Some Properties of Complex Fuzzy Integral

In order to simplify the calculations of complex fuzzy integral (complex Liu integral), this paper 
aims to discuss the properties of this kind of complex fuzzy integral. Firstly, some properties of 
one-dimensional complex fuzzy integral are given. Then the properties above are extended to the 
case of multidimensional complex fuzzy integral.


Introduction
Many problems in the world are hard to be described statically.To deal with dynamic fuzzy phenomena, Liu [1] proposed the concept of fuzzy process with credibility measure.A detailed survey on credibility theory may be found in Liu [2].As for more concepts and results, interested reader may consult Liu [3].Subsequently, Liu process, Liu integral, and Liu formula were presented by Liu [1], which are the bases of fuzzy calculus.Later some researches surrounding Liu process have been done.You et al. [4] studied multidimensional Liu process, Liu formula, and Liu integral.Some properties of Liu integral were discussed by You and Wang [5], then these properties were extended to multidimensional Liu integral by You and Huo [6].Lipschitz continuity of Liu process was proved by Dai [7].Besides, You et al. [8] derived existence and uniqueness theorems for two special fuzzy differential equations.Chen and Qin [9] extended the existence and uniqueness theorem to the general fuzzy differential equations.
As for applications in finance, to describe the variance of the price of a stock in fuzzy market, fuzzy differential equation is an important tool.Based on Liu process, Liu [1] initiated a stock model named Liu's stock model.Considering that the price of a stock is a solution of a fuzzy differential equation, Qin and Li [10] deduced option pricing formula for European option.Other kinds of stock model were presented by Gao [11] and Peng [12].Applying the knowledge of fuzzy differential equations, Qin and Li [13] studied the problems of trading strategies.While in the solving process of fuzzy differential equation, fuzzy integral (Liu integral) is needed.As for applications in control, it is known that classical control system is described by a differential equation, if there exist some fuzzy factors, fuzzy differential equations must be taken into consideration.Based on Liu process, Zhu [14] gave the optimal equation in the field of fuzzy control.Qin et al. [15] applied the fuzzy control to production planning problem.
As we know, there exist complex stochastic processes in many fields, such as signal analysis, thermodynamics, and fluid dynamics.Complex stochastic process is different from the two-dimensional stochastic process since it can reveal the relation of its real and imaginary parts.However, if the uncertainty factor is dominated by fuzzy factors in the above systems, how can we deal with them?In order to answer this question, Yang [16] defined a complex fuzzy variable whose real and imaginary parts are both fuzzy variables.Similarly the complex fuzzy process is introduced.In particular, the properties of complex fuzzy process were given by Qin and Wen [17].
Despite the importance of complex fuzzy process, there are so few literatures on this topic, considering these reasons, in this paper, based on credibility theory, the properties of complex fuzzy integral will be presented.For this purpose, the paper is organized as follows: Some basic definitions and properties of Liu integral and the definitions of complex fuzzy process are recalled in Section 2. Section 3 demonstrates the properties of one-dimensional complex fuzzy integral.The definition of -dimensional complex fuzzy integral is given in Section 4, then some properties are obtained.Finally a brief conclusion is given in Section 5.

Preliminaries
In this section, some basic definitions and properties of credibility theory are recalled.
The triplet (Θ, P, Cr) is called a credibility space in Liu [3].
Definition 1 (Liu [1]).A fuzzy process   is said to be a Liu process if (1)  0 = 0, (2)   has stationary and independent increments, (3) every increment  + −   is a normally distributed fuzzy variable with expected value  and variance  2  2 , whose membership function is The parameters  and  are called drift and diffusion coefficients, respectively.Liu process is said to be standard if  = 0 and  = 1.
Definition 2 (Liu [1]).Let   be a fuzzy process and let   be a standard Liu process.For any partition of closed interval provided that the limit exists almost surely and is a fuzzy variable.
Theorem 3 (You and Wang [5]).Let fuzzy process   be monotonous and bounded with respect to .Definition 7 (Qin and Wen [17]).Let  be an index set and let (Θ, P, Cr) be a credibility space.A complex fuzzy process is a function from  × (Θ, P, Cr) to the set of complex numbers.
Definition 8 (Qin and Wen [17]).If  1 and  2 are independent Liu processes, then   =  1 +  2 is called a complex Liu process.Here  is an imaginary number.
Theorem 9 (Qin and Wen [17]).A process   is a complex fuzzy process if and only if there exist two fuzzy processes  1 and  2 such that   =  1 +  2 .
Definition 10 (Qin and Wen [17]).Let   =  1 +  2 be a complex fuzzy process and let   =  1 +  2 be a standard complex Liu process.Then the complex fuzzy integral of   with respect to   is defined by We will discuss some properties of complex fuzzy integral in next two sections, which will be useful for the calculations of fuzzy integral.

One-Dimensional Complex Fuzzy Integral
The theorem is proved.
The proof is completed.
Remark 15.It follows from Theorems 13 and 14 that for any complex numbers  1 and  2 .Proof.It follows from the definition of complex Liu integral that and by using Theorem 6, we have The theorem holds.
Definition 19.Let C  = ( 1 ,  2 , . . .,   )  be an dimensional standard complex Liu process.Denote by V × , the set of  ×  matrices V  = [V  ], where each entry V  is a Liu integrable complex fuzzy process, V  = V 1 + V 2 ,   =  1 +  2 .Suppose  < .If V  ∈ V × , then the -dimensional complex Liu integral is defined, using matrix notation as the ×1 matrix whose th component is the following sum of complex Liu integral In this case, V  is called a complex

Theorem 11 .
If   =  1 +  2 is a complex fuzzy process,  1 and  2 are two continuous fuzzy processes, and   =  1 + 2 is a complex Liu process, then Liu integral ∫     d  exists.Proof.It follows from the definition of complex Liu integral that Let   be a complex fuzzy process and   =  1 +  2 , where  1 and  2 are monotonous and bounded with respect to .Assume that   =  1 +  2 and  1 ,  2 are both standard Liu processes.Then integral ∫     d  exists.Suppose that   =  1 +  2 is Liu integrable complex fuzzy process and   =  1 +  2 is a complex Liu process.Then 1 d 1 , ∫    1 d 2 , ∫    2 d 1 , and ∫    2 d 2 exist and are finite, then Liu integral ∫     d  exists.Theorem 12. Proof.It follows from Definition 8 that ∫     d  = ∫   ( 1 +  2 ) d ( 1 +  2 ) = ∫    1 d 1 +  ∫