Applied Mathematics
Vol.06 No.13(2015), Article ID:61605,12 pages
10.4236/am.2015.613193
Lebesgues-Stieltjes Integrals of Fuzzy Stochastic Processes with Respect to Finite Variation Processes
Jinping Zhang, Lingli Luo, Xingmei Li, Xiaoying Wang
Department of Mathematics and Physics, North China Electric Power University, Beijing, China
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 28 September 2015; accepted 27 November 2015; published 30 November 2015
ABSTRACT
Let be a fuzzy stochastic process and
be a real valued finite variation process. We define the Lebesgue-Stieltjes integral denoted by
for each
by using the selection method, which is direct, nature and different from the indirect definition appearing in some references. We shall show that this kind of integral is also measurable, continuous in time t and bounded a.s. under the Hausdorff metric.
Keywords:
Fuzzy Stochastic Process, Finite Variation Process, Fuzzy Stochastic Lebesgue-Stieltjes Integral, Measurability
1. Introduction
Recently, the theory of fuzzy functions has been developed quickly due to the measurements of various uncertainties arising not only from the randomness but also from the vagueness in some situations. For example, when considering wave height at time t denoted by, due to the influence of random factors and the limitations of the measurement tools and methods, we may not precisely know the height
. It is reasonable to consider the wave height as a fuzzy random variable on a probability space
.
Since Puri and Ralescu [1] (1986) defined fuzzy random variable, there had been many further topics such as expectations of fuzzy random variables, fuzzy stochastic processes, integrals of fuzzy stochastic processes, fuzzy stochastic differential equations etc. In order to study a fuzzy function u, it is natural and equivalent to study its -level set
for any
, where
is a set-valued function. Therefore, as usual, in order to explore the integrals of fuzzy stochastic processes, at first we can study the integrals of set-valued stochastic processes. Kisielewicz (1997) [2] used all selections to define the integral of a set-valued process as a nonempty closed subset of
, but did not consider its measurability. Based on Kisielewicz’s work (1997) [2] , Kim and Kim (1999) [3] studied some properties of this kind of integral. Jung and Kim (2003) [4] modified the definition in 1-dimensional Euclidean space R so that the integral became a set-valued random variable. After the work [4] , there are some references on set-valued integrals and fuzzy integrals. One may refer to papers such as [5] -[13] etc. and references therein. Zhang and Qi [14] (2013) considered the set-valued integral with respect to a finite variation process directly instead of taking the decomposable closure appearing in [4] [6] and other references. As a further work of [14] , here we shall explore the integrals of fuzzy stochastic processes with respect to finite variation processes and prove the measurability and boundedness of this kind of integral, the continuity with respect to t under the Hausdorff metric and its representation theorem.
This paper is organized as follows: in Section 2, we present some notions on set-valued random variables and fuzzy set-valued random variables; in Section 3, we shall give the definition of integral of fuzzy set-valued stochastic processes with respect to finite variation process and prove the measurability and -boundedness which are necessary to our future work on fuzzy stochastic differential equations.
2. Preliminaries
We denote N the set of all natural numbers, R the set of all real numbers, the d-dimensional Euclidean space with the usual norm
,
the set of all nonnegative numbers. Let
be a complete probability space,
a
-field filtration satisfying the usual conditions such that
includes all P-null sets in
. The filtration is non-decreasing and right continuous. Let
be a Borel field of a topological space E.
Let be a complete probability space.
(or brief
)
the set of all
-valued Borel measurable functions
such that the norm
is finite. f is called -integrable if
.
Let (resp.
,
) be the family of all nonempty, closed (resp. nonempty compact, nonempty compact convex) subsets of
. For any
and
, define the distance between x and A by
. The Hausdorff metric
on
(cf. [15] ) is defined by
Denote
For, the support function of A is defined as follows:
A set-valued function is said to be measurable if for any open set
, the inverse
belongs to
. Such a function F is called a set-valued random variable.
Let (resp.
),
) be the family of all measurable
-valued (resp.
,
-valued) functions, briefly by
(resp.
,
. For
, the family of all
-integrable selections is defined by
(1)
In the following, is denoted briefly by
.
A set-valued random variable F is said to be integrable if is nonempty. F is called
-integrably bounded if there exits
s.t. for all
,
almost surely.
An -valued stochastic process
(or denoted by
) is defined as a function
with the
-measurable section
, for
. We say f is measurable if f is
-measurable. The process
is called
-adapted if
is
-measurable for every
. Let
, where
. We know that
is a
-algebra on
. A function
is measurable and
-adapted if and only if it is
-measurable ([9] ).
In a fashion similar to the -valued stochastic processes, a set-valued stochastic process
is defined as a set-valued function
with
-measurable section
for
. It is called measurable if it is
-measurable, and
-adapted if for any fixed t,
is
-measurable.
is measurable and
-adapted if and only if it is
-measurable.
is called
-integrable if every
is
-integrable.
Let denote the family of all fuzzy sets
which satisfy the following two conditions (cf. [3] [6] ):
1) The level set;
2) Each v is upper semi-continuous function, i.e. for each, the level set
is a closed subset of
.
Let denote the family of all fuzzy sets
which satisfy the above conditions 1), 2), and
3) The support set is a compact set.
A fuzzy set v is convex if
It is know that v is convex if and only if, for any, the level set
is a convex subset of
. Let
denote the family of all convex fuzzy sets in
, and
be the subset of all convex fuzzy
sets in. Define
(cf. [1] ) by the expression
We know that is a metric in
and
a complete metric space (cf. [6] [3] ). Moreover, for every
, we have
Lemma 1. (cf. [16] ) Let B be a set and be a family of subsets of B such that
1);
2) implies
;
3),
implies
.
Then the function defined by
has the property that
for every
.
A mapping is said to be measurable if
is an set-valued random variable for each
. Such a mapping G is called a fuzzy random variable (cf. [17] ). Let
(briefly by
) denote the family of all
-measurable fuzzy random variables. As a similar manner, we have the notations
, and
, or briefly by
(resp.
).
is called a fuzzy stochastic process if for any
,
is a fuzzy random variable. A fuzzy stochastic process
is said to be
-adapted, if for every
, the set-valued function
is
-measurable for all
. It is called measurable, if
is a
-measurable for all
.
A fuzzy stochastic process G is called -integrably bounded, if there exists a real-valued stochastic process
, for any
such that
for any
. It is equivalent to that
.
Let denote the family of all measurable
-valued
-integrably bounded fuzzy functions. Write for brevity by
, where we consider
as identical if
. Let
denote the family of all
-integrably bounded
-valued
-adapted fuzzy stochastic processes.
Let be a fuzzy random variable and
, The following conditions are equivalent (cf. [15] ):
1);
2);
3).
We define as
, where for
, we have
if
and
if
.
3. Lebesgue-Stieltjes Integrals with Respect to Finite Variation Processes
Let be a complete probability space equipped with the usual filtration
. Let
be a real valued
-adapted measurable process with finite variation and continuous sample trajectories a.s. from the origin. That is to say, for each compact interval
and any partition
of
, the total variation
is finite and as. Then for any
, the process
can generate a random measure denoted by
in the space
. For any
, let
where is the decomposition of A.
and
are non-negative and non-de- creasing processes.
.
In the product space, Michta (2011) in [7] defined a measure as follows:
For, where
is the index function. Then the set function v is a finite measure in the measurable space
if and only if
(cf. [7] ). In the following we always assume
.
For, let
be the family of all
-measurable
-valued stochastic processes f such that
For any and
, the stochastic Lebesgue-Stieltjes integral
is defined by the Bochner integral
path-by-path. One can prove that the integral process
is
-measurable.
Lemma 2. (cf. [8] ) Let be a
-finite measure space and X a separable Banach space. If
is separable with respect to
, (i.e. there exists a countably generated sub-sigma algebra
such that for every
, there is
satisfying
), then space
is separable in norm.
From now on, we always assume the sigma-field is separable with respect to P such that the set-valued integral and fuzzy integral can be well defined.
Let be the family of all
-measurable
-valued stochastic processes F such that
where.
For any, set
(2)
Definition 1. (cf. [7] ) For a set-valued stochastic process the set-valued stochastic Lebesgue-Stieltjes integral (over interval
) of F with respect to the finite variation continuous process A is the set
For some fuzzy stochastic process, it is natural to define the fuzzy integral of G with respect to the finite variation process level-wise.
Let (or abbrev. as
) be the family of all
-measu- rable
-valued fuzzy stochastic processes G such that
where.
For a fuzzy stochastic process, according to Lemma 1 and the properties of set-valued stochastic integrals, the Lebesgue-Stieltjes integral of G (over interval
) can be defined level-wise.
Set
(3)
for all.
Definition 2. For a fuzzy stochastic process and any
, the family
defined by Equation (3) can determine an
-valued function denoted
by, such a fuzzy function is called the Lebesgue-Stieltjes integral (over interval
) of G with respect to finite variation process
.
Theorem 1. ([12] ) For,
and
, the Lebesgue- Stieltjes integral
is a compact and convex subset of
.
Lemma 3. (cf. [18] ) Let be a probability space, X a separable Banach space. For random variables
, both the support function
and the metric
are
-measurable.
Lemma 4. (cf. [14] ) Let be an R-valued stochastic process with finite variation. For
and
, we have
1);
2).
Lemma 5. (cf. [18] ) Let be a measurable space, X a separable Banach space. Taking
and for any
, assume
is measurable. Then if one of the following conditions is satisfied:
1) is separable;
2) for any.
We obtain that F is a set-valued random variable.
From Lemma 3 and Lemma 5, when, taking
, then for any
,
is measurable if and only if
is
-measurable.
Lemma 6. (cf. [19] ) Let be a measurable space, X a separable metrizable space, and Y a metrizable space. Then every Caratheodory function
( i.e. for each
, the function
is
-measurable and for each
, the function
is continuous) is
-mea- surable.
Theorem 2. Let. Then for each
, the fuzzy stochastic integral
is
-measurable. Furthermore, the mapping
is
-measurable.
Proof. Taking, then for each
, the mapping
is
-measurable. For any
, by Lemma 3, the support function
is
-measurable too. By Lemma 4, we have
. Since the real-valued Lebesgue-Stieltjes integral
is a Carathedory function, then by Lemma 6, we obtain that
is
-measurable. Therefore, by Lemma 5, for each
, the mapping
is
-measurable and
-adapted, which means the integral
is
-measurable and
-adapted.
Theorem 3. Let. Then for any
,
.
Proof. By Theorem 2, for any,
is
-measurable. We will show that for any
,
,
.
For any,
. (4)
Then
Hence,
(5)
which means.
Theorem 4. Let. Then for any
,
is continuous with respect to t under the metric
.
Proof. Let, for any
, we have
Then
(6)
For any, we have
Then for all,
is left continuous for
under the metric
. Similarly, we can prove that
is a right continuous for
. Therefore it is continuous in t with respect to
.
Lemma 7. Let fuzzy stochastic process. Then for each
, there exists a sequence
, such that for every
,
where the closure is taken in.
Proof. Since is separable with respect to probability measurable P, we have that
is separable with respect to product measure
. By Lemma 2,
is separable. It can be obtained that
is separable under the norm
. So that for any
,
is separable since it is a closed subset of
. Then there exists a sequence
,
Theorem 5. For a fuzzy set-valued stochastic process and any
, there exists a sequence
such that
and for each t
where “cl” denotes the closure in.
Proof. For each, by Lemma 7, there exists a sequence
such that
where the closure is taken in.
For each, by Castaing represent theorem (cf. [15] [20] ), there exists a sequence
such that
At first we will show that
In fact, taking, there exists a sequence
such that
then there exists a subsequence such that
Therefore
On the other hand
since is closed and
, which yields
Since
is closed and, then for each t
(7)
For any, there exists a sequence
such that
Then for each t,
which means
(8)
(7) together with (8) yields
Lemma 8. (cf. [15] ) Let,
satisfy: for fixed
is continuous with respect to x, for fixed
,
is measurable with respect to
, then there exists an
such that
, then we have
Theorem 6. Let. Then for any
,
Proof. Let. By Theorem 5, we can obtain that for each
, there exist sequences
and
such that
,
. For each t,
and
Therefore
(9)
By Lemma 8, we have
(10)
Then
(11)
Then
Similarly, we have
Then for each,
Therefore
Hence
Theorem 7. Let. Then for each
we have
Proof. For any, we have
(12)
by Lemma 8, we have
(13)
Then
(14)
Then
Similarly, we have
Then for each
Moreover
Hence
Remark 1. In Theorem 6 and Theorem 7, the inequalities hold too if we take the expectation on both sides.
4. Conclusion
In [21] , the author studied the Lebesgue-Stieltjes integral of real stochastic processes with respect to fuzzy valued stochastic processes. In some references such as [5] [6] , the integrals of fuzzy stochastic processes with respect to time t and Brownian motion were studied. In order to guarantee measurability of the integral, Kim (2005) Li and Ren (2007) defined the integral indirectly by taking the decomposable closure. Here, when the integrand taked value in compact and convex subsets of, we defined directly the integral of fuzzy stochastic process with respect to real-valued finite variation processes by using selection method, which is different from the above references. Then we proved the measurability (Theorem 2), which was key and guaranteed the reasonability of the definition. Attribute to the good property of finite variation of integrator, the integral was bounded as and
-bounded under the metric
(Theorem 3, Theorem 6 and Theorem 7). This property was much well than the integral with respect to Brownian motion since the latter was of infinite variation. Thanks to the boundedness of the integral, it was possible to do the further work such as exploring solutions of fuzzy stochastic differential equations.
Acknowledgements
We thank the editor and the referees for their comments. This work is partly supported by NSFC (No. 11371135).
Cite this paper
JinpingZhang,LingliLuo,XingmeiLi,XiaoyingWang, (2015) Lebesgues-Stieltjes Integrals of Fuzzy Stochastic Processes with Respect to Finite Variation Processes. Applied Mathematics,06,2199-2210. doi: 10.4236/am.2015.613193
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