nonempty subsets of the space IL²(IR⁺× Ω, Σ_IF, IR^d×m) of all square integrable IF-nonanticipative matrix-valued processes. For a given m-dimensional IF-Brownian motion B = (Bt)_t≥0 de- fined on a filtered probability space P_IF = (Ω, F, IF, P ) and a nonempty sub- set G of the space IL²(IR⁺× Ω, Σ_IF, IR^d×m), a generalized set-valued stochastic integral Rt

0GdBτ is understood as an Ft-measurable set-valued random variable

with values in the d-dimensional Euclidean space IR^d and subtrajectory inte- grals S_Ft(Rt

0GdBτ) equal to dec Jt(G). By Jtwe denote the Itˆo isometry defined on the space IL²(IR⁺ × Ω, ΣIF, IR^d×m) by setting Jt(g) = Rt

0gτdBτ for every g ∈ IL²(IR⁺× Ω, ΣIF, IR^d×m). Subtrajectory integrals S_Ft(Rt

0GdBτ) ofRt

0GdBτ is defined as a set of all Ft-measurable and square integrable selectors of Rt

0GdBτ. It will be also denoted by St(Rt

0GdBτ). In particular, if G is a nonempty decom- posable subset of IL²(IR⁺× Ω, ΣIF, IR^d×m) thenRt

0GdBτ =Rt

0GτdBτ, where G = (Gt)t≥0 is an IF-nonanticipative set-valued process such that SIF(G) = clIL(G), where S_IF(G) = {g ∈ IL²(IR⁺× Ω, ΣIF, IR^d×m) : gt(ω) ∈ Gt(ω) for a.e. (t, ω) ∈ IR⁺ × Ω}. Set-valued stochastic integrals of the form Rt

0G_τdB_τ have been de- fined by E.J. Jung and J.H. Kim in the paper [4], basing on the definition of set-valued functional stochastic integrals defined in the author papers [5] and [6]

(see also [8]).

The generalized set-valued stochastic integrals, presented in this paper have better properties than set-valued stochastic integrals defined in the paper [4]. In particular, they are integrably bounded for some subsets of the space IL²(IR⁺× Ω, Σ_IF, IR^d×m). Integrable boundedness of set-valued stochastic integrals, defined by E.J. Jung and J.H. Kim, has been investigated by the author of the present paper (see [7–9]) without positive results. We were not able to present any example of multifunction, different on a singleton, having integrably bounded set- valued integral defined in [4]. Unfortunately, the result dealing with integrable boundedness of multifunctions with finite representations Castaing, presented in [9] is not true. The problem was also considered by M. Michta, who has showed (see [11]) that in the general case set-valued integrals, defined by E.J. Jung and J.H. Kim, are not integrably bounded.

Apart from the extension of the definition of set-valued stochastic integrals, we shall also extend the definition of the set-valued conditional expectation. It will be defined for every nonempty subsets of the space IL(Ω, F, P, IR^d) and called a generalized set-valued conditional expectation. The present paper is organized as follows. Section 2 contains basic notions of the theory of set-valued stochastic processes, whereas Section 3 is devoted to properties of generalized set-valued stochastic integrals. Integrable boundedness of generalized set-valued stochastic integrals is considered in Section 4. Basic properties of indefinite generalized set-valued stochastic integrals are contained in the last Section of the paper.

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)}. Given a sequence (An)_n≥1 ⊂ Cl(X) ∪ {∅} by Lim An and Lim An we denote its Kuratowski limits inferior and superior defined by Lim An = {x ∈ X : lim d(x, An) = 0} and

Lim An = {x ∈ X : lim d(x, An) = 0}, respectively. It can be verified (see [2, 3]) that Lim An = {x ∈ X : x = lim xn, xn ∈ An, n ≥ 1} and Lim An = {x ∈ X : x = lim x_nk, x_nk ∈ Ank, n₁ < n₂ < · · · < n_k< · · · }. Immediately from the above definions we get Lim An⊂ Lim An. We call a sequence (An)_n≥1 convergent in the Kuratowski sense to A ∈ Cl(X) ∪ {∅} if A = Lim An = Lim An. The limit A is denoted by Lim A_n and said to be Kuratowski’s limit of a sequence (A_n)_n≥1. If A₁ ⊂ A₂ ⊂ A₃ ⊂ . . . , then a sequence (An)_n≥1 is convergent in the Kuratowski sense and Lim An=S

n≥1An.

2. Set-valued stochastic processes

Throughout the paper we shall deal with a complete filtered probability space P_F = (Ω, F, F, P ) with a filtration F = (Ft)_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. Similarly 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-nonanticipative if G is β(IR⁺) ⊗ F-measurable and G(t, · ) is Ft-measurable for every t ≥ 0, where β(IR⁺) denotes the Borel σ-algebra on IR⁺. It is easy to see that the set-valued process G is IF-nonanticipative if and only if the multifunction G : IR⁺× Ω → Cl(IR^r) is Σ_IF-measurable, where Σ_IFis a σ-algebra on IR⁺× Ω defined by Σ_IF= {A ∈ β(IR⁺) ⊗ F : A^t ∈ Ft for t ≥ 0}, where A^t denotes the section of a set A at t ∈ IR⁺. For a given IF-nonanticipative set-valued stochastic process G = (Gt)t≥0 with values in Cl(IR^d×m) defined on a filtered probability space PIF= (Ω, F, IF, P ), its subtrajectory integrals S_IF(G) is defined by S_IF(G) = {g ∈ IL²(IR⁺× Ω, ΣIF, IR^d×m) : g_t(ω) ∈ G_t(ω) for a.e. (t, ω) ∈ IR⁺× Ω}. If SIF(G) 6= ∅ then G is said to be Itˆo integrable.

Properties of set-valued random variables and set-valued stochastic processes follow immediately from properties of measurable multifunctions (see [1] and [2]).

For a given separable Banach space X and a σ-finite complete measure space (T, A, µ), a multifunction Z : T → Cl(X ) is said to be Aumann integrable if its subtrajectory integrals, denoted by S_A(Z) or simply by S(Z) is nonempty.

It can be proved (see [1]) that an Aumann integrable multifunction Z is square integrably bounded if and only if S(Z) is a bounded subset of IL²(T, A, X ). It can be verified (see [3], Corollary 3.5 of Chap. 2) 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 IL(T, A, X ). It is also decomposable, i.e., for every A ∈ A and u, v ∈ S(Z) one has 1IAu + 1I_T\Av ∈ S(Z). If Z is Aumann integrable then there is a Castaing representation (z_n)^∞_n=1 of Z such that (z_n)^∞_n=1 ⊂ S(Z), and therefore (see [1], Lemma 1.3), for every z ∈ S(Z) and ε > 0 there exist a finite A-measurable partition (Ak)^N_k=1 of T and a family (zn_k)^N_k=1 ⊂ {zn: n ≥ 1}

such that R

T|z −PN

k=11IA_kzn_k|²dµ < ε. In what follows the family of all finite A-measurable partitions of T is denoted by Π(T, A). For a given Λ ⊂ IL(T, A, X ) by dec(Λ) we denote the decomposable hull of Λ, i.e., the smallest decomposable subset of IL(T, A, X ) containing Λ. In a similar way the closed decomposable hull dec(Λ) is defined. It can be verified that dec(Λ) = clIL[dec(Λ)], where the closure is taken in the norm topology of IL(T, A, X ).

Immediately from the above properties of Aumann integrable multifunctions it follows that if (zn)^∞_n=1⊂ S(Z) is a Castaing representation of a multifunction Z, then S(Z) = dec{zn : n ≥ 1}. Indeed, it is clear that dec{zn : n ≥ 1} ⊂ S(Z). On the other hand, for every z ∈ S(Z) and ε > 0 there exist a partition (Ak)^N_k=1 ∈ Π(T, A) and a family (znk)^N_k=1 ⊂ {zn : n ≥ 1} such that R

T|z − PN

k=11I_Akz_nk|²dµ ≤ ε, which implies that z ∈ dec{z_n : n ≥ 1}. Thus S(Z) = dec{zn : n ≥ 1}. Finally, let us note (see [3], Th. 3.8 of Chap. 2) that a nonempty closed set K ⊂ IL²(T, A, X ) is decomposable if and only if there exists an A- measurable multifunction F : T → Cl(X ) such that K = S(F ).

For a given integrably bounded set-valued random variable Z : Ω → Cl(IR^r) and a σ-algebra G ⊂ F there exists (see [1], Th. 5.1) a unique G-measurable set- valued random variable E[Z|G] : Ω → Cl(IR^r) such that S(E[Z|G]) = cl_IL{E[f |G] : f ∈ S(Z)}, where the closure is taken in the norm topology of IL(Ω, F, IR^r). A set-valued random variable E[Z|G] is said to be the set-valued conditional expec- tation of Z relative to G. We can extend the above definition to nonempty subsets of the space IL(Ω, F, IR^r). Given a nonempty set Λ ⊂ IL(Ω, F, IR^r) and a σ-algebra G ⊂ F the generalized set-valued condition expectation E[Λ|G] of a set Λ relative to G is defined to be an G-measurable set-valued mapping E[Λ|G] : Ω → Cl(IR^r) such that S(E[Λ|G]) = dec_G{E[z|G] : z ∈ Λ}, where the decomposable hull is taken with respect to a σ-algebra G ⊂ F. It is clear that if Λ = S(Z), where Z is an Aumann integrable set-valued random variable, then E[Λ|G] = E[Z|G] a.s. In- deed, by the above definitions we get S(E[S(Z)|G]) = dec_G{E[z|G] : z ∈ S(Z)} = cl_IL[dec_G{E[z|G] : z ∈ S(Z)}] = clIL{E[z|G] : z ∈ S(Z)} = S(E[Z|G]), because the set {E[z|G] : z ∈ S(Z)} is decomposable with respect to the σ-algebra G ⊂ F.

Therefore, E[S(Z)|G] = E[Z|G] a.s.

3. Some properties of generalized set-valued integrals We present here some general properties of generalized set-valued stochastic in- tegrals. In what follows we shall assume that we have given an m-dimensional

IF-Brownian motion B = (Bt)_t≥0 defined on a complete filtered probability space PIF. Apart from subsets of the space IL²(IR⁺× Ω, ΣIF, IR^d×m) we shall also con- sider subsets of the space IL²(Ω, F_t, IR^d) for every t ≥ 0. The closures of subsets of both IL²(IR⁺× Ω, Σ_IF, IR^d×m) and IL²(Ω, Ft, IR^d) will be denoted by the same way by cl_IL.

Lemma 3.1. For every nonempty set G ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) and t ≥ 0 one has

(i) Jt[cl_IL(G)] = cl_IL[Jt(G)], (ii) Jt(co(G)) = co Jt(G),

(iii) dec{Jt[co(G)]} = co[dec Jt(G)],

(iv) if (Ω, F, P ) is a separable probability space then there is a sequence (gⁿ)^∞_n=1⊂ G such that clILJt(G) = cl_IL{Jt(gⁿ) : n ≥ 1} for every t ≥ 0.

Proof. (i) By continuity of a mapping J_t one has J_t[cl_IL(G)] ⊂ cl_IL[J_t(G)]. For every u ∈ cl_IL[Jt(G)] and every sequence (gⁿ)^∞_n=1 ⊂ G such that E|u−Jt(gⁿ)|² → 0 we have E|Jt(gⁿ) − Jt(g^m)|² → 0 as m, n → ∞. But E|Jt(gⁿ) − Jt(g^m)|² = ERt

0|gⁿ_τ − g^m_τ |²dτ for every t ≥ 0 and every m, n ≥ 1. Then (gⁿ)^∞_n=1 is a Cauchy sequence in the Banach space IL²(IR⁺× Ω, Σ_IF, IR^d×m). Thus there exists g ∈ IL²(IR⁺× Ω, ΣIF, IR^d×m) such that ERt

0|g_τⁿ− gτ|²dτ → 0 as n → ∞, which implies that g ∈ cl_IL(G) and E|J_t(g) − J_t(gⁿ)|² → 0 as n → ∞. Then Jt(g) ∈ J_t[cl_IL(G)]

because Jt[cl_IL(G)] is a closed subset of the space IL²(Ω, Ft, IR^d). Hence it follows that u ∈ Jt[cl_IL(G)] because u = Jt(g). Then cl_IL[Jt(G)] ⊂ Jt[cl_IL(G)].

(ii) By linearity of a mapping Jt we have Jt(co G) = co Jt(G), which implies that cl_IL[J_t(co G)] = co J_t(G). Hence, by (i) it follows that J_t(co(G)) = co [J_t(G)]

because clIL[Jt(co G)] = Jt(co G) .

(iii) It is clear that J_t[co(G)]} ⊂ co[dec J_t(G)] because J_t(co(G)) = co J_t(G).

Let us observe that co [dec Jt(G)] is a decomposable subset of the space IL²(Ω, Ft, IR^d). Indeed, by the properties of a set dec Jt(G) there is an Ft-mea- surable set-valued random variable F = (F_t)_t≥0 with values in Cl(IR^d) and such that decJt(G) = St(F ) for every t ≥ 0. Then, co[decJt(G)] = coSt(F ) = St(co F ), which implies that co[dec Jt(G)] is decomposable. Therefore, dec{Jt[co(G)]}

⊂ co[dec Jt(G)]. On the other hand we have dec J_t(G) ⊂ dec{J_t[co(G)]}. By virtue of ([8], Th. 3.3, Chap. 2) dec{Jt[co(G)]} is a convex subset of the space IL²(Ω, Ft, IR^d). Then co[dec Jt(G)] ⊂ dec {Jt[co(G)]}. Thus, dec{Jt[co(G)]} = co[dec J_t(G)].

(iv) By separability of the space (Ω, F, P ) the space IL²(IR⁺× Ω, ΣIF, IR^d×m) is separable. Then G with its induced topology is a separable subset of IL²(IR⁺× Ω, Σ_IF, IR^d×m). Thus there is a sequence (gⁿ)^∞_n=1 ⊂ G such that G = cl_I{gⁿ: n ≥ 1}, where cl_I denotes the closure in the induced topology of G. But

cl_IL{gⁿ : n ≥ 1} ⊂ cl_ILG and cl_I{gⁿ : n ≥ 1} = G ∩ cl_IL{gⁿ: n ≥ 1}. Then cl_ILG = cl_IL[cl_I{gⁿ : n ≥ 1}] = cl_IL[G ∩ cl_IL{gⁿ : n ≥ 1}] ⊂ cl_ILG ∩ clIL{gⁿ : n ≥ 1} = cl_IL{gⁿ : n ≥ 1}. Hence, and (i) it follows that cl_ILJt(G) = cl_IL{Jt(gⁿ) : n ≥ 1}

for every t ≥ 0.

Lemma 3.2. If (Ω, F, P ) is separable, (X, ρ) is a metric space and Φ : X ∋ x → Φ(x) ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) is l.s.c. and such that Φ(x) is a nonempty closed set for every x ∈ X, then there is a sequence (gⁿ)^∞_n=1 of continuous functions gⁿ : X → IL²(IR⁺ × Ω, ΣIF, IR^d×m) such that gⁿ(x) ∈ co Φ(x) for n ≥ 1 and co Φ(x) = clIL{gⁿ(x) : n ≥ 1} for every x ∈ X.

Proof. The result follows immediately from ([2], Prop. 4.4, Chap. 1), because the space IL²(IR⁺× Ω, ΣIF, IR^d×m) is separable and a set valued mapping X ∋ x → co Φ(x) ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) satisfies the assumptions of ([2], Prop.

4.4, Chap. 1).

Lemma 3.3. If G = {gⁿ : n ≥ 1} ⊂ IL²(IR⁺× Ω, Σ_IF, IR^d×m), then dec Jt(G) = Lim dec Jt(Gp), where Gp = {g¹, . . . , g^p} for p ≥ 1.

Proof. Let us observe that cl_ILG = Lim Gp. Indeed, we have Gp ⊂ G_p+1 for every p ≥ 1. Therefore, the Kuratowski limit Lim Gp exists. Furthermore, Gp ⊂ clILG for every p ≥ 1, which implies that Lim Gp ⊂ clILG, because Lim Gp = cl_IL{S

p≥1 Gp}, where the closures are taken with respect to the norm topology of IL²(IR⁺ × Ω, Σ_IF, IR^d×m). On the other hand, for every g ∈ cl_ILG there is a subsequence (g^nk)^∞_k=1 of (gⁿ)^∞_n=1 such that g^nk → g as k → ∞. For every k ≥ 1 there is p_k ≥ 1 such that g^nk ∈ Gpk. Then g ∈ Lim G_p = Lim G_p. Thus clILG ⊂ Lim Gp.

In a similar way we obtain cl_ILJt(G) = Lim Jt(Gp), which implies that dec J_t(G) = dec[Lim J_t(G_p)]. To the end of the proof, we have to verify that dec[Lim Jt(Gp)] = Lim dec[Jt(Gp)]. Let us observe that Lim dec{Jt(Gp)} ⊂ dec{Lim Jt(Gp)}, because dec{Jt(Gp)} ⊂ dec{Lim Jt(Gp)} for every p ≥ 1 and dec{Lim J_t(G_p)} is a closed subset of IL²(Ω, F, IR^d). For every a ∈ dec{Lim J_t(G_p)}

there is a sequence (ar)^∞_r=1 of dec{Lim Jt(Gp)} = dec{Jt[Lim Gp)]} converg- ing to a in the norm topology of IL²(Ω, F, IR^d). For every r ≥ 1 there are a partition (A^r_k)^N_k=1^r ∈ Π(Ω, Ft) and a family (u^r_k)^N_k=1^r ⊂ Jt[Lim G_p] such that a_r=PNr

k=11I_A^r

ku^r_k. For every r ≥ 1 and k = 1, . . . , N_rthere is a sequence (vp^k,r)^∞_p=1 such that v^k,rp ∈ Gp for every p ≥ 1 and Jt(vp^k,r) → u^r_k in the norm topology of IL²(Ω, F, IR^d) as p → ∞. Then, for every r ≥ 1 we have PNr

k=11IA^r_kJt(v^k,rp ) ∈ dec{Jt(Gp)} for p ≥ 1 and PNr

k=11IA^r_kcjt(v^k,rp ) → PNm

k=11IA^r_ku^r_k = ar as p → ∞.

Therefore, a_r ∈ Lim[dec{Jt(G_p)}] = Lim[dec{J_t(G_p)}] for every r ≥ 1, which implies that ar ∈ Lim[dec{Jt(Gp)}], because decJt(Gp) ⊂ decJt(G_p+1) for every p ≥ 1. Hence it follows that a ∈ Lim[decJt(Gp)] for every a ∈ dec{Lim Jt(Gp)}.

Thus dec{Lim Jt(Gp)} ⊂ Lim[decJt(Gp)], which implies that dec{Lim Jt(Gp)} = Lim dec{Jt(Gp)}.

We present now the basic properties of generalized set-valued stochastic integrals.

Theorem 3.4. For every nonempty setG ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) we have (i) Rt

0clIL(G)dBτ =Rt

0GdBτ and (ii) Rt

0 co(G)dBτ = coRt

0GdBτ a.s. for every t ≥ 0.

Proof. (i) Immediatelly from (i) of Lemma 3.1 we get dec J_t[cl_IL(G)] = dec[Jt(G)]} which, by the definition of generalized set-valued stochastic inte- grals, implies that St(Rt

0cl_IL(G)dBτ) = St(Rt

0GdBτ) for every t ≥ 0. Then Rt

0cl_IL(G)dBτ =Rt

0GdBτ a.s. for every t ≥ 0.

(ii) Similarly, by (iii) of Lemma 3.1 and the definition of generalized set-valued stochastic integrals, we obtain St(Rt

0co (G)dBτ) = co St(Rt

0GdBτ) for every t ≥ 0.

Immediately from (i) it follows thatRt

0co (G)dB_τ =Rt

0 co (G)dB_τ a.s. for every t ≥ 0. Furthermore, co S_t(Rt

0GdBτ) = S_t(coRt

0GdBτ). Therefore, S_t(Rt

0co (G)dB_τ) = S_t(coRt

0GdBτ) for every t ≥ 0. Thus Rt

0co (G)dB_τ = coRt

0GdBτ a.s. for every t ≥ 0.

Theorem 3.5. If (Ω, F, P ) is separable then for every nonempty set G ⊂ IL²(IR⁺×Ω, ΣIF, IR^d×m) there is a sequence (gⁿ)^∞_n=1⊂ G such that (Rt

0 GdBτ)(ω) = cl{(Rt

0g_τⁿdBτ)(ω) : n ≥ 1} for every t ≥ 0 and a.e. ω ∈ Ω.

Proof. By (iv) of Lemma 3.1 there is a sequence (gⁿ)^∞_n=1 ⊂ G such that clILJt(G) = clIL{Jt(gⁿ) : n ≥ 1} for every t ≥ 0, which implies that dec Jt(G) = dec {Jt(gⁿ) : n ≥ 1} for every t ≥ 0. Let Γt(ω) = cl{(Rt

0 g_τⁿdBτ)(ω) : n ≥ 1}

for every t ≥ 0 and ω ∈ Ω. We have St(Γt) = dec {Jt(gⁿ) : n ≥ 1} for every t ≥ 0, because Γt is an Aumann integrably set-valued random variable. On the other hand, by the definition of generalized set-valued stochastic integrals, we have St(Rt

0 GdBτ) = dec Jt(G) for every t ≥ 0. Then St(Rt

0GdBτ) = St(Γt) for every t ≥ 0, which implies that (Rt

0 GdBτ)(ω) = cl{(Rt

0gⁿ_τdBτ)(ω) : n ≥ 1} for every t ≥ 0 and a.e. ω ∈ Ω.

Theorem 3.6. If G = {gⁿ : n ≥ 1} ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m), then Rt

0GdBτ = LimRt

0GpdBτ a.s. for every t ≥ 0, where Gp = {g¹, . . . , g^p} for for p ≥ 1.

Proof. By virtue of Lemma 3.3 we have dec Jt(G) = Lim dec Jt(Gp), for every t ≥ 0, which implies that St(Rt

0 GdBτ) = Lim St(Rt

0GpdBτ) for t ≥ 0. We shall show now that Lim St(Rt

0GpdBτ) = St(LimRt

0 GpdBτ) for t ≥ 0. Indeed, it is clear that Lim St(Rt

0GpdBτ) ⊂ St(LimRt

0GpdBτ) for t ≥ 0. Let a ∈ St(Lim Rt

0 GpdBτ), i.e., let a ∈ LimRt

0GpdBτ = Lim Rt

0GpdBτ a.s. We have, d(a,Rt

0GpdBτ) → 0

a.s. as p → ∞. Let us observe that the sequence {d(a,Rt

0 GpdBτ)}^∞_p=1 is inte- grably bounded by a function ϕ := d(a,RT

0 g_t¹dBt) because d(a,Rt

0Gp+1dBt) ≤ d(a,Rt

0GpdBτ) for every p ≥ 1. Therefore, E{d(a,Rt

0GpdBτ)} → 0 as p → ∞.

Hence, by ([1], Th. 2.2) it follows that d²

 a, S_t

 Z t 0

GpdB_τ



= inf



|| a − u ||² : u ∈ St

 Z t 0

GpdBτ



= E

 d²

 a,

Z t 0

GpdBτ



for every p ≥ 1, where || · || denotes the norm of the space IL²(Ω, F, IR^d).

Then d(a, S_t(Rt

0GpdB_τ)) → 0 as p → ∞. Therefore, a ∈ Lim S_t(Rt

0GpdB_τ) = Lim S_t(Rt

0GpdB_τ) for every a ∈ S_t(LimRt

0GpB_τ). Thus S_t(LimRt

0GpdB_τ) ⊂ LimS_t(Rt

0GpdB_τ), which implies that S_t(LimRt

0GpdB_τ) = Lim S_t(Rt

0GpdB_τ). Now we have S_t(Rt

0GdBτ) = S_t(LimRt

0G^pdB_τ) for every t ≥ 0, which implies that Rt

0GdBτ = LimRt

0GpdB_τ a.s. for every t ≥ 0.

Corollary 3.1. If (Ω, F, P ) is a separable probability space and G is a nonempty subset of the spaceIL²(IR⁺×Ω, ΣIF, IR^d×m) then there exists a sequence (gⁿ)^∞_n=1⊂ G such thatRt

0 GdBτ = LimRt

0GpdBτ a.s. for everyt ≥ 0, where Gp = {g¹, . . . , g^p} for for p ≥ 1.

Proof. By (iv) of Lemma 3.1 there is a sequence (gⁿ)^∞_n=1 ⊂ G such that cl_ILJt(G) = cl_IL{Jt(gⁿ) : n ≥ 1} for every t ≥ 0. Therefore, dec Jt(G) = dec {Jt(gⁿ) : n ≥ 1} for every t ≥ 0, which is equivalent to St(Rt

0GdBτ) = St(Rt

0{gⁿ : n ≥ 1}dBτ) for every t ≥ 0. Hence, similarly as in the proof of Theorem 3.6, it follows that Rt

0 GdBτ = LimRt

0GpdBτ a.s. for every t ≥ 0.

Remark 3.1. If G is a nonempty bounded decomposable subset of the space IL²(IR⁺× Ω, ΣIF, IR^d×m) then the last result is true without the assumption that (Ω, F, P ) is a separable probability space.

Proof. By decomposability of the set G it follows that there is an IF-nonantici- pative process G = (G_t)_t≥0 with values in Cl(IR^d×m) such that cl_ILG = SIF(G).

By ΣIF-measurability of G there is a sequence (gⁿ)^∞_n=1, of ΣIF-measurable pro- cesses gⁿ= (g_tⁿ)_t≥0, a Castaing representation of G, such that Gt(ω) = cl{g_tⁿ(ω) : n ≥ 1} for every (t, ω) ∈ IR⁺× Ω. We have (gⁿ)^∞_n=1 ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) because G is square integrably bounded. Therefore, S_IF(G) = dec{gⁿ : n ≥ 1}.

Thus there is a sequence (gⁿ)^∞_n=1 ⊂ IL²(IR⁺× Ω, Σ_IF, IR^d×m) such that cl_ILG = dec{gⁿ: n ≥ 1}. Hence, similarly as above (see [9], Th. 3.2) we obtainRt

0GdBτ = LimRt

0dec GpdBτ a.s. for every t ≥ 0, where Gp= {g¹, . . . , g^p} for for p ≥ 1.

Remark 3.2. In a similar way as above we can show that if G = {gⁿ : n ≥ 1} ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m), thenRt

0 co GdBτ = LimRt

0co GpdBτ a.s. for every t ≥ 0, where G_p = {g¹, . . . , g^p} for for p ≥ 1.

We shall show now that for every nonempty set G ⊂ IL²(IR⁺× Ω, ΣIF, IR^d) such that a generalized set-valued stochastic integral Rt

0 GdBτ of G with respect to a real IF-Brownian motion B = (Bt)_t≥0 is square integrably bounded, we have σ(p,Rt

0GdBτ) = supRt

0S(p, G)dBτ a.s. for every p ∈ IR^d and t ≥ 0. σ( ·, A) denotes the support function of a set A ⊂ IR^d, S(p, G) = {(p, g) : g ∈ G} and (p, g) denotes for every g ∈ G a real-valued IF-nonanticipative stochastic process defined by the inner product h·, ·i of IR^dby setting (p, g)_t(ω) = hp, g_t(ω)i for every (t, ω) ∈ IR⁺× Ω. Let us note that Rt

0S(p, G)dBτ) is a closed subset of the real line IR.

Theorem 3.7. For every nonempty set G ⊂ IL²(IR⁺× Ω, ΣIF, IR^d) such that a generalized set-valued stochastic integralRt

0GdBτ is square integrably bounded for every t ≥ 0, we have σ(p,Rt

0 GdBτ) = supRt

0S(p, G)dBτ a.s. for every p ∈ IR^d and t ≥ 0.

Proof. Let Rt

0GdBτ be square integrably bounded for fixed t ≥ 0. For every every p ∈ IR^dand A ∈ Ft we have

Z

A

σ

 p,

Z t 0

GdBτ

 dP =

Z

A

sup



hp, xi : x ∈ Z t

0

GdBτ

 dP

= sup

Z

A

hp, ui dP : u ∈ St

Z t 0

GdBτ



= sup

Z

A

hp, ui dP : u ∈ dec Jt(G)



= sup (Z

A

* p,

N

X

k=1

1IAkJt(g^k) +

dP : (Ak)^N_k=1∈ Π(Ω, Ft), (g^k)^N_k=1⊂ G )

= sup (Z

A

" _N X

k=1

1I_AkD

p, J_t(g^k)E

#

dP : (A_k)^N_k=1 ∈ Π(Ω, Ft), (g^k)^N_k=1 ⊂ G )

= sup (Z

A

" _N X

k=1

1IA_kJt

Dp, g^kE

#

dP : (A_k)^N_k=1∈ Π(Ω, Ft), (g^k)^N_k=1 ⊂ G )

= sup

Z

A

u dP : u ∈ dec Jt(S(p, G))



= sup

Z

A

u dP : u ∈ St

Z t 0

S(p, G)dBτ



= Z

A

 sup

Z t 0

S(p, G)dBτ

 dP.

Therefore, σ(p,Rt

0 GdBτ) = supRt

0 S(p, G)dBτ a.s. for every p ∈ IR^d and fixed t ≥ 0.

4. Integrable boundedness of generalized set-valued integrals We present now some results dealing with integrable boundedness of generalized set-valued stochastic integrals. We begin with the following lemma.

Lemma 4.1. For everym-dimensional IF-Brownian motion B = (Bt)_t≥0 defined on a complete filtered probability space PIF, and every set Gp= {gⁱ : 1 ≤ i ≤ p} ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) one has EkRt

0GpdB_τk² ≤ p · ERt

0max_1≤i≤p|gⁱ_τ|²dτ < ∞ for everyt ≥ 0.

Proof. By virtue of ([1], Th. 2.2) we get E

Z t 0

GpdB_τ

2

= sup



E|u|² : u ∈ S_t

 Z t 0

GpdB_τ



= sup



E|u|² : u ∈ dec{Jt(g¹), . . . , Jt(g^p)}



= sup

 E

 ^p X

j=1

1IAj|Jt(g^j)|²



: (Aj)^p_j=1∈ Π(Ω, Ft)



≤

p

X

j=1

E Z t

0

|g^j_τ|²dτ ≤ p · E Z t

0

1≤i≤pmax |g_τⁱ|²dτ < ∞.

We shall prove now the following theorem dealing with integrable boundedness of generalized set-valued stochastic integrals.

Theorem 4.2. Let G be a nonempty subset of IL²(IR⁺ × Ω, ΣIF, IR^d×m). If Rt

0GdBτ is square integrably bounded for every t ≥ 0 then 1I_[0,t]G is a bounded subset of IL²(IR⁺× Ω, ΣIF, IR^d×m) every t ≥ 0.

Proof. Let Rt

0GdBτ be square integrably bounded for fixed t ≥ 0. Then St(Rt

0 GdBτ) is a boundedd subset of IL²(Ω, Ft, IR^d). Thus there exists Mt > 0 such that E|u|² ≤ Mt for every u ∈ St(Rt

0GdBτ). But St(Rt

0GdBτ) = dec Jt(G).

Therefore, Jt(g) ∈ St(Rt

0 GdBτ) for every g ∈ G, which implies that k1I_[0,t]gk² = ERt

0|gτ|²dτ = E|J_t(g)|² ≤ Mt for every g ∈ G. Thus 1I_[0,t]G is a bounded subset of IL²(IR⁺× Ω, ΣIF, IR^d×m).

Let us observe that boundedness of a set 1I_[0,t]G is not sufficient for square inte- grable boundedness of a generalized set-valued integralRt

0GdBτ. Indeed, by virtue of results of [11] there exists an integrably bounded set-valued IF-nonanticipative process G = (Gt)_t≥0 such that S_IF(G) is a bounded subset of IL²(IR⁺× Ω, Σ_IF, IR^d×m) and EkRt

0 GτdBτk² = ∞. Taking G = S_IF(G) we obtain for every fixed

t ≥ 0 a bounded subset 1I_[0,t]G of IL²(IR⁺× Ω, Σ_IF, IR^d×m) such that EkRt

0GdBτk²

= ∞, becauseRt

0GdBτ =Rt

0G_τdB_τ.

We shall present now the following sufficient condition for square integrable boundedness of generalized set-valued stochastic integrals.

Theorem 4.3. LetG ⊂ IL²(IR⁺×Ω, ΣIF, IR^d×m) be a nonempty set and (gⁿ)^∞_n=1⊂ G a sequence such that cl_ILG = cl_IL{gⁿ: n ≥ 1} and P_∞

n=1kgⁿk² < ∞, where k · k is the norm of IL²(IR⁺× Ω, ΣIF, IR^d×m). Then a generalized set-valued stochastic integral Rt

0GdBτ is square integrably bounded for everyt ≥ 0.

Proof. LetP_∞

n=1kgⁿk² < ∞. Similarly as in the proof of Lemma 4.1, for every p ≥ 1 we obtain

E

Z t 0

GpdBτ

2

≤

p

X

j=1

E Z t

0

|g^j_τ|²dτ ≤

∞

X

n=1

kgⁿk²< ∞,

where G_p = {g¹, . . . , g^p}. By Theorem 3.6 we have Rt

0GdBτ = LimRt

0GpdB_τ a.s.

for every t ≥ 0. Therefore, for every t ≥ 0 we get E

Z t 0

GdBτ

2

= sup



E|u|² : u ∈ St

 Lim

Z t 0

GpdBτ



= sup



E|u|² : u ∈ Lim St

 Z t 0

GpdBτ



= sup



E|u|²: u ∈

∞

[

p=1

 St

 Z t 0

GpdBτ



≤ sup

p≥1



E|u|² : u ∈ St

Z t 0

GpdBτ



.

Hence, by ([1], Th. 2.2) it follows that St(Rt

0GdBτ) is a bounded subset in the space IL²(Ω, Ft, IR^d). Thus a set-valued integral Rt

0GτdBτ is square integrably bounded for every t ≥ 0.

Corollary 4.1. For every set Gp = {gⁱ : 1 ≤ i ≤ p} ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) a generalized set-valued stochastic integral Rt

0GpdBτ is square integrably bounded for everyt ≥ 0.

Remark 4.1. Immediately from (iii) of Lemma 3.1 and (ii) of Theorem 3.4 it follows that for every nonempty set G ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) a generalized set-valued stochastic integralRt

0GdBτ is square integrably bounded if and only if Rt

0co GdBτ is square integrably bounded.

We shall prove now some results dealing with estimations of the Hausdorff dis- tance between generalized set-valued stochastic integrals.

Theorem 4.3. For every m-dimensional IF-Brownian motion B = (Bt)_t≥0 de- fined on a complete filtered probability space PIF, and every sets {fⁱ: 1 ≤ i ≤ p}, {gⁱ : 1 ≤ i ≤ p} ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) one has

(4.1) Eh²

 Z t 0

{fⁱ : 1 ≤ i ≤ p}dBτ, Z t

0

{gⁱ: 1 ≤ i ≤ p}dBτ



≤ p · H²({f¹, . . . , f^p}, {g¹, . . . , g^p}),

where H is the Hausdorff metric in IL²(IR⁺× Ω, Σ_IF, IR^d×m).

Proof. Let p ≥ 1 be fixed. By virtue of ([1], Th. 2.2) we have Eh²

 Z t 0

{fⁱ : 1 ≤ i ≤ p}dBτ, Z t

0

{gⁱ: 1 ≤ i ≤ p}dBτ



= sup

 inf



E|u − v|² : v ∈ St

 Z t 0

{gⁱ : 1 ≤ i ≤ p}dBτ



: u ∈ St

 Z t 0

{fⁱ : 1 ≤ i ≤ p}dBτ



= sup{inf{E|u − v|² : v ∈ dec{Jt(gⁱ) : 1 ≤ i ≤ p}} : u ∈ dec{Jt(fⁱ) : 1 ≤ i ≤ p}}.

For every u ∈ dec{J_t(fⁱ) : 1 ≤ i ≤ p} one has

inf{E|u − v|²: v ∈ dec{J_t(gⁱ): 1 ≤ i ≤ p}} ≤ min{E|u − J_t(gⁱ)|²:: 1 ≤ i ≤ p}.

On the other hand, for every u ∈ dec{J_t(fⁱ) : 1 ≤ i ≤ p} there is (A_j)^p_j=1 ∈ Π(Ω, Ft) such that u =Pp

j=11IAjJt(f^j). Then Eh²

 Z t 0

{fⁱ : 1 ≤ i ≤ p}dB_τ, Z t

0

{gⁱ : 1 ≤ i ≤ p}dB_τ



≤ sup



1≤i≤pmin E

 ^p X

j=1

1IAj|Jt(f^j) − Jt(gⁱ)|²



: (Aj)^p_j=1 ∈ Π(Ω, Ft)



≤ p · max

1≤j≤p



1≤i≤pmin E|Jt(f^j) − Jt(gⁱ)|²



≤ p · H²({f¹, . . . , f^p}, {g¹, . . . , g^p}).

In a similar way we also get Eh²(Rt

0{gⁱ : 1 ≤ i ≤ p}dBτ,Rt

0{fⁱ : 1 ≤ i ≤ p}dBτ) ≤ p · H²({f¹, . . . , f^p}, {g¹, . . . , g^p}). Then (4.1) is satisfied.

Corollary 4.2. If the assumptions of Theorem 4.3 are satified then (4.2) Eh²

 Z t 0

co {fⁱ : 1 ≤ i ≤ p}dBτ, Z t

0

co {gⁱ : 1 ≤ i ≤ p}dBτ



≤ Eh²

Z t 0

{fⁱ : 1 ≤ i ≤ p}dBτ, Z t

0

{gⁱ : 1 ≤ i ≤ p}dBτ



≤ p · E Z t

0

1≤j≤pmax |f_τ^j− g_τ^j|²dτ.

Remark 4.2. In a similar way we can verify that for every sequences (fⁿ)^∞_n=1, (gⁿ)^∞_n=1 ⊂ IL²(IR⁺× Ω, Σ_IF, IR^d×m) such thatP∞

n=1kfⁿk²< ∞, andP∞

n=1kgⁿk²

< ∞, one has

(4.2) Eh²

 Z t 0

FdBτ, Z t

0

GdBτ



≤ E Z t

0

sup

j≥1

|f_τ^j− g_τ^j|²dτ

where F = co {fⁿ: n ≥ 1} and G = co {gⁿ: n ≥ 1}.

5. Generalized indefinite set-valued stochastic integrals Throughout this section we shall consider a nonempty set G^p = co {g¹, . . . , g^p}, for a given finite set {g¹, . . . , g^p} ⊂ IL²(IR⁺×Ω, ΣIF, IR^d×m). Given an m-dimensional IF-Brownian motion B = (Bt)t≥0defined on PIF, the set-valued IF-adapted stochas- tic process (Rt

0 G^pdBτ)_t≥0 is called the generalized indefinite set-valued stochastic integral of Gp. Immediately from Lemma 4.1 and the properties of the Haus- dorff metric h we have EkRt

0 G^pdB_τk² ≤ p · ERt

0max_1≤i≤p|g_τⁱ|²dτ . We shall prove that the generalized indefinite set-valued stochastic integral (Rt

0G^pdB_τ)_t≥0 is a set-valued submartingale, i.e., such that Rs

0 G^pdB_τ ⊂ E[Rt

0G^pdB_τ|Fs] a.s.

for every 0 ≤ s < t < ∞. To do that let us observe (see [7], Th. 3.2) that, by square integrable boundedness of a set-valued generalized set-valued stochas- tic integral Rt

0G^pdB_τ, we have Rt

0 G^pdB_τ = Rs

0 G^pdB_τ +Rt

sG^pdB_τ a.s. for every 0 ≤ s < t ≤< ∞.

Lemma 5.1. LetB = (Bt)t≥0 be anm-dimensional IF-Brownian motion defined on P_IF and G^p ⊂ IL²(IR⁺× Ω, Σ_IF, IR^d×m) be such as above. For every 0 ≤ s <

t < ∞ one has Rs

0 G^pdBτ = E[Jt(G^p)|Fs] a.s. for every p ≥ 1.

Proof. By the definitions of generalized set-valued integrals and the generalized set-valued conditional expectations we get

S_Fs

Z s 0

G^pdBτ



= dec_Fs{Js(g) : g ∈ G^p}

= dec_Fs{E[Jt(g)|Fs] : g ∈ G^p} = dec_Fs{E[u|Fs] : u ∈ Jt(G^p)}

= SFs(E[Jt(G^p)|Fs]).

Then Rs

0 G^pdB_τ = E[J_t(G^p)|F_s]] a.s. for every 0 ≤ s < t < ∞.

Corollary 5.1. If the assamptions of Lemma 5.1 are satisfied then a set-valued stochastic process (Rt

0G^pdBτ)_t≥0 is a set-valued submartingale for every p ≥ 1.

Proof. The result follows immediately from Lemma 5.1, an inclusion E[Jt(G^p)|Fs] ⊂ E[dec_Ft{Jt(G^p)}|Fs] for 0 ≤ s < t < ∞ and the equality St(Rt

0 G^pdBτ) = dec_Ft{Jt(G^p)}. Indeed, we have E[dec_Ft{Jt(G^p)}|Fs] = E

 St

 Z t 0

G^pdBτ

   

Fs



= E

 Z t 0

G^pdBτ

Fs

 .

Then E[J_t(G^p)|F_s]⊂E[Rt

0G^pdB_τ|Fs], for 0 ≤ s < t < ∞, which implies that S_Fs

 Z s 0

G^pdBτ



= S_Fs(E[Jt(G^p)|Fs]) ⊂ S_Fs

 E

 Z t 0

G^pdBτ

Fs



for every 0 ≤ s < t ≤ T . Thus Rs

0 G^pdBτ ⊂ E[Rt

0G^pdBτ|Fs] a.s. for every 0 ≤ s < t ≤ T .

Lemma 5.2. LetB = (B_t)_t≥0 be anm-dimensional IF-Brownian motion defined on PIF, and G^p ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) be such that as above. A real-valued process (kRt

0 G^pdBτk)t≥0 is a positive submartingale.

Proof. By Corollary 5.1 the set-valued process (Rt

0 G^pdBτ)_t≥0is a set-valued sub- martingale. Therefore, kRs

0 G^pdBτk ≤ kE[Rt

0 G^pdBτ|Fs]k a.s. for every 0 ≤ s <

t < ∞. By ([1], Th. 5.2) it follows that kE[Rt

0G^pdBτ|Fs]k ≤ E[kRt

0G^pdBτk|Fs] a.s. for every 0 ≤ s < t < ∞. Then, kRs

0 G^pdBτk ≤ E[kRt

0G^pdBτk|Fs] a.s. for every 0 ≤ s < t < ∞.

Immediately from the definition of a generalized set-valued integrals it follows that they are IF-adapted. We shall prove now that by the assumptions of Lemma 5.2, they are also continuous. We begin with the following lemma.

Lemma 5.3. Let B = (Bt)_t≥0 be an m-dimensional IF-Brownian motion, T > 0 and G^p ⊂ IL²(IR⁺× Ω, ΣIF, IR^d×m) be such as above. For every λ > 0 one has