Set-Valued Stochastic Integrals with Respect to Finite Variation Processes ()
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.
NOTES
#Corresponding author.