Keywords

1 Introduction

In signal processing and process control, we often use set-valued stochastic integral (see [3, 4] e.g.). Fuzzy random Lebesgue integral is applied to equations and stochastic inclusions (see [8] e.g.). Some papers [1, 2] have studied the Aumann type integral. Jung and Kim [1] used decomposable closure to give definitions of the stochastic integral, we have the integral is measurable. Li et. al. [7] gave set-valued square integrable martingale integral. Kisielewicz discussed the boundedness of the integral in [2]. We discussed set-valued random Lebesgue integral in [2]. An almost everywhere problem is solved in [5]. Our paper is organized as following: a nonzero set problem is pointed out with the set-valued Lebesgue integral. Aumann integral theorem is proved. We shall also discuss its boundedness, convexity, an important integral inequality etc.

2 Set-Valued Random Processes

First, we provide some definitions and symbols of closed set spaces. A set of real numbers \(R\), natural numbers set \(N\), the \(d\)-dimensional Euclidean space \(R^{d}\). \(K(R^{d} )\) is the all non -empty, closed subsets family of \(R^{d}\), and \(K_{k} (R^{d} )\) (resp.\(K_{kc} (R^{d} )\)) the all nonempty compact (resp. compact convex) subsets family of \(R^{d}\). For \(x \in R^{d}\) and \(A \in K(R^{d} )\),

\(h(x,A) = \inf_{y \in A} ||x - y||\). Define the Hausdorff metric \(h_{d}\) on \(K(R^{d} )\) as

\(h_{d} (A,B) = \max \{ \sup_{a \in A} h(a,B),\sup_{b \in B} h(b,A)\} .\) For \(A \in K(R^{d} )\), denote

$$ ||A||_{K} = h_{d} (\{ 0\} ,A) = \sup_{a \in A} ||a||. $$

Then some properties of set-valued random processes shall be discussed. From first to last, we assume \(T > 0\), \(W = [0,T]\) and \(p \ge 1.\) A complete atomless probability space \((\Omega ,\text{C},P)\), a \(\sigma\)-field filtration \(\{ \text{C}_{t} :t \in [0,T]\}\), and the topological Borel field of a topological space \(E\) is \({\mathcal{B}}(E).\) Assume that \(f = \{ f(t),\text{C}_{t} :t \in [0,T]\}\) is a \(R^{d}\)-valued adapted random process. If for any \(t \in [0,T]\), the mapping \((s,\omega ) \to f(s,\omega )\) from \([0,t] \times \Omega\) to \(R^{d}\) is \({\mathcal{B}}([0,t]) \times \text{C}_{t}\)-measurable,

then \(f\) is sequential measurable.

If

$$ \text{D} = \{ B \subset [0,T] \times \Omega :\forall t \in [0,T],B \cap ([0,t] \times \Omega ) \in {\mathcal{B}}([0,t]) \times \text{C}_{t} \} , $$

we have that \(f\) is \(\text{D}\)-measurable if and only if \(f\) is sequential measurable.

Denote \({\text{SM}}\left( {K\left( {R^{d} } \right)} \right)\) the set of all sequential measurable set-valued random process. Similarly, we know notations \({\text{SM}}\left( {K_{c} \left( {R^{d} } \right)} \right)\), \({\text{SM}}\left( {K_{k} \left( {R^{d} } \right)} \right)\) and \({\text{SM}}\left( {K_{kc} \left( {R^{d} } \right)} \right)\). Sequential measurable \(F\) is adapted and measurable. For \(f_{1} ,f_{2} \in S{\text{M}}\left( {R^{d} } \right)\), define metric \(\Delta_{{\text{M}}} \left( {f_{1} ,f_{2} } \right) = E\int_{0}^{T} {\frac{{\left\| {f_{1} \left( s \right) - f_{2} \left( s \right)} \right\|}}{{1 + \left\| {f_{1} \left( s \right) - f_{2} \left( s \right)} \right\|}}} ds\), we have norm \(\left| {\left\| f \right\|} \right|_{{\text{M}}} = E\int_{0}^{T} {\frac{{\left\| {f\left( s \right)} \right\|}}{{1 + \left\| {f\left( s \right)} \right\|}}}\), then \(\left( {{\text{SM}}\left( {R^{d} } \right), \, \Delta_{{\text{M}}} } \right)\) is a complete space (cf. [6]).

Definition 2.1

\(g\left( {t,\omega } \right) \in G\left( {t,\omega } \right)\) for a.e. \(\left( {t,\omega } \right) \in \left[ {0,T} \right] \times \Omega\), we call the \(R^{d}\)-valued sequential measurable random process \(\left\{ {f\left( t \right),\text{C}_{t} :t \in \left[ {0,T} \right]} \right\} \in S{\text{M}}\left( {R^{d} } \right)\) is a selection of

$$ \left\{ {G\left( t \right),\text{C}_{t} :t \in \left[ {0,T} \right]} \right\}. $$

Let \(S\left\{ {G\left( \cdot \right)} \right\}\) or \(S\left( G \right)\) denote the family of all sequential measurable selections, i.e. \(S\left( G \right) = \left\{ {\left\{ {g\left( t \right):t \in \left[ {0,T} \right]} \right\} \in S{\text{M}}\left( {R^{d} } \right):g\left( {t,\omega } \right) \in G\left( {t,\omega } \right),{\text{ for a}}{\text{.e}}{. }\left( {t,\omega } \right) \in \left[ {0,T} \right] \times \Omega } \right\}\). There are many definitions and results on set-valued theory, we can read this paper [9]. In this paper, the Aumann type Lebesgue integral is given.

Definition 2.2

(cf. [4]): A set-valued random process \(G = \left\{ {G\left( t \right),t \in W} \right\} \in S{\text{M}}\left( {K\left( {R^{d} } \right)} \right)\). Define \(I_{t} \left( G \right)\left( \omega \right) = \left( A \right)\int_{0}^{t} G \left( {s,\omega } \right)ds = \left\{ {\int_{0}^{t} g \left( {s,\omega } \right)ds:f \in S\left( G \right)} \right\}\), for \(t \in W\), \(\omega \in \Omega\),

where \(\int_{0}^{t} g \left( {s,\omega } \right)ds\) is Lebesgue integral. We call \(\left( A \right)\int_{0}^{t} G \left( {s,\omega } \right)ds\) Aumann type Lebesgue integral of set-valued random process \(G\) with respect to \(t\).

Remark 2.3:

The elements of \(S\left( G \right)\) in Definition 2.2 are integrable. By the definition of \(S\left( G \right)\), \(g\left( {t,\omega } \right) \in G\left( {t,\omega } \right)\) is defined for a.e. \(\left( {t,\omega } \right) \in \left[ {0,T} \right] \times \Omega\), and the number of selections is uncountable. The union of uncountable a.e. zero measurable sets is NOT a zero measurable set in general, denoted by \(A_{{_{\left( F \right)} \left[ {0,T} \right] \times \Omega }}\). This helps to solve the boundedness problems in stochastic integral (see [2]). In fact, it may be unmeasurable. Let \(\text{D}_{1} = \left\{ {B_{{_{\left( F \right)} \left[ {0,T} \right] \times \Omega }} \subset B \subset \left[ {0,T} \right] \times \Omega :\forall t \in \left[ {0,T} \right],B \cap \left( {\left[ {0,t} \right] \times \Omega } \right) \in {\mathcal{B}}\left( {\left[ {0,t} \right]} \right) \times \text{C}_{t} } \right\}\), denote \(\min {\text{MB}}_{{_{G} \left[ {0,T} \right] \times \Omega }} = \bigcap\nolimits_{i = 1}^{\infty } {B_{i} }\), for any \(B_{i} \in \text{D}_{1}\). Let \(\Pr_{\Omega } \left( {\min {\text{MB}}_{{_{\left( G \right)} \left[ {0,T} \right] \times \Omega }} } \right) = B_{\left( G \right)} \Omega\), the project set on \(\Omega\) of \(\min {\text{MB}}_{{_{\left( G \right)} \left[ {0,T} \right] \times \Omega }}\), \(\Pr_{{\left[ {0,T} \right]}} \left( {\min {\text{MB}}_{{_{\left( G \right)} \left[ {0,T} \right] \times \Omega }} } \right) = B_{{_{\left( G \right)} \left[ {0,T} \right]}}\), the project set on \(\left[ {0,T} \right]\) of \(\min {\text{MB}}_{{_{\left( G \right)} \left[ {0,T} \right] \times \Omega }}\). In the following, we denote \(\min {\text{MB}}_{{_{\left( G \right)} \left[ {0,T} \right] \times \Omega }}\) as \(B_{{_{\left( G \right)} \left[ {0,T} \right] \times \Omega }}\) for convenience. Thus, \(B_{{_{\left( G \right)} \left[ {0,T} \right] \times \Omega }}\), \(B_{{_{\left( G \right)} \left[ {0,T} \right]}}\) and \(B_{{{}_{(G)}\Omega }}\) are all measurable.

Definition 2.4

Let a set-valued random process \(G = \left\{ {G\left( t \right),t \in W} \right\} \in S{\text{M}}\left( {K\left( {R^{d} } \right)} \right)\). \(t \in \left[ {0,T} \right]\backslash B_{{_{\left( G \right)} \left[ {0,T} \right]}}\), define the integral \(L_{t} \left( G \right)\left( \omega \right)\) by.

\(L_{t} \left( G \right)\left( \omega \right) = \left\{ {\begin{array}{*{20}l} {\left\{ {g\left( s \right)ds:g \in S_{T} \left( G \right)\left( \omega \right)} \right\},} \hfill & {\left( {s,\omega } \right) \notin B_{{_{{\left( G \right)\left[ {0,T} \right] \times \Omega }} }} } \hfill \\ {\left\{ 0 \right\},} \hfill & {\left( {s,\omega } \right) \in B_{{_{{\left( G \right)\left[ {0,T} \right] \times \Omega }} }} } \hfill \\ \end{array} } \right..\) .

We call it Aumann type Lebesgue integral.

Now let's discuss the following Auman theorem and representation theorem.

3 Theorem and Proof

Theorem 3.1:

A set-valued random process \(G \in S{\text{M}}\left( {K\left( {R^{d} } \right)} \right)\), \(t \in \left[ {0,T} \right]\backslash B_{{_{\left( G \right)} \left[ {0,T} \right]}}\), \(\left( A \right)\int_{0}^{t} G \left( s \right)ds\) is a nonempty subset of \({\text{SM}}\left( {K\left( {R^{d} } \right)} \right)\).

Proof.

\(S\left( G \right)\) is not null, \(g \in S\left( G \right)\), \(\int_{0}^{t} g \left( {s,\omega } \right)ds\) is sequential measurable. So \(\left( A \right)\int_{0}^{t} G \left( s \right)ds\) is nonempty.

In the following, a new definition will be given. First, we will define a decomposable closure.

Definition 3.2:

Nonempty subset \(\Xi \subset S{\text{M}}\left[ {\left[ {0,T} \right] \times \Omega ,\text{C},\lambda \times \mu ;R^{d} } \right]\), \(\overline{de} \Xi = \left\{ {\{ g\left( {s,\omega } \right):t \in \left[ {0,T} \right]} \right\}\), \(\varepsilon > 0\), there exist a \(\text{D}\)-measurable finite partition \(\left\{ {A_{1} , \cdots ,A_{n} } \right\}\) of \(\left[ {0,T} \right] \times \Omega\) and \(f_{1} , \cdots ,f_{n} \in \Xi\) such that \(\left| {\left\| {g - \sum\nolimits_{i = 1}^{n} {I_{{A_{i} }} f_{i} } } \right\|} \right|_{{\text{M}}} < \varepsilon \}\) is called the decomposable closure of \(\Xi\) with respect to \(\text{D}\),

Theorem 3.3:

\(\left\{ {G(t):t \in [0,T]} \right\} \in S{\text{M}}(K(R^{d} ))\), \(\Xi (t) = (A)\int_{0}^{t} {G(s)ds}\), there exists a \(\text{D}\)-measurable process \(L(G) = \left\{ {L_{t} (G):t \in [0,T]} \right\} \in S{\text{M}}(K(R^{d} ))\), we have \(S(L(G)) = \overline{de} \{ \Xi (t):t \in [0,T]\}\). In addition, the decomposable of \(\Xi (t) = (A)\int_{0}^{t} {G(s)ds}\) is bounded by a constant \(C\) using the norm in space \({\text{SM}}(R^{d} )\).

Proof

From Theorem 3.1, we know, \(t \in [0,T]\backslash B_{{{}_{(G)}[0,T]}}\), \(\Xi (t) = (A)\int_{0}^{t} {G(s)ds}\) is nonempty in space \({\text{SM}}(R^{d} )\). Let

$$ \begin{aligned} M = & \overline{de} \{ \Xi (t):t \in W\backslash B_{{{}_{(G)}[0,T]}} \} \\ = & \overline{de} \left\{ {h = \{ h(t):t \in [0,t]\backslash B_{{{}_{(F)}[0,T]}} \} :h(t)(\omega ) = \int_{0}^{t} {g(s,\omega )ds,g \in S(G),(s,\omega ) \notin B_{{{}_{(G)}[0,T] \times \Omega }} } } \right\} \\ \end{aligned} $$

\(M\) is a closed subset in \({\text{SM}}[W \times \Omega \backslash A_{{{}_{(G)}[0,T] \times \Omega }} ,\text{D},\lambda \times \mu ;R^{d} ]\). According to Theorem 2.7 in [6], it shows that there is \(L(G) =\) \(\left\{ {L_{t} (G):t \in [0,T]} \right\} \in S{\text{M}}(K(R^{d} ))\), we have \(S(L(G)) = M\).

Now we shall prove boundedness. That is,

$$ \left\| {\sum\limits_{{{\text{i}} = {1}}}^{{\text{n}}} {{\text{I}}_{{{\text{A}}_{{\text{i}}} }} } \int_{0}^{t} {g_{i} (s,\omega )ds} } \right\| \le \sum\limits_{i = 1}^{n} {\left\| {{\text{I}}_{{{\text{A}}_{{\text{i}}} }} \int_{0}^{t} {g_{i} (s,\omega )ds} } \right\|} $$
$$ \le \sum\limits_{i = 1}^{n} {I_{{A_{i} }} \int_{0}^{t} {\left\| {g_{i} (s,\omega )} \right\|} } ds $$

\(\le \sum\limits_{i = 1}^{n} {I_{{A_{i} }} \int_{0}^{t} {\left\| {G(s,\omega )} \right\|_{{\mathbf{K}}} } } ds\) .

Since \(\phi (r) = \frac{r}{1 + r} > 0\) is increasing, we have

$$ \begin{gathered} \left| {\left\| {\sum\limits_{i = 1}^{n} {I_{{A_{i} }} \int_{0}^{t} {g_{i} (s,\omega )ds} } } \right\|} \right|_{\rm{M}} = E\int_{0}^{T} {\frac{{\left\| {\sum\nolimits_{i = 1}^{n} {I_{{A_{i} }} \int_{0}^{t} {g_{i} (s,\omega )ds} } } \right\|}}{{1 + \left\| {\sum\nolimits_{i = 1}^{n} {I_{{A_{i} }} \int_{0}^{t} {g_{i} (s,\omega )ds} } } \right\|}}} dt \\ \le E\int_{0}^{T} {\frac{{\sum\nolimits_{i = 1}^{n} {I_{{A_{i} }} \int_{0}^{t} {\left\| {G(s,\omega )} \right\|_{{\mathbf{K}}} ds} } }}{{1 + \sum\nolimits_{i = 1}^{n} {I_{{A_{i} }} \int_{0}^{t} {\left\| {G(s,\omega )} \right\|_{{\mathbf{K}}} ds} } }}} dt \\ \le E\int_{0}^{T} {\frac{{\int_{0}^{T} {\left\| {G(s,\omega )} \right\|_{{\mathbf{K}}} ds} }}{{1 + \int_{0}^{T} {\left\| {G(s,\omega )} \right\|_{{\mathbf{K}}} ds} }}} dt \\ \le C \\ \end{gathered} $$

This constant \(C\) is not relative to \(n\).

Theorem 3.4 (Aumann Representation Theorem):

\(G = \left\{ {G(t):t \in [0,T]} \right\} \in S{\text{M}}(K(R^{d} ))\), a sequence of \(R^{d}\)-valued random processes \(\left\{ {g^{i} = \left\{ {g^{i} (t):t \in [0,T]} \right\}:i \ge 1} \right\} \subset S(G)\) exists, we have

$$ L_{t} (G)(\omega ) = cl\left\{ {\int_{0}^{t} {g^{i} (s,\omega )ds:i \ge 1} } \right\}a.e.(t,\omega ),\,(s,\omega ) \in [0,T] \times \Omega \backslash B_{{{}_{(G)}[0,T] \times \Omega }} $$

In addition, we have

$$ L_{t} (G)(\omega ) = cl\left\{ {\int_{0}^{t} {g(s,\omega )ds:g \in S(G)} } \right\}a.e.(t,\omega ),\,(s,\omega ) \in [0,T] \times \Omega \backslash B_{{{}_{(G)}[0,T] \times \Omega }} $$

Proof

By Theorem 3.9 in [5], we know, a series of \(\left\{ {\varphi_{n} = \left\{ {\varphi_{n} (t):t \in I} \right\}:n \ge 1} \right\} \subset S(L(G))\) exist,

\(L_{t} (G)(\omega ) = cl\left\{ {\varphi_{n} (t,\omega ):n \ge 1} \right\},a.e.(t,\omega ) \in W \times \Omega \backslash B_{{{}_{(G)}[0,T] \times \Omega }}\) holds.

Since

$$ \begin{aligned} S(L(G)) = & \overline{de} \{ \Xi (t):t \in [0,T]\} \\ = & \overline{de} \{ \left\{ {h(t):t \in I} \right\}:h(t) = \int_{0}^{t} {g(s)ds,} \left\{ {g( \cdot )} \right\} \in S(G)\} \\ = & cl\{ \left\{ {k(t):t \in I} \right\}:k(t) = \sum\limits_{k = 1}^{n} {I_{{A_{k} }} } \int_{0}^{t} {g_{k} (s)ds,} \left\{ {A_{k} :K = 1,2,...,l} \right\} \subset \text{D,}\,{\rm{is}}\,{\rm{a}}\,{\rm{finite}}\,{\rm{partition}}\,{\rm{of}} \\ & {\rm{W}} \times \Omega \backslash B_{{{}_{(G)}[0,T] \times \Omega }} {\rm{and }}\left\{ {\{ g_{k} ( \cdot )\} :k = 1,2,..,l} \right\} \subset S(G),1 \le l\} , \\ \end{aligned} $$

then for any \(1 \le n\), there exists \(\left\{ {k_{n}^{i} :1 \le i} \right\}\) such that \(\begin{array}{*{20}c} {\left\{ {\varphi_{n} = \left\{ {\varphi_{n} (t):t \in I} \right\}:1 \le n} \right\} \subset S(L(G)),} & {\left| {\left\| {\varphi_{n} (t) - k_{n}^{i} (t)} \right\|} \right|_{\rm{M}} \to 0(i \to \infty )} \\ \end{array}\), and \(k_{n}^{i} (t) = \sum\nolimits_{k = 1}^{l(i,n)} {{\text{I}}_{{{\text{A}}_{{\text{k}}}^{(i,n)} }} \int_{0}^{t} {g_{k}^{(i,n)} (s)ds} }\), where \(\left\{ {{\text{A}}_{{\text{k}}}^{(i,n)} :k = 1,...,l(i,n)} \right\} \subset \text{D}\) is a finite partition of \([0,T] \times \Omega \backslash B_{{{}_{(G)}[0,T] \times \Omega }}\), \(\left\{ {\left\{ {g_{k}^{(i,n)} (t):t \in I} \right\}:K = 1,2,...,l(i,n)} \right\}\) \(\subset S(G)\). Therefore there is a subsequence \(\left\{ {i_{j} :1 \le j} \right\}\) of \(\left\{ {1,2,...} \right\}\) such that

$$ \left\| {\varphi_{n} (t,\omega ) - k_{n}^{{i_{j} }} (t,\omega )} \right\| \to 0{\rm{ a}}{\rm{.e}}{\rm{. (}}t,\omega {\rm{)}} \in {\rm{[0,T]}} \times \Omega {\rm{\backslash B}}_{{{}_{{\rm{(G)}}}[0,T] \times \Omega }} (j \to \infty ) $$

Thus for \({\rm{ a}}{\rm{.e}}{\rm{. (}}t,\omega {\rm{)}} \in {\rm{[0,T]}} \times \Omega {\rm{\backslash B}}_{{{}_{{\rm{(G)}}}[0,T] \times \Omega }} (j \to \infty )\), we have that

$$ \begin{gathered} L_{t} (G)(\omega ) = cl\left\{ {k_{n}^{{i_{j} }} (t,\omega ):n,j \ge 1} \right\} \\ \subset cl\left\{ {\int_{0}^{t} {g_{k}^{{(i_{j} ,n)}} (s,\omega )ds} :n,j \ge 1,k = 1,...,l(i_{j} ,n)} \right\} \\ \subset L_{t} (G)(\omega ) \\ \end{gathered} $$

This means that for \({\rm{ a}}{\rm{.e}}{\rm{. (}}t,\omega {\rm{)}},{\rm{(}}s,\omega {\rm{)}} \in {\rm{[0,T]}} \times \Omega {\rm{\backslash B}}_{{{}_{{\rm{(G)}}}[0,T] \times \Omega }}\), we have

$$ L_{t} (G)(\omega ) = {\text{cl}} \left\{ {\int_{0}^{t} {g_{k}^{{\left( {i_{j} ,n} \right)}} } (s,\omega )ds:n,j \ge 1,k = 1, \ldots ,l\left( {i_{j} ,n} \right)} \right\} $$

Without losing generality, we have

$$ L_{t} (G)(\omega ) = {\text{cl}} \left\{ {\int_{0}^{t} {g^{i} } (s,\omega )ds:g^{i} \in S(G),i \ge 1} \right\}. $$

In addition,

$$ {\text{cl}} \left\{ {\int_{0}^{t} {g^{i} } (s,\omega )ds:g^{i} \in S(G),i \ge 1} \right\} \subseteq {\text{cl}} \left\{ {\int_{0}^{t} g (s,\omega )ds:g \in S(G)} \right\}. $$

Since \(\Gamma \subseteq \overline{de} \Gamma = S(L(F)),\) then we have

$$ {\text{cl}} \left\{ {\int_{0}^{t} g (s,\omega )ds:g \in S(G)} \right\} \subseteq {\text{cl}} \left\{ {\int_{0}^{t} {g^{i} } (s,\omega )ds:g^{i} \in S(G),i \ge 1} \right\}. $$

Therefore,

$$ L_{t} (G)(\omega ) = cl\left\{ {\int_{0}^{t} g (s,\omega )ds:g \in S(G)} \right\}. $$

Corollary 3.5 (Representation Theorem):

\(G = \{ G(t):t \in [0,T]\} \in {\text{PM}}\left( {K(R^{d} )} \right){. }\) There is a sequence of \(R^{d}\)-valued random process \(\left\{ {g^{i} = \left\{ {g^{i} (t):t \in [0,T]} \right\}:i \ge 1} \right\} \subset S(G)\) such that

$$ G(t,\omega ) = {\text{cl}} \left\{ {g^{i} (t,\omega ):i \ge 1} \right\}\quad {\text{ a}}{\text{.e}}{. }(t,\omega ) \in [0,T] \times \Omega \backslash B_{(G)} [0,T] \times \Omega , $$
$$ L_{t} (G)(\omega ) = {\text{cl}} \left\{ {\int_{0}^{t} {g^{i} } (s,\omega )ds:i \ge 1} \right\}{\text{ a}}{\text{.e}}{. }(t,\omega ),(s,\omega ) \in [0,T] \times \Omega \backslash B_{(G)} [0,T] \times \Omega . $$

Remark 3.6:

Since \(\left\{ {G(t):t \in [0,T]\backslash B_{(G)} [0,T]} \right\} \in S{\text{M}}\left( {K\left( {R^{d} } \right)} \right)\) is measurable wih respect to \(t \in\) \({\rm{\ominus \mathcal{B}}}\left( {[0,T]\backslash B_{(G)} [0,T]} \right)\) for fixed \(\omega \in \Omega \backslash B_{{{}_{(G)}\Omega }}\). If.

$$ s \in [0,t]\backslash B_{{{}_{(G)}[0,T]}} \subset [0,T],G(s,\omega ) \subseteq R_{ + }^{d} , $$

by Remark 3.11 in [6], we have.

\((A)\int_{0}^{t} G (s,\omega )ds = (A)\int_{0}^{t} {{\text{conv}} G} (s,\omega )ds = {\text{conv}} \left( {(A)\int_{0}^{t} G (s,\omega )} \right)\) .

Therefore, theAumann random Lebesgue integral \((A)\int_{0}^{t} G (s,\omega )\) is convex by using Aumann representation theorem.

Theorem 3.7:

For \(p \ge 1\), \(F,G \in L^{p} \left( {[0,T] \times \Omega ;K\left( {R^{d} } \right)} \right)\), a.e.

\((s,\omega ) \in ([0,t] \times \Omega ) \cap \overline{B}_{{{}_{(F)}[0,T] \times \Omega }} \cap \overline{B}_{{{}_{(G)}[0,T] \times \Omega }}\), we have.

$$ h_{d} \left( {L_{t} (F)(\omega ),L_{t} (G)(\omega )} \right) \le \int_{0}^{t} {h_{d} } (F_{s} (\omega ),G_{s} (\omega ))ds. $$
(1)

Proof

Since \(F,\quad G \in L^{p} \left( {[0,T] \times \Omega ;K\left( {R^{d} } \right)} \right)\), that is \(\left( {E\int_{0}^{T} {\left\| {F(s,\omega )} \right\|^{p} ds} } \right)^{\frac{1}{p}} < + \infty\). Thus, there exists \(\Omega _{F}\) such that \(P(\Omega _{F} ) = 1\), for any \(\omega \in \Omega_{F} ,\left( {\int_{0}^{T} {\left\| {F(s,\omega )} \right\|^{p} ds} } \right)^{\frac{1}{p}} < + \infty\). In the same way, we have \(\Omega _{G}\). Assume \(\omega \in \left( {\Omega_{F} \backslash B_{{{}_{(F)}\Omega }} } \right) \cap \left( {\Omega_{G} \backslash B_{{{}_{(G)}\Omega }} } \right)\) in the following proof. Take an \(f \in S_{T} (F)(\omega )\). Then, for \(t,s \in [0,T] \cap \overline{B}_{{{}_{(F)}[0,T]}} \cap \overline{B}_{{{}_{(G)}[0,T]}}\), we have.

$$ \begin{gathered} h\left( {\int_{0}^{t} f_{s} ds,L_{t} (G)(\omega )} \right) = \mathop {\inf }\limits_{{g \in S_{t}^{1} (G)(\omega )}} \left\| {\int_{0}^{t} f_{s} ds - \int_{0}^{t} g (s)ds} \right\| \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le \mathop {\inf }\limits_{{g \in S_{t}^{1} (G)(\omega )}} \int_{0}^{t} {\left\| {f_{s} - g_{s} } \right\|} ds \hfill \\ \end{gathered} $$

Further, by proving the same point of [8, Theorem 4],

$$ \begin{gathered} \mathop {\inf }\limits_{{g \in S_{t}^{1} (G)(\omega )}} \int_{0}^{t} {\left\| {f_{s} - g_{s} } \right\|ds} \hfill \\ = \int_{0}^{t} {\mathop {\inf }\limits_{y \in G(s,\omega )} } \left\| {f_{s} - y} \right\|ds = \int_{0}^{t} h (f_{s} ,G_{s} (\omega ))ds \hfill \\ \le \int_{0}^{t} {\mathop {\sup }\limits_{{x \in F_{s} (\omega )}} } h(x,G_{s} (\omega ))ds \le \int_{0}^{t} {h_{d} } (F_{s} (\omega ),G_{s} (\omega ))ds \hfill \\ \end{gathered} $$

Thus,

$$ h\left( {\int_{0}^{t} f (s)ds,L_{t} (G)(\omega )} \right) \le \int_{0}^{t} {h_{d} } (F_{s} (\omega ),G_{s} (\omega ))ds $$

We know \(f \in S_{T} (F)(\omega )\), by Definition 2.4 we have that

\(\mathop {\sup }\limits_{{x \in L_{t} (F)(\omega )}} h\left( {x,L_{t} (G)(\omega )} \right) \le \int_{0}^{t} {h_{d} } (F_{s} (\omega ),G_{s} (\omega ))ds\) .

Similarly, we have

$$ \mathop {\sup }\limits_{{x \in L_{t} (G)(\omega )}} h\left( {x,L_{t} (F)(\omega )} \right) \le \int_{0}^{t} {h_{d} } (F_{s} (\omega ),G_{s} (\omega ))ds. $$

The two inequalities above yield

$$ h_{d} \left( {L_{t} (F)(\omega ),L_{t} (G)(\omega )} \right) \le \int_{0}^{t} {h_{d} } (F_{s} (\omega ),G_{s} (\omega ))ds. $$

We obtain (1).