CHAPTER FIVE STOCHASTIC COMPARISONS OF FUZZY STOCHASTIC PROCESSES Abstract Chapter 5 investigates the stochastic comparison of fuzzy stochastic processes. This chapter introduce the concept of stochastic compar- ison of fuzzy stochastic processes. The condition that manifests the stochastic inequality is realized in terms of an increasing functional f . Chapter 5 ends with the concluding section. This section of con- clusion includes the summary of the results of this thesis. The contents of this chapter form the substance of the paper entitled ”Stochastic comparisons of fuzzy stochastic processes”, accepted for publication in the International Journal Reflection des ERA-Journal of Mathematical Sciences, India. 111
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CHAPTER
FIVE
STOCHASTIC COMPARISONS OF FUZZY
STOCHASTIC PROCESSES
Abstract
Chapter 5 investigates the stochastic comparison of fuzzy stochastic
processes. This chapter introduce the concept of stochastic compar-
ison of fuzzy stochastic processes. The condition that manifests the
stochastic inequality is realized in terms of an increasing functional
f . Chapter 5 ends with the concluding section. This section of con-
clusion includes the summary of the results of this thesis.
The contents of this chapter form the substance of the paper
entitled ”Stochastic comparisons of fuzzy stochastic processes”,
accepted for publication in the International Journal Reflection
des ERA-Journal of Mathematical Sciences, India.
111
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 112
5.1 Introduction
The theory of fuzzy random variables is a natural extension
of classical real valued random variables or random vectors.
Fuzzy random variables have many special properties. This
allows new meanings for the classical probability theory. As
a result of advancement in this area in the past three decades
the theory of fuzzy random variables with diverse applications
has become one of new and active branches in probability the-
ory. In reality we often come across with random experiments
whose outcomes are not numbers but are expressed in inexact
linguistic terms, which varies with time t. Such linguistic terms
will be represented by a dynamic fuzzy set [49]. This is a typ-
ical fuzzy stochastic phenomenon with prolonged time. Fuzzy
random variables [33, 34, 44, 67] are mathematical characteriza-
tions for fuzzy stochastic phenomena, but only one point of time
description. For the formulation of a fuzzy stochastic process,
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 113
fuzzy random variables should be considered repeatedly and
even continuously to describe and investigate the structure of
their family. As a desideratum, the study of fuzzy stochastic
processes is essential.
Kwakernaak [33, 34] introduced the notion of a fuzzy ran-
dom variable as a measurable functions F : Ω → F(R), where
(Ω,A ,P) is a probability space and F(R) denotes all piecewise
continuous functions u : R → [0, 1]. Puri and Ralescu [44]
defined the concept of a fuzzy random variable as a function
F : Ω → F(Rn) where (Ω,A ,P) is a probability space and F(Rn)
denotes all functions u : Rn→ [0, 1] such that x ∈ Rn; u(x) ≥ α
is a non-empty and compact for each α ∈ (0, 1]. In this chapter, a
concept of fuzzy random variable, slightly different than that of
Kwakernaak [33, 34] and Puri [44] is introduced. It is defined as
a measurable fuzzy set valued function X : Ω→ F0(R), where R
is the real line, (Ω,A ,P) is a probability space,
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 114
F0(R) = A : R→ [0, 1] and x ∈ R; A(x) ≥ α is a bounded closed
interval for each α ∈ (0, 1]. Guangyuan Wang et.al., [18] have in-
troduced the general theory of fuzzy stochastic processes, which
include the definitions of fuzzy random function, fuzzy stochas-
tic processes. Earnest Lazarus Piriyakuar et.al., [12] have stud-
ied various stochastic comparison of fuzzy random variables.
In this chapter the concept of stochastic comparison is extended
to fuzzy stochastic processes. Congruous to stochastic compar-
isons of classical random variables, stochastic comparisons for
functionals of fuzzy stochastic processes, which are of practical
importance are derived.
The stochastic comparison of two fuzzy random variables
whose end points of each α-cut is univariate in nature can be
generalized and the resulting stochastic comparison is nothing
but the stochastic comparison of two fuzzy stochastic processes.
In many applied problems the exact calculation of quantities of
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 115
interest which are obscured by perceptional deficiencies the re-
sulting fuzzy stochastic process pose a variety of complexities.
In such cases the only remedy is to compute bounds on these
parameters by comparing the given fuzzy stochastic process
with a simpler fuzzy stochastic process. This kind of stochastic
comparison has great relevance in reliability problems. In this
chapter, stronger type of comparison of two fuzzy stochastic pro-
cesses is introduced. Gordon Pledger et. al [15] have discussed
stochastic comparison of random processes with applications in
reliability.
In Section 5.2, some results related to dynamic fuzzy sets,
fuzzy random variables, fuzzy random vectors, fuzzy random
function and fuzzy stochastic processes are introduced.
In Section 5.3, the concept of stochastic comparison of fuzzy
stochastic processes is introduced. Conditions are obtained un-
der which the fuzzy stochastic process X(t); t ≥ 0 stochas-
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 116
tically larger than its counter part Y(t); t ≥ 0 implies that
f X(t); t ≥ 0 ≥st f Y(t); t ≥ 0 for increasing functional f , and
other properties of stochastic comparison are introduced.
5.2 Preliminaries
Let R be the real line and (R,B) be the Borel measurable space.
Let F0(R) denote the set of fuzzy subsets A : R→ [0, 1] with the
following properties:
1. x ∈ R; A(x) = 1 , φ.
2. Aα = x ∈ R; A(x) ≥ α is a bounded closed interval in R for
each α ∈ (0, 1]. i.e., Aα =[(Aα)L , (Aα)U
]where
(Aα)L = inf Aα and (Aα)U = sup Aα
(Aα)L, (Aα)U∈ Aα, −∞ < (Aα)L and (Aα)U < ∞ for each
α ∈ (0, 1]. A ∈ F0(R) is called a bounded closed fuzzy
number.
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 117
Definition 5.2.1 ([49]). A(t), t ∈ T ⊂ R is known as a dynamic
fuzzy set in U where U is a non empty set with respect to T if
A(t) ∈ F(U), the set of all fuzzy subsets of U, for each t ∈ T. In
particular A(t); t ∈ T is called a normal dynamic fuzzy set if
A(t) ∈ F0(R) for each t ∈ T.
Definition 5.2.2. Let A(t) be a normal dynamic fuzzy set with
respect to T and I(R) = [x, y]; x, y ∈ R, x ≤ y.
Let Aα : T→ I(R) defined as
t 7→ Aα(t) =(A(t)
)α
=[(Aα)L (t), (Aα)U (t)
].
Then Aα(t) is known as the level function of A(t). Aα is an
interval valued mapping on T.
Definition 5.2.3 ([18]). Let (Ω,A ,P) be a probability space. A
fuzzy set valued mapping X : Ω → F0(R) is called a fuzzy
random variable if for each B ∈B and every α ∈ (0, 1],
X−1α (B) = ω ∈ Ω; Xα(ω) ∩ B , φ ∈ A .
A fuzzy set valued mapping X : Ω→ Fm0 (R) = F0(R)× · · · ×F0(R)
Ch: 5 Stochastic Comparisons of Fuzzy Stochastic Processes 118