Uncertainty Theory (Studies in Fuzziness and Soft Computing) 3rd Edition

Uncertainty Theory Third Edition Baoding Liu Uncertainty Theory Laboratory Department of Mathematical Sciences Tsinghua...

Author: Baoding Liu

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Uncertainty Theory Third Edition

Baoding Liu Uncertainty Theory Laboratory Department of Mathematical Sciences Tsinghua University Beijing 100084, China [email protected] http://orsc.edu.cn/liu

c 2008 by UTLAB 3rd Edition c 2008 by WAP Japanese Translation Version c 2007 by Springer-Verlag Berlin 2nd Edition c 2004 by Springer-Verlag Berlin 1st Edition

Reference to this book should be made as follows: Liu B, Uncertainty Theory, 3rd ed., http://orsc.edu.cn/liu/ut.pdf

Contents Preface

ix

1 Probability Theory 1.1 Probability Space . . . . . . . . 1.2 Random Variables . . . . . . . . 1.3 Probability Distribution . . . . . 1.4 Independence . . . . . . . . . . . 1.5 Identical Distribution . . . . . . 1.6 Expected Value . . . . . . . . . 1.7 Variance . . . . . . . . . . . . . 1.8 Moments . . . . . . . . . . . . . 1.9 Critical Values . . . . . . . . . . 1.10 Entropy . . . . . . . . . . . . . . 1.11 Distance . . . . . . . . . . . . . 1.12 Inequalities . . . . . . . . . . . . 1.13 Convergence Concepts . . . . . . 1.14 Conditional Probability . . . . . 1.15 Stochastic Process . . . . . . . . 1.16 Stochastic Calculus . . . . . . . 1.17 Stochastic Differential Equation

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2 Credibility Theory 2.1 Credibility Space . . . . 2.2 Fuzzy Variables . . . . 2.3 Membership Function . 2.4 Credibility Distribution 2.5 Independence . . . . . . 2.6 Identical Distribution . 2.7 Expected Value . . . . 2.8 Variance . . . . . . . . 2.9 Moments . . . . . . . . 2.10 Critical Values . . . . . 2.11 Entropy . . . . . . . . .

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1 1 4 7 11 13 14 23 25 26 28 32 32 35 40 43 47 50

vi

Contents

2.12 2.13 2.14 2.15 2.16 2.17 2.18

Distance . . . . . . . . . . Inequalities . . . . . . . . . Convergence Concepts . . . Conditional Credibility . . Fuzzy Process . . . . . . . Fuzzy Calculus . . . . . . . Fuzzy Differential Equation

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106 107 110 114 119 124 128

3 Chance Theory 3.1 Chance Space . . . . . . . . 3.2 Hybrid Variables . . . . . . . 3.3 Chance Distribution . . . . . 3.4 Expected Value . . . . . . . 3.5 Variance . . . . . . . . . . . 3.6 Moments . . . . . . . . . . . 3.7 Independence . . . . . . . . . 3.8 Identical Distribution . . . . 3.9 Critical Values . . . . . . . . 3.10 Entropy . . . . . . . . . . . . 3.11 Distance . . . . . . . . . . . 3.12 Inequalities . . . . . . . . . . 3.13 Convergence Concepts . . . . 3.14 Conditional Chance . . . . . 3.15 Hybrid Process . . . . . . . . 3.16 Hybrid Calculus . . . . . . . 3.17 Hybrid Differential Equation

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129 129 137 143 146 149 151 151 153 153 156 157 157 160 164 169 171 174

4 Uncertainty Theory 4.1 Uncertainty Space . . . . 4.2 Uncertain Variables . . . 4.3 Identification Function . 4.4 Uncertainty Distribution 4.5 Expected Value . . . . . 4.6 Variance . . . . . . . . . 4.7 Moments . . . . . . . . . 4.8 Independence . . . . . . . 4.9 Identical Distribution . . 4.10 Critical Values . . . . . . 4.11 Entropy . . . . . . . . . . 4.12 Distance . . . . . . . . . 4.13 Inequalities . . . . . . . . 4.14 Convergence Concepts . . 4.15 Conditional Uncertainty . 4.16 Uncertain Process . . . . 4.17 Uncertain Calculus . . .

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177 177 181 184 185 187 190 191 193 193 194 195 196 197 200 201 205 209

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Contents

vii

4.18 Uncertain Differential Equation . . . . . . . . . . . . . . . . 211 A Measurable Sets

213

B Classical Measures

216

C Measurable Functions

218

D Lebesgue Integral

221

E Euler-Lagrange Equation

224

F Maximum Uncertainty Principle

225

G Uncertainty Relations

226

Bibliography

229

List of Frequently Used Symbols

244

Index

245

viii

Contents

Preface There are various types of uncertainty in the real world. Randomness is a basic type of objective uncertainty, and probability theory is a branch of mathematics for studying the behavior of random phenomena. The study of probability theory was started by Pascal and Fermat (1654), and an axiomatic foundation of probability theory was given by Kolmogoroff (1933) in his Foundations of Probability Theory. Probability theory has been widely applied in science and engineering. Chapter 1 will provide the probability theory. Fuzziness is a basic type of subjective uncertainty initiated by Zadeh (1965). Credibility theory is a branch of mathematics for studying the behavior of fuzzy phenomena. The study of credibility theory was started by Liu and Liu (2002), and an axiomatic foundation of credibility theory was given by Liu (2004) in his Uncertainty Theory. Chapter 2 will introduce the credibility theory. Sometimes, fuzziness and randomness simultaneously appear in a system. A hybrid variable was proposed by Liu (2006) as a tool to describe the quantities with fuzziness and randomness. Both fuzzy random variable and random fuzzy variable are instances of hybrid variable. In addition, Li and Liu (2007) introduced the concept of chance measure for hybrid events. After that, chance theory was developed steadily. Essentially, chance theory is a hybrid of probability theory and credibility theory. Chapter 3 will offer the chance theory. In order to deal with general uncertainty, Liu (2007) founded an uncertainty theory in his Uncertainty Theory, and had it become a branch of mathematics based on normality, monotonicity, self-duality, and countable subadditivity axioms. Probability theory, credibility theory and chance theory are three special cases of uncertainty theory. Chapter 4 is devoted to the uncertainty theory. For this new edition the entire text has been totally rewritten. More importantly, uncertain process and uncertain calculus as well as uncertain differential equations have been added. The book is suitable for mathematicians, researchers, engineers, designers, and students in the field of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, and management science. The readers will learn the axiomatic approach of uncertainty theory, and find this work a stimulating and useful reference.

x

Preface

Baoding Liu Tsinghua University http://orsc.edu.cn/liu March 5, 2008

Chapter 1

Probability Theory Probability measure is essentially a set function (i.e., a function whose argument is a set) satisfying normality, nonnegativity and countable additivity axioms. Probability theory is a branch of mathematics for studying the behavior of random phenomena. The emphasis in this chapter is mainly on probability space, random variable, probability distribution, independence, identical distribution, expected value, variance, moments, critical values, entropy, distance, convergence almost surely, convergence in probability, convergence in mean, convergence in distribution, conditional probability, stochastic process, renewal process, Brownian motion, stochastic calculus, and stochastic differential equation. The main results in this chapter are well-known. For this reason the credit references are not provided.

1.1

Probability Space

Let Ω be a nonempty set, and A a σ-algebra over Ω. If Ω is countable, usually A is the power set of Ω. If Ω is uncountable, for example Ω = [0, 1], usually A is the Borel algebra of Ω. Each element in A is called an event. In order to present an axiomatic definition of probability, it is necessary to assign to each event A a number Pr{A} which indicates the probability that A will occur. In order to ensure that the number Pr{A} has certain mathematical properties which we intuitively expect a probability to have, the following three axioms must be satisfied: Axiom 1. (Normality) Pr{Ω} = 1. Axiom 2. (Nonnegativity) Pr{A} ≥ 0 for any event A. Axiom 3. (Countable Additivity) For every countable sequence of mutually

2

Chapter 1 - Probability Theory

disjoint events {Ai }, we have (∞ ) ∞ [ X Pr Ai = Pr{Ai }. i=1

(1.1)

i=1

Definition 1.1 The set function Pr is called a probability measure if it satisfies the normality, nonnegativity, and countable additivity axioms. Example 1.1: Let Ω = {ω1 , ω2 , · · · }, and let A be the power set of Ω. Assume that p1 , p2 , · · · are nonnegative numbers such that p1 + p2 + · · · = 1. Define a set function on A as X Pr{A} = pi , A ∈ A. (1.2) ωi ∈A

Then Pr is a probability measure. Example 1.2: Let Ω = [0, 1] and let A be the Borel algebra over Ω. If Pr is the Lebesgue measure, then Pr is a probability measure. Theorem 1.1 Let Ω be a nonempty set, A a σ-algebra over Ω, and Pr a probability measure. Then we have (a) Pr{∅} = 0; (b) Pr is self-dual, i.e., Pr{A} + Pr{Ac } = 1 for any A ∈ A; (c) Pr is increasing, i.e., Pr{A} ≤ Pr{B} whenever A ⊂ B. Proof: (a) Since ∅ and Ω are disjoint events and ∅ ∪ Ω = Ω, we have Pr{∅} + Pr{Ω} = Pr{Ω} which makes Pr{∅} = 0. (b) Since A and Ac are disjoint events and A ∪ Ac = Ω, we have Pr{A} + Pr{Ac } = Pr{Ω} = 1. (c) Since A ⊂ B, we have B = A ∪ (B ∩ Ac ), where A and B ∩ Ac are disjoint events. Therefore Pr{B} = Pr{A} + Pr{B ∩ Ac } ≥ Pr{A}. Probability Continuity Theorem Theorem 1.2 (Probability Continuity Theorem) Let Ω be a nonempty set, A a σ-algebra over Ω, and Pr a probability measure. If A1 , A2 , · · · ∈ A and limi→∞ Ai exists, then n o lim Pr{Ai } = Pr lim Ai . (1.3) i→∞

i→∞

Proof: Step 1: Suppose {Ai } is an increasing sequence. Write Ai → A and A0 = ∅. Then {Ai \Ai−1 } is a sequence of disjoint events and ∞ [ i=1

(Ai \Ai−1 ) = A,

k [ i=1

(Ai \Ai−1 ) = Ak

3

Section 1.1 - Probability Space

for k = 1, 2, · · · Thus we have ∞ ∞ S P Pr{A} = Pr (Ai \Ai−1 ) = Pr {Ai \Ai−1 } i=1

= lim

i=1

k P

k→∞ i=1

Pr {Ai \Ai−1 } = lim Pr k→∞

k S

(Ai \Ai−1 )

i=1

= lim Pr{Ak }. k→∞

Step 2: If {Ai } is a decreasing sequence, then the sequence {A1 \Ai } is clearly increasing. It follows that n o Pr{A1 } − Pr{A} = Pr lim (A1 \Ai ) = lim Pr {A1 \Ai } i→∞

i→∞

= Pr{A1 } − lim Pr{Ai } i→∞

which implies that Pr{Ai } → Pr{A}. Step 3: If {Ai } is a sequence of events such that Ai → A, then for each k, we have ∞ ∞ [ \ Ai . Ai ⊂ Ak ⊂ i=k

i=k

Since Pr is increasing, we have (∞ ) (∞ ) \ [ Pr Ai ≤ Pr{Ak } ≤ Pr Ai . i=k

Note that

i=k

∞ \ i=k

Ai ↑ A,

∞ [

Ai ↓ A.

i=k

It follows from Steps 1 and 2 that Pr{Ai } → Pr{A}. Probability Space Definition 1.2 Let Ω be a nonempty set, A a σ-algebra over Ω, and Pr a probability measure. Then the triplet (Ω, A, Pr) is called a probability space. Example 1.3: Let Ω = {ω1 , ω2 , · · · }, A the power set of Ω, and Pr a probability measure defined by (1.2). Then (Ω, A, Pr) is a probability space. Example 1.4: Let Ω = [0, 1], A the Borel algebra over Ω, and Pr the Lebesgue measure. Then ([0, 1], A, Pr) is a probability space, and sometimes is called Lebesgue unit interval. For many purposes it is sufficient to use it as the basic probability space.

4

Chapter 1 - Probability Theory

Product Probability Space Let (Ωi , Ai , Pri ), i = 1, 2, · · · , n be probability spaces, and Ω = Ω1 × Ω2 × · · · × Ωn , A = A1 × A2 × · · · × An . Note that the probability measures Pri , i = 1, 2, · · · , n are finite. It follows from the product measure theorem that there is a unique measure Pr on A such that Pr{A1 × A2 × · · · × An } = Pr1 {A1 } × Pr2 {A2 } × · · · × Prn {An } for any Ai ∈ Ai , i = 1, 2, · · · , n. This conclusion is called the product probability theorem. The measure Pr is also a probability measure since Pr{Ω} = Pr1 {Ω1 } × Pr2 {Ω2 } × · · · × Prn {Ωn } = 1. Such a probability measure is called the product probability measure, denoted by Pr = Pr1 × Pr2 × · · · × Prn . Definition 1.3 Let (Ωi , Ai , Pri ), i = 1, 2, · · · , n be probability spaces, and Ω = Ω1 × Ω2 × · · · × Ωn , A = A1 × A2 × · · · × An , Pr = Pr1 × Pr2 × · · · × Prn . Then the triplet (Ω, A, Pr) is called the product probability space. Infinite Product Probability Space Let (Ωi , Ai , Pri ), i = 1, 2, · · · be an arbitrary sequence of probability spaces, and Ω = Ω1 × Ω2 × · · · , A = A1 × A2 × · · · (1.4) It follows from the infinite product measure theorem that there is a unique probability measure Pr on A such that Pr {A1 × · · · × An × Ωn+1 × Ωn+2 × · · · } = Pr1 {A1 } × · · · × Prn {An } for any measurable rectangle A1 × · · · × An × Ωn+1 × Ωn+2 × · · · and all n = 1, 2, · · · The probability measure Pr is called the infinite product of Pri , i = 1, 2, · · · and is denoted by Pr = Pr1 × Pr2 × · · ·

(1.5)

Definition 1.4 Let (Ωi , Ai , Pri ), i = 1, 2, · · · be probability spaces, and Ω = Ω1 × Ω2 × · · · , A = A1 × A2 × · · · , Pr = Pr1 × Pr2 × · · · Then the triplet (Ω, A, Pr) is called the infinite product probability space.

1.2

Random Variables

Definition 1.5 A random variable is a measurable function from a probability space (Ω, A, Pr) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {ω ∈ Ω ξ(ω) ∈ B} (1.6) is an event.

Section 1.2 - Random Variables

5

Example 1.5: Take (Ω, A, Pr) to be {ω1 , ω2 } with Pr{ω1 } = Pr{ω2 } = 0.5. Then the function ( 0, if ω = ω1 ξ(ω) = 1, if ω = ω2 is a random variable. Example 1.6: Take (Ω, A, Pr) to be the interval [0, 1] with Borel algebra and Lebesgue measure. We define ξ as an identity function from Ω to [0,1]. Since ξ is a measurable function, it is a random variable. Example 1.7: A deterministic number c may be regarded as a special random variable. In fact, it is the constant function ξ(ω) ≡ c on the probability space (Ω, A, Pr). Definition 1.6 A random variable ξ is said to be (a) nonnegative if Pr{ξ < 0} = 0; (b) positive if Pr{ξ ≤ 0} = 0; (c) continuous if Pr{ξ = x} = 0 for each x ∈ 0, there exists an integer n and real numbers x1 , x2 , · · · , xn with x1 + x2 + · · · + xn ≥ r such that ( n˜ ) X Cr ξi ≥ r − ε ≤ Cr{˜ n = n} ∧ Cr{ξi = xi } i=1

for each i with 1 ≤ i ≤ n. Without loss of generality, we assume that nx1 ≥ r. Then we have ( n˜ ) X Cr ξi ≥ r − ε ≤ Cr{˜ n = n} ∧ Cr{ξ1 = x1 } ≤ Cr {˜ nξ1 ≥ r} . i=1

Letting ε → 0, we get Cr

( n˜ X

) ξi ≥ r

≤ Cr {˜ nξ1 ≥ r} .

i=1

It follows that Cr

( n˜ X

) ξi ≥ r

= Cr {˜ nξ1 ≥ r} .

i=1

Similarly, the above identity still holds if the symbol “≥” is replaced with “≤”. Finally, by the definition of expected value operator, we have " n˜ # Z ( n˜ ) ( n˜ ) Z 0 +∞ X X X E ξi = Cr ξi ≥ r dr − Cr ξi ≤ r dr 0

i=1

Z

−∞

i=1 +∞

Z Cr {˜ nξ1 ≥ r} dr −

= 0

i=1

0

Cr {˜ nξ1 ≤ r} dr = E [˜ nξ1 ] . −∞

The theorem is proved.

2.8

Variance

Definition 2.20 (Liu and Liu [126]) Let ξ be a fuzzy variable with finite expected value e. Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2 ]. The variance of a fuzzy variable provides a measure of the spread of the distribution around its expected value. Example 2.44: Let ξ be an equipossible fuzzy variable (a, b). Then its expected value is e = (a + b)/2, and for any positive number r, we have ( 1/2, if r ≤ (b − a)2 /4 2 Cr{(ξ − e) ≥ r} = 0, if r > (b − a)2 /4.

93

Section 2.8 - Variance

Thus the variance is Z

+∞

Z

2

(b−a)2 /4

Cr{(ξ − e) ≥ r}dr =

V [ξ] =

0

0

1 (b − a)2 dr = . 2 8

Example 2.45: Let ξ = (a, b, c) be a symmetric triangular fuzzy variable, i.e., b − a = c − b. Then its variance is V [ξ] = (c − a)2 /24. Example 2.46: Let ξ = (a, b, c, d) be a symmetric trapezoidal fuzzy variable, i.e., b − a = d − c. Then its variance is V [ξ] = ((d − a)2 + (d − a)(c − b) + (c − b)2 )/24. Example 2.47: A fuzzy variable ξ is called normally distributed if it has a normal membership function −1 π|x − e| √ , µ(x) = 2 1 + exp 6σ

x ∈ 0.

(2.60)

The expected value is e and variance is σ 2 . Let ξ1 and ξ2 be independently and normally distributed fuzzy variables with expected values e1 and e2 , variances σ12 and σ22 , respectively. Then for any real numbers a1 and a2 , the fuzzy variable a1 ξ1 + a2 ξ2 is also normally distributed with expected value a1 e1 + a2 e2 and variance (|a1 |σ1 + |a2 |σ2 )2 . µ(x) ..... ....... ..

1 ........ . . . . . . . . . . . . . . . . ............................

. . .... . .... .... .... .. ........ ... .... .... . .. .... .... . ..... .... . . ... . . ..... .. . ... .. ..... ... . . . . . . . . ............ . . . . . . . .. . . . . . . . ........... . . . ......... . ... . . .... . . ....... . . . . ... . .. ....... . . . ... ............ ....... . . . .......... ......... . . . . ............ ..... . . . . . . . . . . . . . . .................... . . . .. .................... . . . . . . . . . . . . . ................................................................................................................................................................................................................................................................... .. .. ...

0.434 0

e−σ

e

e+σ

x

Figure 2.3: Normal Membership Function

Theorem 2.37 If ξ is a fuzzy variable whose variance exists, a and b are real numbers, then V [aξ + b] = a2 V [ξ]. Proof: It follows from the definition of variance that V [aξ + b] = E (aξ + b − aE[ξ] − b)2 = a2 E[(ξ − E[ξ])2 ] = a2 V [ξ]. Theorem 2.38 Let ξ be a fuzzy variable with expected value e. Then V [ξ] = 0 if and only if Cr{ξ = e} = 1.

94

Chapter 2 - Credibility Theory

Proof: If V [ξ] = 0, then E[(ξ − e)2 ] = 0. Note that Z +∞ Cr{(ξ − e)2 ≥ r}dr E[(ξ − e)2 ] = 0 2

which implies Cr{(ξ−e) ≥ r} = 0 for any r > 0. Hence we have Cr{(ξ−e)2 = 0} = 1, i.e., Cr{ξ = e} = 1. Conversely, if Cr{ξ = e} = 1, then we have Cr{(ξ − e)2 = 0} = 1 and Cr{(ξ − e)2 ≥ r} = 0 for any r > 0. Thus Z +∞ Cr{(ξ − e)2 ≥ r}dr = 0. V [ξ] = 0

Maximum Variance Theorem Let ξ be a fuzzy variable that takes values in [a, b], but whose membership function is otherwise arbitrary. When its expected value is given, the maximum variance theorem will provide the maximum variance of ξ, thus playing an important role in treating games against nature. Theorem 2.39 (Li and Liu [104]) Let f be a convex function on [a, b], and ξ a fuzzy variable that takes values in [a, b] and has expected value e. Then E[f (ξ)] ≤

b−e e−a f (a) + f (b). b−a b−a

(2.61)

Proof: For each θ ∈ Θ, we have a ≤ ξ(θ) ≤ b and ξ(θ) =

ξ(θ) − a b − ξ(θ) a+ b. b−a b−a

It follows from the convexity of f that f (ξ(θ)) ≤

b − ξ(θ) ξ(θ) − a f (a) + f (b). b−a b−a

Taking expected values on both sides, we obtain (2.61). Theorem 2.40 (Li and Liu [104], Maximum Variance Theorem) Let ξ be a fuzzy variable that takes values in [a, b] and has expected value e. Then V [ξ] ≤ (e − a)(b − e) and equality holds if the fuzzy variable ξ has membership function  2(b − e)   ∧ 1, if x = a  b−a µ(x) =    2(e − a) ∧ 1, if x = b. b−a

(2.62)

(2.63)

Proof: It follows from Theorem 2.39 immediately by defining f (x) = (x−e)2 . It is also easy to verify that the fuzzy variable determined by (2.63) has variance (e − a)(b − e). The theorem is proved.

95

Section 2.9 - Moments

2.9

Moments

Definition 2.21 (Liu [128]) Let ξ be a fuzzy variable, and k a positive number. Then (a) the expected value E[ξ k ] is called the kth moment; (b) the expected value E[|ξ|k ] is called the kth absolute moment; (c) the expected value E[(ξ − E[ξ])k ] is called the kth central moment; (d) the expected value E[|ξ −E[ξ]|k ] is called the kth absolute central moment. Note that the first central moment is always 0, the first moment is just the expected value, and the second central moment is just the variance. Example 2.48: A fuzzy variable ξ is called exponentially distributed if it has an exponential membership function −1 πx , x ≥ 0, m > 0. (2.64) µ(x) = 2 1 + exp √ 6m √ The expected value is ( 6m ln 2)/π and the second moment is m2 . Let ξ1 and ξ2 be independently and exponentially distributed fuzzy variables with second moments m21 and m22 , respectively. Then for any positive real numbers a1 and a2 , the fuzzy variable a1 ξ1 +a2 ξ2 is also exponentially distributed with second moment (a1 m1 + a2 m2 )2 . µ(x) .. ......... ... ..... . .... ........ ... ...... .... ... ..... ... .... ..... ... ..... ... .. ... . . . . . . . ......... ..... ... . ........ ... . .. . ........... ... ....... . ... ......... . ............ ... . ................... . ... .. ................................................................................................................................................................................................................... .. .....

1

0.434 0

m

x

Figure 2.4: Exponential Membership Function Theorem 2.41 Let ξ be a nonnegative fuzzy variable, and k a positive number. Then the k-th moment Z +∞ k E[ξ ] = k rk−1 Cr{ξ ≥ r}dr. (2.65) 0

Proof: It follows from the nonnegativity of ξ that Z ∞ Z ∞ Z E[ξ k ] = Cr{ξ k ≥ x}dx = Cr{ξ ≥ r}drk = k 0


0

0

∞

rk−1 Cr{ξ ≥ r}dr.

96


Theorem 2.42 (Li and Liu [104]) Let ξ be a fuzzy variable that takes values in [a, b] and has expected value e. Then for any positive integer k, the kth absolute moment and kth absolute central moment satisfy the following inequalities, b−e k e−a k |a| + |b| , (2.66) E[|ξ|k ] ≤ b−a b−a E[|ξ − e|k ] ≤

b−e e−a (e − a)k + (b − e)k . b−a b−a

(2.67)

Proof: It follows from Theorem 2.39 immediately by defining f (x) = |x|k and f (x) = |x − e|k .

2.10

Critical Values

In order to rank fuzzy variables, we may use two critical values: optimistic value and pessimistic value. Definition 2.22 (Liu [124]) Let ξ be a fuzzy variable, and α ∈ (0, 1]. Then ξsup (α) = sup r Cr {ξ ≥ r} ≥ α (2.68) is called the α-optimistic value to ξ, and ξinf (α) = inf r Cr {ξ ≤ r} ≥ α

(2.69)

is called the α-pessimistic value to ξ. This means that the fuzzy variable ξ will reach upwards of the α-optimistic value ξsup (α) with credibility α, and will be below the α-pessimistic value ξinf (α) with credibility α. In other words, the α-optimistic value ξsup (α) is the supremum value that ξ achieves with credibility α, and the α-pessimistic value ξinf (α) is the infimum value that ξ achieves with credibility α. Example 2.49: Let ξ be an equipossible fuzzy variable on (a, b). Then its α-optimistic and α-pessimistic values are ( ( b, if α ≤ 0.5 a, if α ≤ 0.5 ξinf (α) = ξsup (α) = a, if α > 0.5, b, if α > 0.5. Example 2.50: Let ξ = (a, b, c) be a triangular fuzzy variable. Then its α-optimistic and α-pessimistic values are ( 2αb + (1 − 2α)c, if α ≤ 0.5 ξsup (α) = (2α − 1)a + (2 − 2α)b, if α > 0.5,

97

Section 2.10 - Critical Values

( ξinf (α) =

(1 − 2α)a + 2αb, if α ≤ 0.5 (2 − 2α)b + (2α − 1)c, if α > 0.5.

Example 2.51: Let ξ = (a, b, c, d) be a trapezoidal fuzzy variable. Then its α-optimistic and α-pessimistic values are ( 2αc + (1 − 2α)d, if α ≤ 0.5 ξsup (α) = (2α − 1)a + (2 − 2α)b, if α > 0.5, ( ξinf (α) =

(1 − 2α)a + 2αb, (2 − 2α)c + (2α − 1)d,

if α ≤ 0.5 if α > 0.5.

Theorem 2.43 Let ξ be a fuzzy variable. If α > 0.5, then we have Cr{ξ ≤ ξinf (α)} ≥ α,

Cr{ξ ≥ ξsup (α)} ≥ α.

(2.70)

Proof: It follows from the definition of α-pessimistic value that there exists a decreasing sequence {xi } such that Cr{ξ ≤ xi } ≥ α and xi ↓ ξinf (α) as i → ∞. Since {ξ ≤ xi } ↓ {ξ ≤ ξinf (α)} and limi→∞ Cr{ξ ≤ xi } ≥ α > 0.5, it follows from the credibility semicontinuity law that Cr{ξ ≤ ξinf (α)} = lim Cr{ξ ≤ xi } ≥ α. i→∞

Similarly, there exists an increasing sequence {xi } such that Cr{ξ ≥ xi } ≥ δ and xi ↑ ξsup (α) as i → ∞. Since {ξ ≥ xi } ↓ {ξ ≥ ξsup (α)} and limi→∞ Cr{ξ ≥ xi } ≥ α > 0.5, it follows from the credibility semicontinuity law that Cr{ξ ≥ ξsup (α)} = lim Cr{ξ ≥ xi } ≥ α. i→∞

The theorem is proved. Example 2.52: When α ≤ 0.5, it is possible that the inequalities Cr{ξ ≤ ξinf (α)} < α,

Cr{ξ ≥ ξsup (α)} < α

hold. Let ξ be an equipossible fuzzy variable on (−1, 1). It is clear that ξinf (0.5) = −1. However, Cr{ξ ≤ ξinf (0.5)} = 0 < 0.5. In addition, ξsup (0.5) = 1 and Cr{ξ ≥ ξsup (0.5)} = 0 < 0.5. Theorem 2.44 Let ξ be a fuzzy variable. Then we have (a) ξinf (α) is an increasing and left-continuous function of α; (b) ξsup (α) is a decreasing and left-continuous function of α. Proof: (a) It is easy to prove that ξinf (α) is an increasing function of α. Next, we prove the left-continuity of ξinf (α) with respect to α. Let {αi } be an arbitrary sequence of positive numbers such that αi ↑ α. Then {ξinf (αi )}

98


is an increasing sequence. If the limitation is equal to ξinf (α), then the leftcontinuity is proved. Otherwise, there exists a number z ∗ such that lim ξinf (αi ) < z ∗ < ξinf (α).

i→∞

Thus Cr{ξ ≤ z ∗ } ≥ αi for each i. Letting i → ∞, we get Cr{ξ ≤ z ∗ } ≥ α. Hence z ∗ ≥ ξinf (α). A contradiction proves the left-continuity of ξinf (α) with respect to α. The part (b) may be proved similarly. Theorem 2.45 Let ξ be a fuzzy variable. Then we have (a) if α > 0.5, then ξinf (α) ≥ ξsup (α); (b) if α ≤ 0.5, then ξinf (α) ≤ ξsup (α). Proof: Part (a): Write ξ(α) = (ξinf (α) + ξsup (α))/2. If ξinf (α) < ξsup (α), then we have 1 ≥ Cr{ξ < ξ(α)} + Cr{ξ > ξ(α)} ≥ α + α > 1. A contradiction proves ξinf (α) ≥ ξsup (α). Part (b): Assume that ξinf (α) > ξsup (α). It follows from the definition of ξinf (α) that Cr{ξ ≤ ξ(α)} < α. Similarly, it follows from the definition of ξsup (α) that Cr{ξ ≥ ξ(α)} < α. Thus 1 ≤ Cr{ξ ≤ ξ(α)} + Cr{ξ ≥ ξ(α)} < α + α ≤ 1. A contradiction proves ξinf (α) ≤ ξsup (α). The theorem is proved. Theorem 2.46 Let ξ be a fuzzy variable. Then we have (a) if c ≥ 0, then (cξ)sup (α) = cξsup (α) and (cξ)inf (α) = cξinf (α); (b) if c < 0, then (cξ)sup (α) = cξinf (α) and (cξ)inf (α) = cξsup (α). Proof: If c = 0, then the part (a) is obviously valid. When c > 0, we have (cξ)sup (α) = sup {r | Cr{cξ ≥ r} ≥ α} = c sup {r/c | Cr {ξ ≥ r/c} ≥ α} = cξsup (α). A similar way may prove that (cξ)inf (α) = cξinf (α). In order to prove the part (b), it suffices to verify that (−ξ)sup (α) = −ξinf (α) and (−ξ)inf (α) = −ξsup (α). In fact, for any α ∈ (0, 1], we have (−ξ)sup (α) = sup{r | Cr{−ξ ≥ r} ≥ α} = − inf{−r | Cr{ξ ≤ −r} ≥ α} = −ξinf (α). Similarly, we may prove that (−ξ)inf (α) = −ξsup (α). The theorem is proved.

99

Section 2.11 - Entropy

Theorem 2.47 Suppose that ξ and η are independent fuzzy variables. Then for any α ∈ (0, 1], we have (ξ + η)sup (α) = ξsup (α) + ηsup (α), (ξ + η)inf (α) = ξinf (α) + ηinf (α), (ξη)sup (α) = ξsup (α)ηsup (α), (ξη)inf (α) = ξinf (α)ηinf (α), if ξ ≥ 0, η ≥ 0, (ξ ∨ η)sup (α) = ξsup (α) ∨ ηsup (α), (ξ ∨ η)inf (α) = ξinf (α) ∨ ηinf (α), (ξ ∧ η)sup (α) = ξsup (α) ∧ ηsup (α), (ξ ∧ η)inf (α) = ξinf (α) ∧ ηinf (α). Proof: For any given number ε > 0, since ξ and η are independent fuzzy variables, we have Cr{ξ + η ≥ ξsup (α) + ηsup (α) − ε} ≥ Cr {{ξ ≥ ξsup (α) − ε/2} ∩ {η ≥ ηsup (α) − ε/2}} = Cr{ξ ≥ ξsup (α) − ε/2} ∧ Cr{η ≥ ηsup (α) − ε/2} ≥ α which implies (ξ + η)sup (α) ≥ ξsup (α) + ηsup (α) − ε.

(2.71)

On the other hand, by the independence, we have Cr{ξ + η ≥ ξsup (α) + ηsup (α) + ε} ≤ Cr {{ξ ≥ ξsup (α) + ε/2} ∪ {η ≥ ηsup (α) + ε/2}} = Cr{ξ ≥ ξsup (α) + ε/2} ∨ Cr{η ≥ ηsup (α) + ε/2} < α which implies (ξ + η)sup (α) ≤ ξsup (α) + ηsup (α) + ε.

(2.72)

It follows from (2.71) and (2.72) that ξsup (α) + ηsup (α) + ε ≥ (ξ + η)sup (α) ≥ ξsup (α) + ηsup (α) − ε. Letting ε → 0, we obtain (ξ + η)sup (α) = ξsup (α) + ηsup (α). The other equalities may be proved similarly. Example 2.53: The independence condition cannot be removed in Theorem 2.47. For example, let Θ = {θ1 , θ2 }, Cr{θ1 } = Cr{θ2 } = 1/2, and let fuzzy variables ξ and η be defined as ( ( 0, if θ = θ1 1, if θ = θ1 ξ(θ) = η(θ) = 1, if θ = θ2 , 0, if θ = θ2 . However, (ξ + η)sup (0.6) = 1 6= 0 = ξsup (0.6) + ηsup (0.6).

100

2.11


Entropy

Fuzzy entropy is a measure of uncertainty and has been studied by many researchers such as De Luca and Termini [26], Kaufmann [75], Yager [223], Kosko [85], Pal and Pal [177], Bhandari and Pal [7], and Pal and Bezdek [181]. Those definitions of entropy characterize the uncertainty resulting primarily from the linguistic vagueness rather than resulting from information deficiency, and vanishes when the fuzzy variable is an equipossible one. In order to measure the uncertainty of fuzzy variables, Liu [131] suggested that an entropy of fuzzy variables should meet at least the following three basic requirements: (i) minimum: the entropy of a crisp number is minimum, i.e., 0; (ii) maximum: the entropy of an equipossible fuzzy variable is maximum; (iii) universality: the entropy is applicable not only to finite and infinite cases but also to discrete and continuous cases. In order to meet those requirements, Li and Liu [95] provided a new definition of fuzzy entropy to characterize the uncertainty resulting from information deficiency which is caused by the impossibility to predict the specified value that a fuzzy variable takes. Entropy of Discrete Fuzzy Variables Definition 2.23 (Li and Liu [95]) Let ξ be a discrete fuzzy variable taking values in {x1 , x2 , · · · }. Then its entropy is defined by H[ξ] =

∞ X

S(Cr{ξ = xi })

(2.73)

i=1

where S(t) = −t ln t − (1 − t) ln(1 − t). Remark 2.7: It is easy to verify that S(t) is a symmetric function about t = 0.5, strictly increases on the interval [0, 0.5], strictly decreases on the interval [0.5, 1], and reaches its unique maximum ln 2 at t = 0.5. Remark 2.8: It is clear that the entropy depends only on the number of values and their credibilities and does not depend on the actual values that the fuzzy variable takes. Example 2.54: Suppose that ξ is a discrete fuzzy variable taking values in {x1 , x2 , · · · }. If there exists some index k such that the membership function µ(xk ) = 1, and 0 otherwise, then its entropy H[ξ] = 0. Example 2.55: Suppose that ξ is a simple fuzzy variable taking values in {x1 , x2 , · · · , xn }. If its membership function µ(x) ≡ 1, then its entropy H[ξ] = n ln 2.

101


S(t) ... .......... ... .. ... . . . . . . . . . . . . . . . ....................... ........ . ..... ... ...... ...... . ... ..... ..... . ..... ..... ... . ..... .... . . . ... . .... ... . . . .... ... . . ... .... ... . . ... . . . . ... ... . ... ... ... . . ... .. ... . . ... .. . ... . ... . .. ... . ... . ... .... ... . . ... ... ... . ... ... ... . ... ... ... . ... . ... ... ... . ...... . ... . ...... ... . ..... .................................................................................................................................................................................... .... .. .

ln 2

0

0.5

1

t

Figure 2.5: Function S(t) = −t ln t − (1 − t) ln(1 − t) Theorem 2.48 Suppose that ξ is a discrete fuzzy variable taking values in {x1 , x2 , · · · }. Then H[ξ] ≥ 0 (2.74) and equality holds if and only if ξ is essentially a crisp number. Proof: The nonnegativity is clear. In addition, H[ξ] = 0 if and only if Cr{ξ = xi } = 0 or 1 for each i. That is, there exists one and only one index k such that Cr{ξ = xk } = 1, i.e., ξ is essentially a crisp number. This theorem states that the entropy of a fuzzy variable reaches its minimum 0 when the fuzzy variable degenerates to a crisp number. In this case, there is no uncertainty. Theorem 2.49 Suppose that ξ is a simple fuzzy variable taking values in {x1 , x2 , · · · , xn }. Then H[ξ] ≤ n ln 2 (2.75) and equality holds if and only if ξ is an equipossible fuzzy variable. Proof: Since the function S(t) reaches its maximum ln 2 at t = 0.5, we have H[ξ] =

n X

S(Cr{ξ = xi }) ≤ n ln 2

i=1

and equality holds if and only if Cr{ξ = xi } = 0.5, i.e., µ(xi ) ≡ 1 for all i = 1, 2, · · · , n. This theorem states that the entropy of a fuzzy variable reaches its maximum when the fuzzy variable is an equipossible one. In this case, there is no preference among all the values that the fuzzy variable will take.

102


Entropy of Continuous Fuzzy Variables Definition 2.24 (Li and Liu [95]) Let ξ be a continuous fuzzy variable. Then its entropy is defined by Z +∞ H[ξ] = S(Cr{ξ = x})dx (2.76) −∞

where S(t) = −t ln t − (1 − t) ln(1 − t). For any continuous fuzzy variable ξ with membership function µ, we have Cr{ξ = x} = µ(x)/2 for each x ∈ 0. Proof: The positivity is clear. In addition, when a continuous fuzzy variable tends to a crisp number, its entropy tends to the minimum 0. However, a crisp number is not a continuous fuzzy variable. Theorem 2.51 Let ξ be a continuous fuzzy variable taking values on the interval [a, b]. Then H[ξ] ≤ (b − a) ln 2 (2.78) and equality holds if and only if ξ is an equipossible fuzzy variable (a, b). Proof: The theorem follows from the fact that the function S(t) reaches its maximum ln 2 at t = 0.5.

103


Theorem 2.52 Let ξ and η be two continuous fuzzy variables with membership functions µ(x) and ν(x), respectively. If µ(x) ≤ ν(x) for any x ∈ 0. Thus we have Z +∞ Z E[ξb2 ] = Cr{ξb2 ≥ x}dx ≤ 0

+∞

Cr{ξ 2 ≥ x}dx = E[ξ 2 ] = m2 .

0

It follows from µ(x) ≤ µ b(x) and Step 1 that q π E[ξb2 ] πm b ≤ √ H[ξ] ≤ H[ξ] ≤ √ . 6 6 The theorem is thus proved. Theorem 2.26 (Li and Liu [102]) Let ξ be a continuous fuzzy variable with finite expected value e and variance σ 2 . Then √ 6πσ (2.80) H[ξ] ≤ 3 and the equality holds if ξ is a normally distributed fuzzy variable with expected value e and variance σ 2 .

105


Proof: Let µ be the continuous membership function of ξ. The proof is based on the following two steps. Step 1: Let µ(x) be a unimodal and symmetric function about x = e. For this case, the variance is +∞

Z

Cr{(ξ − e)2 ≥ x}dx =

V [ξ] =

Cr{ξ − e ≥

0

√

x}dx

0 +∞

Z

+∞

Z

Z

+∞

(x − e)µ(x)dx

2(x − e)Cr{ξ ≥ x}dx =

=

e

e

and the entropy is +∞

Z

H[ξ] = −2 e

µ(x) µ(x) µ(x) µ(x) ln + 1− ln 1 − dx. 2 2 2 2

The maximum entropy membership function µ should maximize the entropy subject to the variance constraint. The Lagrangian is Z

+∞

L = −2 e

µ(x) µ(x) µ(x) µ(x) ln + 1− ln 1 − dx 2 2 2 2

+∞

Z −λ

(x − e)µ(x)dx − σ

2

.

e

The maximum entropy membership function meets Euler-Lagrange equation µ(x) µ(x) − ln 1 − + λ(x − e) = 0 ln 2 2 −1

and has the form µ(x) = 2 (1 + exp (λ(x − e))) variance constraint, we get ∗

µ (x) = 2 1 + exp

π|x − e| √ 6σ

. Substituting it into the

−1 ,

x∈
e. y≥x

106


It is easy to verify that µ b(x) is a unimodal and symmetric function about x = e. Furthermore, n o 1 1 Cr (ξb − e)2 ≥ r = b(x) = sup µ sup sup(µ(y) ∨ µ(2e − y)) 2 x≥e+√r 2 x≥e+√r y≥x 1 1 sup√ (µ(y) ∨ µ(2e − y)) = sup µ(y) 2 y≥e+ r 2 (y−e)2 ≥r ≤ Cr (ξ − e)2 ≥ r =

for any r > 0. Thus Z Z +∞ 2 b b Cr{(ξ − e) ≥ r}dr ≤ V [ξ] =

+∞

Cr{(ξ − e)2 ≥ r}dr = σ 2 .

0

0

It follows from µ(x) ≤ µ b(x) and Step 1 that √ q √ b 6π V [ξ] 6πσ b H[ξ] ≤ H[ξ] ≤ ≤ . 3 3 The proof is complete.

2.12

Distance

Distance between fuzzy variables has been defined in many ways, for example, Hausdorff distance (Puri and Ralescu [192], Klement et. al. [81]), and Hamming distance (Kacprzyk [72]). However, those definitions have no identification property. In order to overcome this shortage, Liu [129] proposed a definition of distance as follows. Definition 2.27 (Liu [129]) The distance between fuzzy variables ξ and η is defined as d(ξ, η) = E[|ξ − η|]. (2.81) Example 2.61: Let ξ and η be equipossible fuzzy variables (a1 , b1 ) and (a2 , b2 ), respectively, and (a1 , b1 )∩(a2 , b2 ) = ∅. Then |ξ −η| is an equipossible fuzzy variable on the interval with endpoints |a1 − b2 | and |b1 − a2 |. Thus the distance between ξ and η is the expected value of |ξ − η|, i.e., d(ξ, η) =

1 (|a1 − b2 | + |b1 − a2 |) . 2

Example 2.62: Let ξ = (a1 , b1 , c1 ) and η = (a2 , b2 , c2 ) be triangular fuzzy variables such that (a1 , c1 ) ∩ (a2 , c2 ) = ∅. Then d(ξ, η) =

1 (|a1 − c2 | + 2|b1 − b2 | + |c1 − a2 |) . 4

107

Section 2.13 - Inequalities

Example 2.63: Let ξ = (a1 , b1 , c1 , d1 ) and η = (a2 , b2 , c2 , d2 ) be trapezoidal fuzzy variables such that (a1 , d1 ) ∩ (a2 , d2 ) = ∅. Then d(ξ, η) =

1 (|a1 − d2 | + |b1 − c2 | + |c1 − b2 | + |d1 − a2 |) . 4

Theorem 2.54 (Li and Liu [105]) Let ξ, η, τ be fuzzy variables, and let d(·, ·) be the distance. Then we have (a) (Nonnegativity) d(ξ, η) ≥ 0; (b) (Identification) d(ξ, η) = 0 if and only if ξ = η; (c) (Symmetry) d(ξ, η) = d(η, ξ); (d) (Triangle Inequality) d(ξ, η) ≤ 2d(ξ, τ ) + 2d(η, τ ). Proof: The parts (a), (b) and (c) follow immediately from the definition. Now we prove the part (d). It follows from the credibility subadditivity theorem that Z +∞ d(ξ, η) = Cr {|ξ − η| ≥ r} dr 0

Z

+∞

≤

Cr {|ξ − τ | + |τ − η| ≥ r} dr 0

Z

+∞

≤

Cr {{|ξ − τ | ≥ r/2} ∪ {|τ − η| ≥ r/2}} dr 0

Z

+∞

(Cr{|ξ − τ | ≥ r/2} + Cr{|τ − η| ≥ r/2}) dr

≤ 0

Z

+∞

Z Cr{|ξ − τ | ≥ r/2}dr +

= 0

+∞

Cr{|τ − η| ≥ r/2}dr 0

= 2E[|ξ − τ |] + 2E[|τ − η|] = 2d(ξ, τ ) + 2d(τ, η). Example 2.64: Let Θ = {θ1 , θ2 , θ3 } and Cr{θi } = 1/2 for i = 1, 2, 3. We define fuzzy variables ξ, η and τ as follows, ( ( 1, if θ 6= θ3 −1, if θ 6= θ1 ξ(θ) = η(θ) = τ (θ) ≡ 0. 0, otherwise, 0, otherwise, It is easy to verify that d(ξ, τ ) = d(τ, η) = 1/2 and d(ξ, η) = 3/2. Thus d(ξ, η) =

2.13

3 (d(ξ, τ ) + d(τ, η)). 2

Inequalities

There are several useful inequalities for random variable, such as Markov inequality, Chebyshev inequality, H¨older’s inequality, Minkowski inequality,

108


and Jensen’s inequality. This section introduces the analogous inequalities for fuzzy variable. Theorem 2.55 (Liu [128]) Let ξ be a fuzzy variable, and f a nonnegative function. If f is even and increasing on [0, ∞), then for any given number t > 0, we have E[f (ξ)] Cr{|ξ| ≥ t} ≤ . (2.82) f (t) Proof: It is clear that Cr{|ξ| ≥ f −1 (r)} is a monotone decreasing function of r on [0, ∞). It follows from the nonnegativity of f (ξ) that Z +∞ Cr{f (ξ) ≥ r}dr E[f (ξ)] = 0

Z

+∞

Cr{|ξ| ≥ f −1 (r)}dr

= 0

Z

f (t)

Cr{|ξ| ≥ f −1 (r)}dr

≥ 0

Z ≥

f (t)

dr · Cr{|ξ| ≥ f −1 (f (t))}

0

= f (t) · Cr{|ξ| ≥ t} which proves the inequality. Theorem 2.56 (Liu [128], Markov Inequality) Let ξ be a fuzzy variable. Then for any given numbers t > 0 and p > 0, we have Cr{|ξ| ≥ t} ≤

E[|ξ|p ] . tp

(2.83)

Proof: It is a special case of Theorem 2.55 when f (x) = |x|p . Theorem 2.57 (Liu [128], Chebyshev Inequality) Let ξ be a fuzzy variable whose variance V [ξ] exists. Then for any given number t > 0, we have Cr {|ξ − E[ξ]| ≥ t} ≤

V [ξ] . t2

(2.84)

Proof: It is a special case of Theorem 2.55 when the fuzzy variable ξ is replaced with ξ − E[ξ], and f (x) = x2 . Theorem 2.58 (Liu [128], H¨ older’s Inequality) Let p and q be two positive real numbers with 1/p+1/q = 1, and let ξ and η be independent fuzzy variables with E[|ξ|p ] < ∞ and E[|η|q ] < ∞. Then we have p p E[|ξη|] ≤ p E[|ξ|p ] q E[|η|q ]. (2.85)

109


Proof: The inequality holds trivially if at least one of ξ and η is zero a.s. Now we assume√E[|ξ|p ] > 0 and E[|η|q ] > 0. It is easy to prove that the function √ f (x, y) = p x q y is a concave function on D = {(x, y) : x ≥ 0, y ≥ 0}. Thus for any point (x0 , y0 ) with x0 > 0 and y0 > 0, there exist two real numbers a and b such that f (x, y) − f (x0 , y0 ) ≤ a(x − x0 ) + b(y − y0 ),

∀(x, y) ∈ D.

Letting x0 = E[|ξ|p ], y0 = E[|η|q ], x = |ξ|p and y = |η|q , we have f (|ξ|p , |η|q ) − f (E[|ξ|p ], E[|η|q ]) ≤ a(|ξ|p − E[|ξ|p ]) + b(|η|q − E[|η|q ]). Taking the expected values on both sides, we obtain E[f (|ξ|p , |η|q )] ≤ f (E[|ξ|p ], E[|η|q ]). Hence the inequality (2.85) holds. Theorem 2.59 (Liu [128], Minkowski Inequality) Let p be a real number with p ≥ 1, and let ξ and η be independent fuzzy variables with E[|ξ|p ] < ∞ and E[|η|p ] < ∞. Then we have p p p p E[|ξ + η|p ] ≤ p E[|ξ|p ] + p E[|η|p ]. (2.86) Proof: The inequality holds trivially if at least one of ξ and η is zero a.s. Now we assume √ E[|ξ|p ] > 0 and E[|η|p ] > 0. It is easy to prove that the function √ f (x, y) = ( p x + p y)p is a concave function on D = {(x, y) : x ≥ 0, y ≥ 0}. Thus for any point (x0 , y0 ) with x0 > 0 and y0 > 0, there exist two real numbers a and b such that f (x, y) − f (x0 , y0 ) ≤ a(x − x0 ) + b(y − y0 ),

∀(x, y) ∈ D.

Letting x0 = E[|ξ|p ], y0 = E[|η|p ], x = |ξ|p and y = |η|p , we have f (|ξ|p , |η|p ) − f (E[|ξ|p ], E[|η|p ]) ≤ a(|ξ|p − E[|ξ|p ]) + b(|η|p − E[|η|p ]). Taking the expected values on both sides, we obtain E[f (|ξ|p , |η|p )] ≤ f (E[|ξ|p ], E[|η|p ]). Hence the inequality (2.86) holds. Theorem 2.60 (Liu [129], Jensen’s Inequality) Let ξ be a fuzzy variable, and f : < → < a convex function. If E[ξ] and E[f (ξ)] are finite, then f (E[ξ]) ≤ E[f (ξ)]. Especially, when f (x) = |x|p and p ≥ 1, we have |E[ξ]|p ≤ E[|ξ|p ].

(2.87)

110


Proof: Since f is a convex function, for each y, there exists a number k such that f (x) − f (y) ≥ k · (x − y). Replacing x with ξ and y with E[ξ], we obtain f (ξ) − f (E[ξ]) ≥ k · (ξ − E[ξ]). Taking the expected values on both sides, we have E[f (ξ)] − f (E[ξ]) ≥ k · (E[ξ] − E[ξ]) = 0 which proves the inequality.

2.14

Convergence Concepts

This section discusses some convergence concepts of fuzzy sequence: convergence almost surely (a.s.), convergence in credibility, convergence in mean, and convergence in distribution. Table 2.1: Relations among Convergence Concepts Convergence Almost Surely Convergence in Mean

→

Convergence

%

in Credibility

&

Convergence in Distribution

Definition 2.28 (Liu [128]) Suppose that ξ, ξ1 , ξ2 , · · · are fuzzy variables defined on the credibility space (Θ, P, Cr). The sequence {ξi } is said to be convergent a.s. to ξ if and only if there exists an event A with Cr{A} = 1 such that lim |ξi (θ) − ξ(θ)| = 0 (2.88) i→∞

for every θ ∈ A. In that case we write ξi → ξ, a.s. Definition 2.29 (Liu [128]) Suppose that ξ, ξ1 , ξ2 , · · · are fuzzy variables defined on the credibility space (Θ, P, Cr). We say that the sequence {ξi } converges in credibility to ξ if lim Cr {|ξi − ξ| ≥ ε} = 0

i→∞

(2.89)

for every ε > 0. Definition 2.30 (Liu [128]) Suppose that ξ, ξ1 , ξ2 , · · · are fuzzy variables with finite expected values defined on the credibility space (Θ, P, Cr). We say that the sequence {ξi } converges in mean to ξ if lim E[|ξi − ξ|] = 0.

i→∞

(2.90)

111

Section 2.14 - Convergence Concepts

In addition, the sequence {ξi } is said to converge in mean square to ξ if lim E[|ξi − ξ|2 ] = 0.

i→∞

(2.91)

Definition 2.31 (Liu [128]) Suppose that Φ, Φ1 , Φ2 , · · · are the credibility distributions of fuzzy variables ξ, ξ1 , ξ2 , · · · , respectively. We say that {ξi } converges in distribution to ξ if Φi → Φ at any continuity point of Φ. Convergence in Mean vs. Convergence in Credibility Theorem 2.61 (Liu [128]) Suppose that ξ, ξ1 , ξ2 , · · · are fuzzy variables defined on the credibility space (Θ, P, Cr). If the sequence {ξi } converges in mean to ξ, then {ξi } converges in credibility to ξ. Proof: It follows from Theorem 2.56 that, for any given number ε > 0, Cr {|ξi − ξ| ≥ ε} ≤

E[|ξi − ξ|] →0 ε

as i → ∞. Thus {ξi } converges in credibility to ξ. Example 2.65: Convergence in credibility does not imply convergence in mean. For example, take (Θ, P, Cr) to be {θ1 , θ2 , · · · } with Cr{θ1 } = 1/2 and Cr{θj } = 1/j for j = 2, 3, · · · The fuzzy variables are defined by ( i, if j = i ξi (θj ) = 0, otherwise for i = 1, 2, · · · and ξ = 0. For any small number ε > 0, we have Cr {|ξi − ξ| ≥ ε} =

1 → 0. i

That is, the sequence {ξi } converges in credibility to ξ. However, E [|ξi − ξ|] ≡ 1 6→ 0. That is, the sequence {ξi } does not converge in mean to ξ. Convergence Almost Surely vs. Convergence in Credibility Example 2.66: Convergence a.s. does not imply convergence in credibility. For example, take (Θ, P, Cr) to be {θ1 , θ2 , · · · } with Cr{θj } = j/(2j + 1) for j = 1, 2, · · · The fuzzy variables are defined by ( i, if j = i ξi (θj ) = 0, otherwise

112


for i = 1, 2, · · · and ξ = 0. Then the sequence {ξi } converges a.s. to ξ. However, for any small number ε > 0, we have Cr {|ξi − ξ| ≥ ε} =

i 1 → . 2i + 1 2

That is, the sequence {ξi } does not converge in credibility to ξ. Theorem 2.62 (Wang and Liu [221]) Suppose that ξ, ξ1 , ξ2 , · · · are fuzzy variables defined on the credibility space (Θ, P, Cr). If the sequence {ξi } converges in credibility to ξ, then {ξi } converges a.s. to ξ. Proof: If {ξi } does not converge a.s. to ξ, then there exists an element θ∗ ∈ Θ with Cr{θ∗ } > 0 such that ξi (θ∗ ) 6→ ξ(θ∗ ) as i → ∞. In other words, there exists a small number ε > 0 and a subsequence {ξik (θ∗ )} such that |ξik (θ∗ ) − ξ(θ∗ )| ≥ ε for any k. Since credibility measure is an increasing set function, we have Cr {|ξik − ξ| ≥ ε} ≥ Cr{θ∗ } > 0 for any k. It follows that {ξi } does not converge in credibility to ξ. A contradiction proves the theorem. Convergence in Credibility vs. Convergence in Distribution Theorem 2.63 (Wang and Liu [221]) Suppose that ξ, ξ1 , ξ2 , · · · are fuzzy variables. If the sequence {ξi } converges in credibility to ξ, then {ξi } converges in distribution to ξ. Proof: Let x be any given continuity point of the distribution Φ. On the one hand, for any y > x, we have {ξi ≤ x} = {ξi ≤ x, ξ ≤ y} ∪ {ξi ≤ x, ξ > y} ⊂ {ξ ≤ y} ∪ {|ξi − ξ| ≥ y − x}. It follows from the credibility subadditivity theorem that Φi (x) ≤ Φ(y) + Cr{|ξi − ξ| ≥ y − x}. Since {ξi } converges in credibility to ξ, we have Cr{|ξi − ξ| ≥ y − x} → 0. Thus we obtain lim supi→∞ Φi (x) ≤ Φ(y) for any y > x. Letting y → x, we get lim sup Φi (x) ≤ Φ(x). (2.92) i→∞

On the other hand, for any z < x, we have {ξ ≤ z} = {ξ ≤ z, ξi ≤ x} ∪ {ξ ≤ z, ξi > x} ⊂ {ξi ≤ x} ∪ {|ξi − ξ| ≥ x − z} which implies that Φ(z) ≤ Φi (x) + Cr{|ξi − ξ| ≥ x − z}.


113

Since Cr{|ξi − ξ| ≥ x − z} → 0, we obtain Φ(z) ≤ lim inf i→∞ Φi (x) for any z < x. Letting z → x, we get Φ(x) ≤ lim inf Φi (x). i→∞

(2.93)

It follows from (2.92) and (2.93) that Φi (x) → Φ(x). The theorem is proved. Example 2.67: Convergence in distribution does not imply convergence in credibility. For example, take (Θ, P, Cr) to be {θ1 , θ2 } with Cr{θ1 } = Cr{θ2 } = 1/2, and define −1, if θ = θ1 ξ(θ) = 1, if θ = θ2 . We also define ξi = −ξ for i = 1, 2, · · · Then ξi and ξ are identically distributed. Thus {ξi } converges in distribution to ξ. But, for any small number ε > 0, we have Cr{|ξi − ξ| > ε} = Cr{Θ} = 1. That is, the sequence {ξi } does not converge in credibility to ξ. Convergence Almost Surely vs. Convergence in Distribution Example 2.68: Convergence in distribution does not imply convergence a.s. For example, take (Θ, P, Cr) to be {θ1 , θ2 } with Cr{θ1 } = Cr{θ2 } = 1/2, and define −1, if θ = θ1 ξ(θ) = 1, if θ = θ2 . We also define ξi = −ξ for i = 1, 2, · · · Then {ξi } converges in distribution to ξ. However, {ξi } does not converge a.s. to ξ. Example 2.69: Convergence a.s. does not imply convergence in distribution. For example, take (Θ, P, Cr) to be {θ1 , θ2 , · · · } with Cr{θj } = j/(2j + 1) for j = 1, 2, · · · The fuzzy variables are defined by ( i, if j = i ξi (θj ) = 0, otherwise for i = 1, 2, · · · and ξ = 0. Then the sequence {ξi } converges a.s. to ξ. However, the credibility distributions of ξi are  0, if x < 0   (i + 1)/(2i + 1), if 0 ≤ x < i Φi (x) =   1, if x ≥ i, i = 1, 2, · · · , respectively. The credibility distribution of ξ is ( 0, if x < 0 Φ(x) = 1, if x ≥ 0.

114


It is clear that Φi (x) 6→ Φ(x) at x > 0. That is, the sequence {ξi } does not converge in distribution to ξ.

2.15

Conditional Credibility

We now consider the credibility of an event A after it has been learned that some other event B has occurred. This new credibility of A is called the conditional credibility of A given B. The first problem is whether the conditional credibility is determined uniquely and completely. The answer is negative. It is doomed to failure to define an unalterable and widely accepted conditional credibility. For this reason, it is appropriate to speak of a certain person’s subjective conditional credibility, rather than to speak of the true conditional credibility. In order to define a conditional credibility measure Cr{A|B}, at first we have to enlarge Cr{A ∩ B} because Cr{A ∩ B} < 1 for all events whenever Cr{B} < 1. It seems that we have no alternative but to divide Cr{A ∩ B} by Cr{B}. Unfortunately, Cr{A∩B}/Cr{B} is not always a credibility measure. However, the value Cr{A|B} should not be greater than Cr{A ∩ B}/Cr{B} (otherwise the normality will be lost), i.e., Cr{A|B} ≤

Cr{A ∩ B} . Cr{B}

(2.94)

On the other hand, in order to preserve the self-duality, we should have Cr{A|B} = 1 − Cr{Ac |B} ≥ 1 −

Cr{Ac ∩ B} . Cr{B}

(2.95)

Furthermore, since (A ∩ B) ∪ (Ac ∩ B) = B, we have Cr{B} ≤ Cr{A ∩ B} + Cr{Ac ∩ B} by using the credibility subadditivity theorem. Thus 0≤1−

Cr{Ac ∩ B} Cr{A ∩ B} ≤ ≤ 1. Cr{B} Cr{B}

(2.96)

Hence any numbers between 1 − Cr{Ac ∩ B}/Cr{B} and Cr{A ∩ B}/Cr{B} are reasonable values that the conditional credibility may take. Based on the maximum uncertainty principle, we have the following conditional credibility measure. Definition 2.32 (Liu [132]) Let (Θ, P, Cr) be a credibility space, and A, B ∈ P. Then the conditional credibility measure of A given B is defined by  Cr{A ∩ B} Cr{A ∩ B}   , if < 0.5   Cr{B} Cr{B}   Cr{Ac ∩ B} Cr{Ac ∩ B} (2.97) Cr{A|B} = 1− , if < 0.5    Cr{B} Cr{B}    0.5, otherwise

Section 2.15 - Conditional Credibility

115

provided that Cr{B} > 0. It follows immediately from the definition of conditional credibility that 1−

Cr{Ac ∩ B} Cr{A ∩ B} ≤ Cr{A|B} ≤ . Cr{B} Cr{B}

(2.98)

Furthermore, the value of Cr{A|B} takes values as close to 0.5 as possible in the interval. In other words, it accords with the maximum uncertainty principle. Theorem 2.64 (Liu [132]) Let (Θ, P, Cr) be a credibility space, and B an event with Cr{B} > 0. Then Cr{·|B} defined by (2.97) is a credibility measure, and (Θ, P, Cr{·|B}) is a credibility space. Proof: It is sufficient to prove that Cr{·|B} satisfies the normality, monotonicity, self-duality and maximality axioms. At first, it satisfies the normality axiom, i.e., Cr{Θ|B} = 1 −

Cr{Θc ∩ B} Cr{∅} =1− = 1. Cr{B} Cr{B}

For any events A1 and A2 with A1 ⊂ A2 , if Cr{A1 ∩ B} Cr{A2 ∩ B} ≤ < 0.5, Cr{B} Cr{B} then Cr{A1 |B} = If

Cr{A1 ∩ B} Cr{A2 ∩ B} ≤ = Cr{A2 |B}. Cr{B} Cr{B}

Cr{A2 ∩ B} Cr{A1 ∩ B} ≤ 0.5 ≤ , Cr{B} Cr{B}

then Cr{A1 |B} ≤ 0.5 ≤ Cr{A2 |B}. If 0.5
0. Then for any x ∈ X, the conditional credibility of ξ = x given ξ ∈ X is

Cr {ξ = x|ξ ∈ X} =

       1−      

Cr{ξ = x} , Cr{ξ ∈ X}

if

Cr{ξ = x} < 0.5 Cr{ξ ∈ X}

Cr{ξ 6= x, ξ ∈ X} Cr{ξ 6= x, ξ ∈ X} , if < 0.5 Cr{ξ ∈ X} Cr{ξ ∈ X} 0.5,

otherwise.

Example 2.71: Let ξ and η be two fuzzy variables, and Y a set of real numbers such that Cr{η ∈ Y } > 0. Then we have  Cr{ξ = x, η ∈ Y } Cr{ξ = x, η ∈ Y }   , if < 0.5   Cr{η ∈ Y } Cr{η ∈ Y }   Cr{ξ 6= x, η ∈ Y } Cr{ξ 6= x, η ∈ Y } Cr {ξ = x|η ∈ Y } = 1− , if < 0.5    Cr{η ∈ Y } Cr{η ∈ Y }    0.5, otherwise. Definition 2.33 (Liu [132]) The conditional membership function of a fuzzy variable ξ given B is defined by µ(x|B) = (2Cr{ξ = x|B}) ∧ 1, provided that Cr{B} > 0.

x∈
0 for some x ∈ X. Then the conditional membership function of ξ given ξ ∈ X is  2µ(x)   ∧ 1, if sup µ(x) < 1    sup µ(x) x∈X  x∈X µ(x|X) = (2.100)  2µ(x)   ∧ 1, if sup µ(x) = 1    2 − sup µ(x) x∈X x∈X c

for x ∈ X. Please mention that µ(x|X) ≡ 0 if x 6∈ X. µ(x|X)

µ(x|X) .... ........ ....................................... ... ....................................................................... . ... ... ... ... ..... ... ... ... ... ... ... .. ... ... . ... ... ... ... .. . . ... . ... ... .. .. . ... . . ... . ... .. ... ... . . ... . ... .. ... .. . . . ... ... .. ..... ... .... ... ... ... .. ... ... ... ... ....... ... . ... ... . ... ... ... .... ... .. ... ..... .. ... ... ... .... .. ..... .. ...... ... ... . . . . . .................................................................................................................................................................. .. . .. .............. ............................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . ...

1

0

X

.... ........ ........................................ .......................... . ... ... ...... . ... ..... .. ... ... ... .. ... ... ... .... ... . ... . ... .... .. ... . . ... . .. ... ........ . . ... ... ... .. . . ... . . ... .. ... ..... . . ... ... ... .. . . ... . ... ... .. ... . . .. . ... ... .. . . . ... . ... ..... .. ... . . . ... ... ... . .. ... .. ... ..... ... ... .. ... .... ... .. .. ...... ... ... ..................................................................................................................................................................... .. . . .. ...................... ..................... ... ...

1

x

0

X

x

Figure 2.6: Conditional Membership Function µ(x|X) Example 2.73: Let ξ and η be two fuzzy variables with joint membership function µ(x, y), and Y a set of real numbers. Then the conditional membership function of ξ given η ∈ Y is  2 sup µ(x, y)   y∈Y   ∧ 1, if sup µ(x, y) < 1   sup µ(x, y)  x∈ 0. Definition 2.35 (Liu [132]) The conditional credibility density function φ of a fuzzy variable ξ given B is a nonnegative function such that Z x Φ(x|B) = φ(y|B)dy, ∀x ∈ x}. (2.106) ∆↓0

119

Section 2.16 - Fuzzy Process

The hazard rate tells us the credibility of a failure just after time x when it is functioning at time x. Example 2.75: Let ξ be an exponentially distributed fuzzy variable. Then its hazard rate h(x) ≡ 0.5. In fact, the hazard rate is always 0.5 if the membership function is positive and decreasing. Example 2.76: Let ξ be a triangular fuzzy variable (a, b, c) with a ≥ 0. Then its hazard rate  0, if x ≤ a      x−a    , if a ≤ x ≤ (a + b)/2 b−a h(x) =   0.5, if (a + b)/2 ≤ x < c       0, if x ≥ c.

2.16

Fuzzy Process

Definition 2.38 Let T be an index set and let (Θ, P, Cr) be a credibility space. A fuzzy process is a function from T × (Θ, P, Cr) to the set of real numbers. That is, a fuzzy process Xt (θ) is a function of two variables such that the function Xt∗ (θ) is a fuzzy variable for each t∗ . For each fixed θ∗ , the function Xt (θ∗ ) is called a sample path of the fuzzy process. A fuzzy process Xt (θ) is said to be sample-continuous if the sample path is continuous for almost all θ. Definition 2.39 A fuzzy process Xt is said to have independent increments if Xt1 − Xt0 , Xt2 − Xt1 , · · · , Xtk − Xtk−1 (2.107) are independent fuzzy variables for any times t0 < t1 < · · · < tk . A fuzzy process Xt is said to have stationary increments if, for any given t > 0, the Xs+t − Xs are identically distributed fuzzy variables for all s > 0. Fuzzy Renewal Process Definition 2.40 (Zhao and Liu [251]) Let ξ1 , ξ2 , · · · be iid positive fuzzy variables. Define S0 = 0 and Sn = ξ1 + ξ2 + · · · + ξn for n ≥ 1. Then the fuzzy process Nt = max n Sn ≤ t (2.108) n≥0

is called a fuzzy renewal process.

120


N. t 4 3 2 1 0

... .......... ... .. ........... .............................. .. ... ... .. .. ... .......... ......................................................... .. ... .. .. .... .. .. .. .. .. .......... ....................................... .. ... .. .. .. ... .. .. ... .. .. .. .. ... .. ......................................................... ......... .. .. .. .. .. .... .. .. .. .. ... .. .. .. . .. . ...................................................................................................................................................................................................................................... ... ... ... ... ... .... .... .... .... ... 1 ... 2 3 ... 4 ... ... ... .. .. .. .. ..

ξ

S0

ξ

S1

ξ

S2

ξ

S3

t

S4

Figure 2.7: A Sample Path of Fuzzy Renewal Process If ξ1 , ξ2 , · · · denote the interarrival times of successive events. Then Sn can be regarded as the waiting time until the occurrence of the nth event, and Nt is the number of renewals in (0, t]. Each sample path of Nt is a right-continuous and increasing step function taking only nonnegative integer values. Furthermore, the size of each jump of Nt is always 1. In other words, Nt has at most one renewal at each time. In particular, Nt does not jump at time 0. Since Nt ≥ n if and only if Sn ≤ t, we have t Cr{Nt ≥ n} = Cr{Sn ≤ t} = Cr ξ1 ≤ . (2.109) n Theorem 2.65 Let Nt be a fuzzy renewal process with interarrival times ξ1 , ξ2 , · · · Then we have ∞ ∞ X X t Cr ξ1 ≤ Cr{Sn ≤ t} = E[Nt ] = . (2.110) n n=1 n=1 Proof: Since Nt takes only nonnegative integer values, we have Z ∞ ∞ Z n X E[Nt ] = Cr{Nt ≥ r}dr = Cr{Nt ≥ r}dr 0 ∞ X

n=1 ∞ X

n−1 ∞ X

t Cr ξ1 ≤ = Cr{Nt ≥ n} = Cr{Sn ≤ t} = . n n=1 n=1 n=1 The theorem is proved. Example 2.77: A renewal process Nt is called an equipossible renewal process if ξ1 , ξ2 , · · · are iid equipossible fuzzy variables (a, b) with a > 0. Then for each nonnegative integer n, we have    0, if t < na 0.5, if na ≤ t < nb (2.111) Cr{Nt ≥ n} =   1, if t ≥ nb,

121


1 E[Nt ] = 2

t t + a b

(2.112)

where bxc represents the maximal integer less than or equal to x. Example 2.78: A renewal process Nt is called a triangular renewal process if ξ1 , ξ2 , · · · are iid triangular fuzzy variables (a, b, c) with a > 0. Then for each nonnegative integer n, we have

Cr{Nt ≥ n} =

         

if t ≤ na

0, t − na , 2n(b − a)

if na ≤ t ≤ nb

 nc − 2nb + t   , if nb ≤ t ≤ nc   2n(c − b)     1, if t ≥ nc.

(2.113)

Theorem 2.66 (Zhao and Liu [251], Renewal Theorem) Let Nt be a fuzzy renewal process with interarrival times ξ1 , ξ2 , · · · Then 1 E[Nt ] =E . t→∞ t ξ1 lim

(2.114)

Proof: Since ξ1 is a positive fuzzy variable and Nt is a nonnegative fuzzy variable, we have E[Nt ] = t

Z

∞

Cr

0

Z ∞ 1 1 E = ≥ r dr. Cr ξ1 ξ1 0

Nt ≥ r dr, t

It is easy to verify that Cr

Nt ≥r t

≤ Cr

1 ≥r ξ1

and lim Cr

t→∞

Nt ≥r t

= Cr

1 ≥r ξ1

for any real numbers t > 0 and r > 0. It follows from Lebesgue dominated convergence theorem that Z

∞

lim

t→∞

Cr

0

Hence (2.114) holds.

Z ∞ Nt 1 ≥ r dr = Cr ≥ r dr. t ξ1 0

122


C Process This subsection introduces a fuzzy counterpart of Brownian motion, and provides some basic mathematical properties. Definition 2.41 (Liu [133]) A fuzzy process Ct is said to be a C process if (i) C0 = 0, (ii) Ct has stationary and independent increments, (iii) every increment Cs+t − Cs is a normally distributed fuzzy variable with expected value et and variance σ 2 t2 . The parameters e and σ are called the drift and diffusion coefficients, respectively. The C process is said to be standard if e = 0 and σ = 1. Any C process may be represented by et + σCt where Ct is a standard C process. C. t

..... ....... .... ... ... ... ... ... ... ... . ... ... .. ..... .. ............. ... ..... . . . ... ...... ... ... .... ..... ... ... ... .. ... .. .. .. .. .. ... ... ... ..... .. .... ...... .. .... ........ .. ........ ... . . . .. .... ... . .... . . . . . . .. . . . . . . . ... ... .. ... ...... ..... .. ...... .. ..... .. ............. ... .. .... . ... .. ..... .... . ... ..... ... ....... . ....... ......... ... ... ... .... ... ..... .... ... .. ... ..... ... .. ... ... ... ..... ... .. ....... ... ... ... ... ...... .. .. ...... .... .... ... .... ........ ...... . . .. . . . ... . . . . . . . . . . .. .. ..... ..... ..... .... ..... .. ....... .... .... ... ... ...... ... ...... . ..... .. ... .... ... .. ... . ... ... .. ... ... ... ... ... ... ... ... ...... ... ...... ... .. ....... .. ...... ... . . .... ... .. . . ...... .... ... ... ..... ... ...... .... ...... ... ......... ... ...... ............. ..... ... .. ...... ..... ..... .... ...... .... ... .........................................................................................................................................................................................................................................................................................

t

Figure 2.8: A Sample Path of Standard C Process Perhaps the readers would like to know why the increment is a normally distributed fuzzy variable. The reason is that a normally distributed fuzzy variable has maximum entropy when its expected value and variance are given, just like a normally distributed random variable. Theorem 2.67 (Liu [133], Existence Theorem) There is a C process. Furthermore, each version of C process is sample-continuous. Proof: Without loss of generality, we only prove that there is a standard C process on the range of t ∈ [0, 1]. Let ξ(r) r represents rational numbers in [0, 1] be a countable sequence of independently and normally distributed fuzzy variables with expected value zero and variance one. For each integer n, we define a fuzzy process  k X  i k   1 ξ , if t = (k = 0, 1, · · · , n) n n n Xn (t) = i=1    linear, otherwise.

123


Since the limit lim Xn (t)

n→∞

exists almost surely, we may verify that the limit meets the conditions of C process. Hence there is a standard C process. Remark 2.9: Suppose that Ct is a standard C process. It has been proved that X1 (t) = −Ct , (2.115) X2 (t) = aCt/a ,

(2.116)

X3 (t) = Ct+s − Cs

(2.117)

are each a version of standard C process. Dai [23] proved that almost all C paths are Lipschitz continuous and have a finite variation. Thus almost all C paths are differentiable almost everywhere and have zero squared variation. However, for any number ` ∈ (0, 0.5), there is a sample θ with Cr{θ} = ` such that Ct (θ) is not differentiable in a dense set of t. As a byproduct, C path is an example of Lipschitz continuous function whose nondifferentiable points are dense, while Brownian path is an example of continuous function that is nowhere differentiable. In other words, C path is the worst Lipschitz continuous function and Brownian path is the worst continuous function in the sense of differentiability. Example 2.79: Let Ct be a C process with drift 0. Then for any level x > 0 and any time t > 0, Dai [23] proved the following reflection principle, Cr

max Cs ≥ x = Cr{Ct ≥ x}.

0≤s≤t

In addition, for any level x < 0 and any time t > 0, we have Cr min Cs ≤ x = Cr{Ct ≤ x}. 0≤s≤t

(2.118)

(2.119)

Example 2.80: Let Ct be a C process with drift e > 0 and diffusion coefficient σ. Then the first passage time τ that the C process reaches the barrier x > 0 has the membership function µ(t) = 2 1 + exp

π|x − et| √ 6σt

whose expected value E[τ ] = +∞ (Dai [23]).

−1 ,

t>0

(2.120)

124


Definition 2.42 (Liu [133]) Let Ct be a standard C process. Then et + σCt is a C process, and the fuzzy process Gt = exp(et + σCt )

(2.121)

is called a geometric C process. Geometric C process Gt is employed to model stock prices. For each t > 0, Li [108] proved that Gt has a lognormal membership function −1 π| ln z − et| √ , z≥0 (2.122) µ(z) = 2 1 + exp 6σt whose expected value is √ √ E[Gt ] = exp(et) csc( 6σt) 6σt,

√ ∀t < π/( 6σ).

(2.123)

In addition, the first passage time that a geometric C process Gt reaches the barrier x > 1 is just the time that the C process with drift e and diffusion σ reaches ln x. A Basic Stock Model It was assumed that stock price follows geometric Brownian motion, and stochastic financial mathematics was then founded based on this assumption. Liu [133] presented an alternative assumption that stock price follows geometric C process. Based on this assumption, we obtain a basic stock model for fuzzy financial market in which the bond price Xt and the stock price Yt follow ( Xt = X0 exp(rt) (2.124) Yt = Y0 exp(et + σCt ) where r is the riskless interest rate, e is the stock drift, σ is the stock diffusion, and Ct is a standard C process. It is just a fuzzy counterpart of Black-Scholes stock model [8]. For exploring further development of fuzzy stock models, the interested readers may consult Gao [50], Peng [191], Qin and Li [194], and Zhu [267].

2.17

Fuzzy Calculus

Let Ct be a standard C process, and dt an infinitesimal time interval. Then dCt = Ct+dt − Ct is a fuzzy process such that, for each t, the dCt is a normally distributed fuzzy variable with E[dCt ] = 0, V [dCt ] = dt2 , E[dCt2 ] = dt2 ,

V [dCt2 ] ≈ 7dt4 .

125

Section 2.17 - Fuzzy Calculus

Definition 2.43 (Liu [133]) Let Xt be a fuzzy process and let Ct be a standard C process. For any partition of closed interval [a, b] with a = t1 < t2 < · · · < tk+1 = b, the mesh is written as ∆ = max |ti+1 − ti |. 1≤i≤k

Then the fuzzy integral of Xt with respect to Ct is Z b k X Xt dCt = lim Xti · (Cti+1 − Cti ) ∆→0

a

(2.125)

i=1

provided that the limit exists almost surely and is a fuzzy variable. Example 2.81: Let Ct be a standard C process. Then for any partition 0 = t1 < t2 < · · · < tk+1 = s, we have Z

k X

s

dCt = lim

∆→0

0

(Cti+1 − Cti ) ≡ Cs − C0 = Cs .

i=1

Example 2.82: Let Ct be a standard C process. Then for any partition 0 = t1 < t2 < · · · < tk+1 = s, we have sCs =

k X

ti+1 Cti+1 − ti Cti

i=1

=

k X

ti (Cti+1 − Cti ) +

i=1 Z s

→

k X

Cti+1 (ti+1 − ti )

i=1

Z

s

tdCt + 0

Ct dt 0

as ∆ → 0. It follows that Z

s

Z

s

tdCt = sCs − 0

Ct dt. 0

Example 2.83: Let Ct be a standard C process. Then for any partition 0 = t1 < t2 < · · · < tk+1 = s, we have Cs2 =

k X

Ct2i+1 − Ct2i

i=1

=

k X

Cti+1 − Cti

i=1

2

+2

k X i=1

Z →0+2

s

Ct dCt 0

Cti Cti+1 − Cti

126


as ∆ → 0. That is,

s

Z

Ct dCt = 0

1 2 C . 2 s

This equation shows that fuzzy integral does not behave like Ito integral. In fact, the fuzzy integral behaves like ordinary integrals. Theorem 2.68 (Liu [133]) Let Ct be a standard C process, and let h(t, c) be a continuously differentiable function. Define Xt = h(t, Ct ). Then we have the following chain rule dXt =

∂h ∂h (t, Ct )dt + (t, Ct )dCt . ∂t ∂c

(2.126)

Proof: Since the function h is continuously differentiable, by using Taylor series expansion, the infinitesimal increment of Xt has a first-order approximation ∂h ∂h ∆Xt = (t, Ct )∆t + (t, Ct )∆Ct . ∂t ∂c Hence we obtain the chain rule because it makes Z s Z s ∂h ∂h Xs = X0 + (t, Ct )dt + (t, Ct )dCt ∂t 0 0 ∂c for any s ≥ 0. Remark 2.10: The infinitesimal increment dCt in (2.126) may be replaced with the derived C process dYt = ut dt + vt dCt

(2.127)

where ut and vt are absolutely integrable fuzzy processes, thus producing dh(t, Yt ) =

∂h ∂h (t, Yt )dt + (t, Yt )dYt . ∂t ∂c

(2.128)

Remark 2.11: Assume that C1t , C2t , · · · , Cmt are standard C processes, and h(t, c1 , c2 , · · · , cm ) is a continuously differentiable function. Define Xt = h(t, C1t , C2t , · · · , Cmt ). You [243] proved the following chain rule dXt =

m X ∂h ∂h dt + dCit . ∂t ∂c i i=1

(2.129)

Example 2.84: Applying the chain rule, we obtain the following formula d(tCt ) = Ct dt + tdCt .

127

Section 2.18 - Fuzzy Differential Equation

Hence we have s

Z sCs =

s

Z d(tCt ) =

0

Z

s

Ct dt + 0

tdCt . 0

That is, s

Z

s

Z tdCt = sCs −

Ct dt. 0

0

Example 2.85: Let Ct be a standard C process. By using the chain rule d(Ct2 ) = 2Ct dCt , we get Cs2 =

s

Z

d(Ct2 ) = 2

s

Z

Ct dCt .

0

0

It follows that

s

Z

Ct dCt = 0

1 2 C . 2 s

Example 2.86: Let Ct be a standard C process. Then we have the following chain rule d(Ct3 ) = 3Ct2 dCt . Thus we obtain Cs3

s

Z

d(Ct3 )

=

Z =3

0

That is Z

s

Ct2 dCt .

0 s

Ct2 dCt =

0

1 3 C . 3 s

Theorem 2.69 (Liu [133], Integration by Parts) Suppose that Ct is a standard C process and F (t) is an absolutely continuous function. Then Z s Z s F (t)dCt = F (s)Cs − Ct dF (t). (2.130) 0

0

Proof: By defining h(t, Ct ) = F (t)Ct and using the chain rule, we get d(F (t)Ct ) = Ct dF (t) + F (t)dCt . Thus Z F (s)Cs =

Z d(F (t)Ct ) =

0

which is just (2.130).

s

s

Z Ct dF (t) +

0

s

F (t)dCt 0

128

2.18


Fuzzy Differential Equation

Fuzzy differential equation is a type of differential equation driven by C process. Definition 2.44 (Liu [133]) Suppose Ct is a standard C process, and f and g are some given functions. Then dXt = f (t, Xt )dt + g(t, Xt )dCt

(2.131)

is called a fuzzy differential equation. A solution is a fuzzy process Xt that satisfies (2.131) identically in t. Remark 2.12: Note that there is no precise definition for the terms dXt , dt and dCt in the fuzzy differential equation (2.131). The mathematically meaningful form is the fuzzy integral equation Z s Z s Xs = X0 + f (t, Xt )dt + g(t, Xt )dCt . (2.132) 0

0

However, the differential form is more convenient for us. This is the main reason why we accept the differential form. Example 2.87: Let Ct be a standard C process. Then the fuzzy differential equation dXt = adt + bdCt has a solution Xt = at + bCt which is just a C process with drift coefficient a and diffusion coefficient b. Example 2.88: Let Ct be a standard C process. Then the fuzzy differential equation dXt = aXt dt + bXt dCt has a solution Xt = exp (at + bCt ) which is just a geometric C process. Example 2.89: Let Ct be a standard C process. Then the fuzzy differential equations ( dXt = −Yt dCt dYt = Xt dCt have a solution (Xt , Yt ) = (cos Ct , sin Ct ) which is called a C process on unit circle since Xt2 + Yt2 ≡ 1.

Chapter 3

Chance Theory Fuzziness and randomness are two basic types of uncertainty. In many cases, fuzziness and randomness simultaneously appear in a system. In order to describe this phenomena, a fuzzy random variable was introduced by Kwakernaak [87][88] as a random element taking “fuzzy variable” values. In addition, a random fuzzy variable was proposed by Liu [124] as a fuzzy element taking “random variable” values. For example, it might be known that the lifetime of a modern engine is an exponentially distributed random variable with an unknown parameter. If the parameter is provided as a fuzzy variable, then the lifetime is a random fuzzy variable. More generally, a hybrid variable was introduced by Liu [130] as a tool to describe the quantities with fuzziness and randomness. Fuzzy random variable and random fuzzy variable are instances of hybrid variable. In order to measure hybrid events, a concept of chance measure was introduced by Li and Liu [103]. Chance theory is a hybrid of probability theory and credibility theory. Perhaps the reader would like to know what axioms we should assume for chance theory. In fact, chance theory will be based on the three axioms of probability and four axioms of credibility. The emphasis in this chapter is mainly on chance space, hybrid variable, chance measure, chance distribution, independence, identical distribution, expected value, variance, moments, critical values, entropy, distance, convergence almost surely, convergence in chance, convergence in mean, convergence in distribution, conditional chance, hybrid process, hybrid calculus, and hybrid differential equation.

3.1

Chance Space

Chance theory begins with the concept of chance space that inherits the mathematical foundations of both probability theory and credibility theory.

130

Chapter 3 - Chance Theory

Definition 3.1 (Liu [130]) Suppose that (Θ, P, Cr) is a credibility space and (Ω, A, Pr) is a probability space. The product (Θ, P, Cr) × (Ω, A, Pr) is called a chance space. The universal set Θ × Ω is clearly the set of all ordered pairs of the form (θ, ω), where θ ∈ Θ and ω ∈ Ω. What is the product σ-algebra P × A? What is the product measure Cr × Pr? Let us discuss these two basic problems. What is the product σ-algebra

P × A?

Generally speaking, it is not true that all subsets of Θ × Ω are measurable. Let Λ be a subset of Θ × Ω. Write (3.1) Λ(θ) = ω ∈ Ω (θ, ω) ∈ Λ . It is clear that Λ(θ) is a subset of Ω. If Λ(θ) ∈ A holds for each θ ∈ Θ, then Λ may be regarded as a measurable set. Definition 3.2 (Liu [132]) Let (Θ, P, Cr) × (Ω, A, Pr) be a chance space. A subset Λ ⊂ Θ × Ω is called an event if Λ(θ) ∈ A for each θ ∈ Θ. Example 3.1: Empty set ∅ and universal set Θ × Ω are clearly events. ∈ A. Then X × Y is a subset of Θ × Ω. ( Y, if θ ∈ X (X × Y )(θ) = ∅, if θ ∈ X c

Example 3.2: Let X ∈ Since the set

P and Y

A for each θ ∈ Θ, the rectangle X × Y is an event. Theorem 3.1 (Liu [132]) Let (Θ, P, Cr) × (Ω, A, Pr) be a chance space. The class of all events is a σ-algebra over Θ × Ω, and denoted by P × A. Proof: At first, it is obvious that Θ×Ω ∈ P × A. For any event Λ, we always is in the σ-algebra

have

Λ(θ) ∈ A,

∀θ ∈ Θ.

Thus for each θ ∈ Θ, the set Λc (θ) = ω ∈ Ω (θ, ω) ∈ Λc = (Λ(θ))c ∈ A which implies that Λc ∈ P × A. Finally, let Λ1 , Λ2 , · · · be events. Then for each θ ∈ Θ, we have ! ( ) ∞ ∞ ∞ [ [ [ Λi (θ) = ω ∈ Ω (θ, ω) ∈ Λi = ω ∈ Ω (θ, ω) ∈ Λi ∈ A. i=1

i=1

i=1

That is, the countable union ∪i Λi ∈ P × A. Hence

P × A is a σ-algebra.

131

Section 3.1 - Chance Space

Example 3.3: When Θ × Ω is countable, usually

P × A is the power set.

Example 3.4: When Θ×Ω is uncountable, for example Θ×Ω = 0, there exists a point θ such that Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )(θ)} > 0.5 − ε > Ch{Λ1 } ∨ Ch{Λ2 }, Cr{θ} > 0.5 − ε > Pr{Λ1 (θ)}, Cr{θ} > 0.5 − ε > Pr{Λ2 (θ)}. Thus we have Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )c (θ)} + Cr{θ} ∧ Pr{Λ1 (θ)} + Cr{θ} ∧ Pr{Λ2 (θ)} = Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )c (θ)} + Pr{Λ1 (θ)} + Pr{Λ2 (θ)} ≥ Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )c (θ)} + Pr{(Λ1 ∪ Λ2 )(θ)} ≥ 1 − 2ε because if Cr{θ} ≥ Pr{(Λ1 ∪ Λ2 )c (θ)}, then Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )c (θ)} + Pr{(Λ1 ∪ Λ2 )(θ)} = Pr{(Λ1 ∪ Λ2 )c (θ)} + Pr{(Λ1 ∪ Λ2 )(θ)} = 1 ≥ 1 − 2ε and if Cr{θ} < Pr{(Λ1 ∪ Λ2 )c (θ)}, then Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )c (θ)} + Pr{(Λ1 ∪ Λ2 )(θ)} = Cr{θ} + Pr{(Λ1 ∪ Λ2 )(θ)} ≥ (0.5 − ε) + (0.5 − ε) = 1 − 2ε. Taking supremum on both sides and letting ε → 0, we obtain Ch{Λ1 ∪ Λ2 } = 1 − sup(Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )c (θ)}) θ∈Θ

≤ sup(Cr{θ} ∧ Pr{Λ1 (θ)}) + sup(Cr{θ} ∧ Pr{Λ2 (θ)}) θ∈Θ

θ∈Θ

= Ch{Λ1 } + Ch{Λ2 }. Case 3: Ch{Λ1 ∪ Λ2 } ≥ 0.5 and Ch{Λ1 } ∨ Ch{Λ2 } ≥ 0.5. Without loss of generality, suppose Ch{Λ1 } ≥ 0.5. For each θ, we first have Cr{θ} ∧ Pr{Λc1 (θ)} = Cr{θ} ∧ Pr{(Λc1 (θ) ∩ Λc2 (θ)) ∪ (Λc1 (θ) ∩ Λ2 (θ))} ≤ Cr{θ} ∧ (Pr{(Λ1 ∪ Λ2 )c (θ)} + Pr{Λ2 (θ)}) ≤ Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )c (θ)} + Cr{θ} ∧ Pr{Λ2 (θ)},

136

Chapter 3 - Chance Theory

i.e., Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )c (θ)} ≥ Cr{θ} ∧ Pr{Λc1 (θ)} − Cr{θ} ∧ Pr{Λ2 (θ)}. It follows from Theorem 3.3 that Ch{Λ1 ∪ Λ2 } = 1 − sup(Cr{θ} ∧ Pr{(Λ1 ∪ Λ2 )c (θ)}) θ∈Θ

≤ 1 − sup(Cr{θ} ∧ Pr{Λc1 (θ)}) + sup(Cr{θ} ∧ Pr{Λ2 (θ)}) θ∈Θ

θ∈Θ

≤ Ch{Λ1 } + Ch{Λ2 }. The theorem is proved. Remark 3.1: For any events Λ1 and Λ2 , it follows from the chance subadditivity theorem that the chance measure is null-additive, i.e., Ch{Λ1 ∪ Λ2 } = Ch{Λ1 } + Ch{Λ2 } if either Ch{Λ1 } = 0 or Ch{Λ2 } = 0. Theorem 3.8 Let {Λi } be a decreasing sequence of events with Ch{Λi } → 0 as i → ∞. Then for any event Λ, we have lim Ch{Λ ∪ Λi } = lim Ch{Λ\Λi } = Ch{Λ}.

i→∞

i→∞

(3.14)

Proof: Since chance measure is increasing and subadditive, we immediately have Ch{Λ} ≤ Ch{Λ ∪ Λi } ≤ Ch{Λ} + Ch{Λi } for each i. Thus we get Ch{Λ ∪ Λi } → Ch{Λ} by using Ch{Λi } → 0. Since (Λ\Λi ) ⊂ Λ ⊂ ((Λ\Λi ) ∪ Λi ), we have Ch{Λ\Λi } ≤ Ch{Λ} ≤ Ch{Λ\Λi } + Ch{Λi }. Hence Ch{Λ\Λi } → Ch{Λ} by using Ch{Λi } → 0. Theorem 3.9 (Li and Liu [103], Chance Semicontinuity Law) For events Λ1 , Λ2 , · · · , we have n o lim Ch{Λi } = Ch lim Λi (3.15) i→∞

i→∞

if one of the following conditions is satisfied: (a) Ch{Λ} ≤ 0.5 and Λi ↑ Λ; (b) lim Ch{Λi } < 0.5 and Λi ↑ Λ; i→∞

(c) Ch{Λ} ≥ 0.5 and Λi ↓ Λ;

(d) lim Ch{Λi } > 0.5 and Λi ↓ Λ. i→∞

Proof: (a) Assume Ch{Λ} ≤ 0.5 and Λi ↑ Λ. We first have Ch{Λ} = sup(Cr{θ} ∧ Pr{Λ(θ)}), θ∈Θ

Ch{Λi } = sup(Cr{θ} ∧ Pr{Λi (θ)}) θ∈Θ

for i = 1, 2, · · · For each θ ∈ Θ, since Λi (θ) ↑ Λ(θ), it follows from the probability continuity theorem that lim Cr{θ} ∧ Pr{Λi (θ)} = Cr{θ} ∧ Pr{Λ(θ)}.

i→∞

137

Section 3.2 - Hybrid Variables

Taking supremum on both sides, we obtain lim sup(Cr{θ} ∧ Pr{Λi (θ)}) = sup(Cr{θ} ∧ Pr{Λ(θ}).

i→∞ θ∈Θ

θ∈Θ

The part (a) is verified. (b) Assume limi→∞ Ch{Λi } < 0.5 and Λi ↑ Λ. For each θ ∈ Θ, since Cr{θ} ∧ Pr{Λ(θ)} = lim Cr{θ} ∧ Pr{Λi (θ)}, i→∞

we have sup(Cr{θ} ∧ Pr{Λ(θ)}) ≤ lim sup(Cr{θ} ∧ Pr{Λi (θ)}) < 0.5. i→∞ θ∈Θ

θ∈Θ

It follows that Ch{Λ} < 0.5 and the part (b) holds by using (a). (c) Assume Ch{Λ} ≥ 0.5 and Λi ↓ Λ. We have Ch{Λc } ≤ 0.5 and Λci ↑ Λc . It follows from (a) that lim Ch{Λi } = 1 − lim Ch{Λci } = 1 − Ch{Λc } = Ch{Λ}.

i→∞

i→∞

(d) Assume limi→∞ Ch{Λi } > 0.5 and Λi ↓ Λ. We have lim Ch{Λci } < i→∞

0.5 and Λci ↑ Λc . It follows from (b) that

lim Ch{Λi } = 1 − lim Ch{Λci } = 1 − Ch{Λc } = Ch{Λ}.

i→∞

i→∞

The theorem is proved. Theorem 3.10 (Chance Asymptotic Theorem) For any events Λ1 , Λ2 , · · · , we have lim Ch{Λi } ≥ 0.5, if Λi ↑ Θ × Ω, (3.16) i→∞

lim Ch{Λi } ≤ 0.5,

i→∞

if Λi ↓ ∅.

(3.17)

Proof: Assume Λi ↑ Θ × Ω. If limi→∞ Ch{Λi } < 0.5, it follows from the chance semicontinuity law that Ch{Θ × Ω} = lim Ch{Λi } < 0.5 i→∞

which is in contradiction with Cr{Θ × Ω} = 1. The first inequality is proved. The second one may be verified similarly.

3.2

Hybrid Variables

Recall that a random variable is a measurable function from a probability space to the set of real numbers, and a fuzzy variable is a function from a credibility space to the set of real numbers. In order to describe a quantity with both fuzziness and randomness, we introduce a concept of hybrid variable as follows.

138

Chapter 3 - Chance Theory ............................................................................... ........... ............... .......... ........ ........ ....... ....... ...... . . . . .... ..... ... . ... .... ... .. ... .. .. .... . . . . . . ....... ..... .. . . ........ . . . ...... ........... . . . . . ... ...... ....... .. .... .... .......... .......................... .. .......... ..... ..................................................................................... ... ... .. .. ... . . ... ... .... ... . .. ......................................................... ......................................................... . . ... ... ... . .. ... . ... ... . . ... ... ........ ... . ... .. . ... . ... ... . . ..... ... ... . . . . .. . . . ... ... . . ..... ... .... ... .. ... . . . . . ... ... . . . . ..... ... .... ... .. ... . . . . ........ . . .... . ..... ..................................................................... ... . ............................................................. . . . . . . ..... ... ... . . . .. . . . . . . . ..... ... ..... ..... ... ... ..... ..... ... ... ..... ..... ... . ... ..... ... .................................................................................................................... .. . ... .. .. . . ... .. .. . . ... .. .. ... . . .. ... .. . . ..... ... . .. . . . . ... ... ........ . . .. . . . . ...... ... ..... . .. . . . . . . . ...... ... ............................................ .............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... ...... ............. ..... ... . ......... . . . . . . . . . . . . . . . . . . . . . . . . ...... ... ........ ...... .... .... .... ... . . . . . . . . ..... . . . . . . ... . . . . . . . ........... ......... .. ..... .. ... ... ..... ... .............. .. ... ...... ...... . . ... . ... . . . . . . .... . . ... ... .. .... . .. . . ... ... . . . .... . . .. ... .. ... ... ... . . ... . ... . . ... . . . . ... ... . . . . . . . ..... . . . ..... . ..... ..... ...... ..... ...... ..... ..... ....... ........ ...... ....... ......... ....... ........... ......... ................ ............................................ ...............................

Set of Real Numbers • • •

Fuzzy Variable

Hybrid Variable

• • Credibility Space

Random Variable

• • Probability Space

Figure 3.1: Graphical Representation of Hybrid Variable Definition 3.4 (Liu [130]) A hybrid variable is a measurable function from a chance space (Θ, P, Cr) × (Ω, A, Pr) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {(θ, ω) ∈ Θ × Ω ξ(θ, ω) ∈ B} (3.18) is an event. Remark 3.2: A hybrid variable degenerates to a fuzzy variable if the value of ξ(θ, ω) does not vary with ω. For example, ξ(θ, ω) = θ,

ξ(θ, ω) = θ2 + 1,

ξ(θ, ω) = sin θ.

Remark 3.3: A hybrid variable degenerates to a random variable if the value of ξ(θ, ω) does not vary with θ. For example, ξ(θ, ω) = ω,

ξ(θ, ω) = ω 2 + 1,

ξ(θ, ω) = sin ω.

Remark 3.4: For each fixed θ ∈ Θ, it is clear that the hybrid variable ξ(θ, ω) is a measurable function from the probability space (Ω, A, Pr) to the set of real numbers. Thus it is a random variable and we will denote it by ξ(θ, ·). Then a hybrid variable ξ(θ, ω) may also be regarded as a function from a credibility space (Θ, P, Cr) to the set {ξ(θ, ·)|θ ∈ Θ} of random variables. Thus ξ is a random fuzzy variable defined by Liu [124]. Remark 3.5: For each fixed ω ∈ Ω, it is clear that the hybrid variable ξ(θ, ω) is a function from the credibility space (Θ, P, Cr) to the set of real

139

Section 3.2 - Hybrid Variables

numbers. Thus it is a fuzzy variable and we will denote it by ξ(·, ω). Then a hybrid variable ξ(θ, ω) may be regarded as a function from a probability space (Ω, A, Pr) to the set {ξ(·, ω)|ω ∈ Ω} of fuzzy variables. If Cr{ξ(·, ω) ∈ B} is a measurable function of ω for any Borel set B of real numbers, then ξ is a fuzzy random variable in the sense of Liu and Liu [140]. Model I If a ˜ is a fuzzy variable and η is a random variable, then the sum ξ = a ˜ + η is a hybrid variable. The product ξ = a ˜ · η is also a hybrid variable. Generally speaking, if f : 0, we have (cξ)sup (α) = sup{r M{cξ ≥ r} ≥ α} = c sup{r/c |

M{ξ ≥ r/c} ≥ α}

= cξsup (α). A similar way may prove (cξ)inf (α) = cξinf (α). In order to prove the part (b), it suffices to prove that (−ξ)sup (α) = −ξinf (α) and (−ξ)inf (α) = −ξsup (α). In fact, we have (−ξ)sup (α) = sup{r M{−ξ ≥ r} ≥ α} = − inf{−r |

M{ξ ≤ −r} ≥ α}

= −ξinf (α). Similarly, we may prove that (−ξ)inf (α) = −ξsup (α). The theorem is proved.

195


Theorem 4.27 Let ξ be an uncertain variable. Then we have (a) if α > 0.5, then ξinf (α) ≥ ξsup (α); (b) if α ≤ 0.5, then ξinf (α) ≤ ξsup (α). ¯ Proof: Part (a): Write ξ(α) = (ξinf (α) + ξsup (α))/2. If ξinf (α) < ξsup (α), then we have ¯ ¯ 1 ≥ M{ξ < ξ(α)} + M{ξ > ξ(α)} ≥ α + α > 1. A contradiction proves ξinf (α) ≥ ξsup (α). Part (b): Assume that ξinf (α) > ξsup (α). It follows from the definition ¯ of ξinf (α) that M{ξ ≤ ξ(α)} < α. Similarly, it follows from the definition of ¯ ξsup (α) that M{ξ ≥ ξ(α)} < α. Thus ¯ ¯ 1 ≤ M{ξ ≤ ξ(α)} + M{ξ ≥ ξ(α)} < α + α ≤ 1.

A contradiction proves ξinf (α) ≤ ξsup (α). The theorem is verified. Theorem 4.28 Let ξ be an uncertain variable. Then ξsup (α) is a decreasing function of α, and ξinf (α) is an increasing function of α. Proof: It follows from the definition immediately.

4.11

Entropy

This section provides a definition of entropy to characterize the uncertainty of uncertain variables resulting from information deficiency. Definition 4.20 (Liu [132]) Suppose that ξ is a discrete uncertain variable taking values in {x1 , x2 , · · · }. Then its entropy is defined by H[ξ] =

∞ X

S(M{ξ = xi })

(4.40)

i=1

where S(t) = −t ln t − (1 − t) ln(1 − t). Example 4.9: Suppose that ξ is a discrete uncertain variable taking values in {x1 , x2 , · · · }. If there exists some index k such that M{ξ = xk } = 1, and 0 otherwise, then its entropy H[ξ] = 0. Example 4.10: Suppose that ξ is a simple uncertain variable taking values in {x1 , x2 , · · · , xn }. If M{ξ = xi } = 0.5 for all i = 1, 2, · · · , n, then its entropy H[ξ] = n ln 2. Theorem 4.29 Suppose that ξ is a discrete uncertain variable taking values in {x1 , x2 , · · · }. Then H[ξ] ≥ 0 (4.41) and equality holds if and only if ξ is essentially a deterministic/crisp number.

196

Chapter 4 - Uncertainty Theory

Proof: The nonnegativity is clear. In addition, H[ξ] = 0 if and only if

M{ξ = xi } = 0 or 1 for each i. That is, there exists one and only one index k such that M{ξ = xk } = 1, i.e., ξ is essentially a deterministic/crisp number.

This theorem states that the entropy of an uncertain variable reaches its minimum 0 when the uncertain variable degenerates to a deterministic/crisp number. In this case, there is no uncertainty. Theorem 4.30 Suppose that ξ is a simple uncertain variable taking values in {x1 , x2 , · · · , xn }. Then H[ξ] ≤ n ln 2 and equality holds if and only if

(4.42)

M{ξ = xi } = 0.5 for all i = 1, 2, · · · , n.

Proof: Since the function S(t) reaches its maximum ln 2 at t = 0.5, we have H[ξ] =

n X

S(M{ξ = xi }) ≤ n ln 2

i=1

and equality holds if and only if

M{ξ = xi } = 0.5 for all i = 1, 2, · · · , n.

This theorem states that the entropy of an uncertain variable reaches its maximum when the uncertain variable is an equipossible one. In this case, there is no preference among all the values that the uncertain variable will take.

4.12

Distance

Definition 4.21 (Liu [132]) The distance between uncertain variables ξ and η is defined as d(ξ, η) = E[|ξ − η|].

(4.43)

Theorem 4.31 Let ξ, η, τ be uncertain variables, and let d(·, ·) be the distance. Then we have (a) (Nonnegativity) d(ξ, η) ≥ 0; (b) (Identification) d(ξ, η) = 0 if and only if ξ = η; (c) (Symmetry) d(ξ, η) = d(η, ξ); (d) (Triangle Inequality) d(ξ, η) ≤ 2d(ξ, τ ) + 2d(η, τ ). Proof: The parts (a), (b) and (c) follow immediately from the definition. Now we prove the part (d). It follows from the countable subadditivity axiom

197


that Z

+∞

M {|ξ − η| ≥ r} dr

+∞

M {|ξ − τ | + |τ − η| ≥ r} dr

+∞

M {{|ξ − τ | ≥ r/2} ∪ {|τ − η| ≥ r/2}} dr

+∞

(M{|ξ − τ | ≥ r/2} + M{|τ − η| ≥ r/2}) dr

+∞

M{|ξ − τ | ≥ r/2}dr +

d(ξ, η) = 0

Z ≤ 0

Z ≤ 0

Z ≤ 0

Z = 0

Z

+∞

0

M{|τ − η| ≥ r/2}dr

= 2E[|ξ − τ |] + 2E[|τ − η|] = 2d(ξ, τ ) + 2d(τ, η).

Example 4.11: Let Γ = {γ1 , γ2 , γ3 }. Define M{∅} = 0, M{Γ} = 1 and M{Λ} = 1/2 for any subset Λ (excluding ∅ and Γ). We set uncertain variables ξ, η and τ as follows, ( ξ(γ) =

1, if γ 6= γ3 0, otherwise,

( η(γ) =

−1, if γ 6= γ1 0, otherwise,

τ (γ) ≡ 0.

It is easy to verify that d(ξ, τ ) = d(τ, η) = 1/2 and d(ξ, η) = 3/2. Thus

d(ξ, η) =

4.13

3 (d(ξ, τ ) + d(τ, η)). 2

Inequalities

Theorem 4.32 (Liu [132]) Let ξ be an uncertain variable, and f a nonnegative function. If f is even and increasing on [0, ∞), then for any given number t > 0, we have (ξ)] . M{|ξ| ≥ t} ≤ E[f f (t)

(4.44)

198


Proof: It is clear that M{|ξ| ≥ f −1 (r)} is a monotone decreasing function of r on [0, ∞). It follows from the nonnegativity of f (ξ) that Z

+∞

M{f (ξ) ≥ r}dr

+∞

M{|ξ| ≥ f −1 (r)}dr

f (t)

M{|ξ| ≥ f −1 (r)}dr

f (t)

dr · M{|ξ| ≥ f −1 (f (t))}

E[f (ξ)] = 0

Z = 0

Z ≥ 0

Z ≥ 0

= f (t) · M{|ξ| ≥ t} which proves the inequality. Theorem 4.33 (Liu [132], Markov Inequality) Let ξ be an uncertain variable. Then for any given numbers t > 0 and p > 0, we have p

] M{|ξ| ≥ t} ≤ E[|ξ| . tp

(4.45)

Proof: It is a special case of Theorem 4.32 when f (x) = |x|p . Theorem 4.34 (Liu [132], Chebyshev Inequality) Let ξ be an uncertain variable whose variance V [ξ] exists. Then for any given number t > 0, we have

M {|ξ − E[ξ]| ≥ t} ≤ Vt[ξ] . 2

(4.46)

Proof: It is a special case of Theorem 4.32 when the uncertain variable ξ is replaced with ξ − E[ξ], and f (x) = x2 . Theorem 4.35 (Liu [132], H¨ older’s Inequality) Let p and q be positive real numbers with 1/p + 1/q = 1, and let ξ and η be independent uncertain variables with E[|ξ|p ] < ∞ and E[|η|q ] < ∞. Then we have p p E[|ξη|] ≤ p E[|ξ|p ] q E[|η|q ]. (4.47) Proof: The inequality holds trivially if at least one of ξ and η is zero a.s. Now we assume√E[|ξ|p ] > 0 and E[|η|q ] > 0. It is easy to prove that the function √ f (x, y) = p x q y is a concave function on D = {(x, y) : x ≥ 0, y ≥ 0}. Thus for any point (x0 , y0 ) with x0 > 0 and y0 > 0, there exist two real numbers a and b such that f (x, y) − f (x0 , y0 ) ≤ a(x − x0 ) + b(y − y0 ),

∀(x, y) ∈ D.

199


Letting x0 = E[|ξ|p ], y0 = E[|η|q ], x = |ξ|p and y = |η|q , we have f (|ξ|p , |η|q ) − f (E[|ξ|p ], E[|η|q ]) ≤ a(|ξ|p − E[|ξ|p ]) + b(|η|q − E[|η|q ]). Taking the expected values on both sides, we obtain E[f (|ξ|p , |η|q )] ≤ f (E[|ξ|p ], E[|η|q ]). Hence the inequality (4.47) holds. Theorem 4.36 (Liu [132], Minkowski Inequality) Let p be a real number with p ≥ 1, and let ξ and η be independent uncertain variables with E[|ξ|p ] < ∞ and E[|η|p ] < ∞. Then we have p p p p E[|ξ + η|p ] ≤ p E[|ξ|p ] + p E[|η|p ]. (4.48) Proof: The inequality holds trivially if at least one of ξ and η is zero a.s. Now we assume √ E[|ξ|p ] > 0 and E[|η|p ] > 0. It is easy to prove that the function √ f (x, y) = ( p x + p y)p is a concave function on D = {(x, y) : x ≥ 0, y ≥ 0}. Thus for any point (x0 , y0 ) with x0 > 0 and y0 > 0, there exist two real numbers a and b such that f (x, y) − f (x0 , y0 ) ≤ a(x − x0 ) + b(y − y0 ),

∀(x, y) ∈ D.

Letting x0 = E[|ξ|p ], y0 = E[|η|p ], x = |ξ|p and y = |η|p , we have f (|ξ|p , |η|p ) − f (E[|ξ|p ], E[|η|p ]) ≤ a(|ξ|p − E[|ξ|p ]) + b(|η|p − E[|η|p ]). Taking the expected values on both sides, we obtain E[f (|ξ|p , |η|p )] ≤ f (E[|ξ|p ], E[|η|p ]). Hence the inequality (4.48) holds. Theorem 4.37 (Liu [132], Jensen’s Inequality) Let ξ be an uncertain variable, and f : < → < a convex function. If E[ξ] and E[f (ξ)] are finite, then f (E[ξ]) ≤ E[f (ξ)].

(4.49)

Especially, when f (x) = |x|p and p ≥ 1, we have |E[ξ]|p ≤ E[|ξ|p ]. Proof: Since f is a convex function, for each y, there exists a number k such that f (x) − f (y) ≥ k · (x − y). Replacing x with ξ and y with E[ξ], we obtain f (ξ) − f (E[ξ]) ≥ k · (ξ − E[ξ]). Taking the expected values on both sides, we have E[f (ξ)] − f (E[ξ]) ≥ k · (E[ξ] − E[ξ]) = 0 which proves the inequality.

200

4.14


Convergence Concepts

We have the following four convergence concepts of uncertain sequence: convergence almost surely (a.s.), convergence in measure, convergence in mean, and convergence in distribution. Table 4.1: Relationship among Convergence Concepts Convergence in Mean

Convergence

⇒

in Measure

⇒

Convergence in Distribution

Definition 4.22 (Liu [132]) Suppose that ξ, ξ1 , ξ2 , · · · are uncertain variables defined on the uncertainty space (Γ, L, M). The sequence {ξi } is said to be convergent a.s. to ξ if there exists an event Λ with M{Λ} = 1 such that lim |ξi (γ) − ξ(γ)| = 0

i→∞

(4.50)

for every γ ∈ Λ. In that case we write ξi → ξ, a.s. Definition 4.23 (Liu [132]) Suppose that ξ, ξ1 , ξ2 , · · · are uncertain variables. We say that the sequence {ξi } converges in measure to ξ if lim

i→∞

M {|ξi − ξ| ≥ ε} = 0

(4.51)

for every ε > 0. Definition 4.24 (Liu [132]) Suppose that ξ, ξ1 , ξ2 , · · · are uncertain variables with finite expected values. We say that the sequence {ξi } converges in mean to ξ if lim E[|ξi − ξ|] = 0. (4.52) i→∞

In addition, the sequence {ξi } is said to converge in mean square to ξ if lim E[|ξi − ξ|2 ] = 0.

i→∞

(4.53)

Definition 4.25 (Liu [132]) Suppose that Φ, Φ1 , Φ2 , · · · are the uncertainty distributions of uncertain variables ξ, ξ1 , ξ2 , · · · , respectively. We say that {ξi } converges in distribution to ξ if Φi → Φ at any continuity point of Φ. Theorem 4.26 (Liu [132]) Suppose that ξ, ξ1 , ξ2 , · · · are uncertain variables. If {ξi } converges in mean to ξ, then {ξi } converges in measure to ξ. Proof: It follows from the Markov inequality that for any given number ε > 0, we have M{|ξi − ξ| ≥ ε} ≤ E[|ξiε− ξ|] → 0 as i → ∞. Thus {ξi } converges in measure to ξ. The theorem is proved.

Section 4.15 - Conditional Uncertainty

201

Theorem 4.27 (Liu [132]) Suppose ξ, ξ1 , ξ2 , · · · are uncertain variables. If {ξi } converges in measure to ξ, then {ξi } converges in distribution to ξ. Proof: Let x be a given continuity point of the uncertainty distribution Φ. On the one hand, for any y > x, we have {ξi ≤ x} = {ξi ≤ x, ξ ≤ y} ∪ {ξi ≤ x, ξ > y} ⊂ {ξ ≤ y} ∪ {|ξi − ξ| ≥ y − x}. It follows from the countable subadditivity axiom that Φi (x) ≤ Φ(y) + M{|ξi − ξ| ≥ y − x}. Since {ξi } converges in measure to ξ, we have M{|ξi − ξ| ≥ y − x} → 0 as i → ∞. Thus we obtain lim supi→∞ Φi (x) ≤ Φ(y) for any y > x. Letting y → x, we get lim sup Φi (x) ≤ Φ(x). (4.54) i→∞

On the other hand, for any z < x, we have {ξ ≤ z} = {ξi ≤ x, ξ ≤ z} ∪ {ξi > x, ξ ≤ z} ⊂ {ξi ≤ x} ∪ {|ξi − ξ| ≥ x − z} which implies that Φ(z) ≤ Φi (x) + M{|ξi − ξ| ≥ x − z}. Since M{|ξi − ξ| ≥ x − z} → 0, we obtain Φ(z) ≤ lim inf i→∞ Φi (x) for any z < x. Letting z → x, we get Φ(x) ≤ lim inf Φi (x). i→∞

(4.55)

It follows from (4.54) and (4.55) that Φi (x) → Φ(x). The theorem is proved.

4.15

Conditional Uncertainty

We consider the uncertain measure of an event A after it has been learned that some other event B has occurred. This new uncertain measure of A is called the conditional uncertain measure of A given B. In order to define a conditional uncertain measure M{A|B}, at first we have to enlarge M{A ∩ B} because M{A ∩ B} < 1 for all events whenever M{B} < 1. It seems that we have no alternative but to divide M{A ∩ B} by M{B}. Unfortunately, M{A∩B}/M{B} is not always an uncertain measure. However, the value M{A|B} should not be greater than M{A ∩ B}/M{B} (otherwise the normality will be lost), i.e., {A ∩ B} M{A|B} ≤ MM . {B}

(4.56)

202


On the other hand, in order to preserve the self-duality, we should have ∩ B} M{A|B} = 1 − M{Ac |B} ≥ 1 − M{A M{B} . c

(4.57)

Furthermore, since (A ∩ B) ∪ (Ac ∩ B) = B, we have M{B} ≤ M{A ∩ B} + M{Ac ∩ B} by using the countable subadditivity axiom. Thus

M{Ac ∩ B} ≤ M{A ∩ B} ≤ 1. (4.58) M{B} M{B} Hence any numbers between 1− M{Ac ∩B}/M{B} and M{A∩B}/M{B} are 0≤1−

reasonable values that the conditional uncertain measure may take. Based on the maximum uncertainty principle, we have the following conditional uncertain measure.

Definition 4.28 (Liu [132]) Let (Γ, L, M) be an uncertainty space, and A, B ∈ L. Then the conditional uncertain measure of A given B is defined by  M{A ∩ B} , if M{A ∩ B} < 0.5     M{B} M{B}   c M{A|B} =  1 − M{A ∩ B} , if M{Ac ∩ B} < 0.5 (4.59)   M {B} M {B}   

provided that

0.5,

otherwise

M{B} > 0.

It follows immediately from the definition of conditional uncertain measure that M{Ac ∩ B} ≤ M{A|B} ≤ M{A ∩ B} . 1− (4.60) M{B} M{B} Furthermore, the conditional uncertain measure obeys the maximum uncertainty principle, and takes values as close to 0.5 as possible. Remark 4.9: Conditional uncertain measure coincides with conditional probability, conditional credibility, and conditional chance. Theorem 4.38 (Liu [132]) Let (Γ, L, M) be an uncertainty space, and B an event with M{B} > 0. Then M{·|B} defined by (4.59) is an uncertain measure, and (Γ, L, M{·|B}) is an uncertainty space. Proof: It is sufficient to prove that M{·|B} satisfies the normality, monotonicity, self-duality and countable subadditivity axioms. At first, it satisfies the normality axiom, i.e., c M{∅} = 1. ∩ B} =1− M{Γ|B} = 1 − M{Γ M{B} M{B}

203

Section 4.15 - Conditional Uncertainty

For any events A1 and A2 with A1 ⊂ A2 , if

M{A1 ∩ B} ≤ M{A2 ∩ B} < 0.5, M{B} M{B} then

M{A2 ∩ B} 1 ∩ B} M{A1 |B} = M{A M{B} ≤ M{B} = M{A2 |B}.

If

M{A1 ∩ B} ≤ 0.5 ≤ M{A2 ∩ B} , M{B} M{B} then M{A1 |B} ≤ 0.5 ≤ M{A2 |B}. If M{A1 ∩ B} ≤ M{A2 ∩ B} , 0.5 < M{B} M{B} then we have M{A1 |B} = 1 −

M{Ac1 ∩ B} ∨ 0.5 ≤ 1 − M{Ac2 ∩ B} ∨ 0.5 = M{A |B}. 2 M{B} M{B} This means that M{·|B} satisfies the monotonicity axiom. For any event A, if M{A ∩ B} ≥ 0.5, M{Ac ∩ B} ≥ 0.5, M{B} M{B} c then we have M{A|B} + M{A |B} = 0.5 + 0.5 = 1 immediately. Otherwise,

without loss of generality, suppose

M{A ∩ B} < 0.5 < M{Ac ∩ B} , M{B} M{B} then we have M {A ∩ B} M {A ∩ B} M{A|B} + M{A |B} = M{B} + 1 − M{B} = 1. That is, M{·|B} satisfies the self-duality axiom. Finally, for any countable sequence {Ai } of events, if M{Ai |B} < 0.5 for all i, it follows from the c

countable subadditivity axiom that (∞ ) ∞ X [ (∞ ) M Ai ∩ B M{Ai ∩ B} X ∞ [ i=1 i=1 M Ai ∩ B ≤ ≤ = M{Ai |B}. M{B} M{B} i=1 i=1 Suppose there is one term greater than 0.5, say

M{A1 |B} ≥ 0.5, M{Ai |B} < 0.5,

i = 2, 3, · · ·

204 If


M{∪i Ai |B} = 0.5, then we immediately have M

(∞ [

) Ai ∩ B

≤

i=1

∞ X

M{Ai |B}.

i=1

If M{∪i Ai |B} > 0.5, we may prove the above inequality by the following facts: ! ∞ ∞ [ \ Ac1 ∩ B ⊂ (Ai ∩ B) ∪ Aci ∩ B , i=2

M

{Ac1

∩ B} ≤

∞ X

i=1

M{Ai ∩ B} + M

(∞ \

i=2

M

(∞ [

∩B

,

i=1

M

) Ai |B

) Aci

=1−

i=1

(∞ \

) Aci

i=1

∩B

M{B}

,

∞ X

M{Ai ∩ B} M {Ac1 ∩ B} i=2 M{Ai |B} ≥ 1 − M{B} + M{B} . i=1

∞ X

If there are at least two terms greater than 0.5, then the countable subadditivity is clearly true. Thus M{·|B} satisfies the countable subadditivity axiom. Hence M{·|B} is an uncertain measure. Furthermore, (Γ, L, M{·|B}) is an uncertainty space. Example 4.12: Let ξ and η be two uncertain variables. Then we have       

M {ξ = x|η = y} =      

provided that

M{ξ = x, η = y} , M{η = y} M{ξ 6= x, η = y} , 1− M{η = y} 0.5,

M{ξ = x, η = y} < 0.5 M{η = y} M{ξ 6= x, η = y} < 0.5 if M{η = y} if

otherwise

M{η = y} > 0.

Definition 4.29 (Liu [132]) The conditional uncertainty distribution Φ: < → [0, 1] of an uncertain variable ξ given B is defined by Φ(x|B) = M {ξ ≤ x|B} provided that

M{B} > 0.

(4.61)

205

Section 4.16 - Uncertain Process

Example 4.13: Let ξ and η be uncertain variables. Then the conditional uncertainty distribution of ξ given η = y is

Φ(x|η = y) =

            

provided that

M{ξ ≤ x, η = y} < 0.5 M{η = y} M{ξ > x, η = y} < 0.5 if M{η = y}

M{ξ ≤ x, η = y} , M{η = y} M{ξ > x, η = y} , 1− M{η = y}

if

0.5,

otherwise

M{η = y} > 0.

Definition 4.30 (Liu [132]) The conditional uncertainty density function φ of an uncertain variable ξ given B is a nonnegative function such that Z x Φ(x|B) = φ(y|B)dy, ∀x ∈ 0, the increments Xs+t −Xs are identically distributed uncertain variables for all s > 0. Uncertain Renewal Process Definition 4.34 (Liu [133]) Let ξ1 , ξ2 , · · · be iid positive uncertain variables. Define S0 = 0 and Sn = ξ1 + ξ2 + · · · + ξn for n ≥ 1. Then the uncertain process Nt = max n Sn ≤ t (4.67) n≥0

is called an uncertain renewal process. If ξ1 , ξ2 , · · · denote the interarrival times of successive events. Then Sn can be regarded as the waiting time until the occurrence of the nth event, and Nt is the number of renewals in (0, t]. Each sample path of Nt is a right-continuous and increasing step function taking only nonnegative integer values. Furthermore, the size of each jump of Nt is always 1. In other words, Nt has at most one renewal at each time. In particular, Nt does not jump at time 0. Since Nt ≥ n is equivalent to Sn ≤ t, we immediately have

M{Nt ≥ n} = M{Sn ≤ t}.

(4.68)

Theorem 4.39 (Liu [133]) Let Nt be an uncertain renewal process. Then we have ∞ X E[Nt ] = M{Sn ≤ t}. (4.69) n=1

Proof: Since Nt takes only nonnegative integer values, we have Z ∞ ∞ Z n X E[Nt ] = M{Nt ≥ r}dr = M{Nt ≥ r}dr 0

=

∞ X n=1


M{Nt ≥ n} =

n=1 ∞ X

n−1

M{Sn ≤ t}.

n=1

Section 4.16 - Uncertain Process

207

Canonical Process Definition 4.35 (Liu [133]) An uncertain process Wt is said to be a canonical process if (i) W0 = 0 and Wt is sample-continuous, (ii) Wt has stationary and independent increments, (iii) W1 is an uncertain variable with expected value 0 and variance 1. Theorem 4.40 (Existence Theorem) There is a canonical process. Proof: In fact, standard Brownian motion and standard C process are instances of canonical process. Example 4.14: Let Bt be a standard Brownian motion, and Ct a standard C process. Then for each number a ∈ [0, 1], we may verify that Wt = aBt + (1 − a)Ct

(4.70)

is a canonical process. Theorem 4.41 Let Wt be a canonical process. Then E[Wt ] = 0 for any t. Proof: Let f (t) = E[Wt ]. Then for any times t1 and t2 , by using the property of stationary and independent increments, we obtain f (t1 + t2 ) = E[Wt1 +t2 ] = E[Wt1 +t2 − Wt2 + Wt2 − W0 ] = E[Wt1 ] + E[Wt2 ] = f (t1 ) + f (t2 ) which implies that there is a constant e such that f (t) = et. The theorem is proved via f (1) = 0. Theorem 4.42 Let Wt be a canonical process. If for any t1 and t2 , we have V [Wt1 +t2 ] = V [Wt1 ] + V [Wt2 ], then V [Wt ] = t for any t. Proof: Let f (t) = V [Wt ]. Then for any times t1 and t2 , by using the property of stationary and independent increments as well as variance condition, we obtain f (t1 + t2 ) = V [Wt1 +t2 ] = V [Wt1 +t2 − Wt2 + Wt2 − W0 ] = V [Wt1 ] + V [Wt2 ] = f (t1 ) + f (t2 ) which implies that there is a constant σ such that f (t) = σ 2 t. It follows from f (1) = 1 that σ 2 = 1 and f (t) = t. Hence the theorem is proved.

208


Theorem 4.43 Let Wt be a canonical process. If for any t1 and t2 , we have p p p V [Wt1 +t2 ] = V [Wt1 ] + V [Wt2 ], then V [Wt ] = t2 for any t. p Proof: Let f (t) = V [Wt ]. Then for any times t1 and t2 , by using the property of stationary and independent increments as well as variance condition, we obtain p p f (t1 + t2 ) = V [Wt1 +t2 ] = V [Wt1 +t2 − Wt2 + Wt2 − W0 ] p p = V [Wt1 ] + V [Wt2 ] = f (t1 ) + f (t2 ) which implies that there is a constant σ such that f (t) = σt. It follows from f (1) = 1 that σ = 1 and f (t) = t. The theorem is verified. Definition 4.36 For any partition of closed interval [0, t] with 0 = t1 < t2 < · · · < tk+1 = t, the mesh is written as ∆ = max |ti+1 − ti |. 1≤i≤k

Let α > 0 be a real number. Then the α-variation of uncertain process Wt is lim

∆→0

k X

|Wti − Wti |α

(4.71)

i=1

provided that the limit exists in mean square and is an uncertain process. Especially, the α-variation is called total variation if α = 1; and squared variation if α = 2. Definition 4.37 (Liu [133]) Let Wt be a canonical process. Then et + σWt is called a derived canonical process, and the uncertain process Gt = exp(et + σWt )

(4.72)

is called a geometric canonical process. An Uncertain Stock Model Assume that stock price follows geometric canonical process. Then we have an uncertain stock model in which the bond price Xt and the stock price Yt are determined by ( Xt = X0 exp(rt) (4.73) Yt = Y0 exp(et + σWt ) where r is the riskless interest rate, e is the stock drift, σ is the stock diffusion, and Wt is a canonical process.

209

Section 4.17 - Uncertain Calculus

4.17

Uncertain Calculus

Let Wt be a canonical process, and dt an infinitesimal time interval. Then dWt = Wt+dt − Wt

(4.74)

is an uncertain process with E[dWt ] = 0 and dt2 ≤ E[dWt2 ] ≤ dt. Definition 4.38 (Liu [133]) Let Xt be an uncertain process and let Wt be a canonical process. For any partition of closed interval [a, b] with a = t1 < t2 < · · · < tk+1 = b, the mesh is written as ∆ = max |ti+1 − ti |. 1≤i≤k

Then the uncertain integral of Xt with respect to Wt is Z

k X

b

Xt dWt = lim

∆→0

a

Xti · (Wti+1 − Wti )

(4.75)

i=1

provided that the limit exists in mean square and is an uncertain variable. Example 4.15: Let Wt be a canonical process. Then for any partition 0 = t1 < t2 < · · · < tk+1 = s, we have s

Z

dWt = lim

∆→0

0

k X

(Wti+1 − Wti ) ≡ Ws − W0 = Ws .

i=1

Example 4.16: Let Wt be a canonical process. Then for any partition 0 = t1 < t2 < · · · < tk+1 = s, we have sWs =

k X

ti+1 Wti+1 − ti Wti

i=1

=

k X

ti (Wti+1 − Wti ) +

i=1 Z s

k X

Wti+1 (ti+1 − ti )

i=1

Z

→

tdWt + 0

s

Wt dt 0

as ∆ → 0. It follows that Z

s

Z tdWt = sWs −

0

s

Wt dt. 0

210


Theorem 4.44 (Liu [133]) Let Wt be a canonical process, and let h(t, w) be a twice continuously differentiable function. Define Xt = h(t, Wt ). Then we have the following chain rule dXt =

∂h ∂h 1 ∂2h (t, Wt )dWt2 . (t, Wt )dt + (t, Wt )dWt + ∂t ∂w 2 ∂w2

(4.76)

Proof: Since the function h is twice continuously differentiable, by using Taylor series expansion, the infinitesimal increment of Xt has a second-order approximation ∆Xt =

∂h 1 ∂2h ∂h (t, Wt )(∆Wt )2 (t, Wt )∆t + (t, Wt )∆Wt + ∂t ∂w 2 ∂w2 +

1 ∂2h ∂2h 2 (t, W )(∆t) + (t, Wt )∆t∆Wt . t 2 ∂t2 ∂t∂w

Since we can ignore the terms (∆t)2 and ∆t∆Wt , the chain rule is proved because it makes Z s Z Z s ∂h 1 s ∂2h ∂h (t, Wt )dt + (t, Wt )dWt + (t, Wt )dWt2 Xs = X0 + 2 0 ∂w2 0 ∂w 0 ∂t for any s ≥ 0. Remark 4.10: The infinitesimal increment dWt in (4.76) may be replaced with the derived canonical process dYt = ut dt + vt dWt

(4.77)

where ut is an absolutely integrable uncertain process, and vt is a square integrable uncertain process, thus producing dh(t, Yt ) =

∂h ∂h 1 ∂2h (t, Yt )dt + (t, Yt )dYt + (t, Yt )vt2 dWt2 . ∂t ∂w 2 ∂w2

(4.78)

Example 4.17: Applying the chain rule, we obtain the following formula d(tWt ) = Wt dt + tdWt . Hence we have s

Z sWs =

Z d(tWt ) =

0

That is, Z

s

Z

0

s

tdWt . 0

Z tdWt = sWs −

0

s

Wt dt +

s

Wt dt. 0

211

Section 4.18 - Uncertain Differential Equation

Theorem 4.45 (Liu [133], Integration by Parts) Suppose that Wt is a canonical process and F (t) is an absolutely continuous function. Then Z s Z s Wt dF (t). (4.79) F (t)dWt = F (s)Ws − 0

0

Proof: By defining h(t, Wt ) = F (t)Wt and using the chain rule, we get d(F (t)Wt ) = Wt dF (t) + F (t)dWt . Thus

Z

s

Z

s

Z 0

0

0

s

F (t)dWt

Wt dF (t) +

d(F (t)Wt ) =

F (s)Ws = which is just (4.79).

4.18

Uncertain Differential Equation

Definition 4.39 (Liu [133]) Suppose Wt is a canonical process, and f and g are some given functions. Then dXt = f (t, Xt )dt + g(t, Xt )dWt

(4.80)

is called an uncertain differential equation. A solution is an uncertain process Xt that satisfies (4.80) identically in t. Remark 4.11: Note that there is no precise definition for the terms dXt , dt and dWt in the uncertain differential equation (4.80). The mathematically meaningful form is the uncertain integral equation Z s Z s Xs = X0 + f (t, Xt )dt + g(t, Xt )dWt . (4.81) 0

0

However, the differential form is more convenient for us. This is the main reason why we accept the differential form. Example 4.18: Let Wt be a canonical process. Then the uncertain differential equation dXt = adt + bdWt has a solution Xt = at + bWt .

Appendix A

Measurable Sets Algebra and σ-algebra are very important and fundamental concepts in measure theory. Definition A.1 Let Ω be a nonempty set. A collection A is called an algebra over Ω if the following conditions hold: (a) Ω ∈ A; (b) if A ∈ A, then Ac ∈ A; (c) if Ai ∈ A for i = 1, 2, · · · , n, then ∪ni=1 Ai ∈ A. If the condition (c) is replaced with closure under countable union, then A is called a σ-algebra over Ω. Example A.1: Assume that Ω is a nonempty set. Then {∅, Ω} is the smallest σ-algebra over Ω, and the power set P (all subsets of Ω) is the largest σ-algebra over Ω. Example A.2: Let A be the set of all finite disjoint unions of all intervals of the form (−∞, a], (a, b], (b, ∞) and ∅. Then A is an algebra over t , ω f1 (ω) ∧ f2 (ω) > t = ω f1 (ω) > t ∩ ω f2 (ω) > t . Theorem C.2 Let {fi } be a sequence of measurable functions from (Ω, A) to

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