1INTRODUCTION
In order to describe these nondeterminate 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 selfduality and countable subadditivity, as we all know, selfduality and countable subadditivity are of utmost importance. To define a selfdual 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, $\mathcal{L}$ a σ algebra over Γ, and M an uncertain measure. Then the triplet ( Γ, $\mathcal{L}$, M ) is called an uncertainty space.
Definition 2.2 (Liu, 2007) An uncertain variable ξ is a function from an uncertainty space ( Γ, $\mathcal{L}$, M ) to the set of real numbers such that {ξ ∈ B} 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, $\mathcal{L}$, M ) to the set of complex numbers such that {ζ ∈ B} 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
for any complex number z = x + iy, x, y ∈ R .
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) $\underset{x\to \infty}{\text{lim}}\text{\Phi}\left(x+bi\right)\ne 1,\text{\hspace{0.17em}}\underset{y\to \infty}{\text{lim}}\text{\Phi}\left(a+yi\right)\ne 1,\text{\hspace{0.17em}}\forall a,\text{\hspace{0.17em}}b\in R.$

2) $\underset{x\to +\infty \text{\hspace{0.17em}}y\to +\infty}{\text{lim}}\text{\Phi}\left(x+yi\right)\ne 0$
Definition 2.5 (Liu, 2007) Let ξ be an uncertain variable. Then the expected value of ξ is defined by
provided that at least one of the two integrals is finite.
Definition 2.6If the real and imaginary parts of complex uncertain process Z_{t} exist, then the expected value of Z_{t} is defined by
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, $\mathcal{L}$, M ) be an uncertainty space. An uncertain process is a functionX_{t} (γ) from T × (Γ, $\mathcal{L}$, M) to the set of the real numbers such that {X_{t} ∈ B} 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 X_{t} is an uncertain process if and only if it is an uncertain variable at each time t.
Definition 2.8Let (Γ, $\mathcal{L}$, M) be an uncertain space and let T be an index set. A complex uncertain process is a function from T × (Γ, $\mathcal{L}$, M) to the set of complex numbers.
Furthermore, Z_{t} is a complex uncertain process if and only if there exist two uncertain processes X_{t}, Y_{t} such that Z_{t} = X_{t} + iY_{t}.
Definition 2.9 (Liu, 2009) An uncertain process C_{t} is said to be a canonical Liu process if

(i) C_{t} = 0 and almost all sample paths are Lipschitz continuous,

(ii) C_{t} has stationary and independent increments,

(iii) every increment C_{t+s}  C_{t} is a normal uncertain variable with expected value 0 and variance t^{2}.
Definition 2.10IfC_{1t}, C_{2t}are independent Liu processes, thenC_{t} = C_{1t} + iC_{2t}is a complex Liu process, wherei^{2} = −1. Especially, complex Liu process C_{t} is said to be standard ifC_{1t}, C_{2t}are canonical Liu processes.
Definition 2.11(Liu, 2009)Let X_{t} be an uncertain process and let C_{t} be a canonical Liu process. For any partition of closed interval [a, b] witha = t_{1} < t_{2} < … < t_{k+1} = b, the mesh is written as
Then Liu integral of X_{t} with respect to C_{t} is defined as
provided that the limit exists almost surely and is finite. In this case, the uncertain process X_{t} is said to be integrable.
Theorem 2.2 (Liu, 2011) If X_{t} is a samplecontinuous uncertain process on [a, b], C_{t} is a canonical Liu process, then X_{t} is integrable with respect to C_{t} on [a, b].
Theorem 2.3If uncertain processes X_{t}, Y_{t} are all integrable with respect to complex Liu process , C_{t} then the complex process Z_{t} is said to be integrable with respect to , C_{t} and
where${Z}_{t}={X}_{t}+i{Y}_{t},\text{\hspace{0.17em}}{C}_{t}={C}_{1i}+i{C}_{2t}$
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 Z_{t} is said to have a complex uncertainty distribution Φ_{t} (z) if at each fixed time t^{*}, complex uncertain variable${Z}_{{t}^{\ast}}$has complex uncertainty distribution${\text{\Phi}}_{{t}^{\ast}}\left(z\right)$.
Theorem 3.1A function Φ_{t} (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

$i)\text{\hspace{1em}}\underset{x\to \infty}{\text{lim}}{\Phi}_{t}(x+bi)\ne 1,\text{\hspace{0.17em}}\underset{y\to \infty}{\text{lim}}{\Phi}_{t}(a+yi)\ne 1,\text{\hspace{0.17em}}\forall a,\text{\hspace{0.17em}}b\in R.$

$ii)\text{\hspace{1em}}\underset{x\to +\infty \text{\hspace{0.17em}}y\to +\infty}{\text{lim}}{\Phi}_{t}(x+yi)\ne 0.$
Proof: Since Z_{t} is a complex uncertain process if and only if the real part and imaginary part of Z_{t} 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 processesZ_{1t}, Z_{2t}, …, Z_{nt}are said to be independent, if for any Borel sets of complex numbersB_{1}, B_{2}, …, B_{n}, we have
Theorem 3.2Complex uncertain processesZ_{1t}, Z_{2t}, …, Z_{nt}are independent if and only if for any Borel sets of complex numbersB_{1}, B_{2}, …, B_{n}, we have
Proof: By the selfduality of uncertain measure, Z_{1t}, Z_{2t}, …, Z_{nt} are independent if and only if
Theorem 3.3LetZ_{1t} = X_{1t} + iY_{1t}, Z_{2t} = X_{2t} + iY_{2t}be two independent complex uncertain processes, whereX_{1t}, Y_{1t}, X_{2t}, Y_{2t}are uncertain processes. Then

$i)\text{\hspace{1em}}{X}_{1t}\text{\hspace{1em}}is\text{\hspace{0.17em}}independent\text{\hspace{0.17em}}with\text{\hspace{0.17em}}{X}_{2t}\text{\hspace{0.17em}}or\text{\hspace{0.17em}}{Y}_{2t}$

$ii)\text{\hspace{1em}}{X}_{2t}\text{\hspace{1em}}is\text{\hspace{0.17em}}independent\text{\hspace{0.17em}}with\text{\hspace{0.17em}}{X}_{1t}\text{\hspace{0.17em}}or\text{\hspace{0.17em}}{Y}_{1t}$ .

$iii)\text{\hspace{1em}}{Y}_{1t}\text{\hspace{1em}}is\text{\hspace{0.17em}}independent\text{\hspace{0.17em}}with\text{\hspace{0.17em}}{Y}_{2t}$ .
Proof: For B_{1}, B_{2} ⊆ R , we have
Since Z_{1t}, Z_{2t} are independent, we get
Thus X_{1t}, X_{2t} are independent. Similarly, we can prove the other conclusions.
Theorem 3.4LetZ_{1t}, Z_{2t}be independent complex uncertain processes with finite expected values. Then for any complex numbers a, b, we have
where Z_{jt} = X_{jt} + iY_{jt}, j = 1, 2.
Proof: If Z_{1t}, Z_{2t} are independent complex uncertain processes with finite expected values, then
Since X_{1t} is independent with X_{2t}, Y_{1t} is independent with Y_{2t} we have
Definition 3.3Let Z_{t} be a complex uncertain process with finite expected value Z. Then the variance of Z_{t} is defined by
Some useful properties of complex uncertain processes are illustrated as follows.
Theorem 3.5Let Z_{t} be a complex uncertain process, f a function from C to R. Thenf(Z_{t}) is an uncertain process.
Proof: If Z_{t} 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.
Then we have
Theorem 3.6LetZ_{1t}, Z_{2t}, …, Z_{nt}be complex processes, f a function from C^{n} to C. ThenZ_{t} = f (Z_{1t}, Z_{2t}, …, Z_{nt}) is a complex uncertain process.
Proof: If Z_{1t}, Z_{2t}, …, Z_{nt} are complex processes, f is a function from C^{n} to C, then for any Borel set B of complex numbers, f^{−1} (B) is a Borel set of C^{n} , we get
then f (Z_{1t}, Z_{2t}, …, Z_{nt}) is a complex uncertain process.
Theorem 3.7If X_{t} is a monotone bounded uncertain process on [a, b], and C_{t} is a canonical Liu process, then$\int}_{a}^{b}{X}_{t}d{C}_{t$exists.
Proof: Let uncertain process X_{t} be increasing on [a, b] with respect to t. Thus X_{b} > X_{a}. For any ε > 0, let $\delta =\frac{\epsilon}{{X}_{b}{X}_{a}}$. Then partition closed interval [a, b], such that the mesh
Since almost all sample paths of C_{t} are Lipschitz continuous with respect to t, we have $\left{\text{\Delta C}}_{t}\right<\frac{\epsilon}{{X}_{b}{X}_{a}}$. Let M_{i}, m_{i} be the supremum and in fimum of X_{t} on [t_{i}, t_{i + 1}], respectively. Since X_{t} is increasing on [a, b] with respect to t, then ${M}_{i}={X}_{{t}_{i+1}},\text{\hspace{0.17em}}{m}_{i}={X}_{{t}_{i}}$, thus
In a similar proof of Theorem 2.2, the theorem is proved.
Theorem 3.8IfX_{t}, Y_{t}are monotone bounded uncertain processes on [a, b], then$\int}_{a}^{b}{X}_{t}{Y}_{t}d{C}_{t$exists.
Proof: Let uncertain processes X_{t}, Y_{t} be increasing on [a, b] with respect to t. For any ε_{1} > 0, there is $\delta =\frac{{\epsilon}_{1}}{{X}_{b}{X}_{a}}$. Then partition [a, b] into T_{1} parts. Since almost all sample paths of C_{t} are Lipschitz continuous with respect to time t, we have $\left{\text{\Delta C}}_{t}\right<\frac{{\epsilon}_{1}}{{X}_{b}{X}_{a}}$. Let M^{X} be the supremun of X_{t} and m^{Y} be the infimum of Y_{t} on [a, b]. Let ${M}_{i}^{X},\text{\hspace{0.17em}}{m}_{i}^{X}$ be the supremum and infimum of X_{t} and ${M}_{i}^{Y},\text{\hspace{0.17em}}{m}_{i}^{Y}$ be the supremum and infimum of Y_{t} on [t_{i}, t_{i+1}], respectively. Since X_{t} is increasing on [a, b] with respect to t, then ${M}_{i}^{X}={X}_{{t}_{i+1}},\text{\hspace{0.17em}}{m}_{i}^{X}={X}_{{t}_{i}}$, thus
Similarly, for the partition T_{2} of Y_{t}, we have
Let T = T_{1} + T_{2}, and ε_{1}=ε/2m^{Y}, ε_{2}=ε/2M^{X}. We have
where k = k_{1} + k_{2}.
The theorem is verified.
Theorem 3.9If Z_{t} = X_{t} + iY_{t} is a complex uncertain process, where X_{t}, Y_{t} are all monotone bounded uncertain processes,C_{t} = C_{1t} + iC_{2t}is a complex uncertain Liu process, then Liu integral$\int}_{a}^{b}{Z}_{t}d{C}_{t$exists.
Proof: Since
X_{t}, Y_{t} are all monotone bounded uncertain processes. According to Theorem 3.7, $\int}_{a}^{b}{X}_{t}d{C}_{1t}},{\displaystyle {\int}_{a}^{b}{X}_{t}d{C}_{2t}},{\displaystyle {\int}_{a}^{b}{Y}_{t}d{C}_{1t}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}{\displaystyle {\int}_{a}^{b}{Y}_{t}d{C}_{2t$ are finite, thus Liu integral $\int}_{a}^{b}{Z}_{t}d{C}_{t$ exists.
Theorem 3.10IfZ_{1t} = X_{1t} + iY_{1t} and Z_{2}_{t} =X_{2}_{t} + iY_{2}_{t}are Liu integrable complex uncertain processes, whereX_{1t}, Y_{1t}, X_{2t}, Y_{2t}are all monotone bounded uncertain processes,C_{t} = C_{2t} + iC_{2t}is a complex uncertain Liu process, then$\int}_{a}^{b}{Z}_{1t}{Z}_{2t}d{C}_{t$exists.
Proof: It follows Theorem 3.8 that
The above integrals are all exist, then $\int}_{a}^{b}{Z}_{1t}{Z}_{2t}d{C}_{t$ 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.