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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.15 No.2 pp.143-147
DOI : https://doi.org/10.7232/iems.2016.15.2.143

# Some Properties of Complex Uncertain Process

Cuilian You*, Na Xiang
College of Mathematics and Information Science, Hebei University, Baoding, China
Corresponding Author, yycclian@163.com
April 7, 2016 May 25, 2016 June 1, 2016

## ABSTRACT

Uncertainty appears not only in real quantities but also in complex quantities. Complex uncertain process is essentially a sequence of complex uncertain variables indexed by time. In order to describe complex uncertain process, a formal definition of complex uncertain distribution is given in this paper, as well as the concepts of independence and variance. In addition, some properties of complex uncertain integral are presented.

## 초록

Natural Science Foundation of China
11201110
61374184 utstanding Youth Science Fund of the Education Department of Hebei Province
Y2012021

## 1INTRODUCTION

In order to describe these non-determinate phenolmena in nature, probability theory, fuzzy mathematics and rough set theory have been produced. In 1933, Kolmogorov founded the theory of probability axiomatic theorem. The theory of fuzzy set was established by Zadeh (1965). Rough set theory was presented by Pawlak (1982) for the first time in 1982. In many cases of our real life, classical measure is widely used. Classical measure satisfies nonnegativity, but the measure has no countable additivity. In order to satisfy this demand, capacities was proposed by Choquet in 1954 and fuzzy measures introduced by Sugeno (1974). Although both capacity and fuzzy measures put emphasis on continuity, instead of on self-duality and countable subadditivity, as we all know, self-duality and countable subadditivity are of utmost importance. To define a self-dual measure, the concept of credibility measure was put forward by Liu and Liu (2002). In 2007, Liu (2004) founded uncertainty theory, then it was refined by Liu (2011) which based on an uncertain measure satisfying normality, duality, subadditivity, and product axioms. Thereafter, a lot of explorations were undertaken. For example, You (2009) researched the convergence of uncertain sequences in 2009. For studying dynamic uncertain events, the concept of uncertain process was initiated by Liu (2008).

Up to now, uncertainty theory has been applied to uncertain calculus (Liu, 2009), uncertain differential equation (Liu, 2008; Yao and Chen, 2013), uncertain logic (Liu, 2011), uncertain programming (Liu, 2009; Liu and Chen, 2015) and uncertain finance (Liu, 2009; Chen, 2011), etc.

Since the 1900s, complex stochastic variable and complex stochastic process have been widely applied in information science, electronic systems and physical fields. Inspired by this, complex uncertain variable was discussed by Peng (2013). Considering the increasing applications of complex numbers and the uncertainty involved in dynamic systems, the properties of complex uncertain process are explored in this paper, including definition and theorems of complex uncertain distribution, independence, expected value, variance and integrals of complex uncertain process.

There are four sections in this paper. In Section 2, we will recall some concepts in uncertainty theory, then some properties of complex uncertain process are discussed in Section 3. A brief summary is given in the last section.

## 2PRELIMINARY

In this section, some basic knowledge in uncertainty theory are introduced, which will be used in this paper.

To measure uncertain event, uncertain measure was introduced as a set function satisfying normality axiom, duality axiom and subadditivity axiom.

After the introduction of uncertain measure, we recall the definition of uncertain space.

Definition 2.1 (Liu, 2007) Let Γ be a nonempty set, $L$ a σ -algebra over Γ, and M an uncertain measure. Then the triplet ( Γ, $L$, M ) is called an uncertainty space.

Definition 2.2 (Liu, 2007) An uncertain variable ξ is a function from an uncertainty space ( Γ, $L$, M ) to the set of real numbers such that {ξB} is an event for any Borel set B of real numbers.

Definition 2.3 (Peng, 2013) A complex uncertain variable ζ is a function from an uncertainty space ( Γ, $L$, M ) to the set of complex numbers such that {ζB} is an event for any Borel set B of complex numbers.

Definition 2.4 (Peng, 2013) The complex uncertain distribution Φ of a complex uncertain variable ζ is defined by

$Φ ( z ) = M { Re ( ζ ) ≤ x , Im ( ζ ) ≤ y } ,$

for any complex number z = x + iy, x, yR .

Theorem 2.1 (Peng, 2013) A function Φ(c) : C ↦ [0, 1] is a complex uncertainty distribution of complex uncertain variable if and only if it is a monotone increasing function with respect to the real part and imaginary part of c, respectively, and

• 1) $lim x → ∞ Φ ( x + b i ) ≠ 1 , lim y → ∞ Φ ( a + y i ) ≠ 1 , ∀ a , b ∈ R .$

• 2) $lim x → + ∞ y → + ∞ Φ ( x + y i ) ≠ 0$

Definition 2.5 (Liu, 2007) Let ξ be an uncertain variable. Then the expected value of ξ is defined by

$E [ ξ ] = ∫ 0 + ∞ M { ξ ≥ x } d x − ∫ − ∞ 0 M { ξ ≤ x } d x ,$

provided that at least one of the two integrals is finite.

Definition 2.6If the real and imaginary parts of complex uncertain process Zt exist, then the expected value of Zt is defined by

$E [ Z t ] = E [ Re [ Z t ] ] + i E [ Im [ Z t ] ] .$

In order to describe dynamic uncertain systems, uncertain processes are introduced.

Definition 2.7 (Liu, 2008) Let T be an index set and let ( Γ, $L$, M ) be an uncertainty space. An uncertain process is a functionXt (γ) from T × (Γ, $L$, M) to the set of the real numbers such that {XtB} is an event for any Borel set B of real numbers at each time t.

Since uncertain process is used to describe uncertain phenomena changing over time, it is a series of uncertain variables changing over time.

Remark 2.1 The above definition says Xt is an uncertain process if and only if it is an uncertain variable at each time t.

Definition 2.8Let (Γ, $L$, M) be an uncertain space and let T be an index set. A complex uncertain process is a function from T × (Γ, $L$, M) to the set of complex numbers.

Furthermore, Zt is a complex uncertain process if and only if there exist two uncertain processes Xt, Yt such that Zt = Xt + iYt.

Definition 2.9 (Liu, 2009) An uncertain process Ct is said to be a canonical Liu process if

• (i) Ct = 0 and almost all sample paths are Lipschitz continuous,

• (ii) Ct has stationary and independent increments,

• (iii) every increment Ct+s - Ct is a normal uncertain variable with expected value 0 and variance t2.

Definition 2.10IfC1t, C2tare independent Liu processes, thenCt = C1t + iC2tis a complex Liu process, wherei2 = −1. Especially, complex Liu process Ct is said to be standard ifC1t, C2tare canonical Liu processes.

Definition 2.11(Liu, 2009)Let Xt be an uncertain process and let Ct be a canonical Liu process. For any partition of closed interval [a, b] witha = t1 < t2 < … < tk+1 = b, the mesh is written as

$Δ= max 1 ≤ i ≤ k | t i + 1 − t i | .$

Then Liu integral of Xt with respect to Ct is defined as

$∫ a b X t d C t = lim Δ → ∞ ∑ i = 1 k X t i ⋅ ( C t i + 1 − C t i ) , ,$

provided that the limit exists almost surely and is finite. In this case, the uncertain process Xt is said to be integrable.

Theorem 2.2 (Liu, 2011) If Xt is a sample-continuous uncertain process on [a, b], Ct is a canonical Liu process, then Xt is integrable with respect to Ct on [a, b].

Theorem 2.3If uncertain processes Xt, Yt are all integrable with respect to complex Liu process , Ct then the complex process Zt is said to be integrable with respect to , Ct and

$∫ a b Z t d C t = ( ∫ a b X t d C 1 t − ∫ a b Y t d C 2 t ) + i ( ∫ a b X t d C 2 t + ∫ a b Y t d C 1 t ) ,$

where$Z t = X t + i Y t , C t = C 1 i + i C 2 t$

## 3THE PROPERTIES OF COMPLEX UNCERTAIN PROCESS

In this section some concepts and theorems of complex uncertain process will be discussed.

As uncertainty distribution is an important method to describe uncertain variable, the complex uncertainty distribution of a complex uncertain process is of prime importance.

Definition 3.1Complex uncertain process Zt is said to have a complex uncertainty distribution Φt (z) if at each fixed time t*, complex uncertain variable$Z t ∗$has complex uncertainty distribution$Φ t ∗ ( z )$.

Theorem 3.1A function Φt (z) ↦ [0, 1] is a complex uncertainty distribution of complex uncertain process if and only if at each time t, it is monotone increasing with respect to the real part Re(z) and imaginary part Im(z), respectively, and

• $i ) lim x → ∞ Φ t ( x + b i ) ≠ 1 , lim y → ∞ Φ t ( a + y i ) ≠ 1 , ∀ a , b ∈ R .$

• $i i ) lim x → + ∞ y → + ∞ Φ t ( x + y i ) ≠ 0.$

Proof: Since Zt is a complex uncertain process if and only if the real part and imaginary part of Zt are complex uncertain variables at each time t, by Theorem 2.1, the theorem is proved.

Next, we introduce the definition of independence of complex uncertain processes.

Definition 3.2Complex uncertain processesZ1t, Z2t, …, Zntare said to be independent, if for any Borel sets of complex numbersB1, B2, …, Bn, we have

$M { ∩ j = 1 n Z j t ∈ B j } = Λ j = 1 n M { Z j t ∈ B j } .$

Theorem 3.2Complex uncertain processesZ1t, Z2t, …, Zntare independent if and only if for any Borel sets of complex numbersB1, B2, …, Bn, we have

$M { ∪ j = 1 n Z j t ∈ B j } = ∨ j = 1 n M { Z j t ∈ B j } .$

Proof: By the self-duality of uncertain measure, Z1t, Z2t, …, Znt are independent if and only if

$M { ∪ j = 1 n Z j t ∈ B j } = 1 − M { ∩ j = 1 n Z j t ∈ B j c } = 1 − ∧ j = 1 n M { Z j t ∈ B j c } = ∨ j = 1 n M { Z j t ∈ B j } .$

Theorem 3.3LetZ1t = X1t + iY1t, Z2t = X2t + iY2tbe two independent complex uncertain processes, whereX1t, Y1t, X2t, Y2tare uncertain processes. Then

• $i ) X 1 t i s i n d e p e n d e n t w i t h X 2 t o r Y 2 t$

• $i i ) X 2 t i s i n d e p e n d e n t w i t h X 1 t o r Y 1 t$ .

• $i i i ) Y 1 t i s i n d e p e n d e n t w i t h Y 2 t$ .

Proof: For B1, B2R , we have

${ X 1 t ∈ B 1 } ∩ { X 2 t ∈ B 2 } = { X 1 t ∈ B 1 , Y 1 t ∈ R } ∩ { X 2 t ∈ B 2 , Y 2 t ∈ R } = { Z 1 t ∈ { x t + i y t | x t ∈ B 1 } } ∩ { Z 2 t ∈ { x t + i y t | x t ∈ B 2 } } .$

Since Z1t, Z2t are independent, we get

$M { { X 1 t ∈ B 1 } ∩ { X 2 t ∈ B 2 } } = M { Z 1 t ∈ { x t + i y t | x t ∈ B 1 } } ∧ M { Z 2 t ∈ { x t + i y t | x t ∈ B 2 } } = M { X 1 t ∈ B 1 , Y 1 t ∈ R } ∧ M { X 2 t ∈ B 2 , Y 2 t ∈ R } = M { X 1 t ∈ B 1 } ∧ M { X 2 t ∈ B 2 } .$

Thus X1t, X2t are independent. Similarly, we can prove the other conclusions.

Theorem 3.4LetZ1t, Z2tbe independent complex uncertain processes with finite expected values. Then for any complex numbers a, b, we have

$E [ a Z 1 t + b Z 2 t ] = a E [ Z 1 t ] + b E [ Z 2 t ] ,$

where Zjt = Xjt + iYjt, j = 1, 2.

Proof: If Z1t, Z2t are independent complex uncertain processes with finite expected values, then

$Re [ Z 1 t + Z 2 t ] = X 1 t + X 2 t , Im [ Z 1 t + Z 2 t ] = Y 1 t + Y 2 t .$

Since X1t is independent with X2t, Y1t is independent with Y2t we have

$E [ a Z 1 t + b Z 2 t ] = E [ a Re ( Z 1 t ) + b Re ( Z 2 t ) ] + i E [ a Im ( Z 1 t ) + b Im ( Z 2 t ) ] = E [ a X 1 t + b X 2 t ] + i E [ a Y 1 t + b Y 2 t ] = E [ a X 1 t ] + i E [ a Y 1 t ] + E [ b X 2 t ] + i E [ b Y 2 t ] = a E [ X 1 t ] + i a E [ Y 1 t ] + b E [ X 2 t ] + i b E [ Y 2 t ] = a E [ Z 1 t ] + b E [ Z 2 t ] .$

Definition 3.3Let Zt be a complex uncertain process with finite expected value Z. Then the variance of Zt is defined by

$V [ Z t ] = E [ ( Z t − Z ) 2 ] .$

Some useful properties of complex uncertain processes are illustrated as follows.

Theorem 3.5Let Zt be a complex uncertain process, f a function from C to R. Thenf(Zt) is an uncertain process.

Proof: If Zt is a complex process, f is a function from C to R, then f−1 (B) is a Borel set of real numbers, i.e.

${ Z t ∈ f − 1 ( B ) } = { γ ∈ Γ | Z t ∈ f − 1 ( B ) } .$

Then we have

${ f ( Z t ) ∈ B } = { γ ∈ Γ | f ( Z t ) ∈ B } .$

Theorem 3.6LetZ1t, Z2t, …, Zntbe complex processes, f a function from Cn to C. ThenZt = f (Z1t, Z2t, …, Znt) is a complex uncertain process.

Proof: If Z1t, Z2t, …, Znt are complex processes, f is a function from Cn to C, then for any Borel set B of complex numbers, f−1 (B) is a Borel set of Cn , we get

${ ( Z 1 t , Z 2 t , ⋯ , Z n t ) ∈ f − 1 ( B ) } = { f ( Z 1 t , Z 2 t , ⋯ , Z n t ) ∈ B } ,$

then f (Z1t, Z2t, …, Znt) is a complex uncertain process.

Theorem 3.7If Xt is a monotone bounded uncertain process on [a, b], and Ct is a canonical Liu process, then$∫ a b X t d C t$exists.

Proof: Let uncertain process Xt be increasing on [a, b] with respect to t. Thus Xb > Xa. For any ε > 0, let $δ = ε X b − X a$. Then partition closed interval [a, b], such that the mesh

$Δ= max 1 ≤ i ≤ k | t i + 1 − t i | < ε .$

Since almost all sample paths of Ct are Lipschitz continuous with respect to t, we have $| ΔC t | < ε X b − X a$. Let Mi, mi be the supremum and in fimum of Xt on [ti, ti + 1], respectively. Since Xt is increasing on [a, b] with respect to t, then $M i = X t i + 1 , m i = X t i$, thus

$∑ i = 1 k | ΔC t i | ⋅ ( M i − m i ) < ε X b − X a ⋅ ∑ i = 1 k ( X t i + 1 − X t i ) = ε X b − X a ( X b − X a ) = ε .$

In a similar proof of Theorem 2.2, the theorem is proved.

Theorem 3.8IfXt, Ytare monotone bounded uncertain processes on [a, b], then$∫ a b X t Y t d C t$exists.

Proof: Let uncertain processes Xt, Yt be increasing on [a, b] with respect to t. For any ε1 > 0, there is $δ = ε 1 X b − X a$. Then partition [a, b] into T1 parts. Since almost all sample paths of Ct are Lipschitz continuous with respect to time t, we have $| ΔC t | < ε 1 X b − X a$. Let MX be the supremun of Xt and mY be the infimum of Yt on [a, b]. Let $M i X , m i X$ be the supremum and infimum of Xt and $M i Y , m i Y$ be the supremum and infimum of Yt on [ti, ti+1], respectively. Since Xt is increasing on [a, b] with respect to t, then $M i X = X t i + 1 , m i X = X t i$, thus

$∑ i = 1 k 1 | ΔC t i | ⋅ ( M i X − m i X ) < ε 1 X b − X a ⋅ ∑ i = 1 k 1 ( M i X − m i X ) = ε 1 X b − X a ( X b − X a ) = ε 1 .$

Similarly, for the partition T2 of Yt, we have

$∑ i = 1 k 2 | ΔC t i | ⋅ ( M i Y − m i Y ) < ε 2 Y b − Y a ⋅ ∑ i = 1 k 2 ( M i Y − m i Y ) = ε 2 Y b − Y a ( Y b − Y a ) = ε 2 .$

Let T = T1 + T2, and ε1=ε/2mY, ε2=ε/2MX. We have

$∑ i = 1 k | ΔC t i | ⋅ ( M i X M i Y − m i X m i Y ) = ∑ i = 1 k | ΔC t i | ⋅ ( M i X M i Y − M i X M i Y + M i X M i Y − m i X m i Y ) = ∑ i = 1 k | ΔC t i | ⋅ [ M i X ( M i Y − m i Y ) + ( M i X − m i X ) m i Y ] ≤ M X ∑ i = 1 k | ΔC t i | ⋅ ( M i Y − m i Y ) + m Y ∑ i = 1 k | ΔC t i | ⋅ ( M i X − m i X ) < M X ε 2 + m Y ε 1 = ε ,$

where k = k1 + k2.

The theorem is verified.

Theorem 3.9If Zt = Xt + iYt is a complex uncertain process, where Xt, Yt are all monotone bounded uncertain processes,Ct = C1t + iC2tis a complex uncertain Liu process, then Liu integral$∫ a b Z t d C t$exists.

Proof: Since

$∫ a b Z t d C t = ∫ a b X t d C 1 t + i ∫ a b X t d C 2 t + i ∫ a b Y t d C 1 t − ∫ a b Y t d C 2 t ,$

Xt, Yt are all monotone bounded uncertain processes. According to Theorem 3.7, $∫ a b X t d C 1 t , ∫ a b X t d C 2 t , ∫ a b Y t d C 1 t a n d ∫ a b Y t d C 2 t$ are finite, thus Liu integral $∫ a b Z t d C t$ exists.

Theorem 3.10IfZ1t = X1t + iY1t and Z2t =X2t + iY2tare Liu integrable complex uncertain processes, whereX1t, Y1t, X2t, Y2tare all monotone bounded uncertain processes,Ct = C2t + iC2tis a complex uncertain Liu process, then$∫ a b Z 1 t Z 2 t d C t$exists.

Proof: It follows Theorem 3.8 that

$∫ a b Z 1 t Z 2 t d C t = ∫ a b ( ( X 1 t X 2 t − Y 1 t Y 2 t ) + i ( X 1 t Y 2 t + Y 1 t X 2 t ) ) d C t = ∫ a b ( X 1 t X 2 t − Y 1 t Y 2 t ) d C 1 t − ∫ a b ( X 1 t Y 2 t + Y 1 t X 2 t ) d C 2 t + i ∫ a b ( X 1 t X 2 t − Y 1 t Y 2 t ) d C 2 t + i ∫ a b ( X 1 t Y 2 t + Y 1 t X 2 t ) d C 1 t = ∫ a b X 1 t X 2 t d C 1 t − ∫ a b Y 1 t Y 2 t d C 1 t − ∫ a b X 1 t Y 2 t d C 2 t − ∫ a b Y 1 t X 2 t d C 2 t + i ∫ a b X 1 t X 2 t d C 2 t − i ∫ a b Y 1 t Y 2 t d C 2 t + i ∫ a b X 1 t Y 2 t d C 1 t + ∫ a b Y 1 t X 2 t d C 1 t .$

The above integrals are all exist, then $∫ a b Z 1 t Z 2 t d C t$ exists.

## 4CONCLUSIONS

In this paper, distribution, independence and variance of complex uncertain process were defined which broadened the scope of uncertain process. Based on the results of uncertain theory, sufficient and necessary condition of complex uncertain distribution, some theorems about independence, expected value and integrals of complex uncertain process were deduced. These theorems provided theoretical basis for complex uncertain process and promoted the developments of uncertainty theory in complex field.

## ACKNOWLEDGMENTS

This work was supported by Natural Science Foundation of China Grant No. 11201110, 61374184, and Outstanding Youth Science Fund of the Education Department of Hebei Province No. Y2012021.

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