Advances in Pure Mathematics
Vol.3 No.9A(2013), Article ID:40869,5 pages DOI:10.4236/apm.2013.39A1003
Set-Valued Stochastic Integrals with Respect to Finite Variation Processes*
Department of Mathematics and Physics, North China Electric Power University, Beijing, China
Email: #zhangjinpingxzy@gmail.com, bycqjj@126.com
Copyright © 2013 Jinping Zhang, Jiajia Qi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In accordance of the Creative Commons Attribution License all Copyrights © 2013 are reserved for SCIRP and the owner of the intellectual property Jinping Zhang, Jiajia Qi. All Copyright © 2013 are guarded by law and by SCIRP as a guardian.
Received October 31, 2013; revised November 30, 2013; accepted December 6, 2013
Keywords: Set-valued stochastic process; Finite variation process; Measurability
ABSTRACT
In a Euclidean space, the Lebesgue-Stieltjes integral of set-valued stochastic processes
with respect to real valued finite variation process
is defined directly by employing all integrably bounded selections instead of taking the decomposable closure appearing in some existed references. We shall show that this kind of integral is measurable, continuous in
under the Hausdorff metric and
-bounded.
1. Introduction
Recently, integrals for set-valued stochastic processes with respect to Brownian motion, martingales and the Lebesgue measure have received much attention.
In 1997, Kisielewicz ([1]) defined the integral of setvalued process as a subset of space, but he didn’t consider the measurability of the integral. In 1999, Kim and Kim [2] used the definition of stochastic integrals of set-valued stochastic process with respect to the Brownian motion. They called it Aumann ([3]) type It
integrals. In [4], Jung and Kim modified the definition by taking the decomposable closure such that the integral is measurable. Li and Ren [5] modified Jung and Kim’s definition by considering the predictable set-valued stochastic process as a set-valued random variable in the product space
, and the measurability and decomposability also were based on product
-algebra. After that, Zhang et al. ([6,7]) studied the set-valued integrals with respect to the martingale and Brownian motion.
Stochastic differential inclusions and set-valued stochastic differential (or integral) equations are employed to model the problems with not only randomness but also impreciseness. Recently, there are some references related to set-valued differential equations such as [8-13] etc.
Concerning to the integral with respect to finite variation processes, Malinowski and Michta [12] give the notion of set-valued integral with respect to single valued finite variation but without considering the measurability. Z.Wang and R.Wang [14] defined the Lebesgue-Stieltjes stochastic integral of single valued stochastic processes with respect to set-valued finite variation processes (refer to [14] for the detail).
In this paper, different from the definition in [14], based on the Definition 3.1 in [12], we will study the Lebesgue-Stieltjes integral of set-valued stochastic processes with respect to single valued finite variation process. We shall prove the measurability of integral, namely, it is a set-valued random, which is similar to the classical stochastic integral.
This paper is organized as follows: in section 2, we present some notions and facts on set-valued random variables; in section 3, we shall give the definition of integral of set-valued stochastic processes with respect to finite variation process and then prove the measurability and -boundedness.
2. Preliminaries
We denote the set of all natural numbers,
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. Let
be a Borel field of a topological space
.
Let (resp.
) be the family of all nonempty, closed (resp. nonempty compact, nonempty compact convex) subsets of
. For any
and
, define the distance between
and A by
. The Hausdorff metric
on
(see e.g. [15]) is defined by
(1)
.
Denote. For
, we have
For the support function of
is defined as follows:
: the set of all
—valued Borel measurable functions
such that the norm
is finite. is called
-integrable if
.
A set-valued function is said to be measurable if for any open set
, the inverse
belongs to
. Such a function
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
(2)
In the following, is denoted briefly by
.
A set-valued random variable is said to be integrable if
is nonempty.
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
is measurable if
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 ([8]).
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
,
is
-measurable.
is measurable and
-adapted if and only if it is
-measurable.
is called
-integrable if every
is
-integrable.
3. Set-Valued Stochastic Integral w.r.t Finite Variation Processes
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 a.s. Then for any
, the process
can generate a random measure denoted by
in the space
. For any
, let
where is the decomposition of
,
and
are non-negative and nondecreasing processes,
. In the product space
, set
(3)
for, where
is the index function. Then the set function
is a finite measure in the measurable space
if and only if
. In the following we always assume
.
Let be the family of all
-measurable
-valued stochastic processes
such that
For any and
, the stochastic Lebesgue-Stieltjes integral
is defined by the Bochner integral
pathby-path. One can show that the integral process
is
-measurable.
Note: in [12], the integrand is assumed being predictable, in fact the integrand can be relaxed to the - measurable class since the integrator
is continuous.
Let be the family of all
-measurable
-valued stochastic processes
such that
where. For any
, set
(4)
Definition 1. (see [12]) For a set-valued stochastic process the set-valued stochastic Lebesgue-Stieltjes integral (over interval
) of
with respect to the finite variation continuous process
is the set
In [12], the authors call this kind of integral as trajectory integral since they consider it as a
-valued random variable. Here, we shall consider it as a subset of
and show the measurability with respect to
, which is very different from the way in [12], also different from other references such as [10,16,17] etc. In fact, for almost every
, the above integral
is a subset of
. In the following, we shall assume the
- algebra
is separable w.r.t
. In addition,
is separable and
, then one can get
is separable. Therefore we can find an
- measurable set
, such that
and for every
, the integral
is defined path-bypath. For
, set
, therefore it is well defined for every
.
since the continuity of
. In the sequel, we shall denote the integral by
instead of
. For any
denote
by
.
Theorem 1. For,
and
, the Lebesgue-Stieltjes integral
is a compact and convex subset of
.
Proof 1. In fact, is a bounded and convex subset of
, since
is convex and compact,moreover, it is weakly compact since
is reflexive. The convexity of the integral is obvious.
We shall show the linear operator :
is bounded.
For any,
,
(5)
which implies the linear operator is bounded. Therefore the integral
is weakly compact since the bounded linear operator mapping a weakly compact set to a weakly compact one. In
space, a weakly compact set is compact.
Lemma 1. (see [16] Corollary 2.1.1 (5)) Assume is a measurable space,
is a separable Banach space,
, and F is a set-valued random variable, then
is measurable.
By using Lemma 1, as a manner similar to Theorem 1 in [17], we have the following result:
Lemma 2. Assume is the corresponding stochastic process,
for any
, we have 1)
;
2)
Lemma 3. (see [16] Theorem 2.1.16) Assume is a measurable space,
is a separable Banach space,
, and for any fixed
is measurable, if one of the following conditions is satisfied:
1) is separable;
2) for any.
Then is a set-valued random variable.
From Lemma 1 and Lemma 3, when, for any
,
is
-measurable if and only if
is
-measurable.
Lemma 4. ([16] Theorem 1.7.7) If is a separable space,
are separable metric space
satisfy:
(a) for any is measurable;
(b) for any is continuous or is continuous with respect to Hausdorff metricThen
is jointly measurable.
Then by Lemma 1 we have the following:
Lemma 5. Assume. Then
is
-measurable.
Theorem 2. Assume . Then
for each
. Furthermore, the mapping
is
-measurable.
Proof 2. Step 1. We will show that is
- measurable for each
,
is
-measurable.
By Theorem 1, we have
(6)
for all. Furthermore, we obtain
for all. Moreover, since
is
-measurable, from the Lemma 5 we can obtain that the function
is
measurable. By Fubini theorem,
is
-measurable, based on Lemma 3,
is
-measurable.
Finally, in the argument above, the function is
-measurable for each
. Since it is continuous in
for all
, so it is
-measurable. From Lemma 4, we obtain that
is
- measurable.
Step 2. In this step, we will show that for each
.
For each and
, we have
(7)
then
Hence,
(8)
which implies
As a manner similar to Theorem 3.8. in [8], we have the Castaing representation as following:
Theorem 3. For a set-valued stochastic process, there exists a sequence
such that
and, for,
where cl denotes the closure in.
Theorem 4. For each
is continuous a.s. with respect to the Hausdorff metric
.
Proof 3. Let and
. We then have
(9)
Hence,
(10)
since for each,
. Hence,
. So
is leftcontinuous in
for all a.s. In a similar way, we see that
is right-continuous in
a.s.
Similar to the proof of Theorem 3.15 in [8], we have the following theorem:
Theorem 5. Let, for any
, we have
and
4. Conclusion
When the integrand takes values in compact and convex subsets of, we defined the integral with respect to real-valued variation processes. And then we proved some properties of this kind of integral such as measurability,
-boundedness and continuity under the Hausdorff metric.
5. Acknowledgements
We would like to thank the referees for their valuable comments. Moreover, we express special thanks to our editor of the journal APM for his(her) efficiency and support.
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NOTES
*This work is partly supported by The Co-Construction Project of Beijing Municipal Commission of Education, The Project Sponsored by SRF for ROCS, SEM and The Fundamental Research Funds for the Central Universities, No 12MS81.
#Corresponding author.