Complex uncertain differential equations with application to time integral

Abstract

In this paper, we propose complex uncertain differential equations (CUDEs) based on uncertainty theory. In order to describe the evolution of complex uncertain phenomenon related to belief degrees, we apply the complex Liu process to CUDEs. Firstly, we pose a concept of a linear CUDE and prove that homogeneous linear CUDE and general linear CUDE have solutions. Then, we prove existence and uniqueness theorem of a special CUDE. Further, we design a numerical algorithm to obtain inverse uncertainty distribution of the solution. Finally, as an application, we analyse the inverse uncertainty distributions of time integral of CUDEs and design numerical algorithms to obtain inverse uncertainty distributions of time integral.

Keywords

Complex uncertain differential equations existence and uniqueness theorem time integral

1 Introduction

Differential equation [1, 2] is a classic mathematical branch and plays a very important role in formulation and analysis in modern nature and social sciences. The existence of randomness and fuzziness in dynamic systems brought stochastic differential equations and fuzzy differential equations which have been widely applied in various sciences, such as physical [3], mathematics [4, 5], control model [6], epidemic model [7], economics [8] and finance [9] and other fields. However, the complexity of the world makes the events we face uncertain in various forms. In the real life, some information and knowledge, which usually are represented by human language like about 100km, approximately 39°C, roughly 80kg, low speed, middle age, and big size, behave neither like randomness nor like fuzziness. In order to deal with this type of uncertainty, we need a new theory called uncertainty theory to replace randomness and fuzziness.

Uncertainty theory goes back to a series of papers [10, 11] by the mathematician B. Liu in the past decade. Uncertainty theory as a perfect axiomatic system provides us an alternative for modeling human uncertainty without the scope of probability theory. Probability is interpreted as a frequency that requires enough historical data for probabilistic reasoning, while uncertainty is interpreted as personal belief degree that is from domain experts for lack of samples. Until now, a number of authors have considered the effects of uncertainty theory in various fields (for example, Liu [12]; Gao et al. [13]; Wang et al. [14]; Li et al. [15]).

In 2008, in order to describe the evolution of uncertain phenomenon related to belief degrees, Liu [16] first proposed uncertain differential equation for describing dynamic uncertain systems. Later, in 2009, Liu [17] claimed Liu process is an uncertain process with stationary and independent normal uncertain increments. Chen and Liu [18] considered the analytic solution of a linear uncertain differential equation in 2010. After that, Gao [19] proved the existence and uniqueness theorem of uncertain differential equation. Furthermore, the special nonlinear uncertain differential equations were solved in 2013(for example, Liu et al. [20]; Yao et al. [21]; Liu et al. [22]; Wang et al. [23]). More importantly, Yao and Chen [24, 25] developed a numerical method to obtain the inverse uncertainty distributions of solution of an uncertain differential equation. Based on above theoretical development, uncertain differential equation has been successfully applied in many fields (for example, Yang et al. [26, 27]; Gao et al. [28 –30]; Jia et al. [31 –33]). Of course, there are still many literatures about the fields of uncertain differential equations, which will not be listed here.

In the field of systems analysis, signal analysis, anomalous integral, finance analysis, quantum mechanics, relativity, fluid mechanics and so on, some quantities needed to be characterized by complex numbers, such as root locus method of systems analysis, periodic signals of signal analysis, metric equation of relativity, and two-dimensional potential flow of fluid mechanics. When we face with the lack of measuring methods, experimental conditions etc., we tend to determine the quantities by subjective methods. Complex random variable and fuzzy complex number have been employed to model these quantities. However, a lot of literatures (such as Liu [10, 11]) show that probability theory and fuzzy theory are not enough for all the problems. Under the framework of uncertainty theory, Peng [34] proposed the concept of complex uncertain variable and studied some properties including independence, distribution, expected value and variance. In order to describe complex uncertain process, the concept of complex uncertain distribution and some properties of complex uncertain integral were presented by You et al. [35]. After that, Chen et al. [36] investigated the convergence of complex uncertain sequences including convergence almost surely, in measure,in mean, convergence in distribution and uniformly almost surely and relationships among them. Chen et al. [37] further proposed the concepts of pseudo-variance of a complex uncertain variable and calculated the variance and the pseudo-variance by inverse uncertainty distributions of the real and imaginary parts of complex uncertain variable. In the same year, Gao et al. [38] extended complex uncertain variable to complex uncertain random variables. However, to our best knowledge, the research on complex uncertain differential equation has not been carried out so far.

The paper is organized as follows: In Section 2, we recall some basic definitions and theorems of uncertainty theory, the definitions of complex uncertain process, the expected value and so on. Section 3 demonstrates the existence and uniqueness theorem of complex uncertain differential equation. Section 4 give inverse uncertainty distribution of time integral of complex uncertain differential equation and two numerical examples. Finally a brief conclusion is given in Section 5.

2 Preliminaries

In this section, we will recall some fundamental concepts and properties concerning uncertain variables, uncertain expected value, uncertain processes, complex uncertain expected value, complex uncertain processes and so on.

Definition 2.1. [11] An uncertain variable is a measurable function ξ from an uncertainty space $(Γ, L, M)$ to the set of real numbers, i.e., for any Borel set B of real numbers, the set ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ is an event.

Definition 2.2. [11] Let ξ be an uncertain variable. Then the expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} M {ξ \geq x} dx - \int_{- \infty}^{0} M {ξ \leq x} dx$ provided that at least one of the two integrals is finite.

An uncertain process is essentially a sequence of uncertain variables indexed by time.

Definition 2.3. [11] Let T be an index set and let $(Γ, L, M)$ be an uncertainty space. An uncertain process is a measurable function from $T \times (Γ, L, M)$ to the set of real numbers such that {Z_t ∈ B} is an event for any Borel set B for each time t.

Definition 2.4. [11] An uncertain process C_t with respect to time t is said to be a Liu process if

(i) C₀ = 0 and almost all sample paths are Lipschitz continuous,

(ii) C_t has stationary and independent increments,

(iii) every increment C_r+t - C_r is a normal uncertain variable with expected value 0 and variance t², whose uncertainty distribution is $Φ (x) = {(1 + exp (\frac{- π x}{\sqrt{3} t}))}^{- 1}, x \in R .$

Definition 2.5. [11] Let Z_t be an uncertain process and C_t be a Liu process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_n+1 = b, the mesh is written as $Δ = max_{1 \leq i \leq n} | t_{i + 1} - t_{i} | .$ Then uncertain integral of Z_t with respect to C_k is $\int_{a}^{b} Z_{t} d C_{t} = lim_{Δ \to 0} \sum_{i = 1}^{n} Z_{t_{i}} \cdot (C_{t_{i + 1}} - C_{t_{i}})$ provided that the limit exists almost surely and is finite. In this case, the uncertain process Z_t is said to be integrable.

Definition 2.6. [11] Let Z_t be an uncertain process and C_t be a Liu process. If there exist two processes μ_t and σ_t such that $Z_{t} = Z_{0} + \int_{0}^{t} μ_{r} dr + \int_{0}^{t} σ_{r} d C_{r}, \forall t \geq 0 .$ Then Z_t is called a uncertain process with drift μ_t and diffusion σ_t. Furthermore, Z_t has an uncertain differential $d Z_{t} = μ_{t} dt + σ_{t} d C_{t} .$

Definition 2.7. [11] Suppose that C_t is a Liu process, f and g are continuous functions. Given an initial value Z₀, the uncertain differential equation $d Z_{t} = f (t, Z_{t}) dt + g (k, Z_{t}) d C_{t}$ is called an uncertain differential equation with an initial value Z₀.

Theorem 2.1. [16] Let Z_t and $Z_{t}^{α}$ be the solution and α-path of the uncertain differential equation $d Z_{t} = f (t, Z_{t}) dt + g (t, Z_{t}) d C_{t}$ Then $M {Z_{t} \leq Z_{t}^{α}, \forall t} = α, M {Z_{t} > Z_{t}^{α}, \forall t} = 1 - α .$

Lemma 2.1. [18] Suppose that C_t is a Liu process, and Z_t is an integrable uncertain process on [a, b] with respect to t. Then the inequality $| \int_{a}^{b} Z_{t} (γ) d C_{t} | \leq K (γ) \int_{a}^{b} | Z_{t} (γ) | dt$ holds, where K (γ) is the Lipschitz constant of the sample path Z_t (γ).

Definition 2.8. [34] An complex uncertain variable is a function from an uncertainty space $(Γ, L, M)$ to the set of complex numbers.

Definition 2.9. [35] Let T be an index set and let $(Γ, L, M)$ be an uncertainty space. A complex uncertain process is a function from $(Γ, L, M)$ to the set of complex numbers for each time t.

Theorem 2.2. [35] A uncertain process Z_t is complex if and only if there exist two uncertain processes Z_1k and Z_2k such that $\begin{matrix} Z_{t} = Z_{1 t} + {iZ}_{2 t} . \end{matrix}$

Definition 2.10. [35] Let C_1k and C_2t be two independent Liu processes. Then the complex uncertain process C_t = C_1t + iC_2t is called a complex Liu process. In particular, a complex Liu process C_t = C_1t + iC_2t is said to be standard if C_1t and C_2t are both standard Liu processes.

Definition 2.11. [35] Let Z_t = Z_1t + iZ_2t be a complex uncertain process and let $\begin{matrix} C_{t} = C_{1 t} + {iC}_{2 t} \end{matrix}$

be a standard complex Liu process. Then the complex uncertain integral of Z_t with respect to C_t is defined by $\begin{matrix} \int_{a}^{b} Z_{t} {dC}_{t} = \int_{a}^{b} (Z_{1 t} + {iZ}_{2 t}) d (C_{1 t} + {iC}_{2 t}) \end{matrix}$

Theorem 2.3. [35] If Z_t = Z_1t + iZ_2t is a complex uncertain process, Z_1t and Z_2t are two continuous uncertain processes, and C_t = C_1t + iC_2t is a complex Liu process, then $\int_{a}^{b} Z_{t} {dC}_{t}$ exists and $\begin{matrix} \int_{a}^{b} Z_{t} {dC}_{t} = \int_{a}^{b} (Z_{1 t} {dC}_{1 t} - Z_{2 t} {dC}_{2 t}) \\ + i \int_{a}^{b} (Z_{1 t} {dC}_{2 t} + Z_{2 t} {dC}_{1 t}) \end{matrix}$

Definition 2.12. [35] The complex uncertain distribution ϒ of a complex uncertain variable ξ is defined by $\begin{matrix} ϒ (z) = M {Re (ξ) \leq z_{1}, Im (ξ) \leq z_{2}} \end{matrix}$ for any complex number z = z₁ + iz₂, $z_{1}, z_{1} \in R .$ Definition 2.13. [35] Complex uncertain process Z_t is said to have a complex uncertainty distribution ϒ_t (z) if at each fixed time $\hat{t}$ , complex uncertain variable $Z_{\hat{t}}$ has complex uncertainty distribution $ϒ_{\hat{t}} (z)$ , the inverse function $ϒ_{t}^{- 1} (α)$ is called the inverse uncertainty distribution of Z_t.

Theorem 2.4. [35] A function ϒ_t (z) : T × C ↦ [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 $\begin{matrix} lim_{m \to - \infty} ϒ_{t} (m + β i) \neq 1, lim_{n \to - \infty} ϒ_{t} (α + ni) \neq 1, \\ \forall α, β \in R . \end{matrix}$ and $\begin{matrix} lim_{m \to - \infty, n \to + \infty} ϒ_{t} (m + ni) \neq 0 . \end{matrix}$

Definition 2.14. [37] If the real and imaginary parts of complex uncertain process Z_t exist, then the expected value of Z_t is defined by $\begin{matrix} E [Z_{t}] = E [Re [Z_{t}]] + iE [Im [Z_{t}]] \end{matrix}$

Theorem 2.5. [37] Let Z_t be a complex uncertain process. Assume Re [Z_t] and Im [Z_t] are independent uncertain variables with regular uncertainty distributions Φ and Ψ, respectively. Then $\begin{matrix} E [Z_{t}] = \int_{0}^{1} [Φ^{- 1} (α) + i Ψ^{- 1} (α)] d α \end{matrix}$

3 Some classes of CUDEs

3.1 Linear CUDEs

Definition 3.1. Let Z_t = Z_1t + iZ_2t be a complex uncertain process, let C_t = C_1t + iC_2t be a complex Liu process. Then $\begin{matrix} {dZ}_{t} = & (δ_{t} + ζ_{t} Z_{t}) dt + (σ_{t} + η_{t} Z_{t}) {dC}_{t} \\ = & (δ_{t} + ζ_{t} Z_{t}) dt + (σ_{t} + η_{t} Z_{t}) {dC}_{1 t} \\ + i (σ_{t} + η_{t} Z_{t}) {dC}_{2 t} \end{matrix}$ is called a linear CUDE, where δ_t, ζ_t, σ_t, η_t are given complex uncertain processes. If δ_t = 0, σ_t = 0, we call it homogeneous.

Theorem 3.1. Suppose C_t = C_1t + iC_2t is a complex Liu process. If ζ_t, η_t are complex uncertain processes, then homogeneous linear CUDE

${dZ}_{t} = ζ_{t} Z_{t} dt + η_{t} Z_{t} {dC}_{1 t} + i η_{t} Z_{t} {dC}_{2 t}$ (1) has a solution $\begin{matrix} Z_{t} = & Z_{0} exp (\int_{0}^{t} ζ_{r} dr + \int_{0}^{t} η_{r} {dC}_{1 r} \\ + \int_{0}^{t} i η_{r} {dC}_{2 r}) . \end{matrix}$

Proof. By Theorem 4 of [17], it holds that $\begin{matrix} d ln Z_{t} = \frac{1}{Z_{t}} {dZ}_{t} = ζ_{t} dt + η_{t} {dC}_{1 t} + i η_{t} {dC}_{2 t} . \end{matrix}$ Taking integration on both sides of the above equation, we can get $\begin{matrix} ln Z_{t} - ln Z_{0} = & \int_{0}^{t} ζ_{r} dr + \int_{0}^{t} η_{r} {dC}_{1 r} \\ + \int_{0}^{t} i η_{r} {dC}_{2 r} . \end{matrix}$ So, the solution of (1) is $\begin{matrix} Z_{t} = & Z_{0} exp (\int_{0}^{t} ζ_{r} dr \\ + \int_{0}^{t} η_{r} {dC}_{1 r} + \int_{0}^{t} i η_{r} {dC}_{2 r}) . \end{matrix}$ □

Corollary 3.1. Linear CUDE

$\begin{matrix} {dZ}_{t} = & (δ_{t} + ζ_{t} Z_{t}) dk + (σ_{t} + η_{t} Z_{t}) {dC}_{1 t} \\ + i (σ_{t} + η_{t} Z_{t}) {dC}_{2 t} \end{matrix}$ (2) has a solution $\begin{matrix} Z_{t} = & X_{t} (Z_{0} + \int_{0}^{t} \frac{δ_{r}}{U_{r}} ds \\ + \int_{0}^{t} \frac{γ_{r}}{U_{r}} {dC}_{1 r} + \int_{0}^{t} i \frac{γ_{r}}{U_{r}} {dC}_{2 r}), \end{matrix}$ $\begin{matrix} X_{t} = & exp (\int_{0}^{t} ζ_{r} dr \\ + \int_{0}^{t} η_{r} {dC}_{1 r} + \int_{0}^{t} i η_{r} {dC}_{2 r}), X_{0} = 1 . \end{matrix}$

Proof. Let $\begin{matrix} Z_{t} = X_{t} Y_{t}, \end{matrix}$ where $\begin{matrix} {dX}_{t} = ζ_{t} X_{t} d_{t} + η_{t} X_{t} {dC}_{1 t} + i η_{t} X_{t} {dC}_{2 t}, \end{matrix}$ $\begin{matrix} {dY}_{t} = \frac{δ_{t}}{X_{t}} d_{t} + \frac{σ_{t}}{X_{t}} {dC}_{1 t} + i \frac{σ_{t}}{X_{t}} {dC}_{2 t} . \end{matrix}$

Set X₀ = 1 and Y₀ = Z₀ . According to Theorem 4 of [17], taking the differentials on both sides of Z_t = X_tY_t, it holds that $\begin{matrix} {dZ}_{t} = & X_{t} {dY}_{t} + Y_{t} {dX}_{t} \\ = & δ_{t} dt + σ_{t} {dC}_{1 t} + i σ_{t} {dC}_{2 t} + ζ_{t} X_{t} Y_{t} dt \\ + η_{t} X_{t} Y_{t} {dC}_{1 t} + i η_{t} X_{t} Y_{t} {dC}_{2 t} \\ = & δ_{t} dt + σ_{t} {dC}_{1 t} + i σ_{t} {dC}_{2 t} + ζ_{t} Z_{t} dt \\ + η_{t} Z_{t} {dC}_{1 t} + i η_{t} Z_{t} {dC}_{2 t} \\ = & (δ_{t} + ζ_{t} Z_{t}) dt + (σ_{t} \\ + η_{t} Z_{t}) {dC}_{1 t} + i (σ_{t} + η_{t} Z_{t}) {dC}_{2 t} . \end{matrix}$

It follows from the solution of (1) in Theorem 3.1 that $\begin{matrix} X_{t} = exp (\int_{0}^{t} ζ_{r} dr + \int_{0}^{t} η_{r} {dC}_{1 r} + \int_{0}^{t} i η_{r} {dC}_{2 r}), \end{matrix}$ which means $\begin{matrix} Z_{t} = & X_{t} (Z_{0} + \int_{0}^{t} \frac{δ_{r}}{X_{r}} dr + \int_{0}^{t} \frac{γ_{r}}{X_{r}} {dC}_{1 r} \\ + \int_{0}^{t} i \frac{γ_{r}}{X_{r}} {dC}_{2 r}) . \end{matrix}$ □

Example 3.1. Let C_t be a standard complex Liu process. And complex uncertain process Z_t is given as follows

${dZ}_{t} = \frac{1}{1 - t} Z_{t} dt + {tZ}_{t} {dC}_{1 t} + {itZ}_{t} {dC}_{2 t} .$ (3)

By Theorem 3.1, we get $\begin{matrix} Z_{t} = & Z_{0} exp (\int_{0}^{t} \frac{1}{1 - r} dr + \int_{0}^{t} {rdC}_{1 r} \\ + \int_{0}^{t} {irdC}_{2 r}) . \end{matrix}$ that is $\begin{matrix} Z_{t} = Z_{0} exp (ln \frac{1}{1 - t} + \int_{0}^{t} {rdC}_{1 r} + \int_{0}^{t} {irdC}_{2 r}) . \end{matrix}$

Example 3.2. Let η, a, b be real number. For the following nonlinear CUDE,

${dZ}_{t} = (η - {aZ}_{t}) dt + {bdC}_{1 t} + {ibdC}_{2 t}$ (4)

At first,we have $\begin{matrix} X_{t} = & exp (\int_{0}^{t} - adr + \int_{0}^{t} 0 {dC}_{1 r} + i \int_{0}^{t} 0 {dC}_{2 r}) \\ = & exp (- at) . \end{matrix}$

It follows from Corollary 3.1 that the solution is $\begin{matrix} Z_{t} = & X_{t} (Z_{0} + \int_{0}^{t} η exp (ar) dr \\ + \int_{0}^{t} b exp (ar) {dC}_{1 r} + i \int_{0}^{t} b exp (ar) {dC}_{2 r}) \end{matrix}$

That is, $\begin{matrix} Z_{t} = & \frac{η}{a} + exp (- at) (Z_{0} - \frac{η}{a}) \\ + b exp (- at) \int_{0}^{t} exp (ar) {dC}_{1 r} \\ + ib exp (- at) \int_{0}^{t} exp (ar) {dC}_{2 r} \end{matrix}$ provided that a ≠ 0. Note that Z_t is a normal complex uncertain variable,i.e., $\begin{matrix} Re (Z_{t}) \sim & N (\frac{η}{a} + exp (- at) (Z_{0} - \frac{η}{a}), \\ \frac{b}{a} - exp (- at) \frac{b}{a}) \end{matrix}$ and $\begin{matrix} Im (Z_{t}) \sim N (0, \frac{b}{a} - exp (- at) \frac{b}{a}) \end{matrix}$

3.2 Existence and uniqueness theorem of CUDEs

Lemma 3.1. Suppose that C_t = C_1t + iC_2t is a complex Liu process, and Z_t is an integrable uncertain process on [a, b] with respect to t. Then the inequality $| \int_{a}^{b} Z_{t} (γ) d C_{t} | \leq 2 K (γ) \int_{a}^{b} | Z_{t} (γ) | dt$ holds, where K (γ) is the larger Lipschitz constant of C_1t and C_2t.

Proof. Let a = t₁ < t₂ < ⋯ < t_n+1 = b, $Δ = max_{1 \leq i \leq n} | t_{i + 1} - t_{i} | .$ Then uncertain integral of Z_t with respect to C_t is $| \int_{a}^{b} Z_{t} (γ) d C_{t} | = | \int_{a}^{b} Z_{t} (γ) d (C_{1 t} + {iC}_{2 t}) |$ . By the definition of uncertain integral and complex uncertain integral, we get

$\begin{matrix} | \int_{a}^{b} Z_{t} (γ) d C_{t} | \\ = & | \int_{a}^{b} Z_{t} (γ) d (C_{1 t} + {iC}_{2 t}) | \\ \leq & | \int_{a}^{b} Z_{t} (γ) d C_{1 t} | + | \int_{a}^{b} Z_{t} (γ) d C_{2 t} | \\ = & | lim_{Δ \to 0} \sum_{i = 1}^{n} Z_{t_{i}} \cdot (C_{{1 t}_{i + 1}} - C_{{1 t}_{i}}) | \\ + | lim_{Δ \to 0} \sum_{i = 1}^{n} Z_{t_{i}} \cdot (C_{{2 t}_{i + 1}} - C_{{2 t}_{i}}) | . \end{matrix}$ (5)

For each sample γ, it follows from the definition of Liu process that C_t (γ) is Lipschitz continuous with respect to time t . Then there exists a larger Lipschitz constant K (γ) of C_1t and C_2t such that $\begin{matrix} | C_{t_{1}} (γ) - C_{t_{2}} (γ) | \leq K (γ) | t_{1} - t_{2} | . \end{matrix}$

Thus

$\begin{matrix} | lim_{Δ \to 0} \sum_{i = 1}^{n} Z_{t_{i}} \cdot (C_{{1 t}_{i + 1}} - C_{{1 t}_{i}}) | \\ + | lim_{Δ \to 0} \sum_{i = 1}^{n} Z_{t_{i}} \cdot (C_{{2 t}_{i + 1}} - C_{{2 t}_{i}}) | \\ \leq 2 K (γ) lim_{Δ \to 0} \sum_{i = 1}^{n} | Z_{t_{i}} | \cdot | t_{i + 1} - t_{i} | \\ \leq 2 K (γ) \int_{a}^{b} | Z_{t} | dt . \end{matrix}$ (6) □

Theorem 3.2. The following CUDE

${dZ}_{t} = f (Z_{t}, t) dt + g (Z_{t}, t) {dC}_{t}$ (7) has a unique solution and the solution is sample-continuous, where C_t = C_1t + iC_2t and the coefficients f (z, t) and g (z, t) satisfy the following Lipschitz condition and linear growth condition for some constants L, T, $\begin{matrix} | f (z, t) - f (\hat{z}, t) | + | g (z, t) - g (\hat{z}, t) | \leq L | z - \hat{z} |, \\ \forall z, \hat{z} \in R, 0 \leq t \leq T, \end{matrix}$ and $\begin{matrix} | f (z, t) | + | g (z, t) | \leq L (1 + | z |), \forall z \in R, \\ 0 \leq t \leq T . \end{matrix}$

Proof. Step 1. (existence), we consider approximation method to construct a solution of the CUDE (7). Define $Z_{t}^{(0)} = Z_{0},$ and $\begin{matrix} Z_{t}^{(n + 1)} = Z_{0} + \int_{0}^{t} f (Z_{r}^{n}, r) dr + \int_{0}^{t} g (Z_{r}^{n}, r) {dC}_{r} . \end{matrix}$

For any sample γ, we define $\begin{matrix} D_{t}^{(n)} (γ) = max_{0 \leq r \leq k} | Z_{t}^{(n + 1)} - Z_{t}^{(n)} |, n = 0, 1, 2, \dots \end{matrix}$

We claim that $\begin{matrix} D_{t}^{(n)} (γ) \leq (1 + | Z_{0} |) \frac{(L (1 + 2 K (γ)))^{n + 1}}{(n + 1)!} t^{n + 1}, \\ n = 0, 1, 2 \dots, t \in [0, T] . \end{matrix}$ where T is a constant. Indeed for n = 0, it follows from Lemma 3.1 that $\begin{matrix} D_{t}^{(0)} (γ) = & max_{0 \leq r \leq t} | \int_{0}^{r} f (Z_{0}, μ) d μ \\ + \int_{0}^{r} g (Z_{0}, μ) {dC}_{μ} (γ) | \\ \leq & max_{0 \leq r \leq t} | \int_{0}^{r} f (Z_{0}, μ) d μ | \\ + max_{0 \leq r \leq t} | \int_{0}^{r} g (Z_{0}, μ) {dC}_{μ} (γ) | \\ \leq & max_{0 \leq r \leq t} | \int_{0}^{r} f (Z_{0}, μ) | d μ \\ + 2 K (γ) max_{0 \leq r \leq t} \int_{0}^{r} | g (Z_{0}, μ) | d μ \\ \leq & \int_{0}^{r} | f (Z_{0}, μ) | d μ \\ + 2 K (γ) \int_{0}^{r} | g (Z_{0}, μ) | d μ . \\ \leq & (1 + 2 K (γ)) L (1 + | Z_{0} |) t . \end{matrix}$

This confirms the claim for n = 0 . Next we assume the claim is true for some n - 1 . Then $\begin{matrix} D_{t}^{(n)} (γ) \\ = & max_{0 \leq r \leq t} | \int_{0}^{r} (f (Z_{μ}^{(n)}, μ) - f (Z_{μ}^{(n - 1)}, μ)) d μ \\ + \int_{0}^{r} (g (Z_{μ}^{(n)}, μ) - g (Z_{μ}^{(n - 1)}, μ)) {dC}_{μ} (γ) \\ \leq & max_{0 \leq r \leq t} | \int_{0}^{r} (f (Z_{μ}^{(n)}, μ) - f (Z_{μ}^{(n - 1)}, μ)) d μ | \\ + max_{0 \leq r \leq t} | \int_{0}^{r} (g (Z_{μ}^{(n)}, μ) - g (Z_{μ}^{(n - 1)}, μ)) {dC}_{μ} (γ) | \\ \leq & L \int_{0}^{r} | Z_{μ}^{(n)} - Z_{μ}^{(n - 1)} | d μ + \\ 2 K (γ) max_{0 \leq r \leq t} \int_{0}^{r} | (g (Z_{μ}^{(n)}, μ) - g (Z_{μ}^{(n - 1)}, μ)) | d μ \\ \leq & L \int_{0}^{r} | Z_{μ}^{(n)} - Z_{μ}^{(n - 1)} | d μ + \\ 2 K (γ) L \int_{0}^{r} | Z_{μ}^{(n)} - Z_{μ}^{(n - 1)} | d μ \\ \leq & (L + 2 K (γ) L) \int_{0}^{t} | Z_{μ}^{(n)} - Z_{μ}^{(n - 1)} | d μ \\ \leq & (L + 2 K (γ) L) \int_{0}^{t} (1 + | Z_{0} |) \frac{((L + 2 K (γ) L) μ)^{n}}{n!} d μ \\ \leq & (1 + | Z_{0} |) \frac{((L + 2 K (γ) L))^{n + 1}}{(n + 1)!} t^{n + 1} . \end{matrix}$ Since $\begin{matrix} Z_{t}^{(0)} (γ) + \sum_{m = 0}^{n - 1} (Z_{t}^{(m + 1)} (γ) - Z_{t}^{(m)} (γ)) \\ = & Z_{t}^{(n)} (γ), \end{matrix}$ we have $Z_{t}^{(n)} (γ)$ is convergent uniformly in t ∈ [0, T] . Denote the limit by Z_t (γ) , we have $\begin{matrix} lim_{n \to \infty} Z_{t}^{(n)} (γ) = Z_{t} (γ) . \end{matrix}$

By the proof above, for each γ with C_t (γ) >0, we get $\begin{matrix} Z_{t} (γ) \leq & \sum_{n = 0}^{\infty} (1 + | Z_{0} |) \frac{((L + 2 K (γ) L) t)^{n}}{n!} \\ = & (1 + | Z_{0} |) exp ((L + 2 K (γ) L) t) . \end{matrix}$ Step 2. (sample continuity), for ɛ > 0, there exists $\begin{matrix} δ = & ε / [(1 + 2 K (γ)) (1 + | Z_{0} |) \\ exp ((L + 2 K (γ) L) t)] \end{matrix}$ when t > r and |t - r| < δ, we have $\begin{matrix} | Z_{t} (γ) - Z_{r} (γ) | \\ = & | \int_{r}^{t} f (μ, Z_{μ}) d μ + \int_{r}^{t} g (μ, Z_{μ}) {dC}_{μ} | \\ \leq & | \int_{r}^{t} f (μ, Z_{μ}) d μ | + | \int_{r}^{t} g (μ, Z_{μ}) {dC}_{μ} | \\ \leq & \int_{r}^{t} | f (μ, Z_{μ}) | d μ + 2 K (γ) \int_{r}^{t} | g (μ, Z_{μ}) | d μ \\ \leq & (t - r) (1 + 2 K (γ)) (1 \\ + | Z_{0} |) exp ((L + 2 K (γ) L) t) \\ \leq & ε . \end{matrix}$

Thus the solution of (7) is sample-continuous.

Step 3. (uniqueness), we assume that both of Z_t and $Z_{t}^{*}$ are solutions of (7). Then for any γ ∈ Γ with C_t (γ) >0, we have $\begin{matrix} | Z_{t} (γ) - Z_{t}^{*} (γ) | \\ = & | \int_{0}^{t} (f (μ, Z_{μ}) - f (μ, Z_{μ}^{*})) d μ \\ + \int_{0}^{t} (g (μ, Z_{μ}) - g (μ, Z_{μ}^{*})) {dC}_{μ} \\ \leq & | \int_{0}^{t} (f (μ, Z_{μ}) - f (μ, Z_{μ}^{*})) d μ | \\ + | \int_{0}^{t} (g (μ, Z_{μ}) - g (μ, Z_{μ}^{*})) {dC}_{μ} | \\ \leq & L \int_{0}^{t} | Z_{μ} - Z_{μ}^{*} | d μ \\ + 2 K (γ) \int_{0}^{t} | g (μ, Z_{μ}) - g (μ, Z_{μ}^{*}) | d μ \\ \leq & L \int_{0}^{t} | Z_{μ} - Z_{μ}^{*} | d μ + 2 K (γ) L \int_{0}^{t} | Z_{μ} - Z_{μ}^{*} | d μ \\ \leq & (L + 2 K (γ) L) \int_{0}^{t} | Z_{μ} - Z_{μ}^{*} | d μ . \end{matrix}$

Gronwall inequality implies the uniqueness. The proof is completed.□

Example 3.3. Considering the following CUDE

${dZ}_{t} = adt + {dC}_{1 t} + {idC}_{2 t},$ (8) where C_1t, C_2t are standard Liu processes, $a \in R .$

In this example, f = a . It is obvious that f = a is Lipschitz continuous function. According to Theorem 3.2, the solution of (8) exists and is unique. In fact, we have $\begin{matrix} Z_{t} = Z_{0} + at + C_{1 t} + {iC}_{2 t} \end{matrix}$

Example 3.4. Considering the following CUDE

${dZ}_{t} = {aZ}_{t} dt + {dC}_{1 t} + {idC}_{2 t}$ (9) where C_1t, C_2t are standard Liu processes, $a \in R .$

In this example, f = az . Since $| az - a \hat{z} | = | a | | z - \hat{z} |, \forall z, \hat{z} \in C,$ we know f = az is Lipschitz continuous function. According to Theorem 3.2, the solution of (9) exists and is unique. In fact, we have $\begin{matrix} Z_{t} = & exp (at) (Z_{0} + \int_{0}^{t} exp (- 2 r) {dC}_{1 r} \\ + \int_{0}^{t} i exp (- 2 r) {dC}_{2 r}) \end{matrix}$

Example 3.5. Considering the following linear CUDE $\begin{matrix} {dZ}_{t} = & (δ_{t} + ζ_{t} Z_{t}) dt + (σ_{t} + η_{t} Z_{t}) {dC}_{1 t} \\ + i (σ_{t} + η_{t} Z_{t}) {dC}_{2 t} \end{matrix}$

According to Theorem 3.2, in this example, $\begin{matrix} f (z, t) = δ_{t} + ζ_{t} z, g (z, t) = γ_{t} + η_{t} z . \end{matrix}$

Thus $\begin{matrix} | f (z, t) - f (\hat{z}, t) | + | g (z, t) - g (\hat{z}, t) | \\ = & (| ζ_{t} | + | η_{t} |) | z - \hat{z} |, \forall z, \hat{z} \in R, \\ 0 \leq t \leq T, \end{matrix}$ and $\begin{matrix} | f (z, t) | + | g (z, t) | \\ = & | δ_{t} + ζ_{t} z | + | γ_{t} + η_{t} z | \\ \leq & (| δ_{t} | + | γ_{t} |) + (| ζ_{t} | + | η_{t} |) | z |, \forall z \in R, \\ 0 \leq t \leq T . \end{matrix}$

Taking $\begin{matrix} L = max {| δ_{t} | + | γ_{t} |, | ζ_{t} | + | η_{t} |} \leq L | z - \hat{z} |, \end{matrix}$ and $\begin{matrix} f (z, t) | + | g (z, t) | \leq L (1 + | z |) . \end{matrix}$

So, the solution of linear CUDE of this example exists and is unique. It follows from Corollary 3.1 that the unique solution is $\begin{matrix} Z_{t} = & X_{t} (Z_{0} + \int_{0}^{t} \frac{δ_{r}}{X_{r}} dr + \int_{0}^{t} \frac{γ_{r}}{X_{r}} {dC}_{1 r} \\ + \int_{0}^{t} i \frac{γ_{r}}{X_{r}} {dC}_{2 r}) \end{matrix}$ $\begin{matrix} X_{t} = & exp (\int_{0}^{t} ζ_{r} dr + \int_{0}^{t} η_{r} {dC}_{1 r} \\ + \int_{0}^{t} i η_{r} {dC}_{2 r}), X_{0} = 1 . \end{matrix}$

3.3 Numerical method

In this subsection, we generalize the Yao-Chen formula [24] to the CUDE. Let α be a number with 0 < α < 1, and let C_t = C_1t + iC_2t be complex Liu process. An CUDE $\begin{matrix} {dZ}_{t} = f (t, Z_{t}) dt + g (t, Z_{t}) {dC}_{1 t} + ig (t, Z_{t}) {dC}_{2 t} \end{matrix}$

is said to have an α-path $Z_{t}^{α}$ if it solves the corresponding complex ordinary differential equation: $\begin{matrix} {dZ}_{t}^{α} = & f (t, Z_{t}^{α}) dt + ∣ g (t, Z_{t}^{α}) ∣ Φ^{- 1} (α) dt \\ + i ∣ g (t, Z_{t}^{α}) ∣ Φ^{- 1} (α) dt, \end{matrix}$

where Φ^-1 (α) is the inverse uncertainty distribution of standard normal uncertain variable $N (0, 1),$ i.e., $\begin{matrix} Φ^{- 1} (α) = \frac{\sqrt{3}}{π} ln \frac{α}{1 - α}, 0 < α < 1 . \end{matrix}$

Example 3.6. Let a, b be real numbers. The CUDE: $\begin{matrix} {dZ}_{t} = adt + {bdC}_{1 t} + {ibdC}_{2 t}, Z_{0} = 0 \end{matrix}$

has an α-path: $\begin{matrix} Z_{t}^{α} = at + (∣ b ∣ + i ∣ b ∣) Φ^{- 1} (α) t . \end{matrix}$

Lemma 3.2. Assume that f, g are continuous functions of two variables and C_t = C_1t + iC_2t is complex Liu process. Let Z_t and $Z_{t}^{α}$ be the solution and α-path of the CUDE: $\begin{matrix} {dZ}_{t} = f (t, Z_{t}) dt + g (t, Z_{t}) {dC}_{1 t} + ig (t, Z_{t}) {dC}_{2 t}, \end{matrix}$ respectively. Then we get $\begin{matrix} M {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t} \\ = & α, \end{matrix}$ $\begin{matrix} M {Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α}), \forall t} \\ = & 1 - α . \end{matrix}$

Proof. For each α-path $Z_{t}^{α},$ we construct sets as follows, $\begin{matrix} T^{+} = {t ∣ g (t, Z_{t}^{α}) \geq 0}, \end{matrix}$ $\begin{matrix} T^{-} = {t ∣ g (t, Z_{t}^{α}) < 0}, \end{matrix}$

It is obvious that T⁺∩ T^- = ∅ and T⁺ ∪ T^- = [0, + ∞).

Write $\begin{matrix} Λ_{j 1}^{+} = & {γ ∣ \frac{{dC}_{jt} (γ)}{dt} \leq Φ^{- 1} (α) for t \in T^{+}}, \\ j = 1, 2, \end{matrix}$ $\begin{matrix} Λ_{j 1}^{-} = & {γ ∣ \frac{{dC}_{jk} (γ)}{dt} \geq Φ^{- 1} (1 - α) for t \in T^{-}}, \\ j = 1, 2, \end{matrix}$

where Φ^-1 (α) is the inverse uncertainty distribution of $N (0, 1) .$ Since T⁺ and T^- are disjoint sets and C_jt have independent increments, we get

$\forall γ \in Λ_{j 1}^{+} \cap Λ_{j 1}^{-},$ we always have $\begin{matrix} g (t, Z_{t} (γ)) \frac{{dC}_{jt} (γ)}{dt} \leq & ∣ g (t, Z_{t}^{α}) ∣ Φ^{- 1} (α), \\ \forall t, j = 1, 2 . \end{matrix}$

Let $Λ_{1}^{+} \cap Λ_{1}^{-} = ⋂_{j = 1}^{2} (Λ_{j 1}^{+} \cap Λ_{j 1}^{-})$ . Because C_t = C_1t + iC_2t, then, $\forall γ \in Λ_{1}^{+} \cap Λ_{1}^{-},$ we have $\begin{matrix} {\begin{matrix} Re (g (t, Z_{t} (γ)) \frac{{dC}_{t} (γ)}{dt}) \leq ∣ g (t, Z_{t}^{α}) ∣ Φ^{- 1} (α) \\ Im (g (t, Z_{t} (γ)) \frac{{dC}_{t} (γ)}{dt}) \leq ∣ g (t, Z_{t}^{α}) ∣ Φ^{- 1} (α) . \end{matrix} \end{matrix}$

That is, $\begin{matrix} {\begin{matrix} Re (Z_{t}) \leq Re (Z_{t}^{α}), \forall t, \\ Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t . \end{matrix} \end{matrix}$

So, it holds that $Λ_{1}^{+} \cap Λ_{1}^{-} \subset {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t} .$

Hence, we have

$\begin{matrix} M {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t} \\ \geq M {Λ_{1}^{+} \cap Λ_{1}^{-}} = α, \end{matrix}$ (10)

On the other hand, write $\begin{matrix} Λ_{j 2}^{+} = & {γ ∣ \frac{{dC}_{jt} (γ)}{dt} > Φ^{- 1} (α) for t \in T^{+}}, \\ j = 1, 2 . \end{matrix}$

$\begin{matrix} Λ_{j 2}^{-} = & {γ ∣ \frac{{dC}_{jt} (γ)}{dt} < Φ^{- 1} (1 - α) for t \in T^{-}}, \\ j = 1, 2 . \end{matrix}$

Since T⁺ and T^- are disjoint sets and C_jt have independent increments, we get

$\forall γ \in Λ_{j 2}^{+} \cap Λ_{j 2}^{-},$ we always have

$g (t, Z_{t} (γ)) \frac{{dC}_{jt} (γ)}{dt} > ∣ g (t, Z_{t}^{α}) ∣ Φ^{- 1} (α), \forall t, j = 1, 2 .$

Let $Λ_{2}^{+} \cap Λ_{2}^{-} = ⋂_{j = 1}^{2} (Λ_{i 2}^{+} \cap Λ_{i 2}^{-}) .$ Because C_t = C_1t + iC_2t, then, $\forall γ \in Λ_{2}^{+} \cap Λ_{2}^{-},$ we have $\begin{matrix} {\begin{matrix} Re (g (t, Z_{t} (γ)) \frac{{dC}_{t} (γ)}{dt}) > ∣ g (t, Z_{t}^{α}) ∣ Φ^{- 1} (α) \\ Im (g (t, Z_{t} (γ)) \frac{{dC}_{t} (γ)}{dt}) > ∣ g (t, Z_{t}^{α}) ∣ Φ^{- 1} (α) . \end{matrix} \end{matrix}$

That is $\begin{matrix} {\begin{matrix} Re (Z_{t}) > Re (Z_{t}^{α}), \forall t, \\ Im (Z_{t}) > Im (Z_{t}^{α}), \forall t . \end{matrix} \end{matrix}$

So, it holds that $Λ_{2}^{+} \cap Λ_{2}^{-} \subset {Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α}), \forall t} .$

Hence, we have

$\begin{matrix} M {Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α}), \forall t} \\ \geq M {Λ_{2}^{+} \cap Λ_{2}^{-}} = 1 - α, \end{matrix}$ (11)

Since ${Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t}$ and ${Re (Z_{t}) ≰ Re (Z_{t}^{α}), Im (Z_{t}) ≰ Im (Z_{t}^{α}), \forall t}$ are opposite events with each other. It follows from the duality axiom that $\begin{matrix} M {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t} \\ + M {Re (Z_{t}) ≰ Re (Z_{t}^{α}), Im (Z_{t}) ≰ Im (Z_{t}^{α}), \\ \forall t} = 1 . \end{matrix}$

It follows from $\begin{matrix} {Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α}), \forall t} \\ \subset {Re (Z_{t}) ≰ Re (Z_{t}^{α}), Im (Z_{t}) ≰ Im (Z_{t}^{α}), \forall t} \end{matrix}$ and monotonicity theorem that

$\begin{matrix} M {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t} \\ + M {Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α}), \\ \forall t} \leq 1 . \end{matrix}$ (12)

Thus, the results follow from (1)0, (1)1, and (1)21. □

Theorem 3.3. Assume that f, g are continuous functions of two variables and C_t = C_1t + iC_2t. Let Z_t and $Z_{t}^{α}$ be the solution and α-path of the CUDE: ${dZ}_{t} = f (t, Z_{t}) dt + g (t, Z_{t}) {dC}_{1 t} + ig (t, Z_{t}) {dC}_{2 t},$ respectively. Then, the solution Z_t has an inverse uncertainty distribution: $ϒ_{t}^{- 1} (α) = Z_{t}^{α} .$

Proof. Obviously, $\begin{matrix} {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α})} \\ \supset & {Re (Z_{r}) \leq Re (Z_{r}^{α}), Im (Z_{r}) \leq Im (Z_{r}^{α}), \forall r} . \end{matrix}$

It follows from the monotonicity theorem and Lemma 3.2 that:

$\begin{matrix} M {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α})} \\ \geq & M {Re (Z_{r}) \leq Re (Z_{r}^{α}), Im (Z_{r}) \leq Im (Z_{r}^{α}), \\ \forall r} = α \end{matrix}$ (13)

Similarly, we also obtain

$\begin{matrix} M {Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α})} \\ \geq & M {Re (Z_{r}) > Re (Z_{r}^{α}), Im (Z_{r}) > Im (Z_{r}^{α}), \\ \forall r} = 1 - α . \end{matrix}$ (14)

Since ${Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t}$ and ${Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α}), \forall t}$ are opposite events with each other. Besides, by using the duality axiom, we have:

$\begin{matrix} M {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α})} \\ + M {Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α})} \\ = 1 . \end{matrix}$ (15)

It follows from (1)2, (1)3, and (1)4 that: $M {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α})} = α,$ that is, $ϒ_{t}^{- 1} (α) = Z_{t}^{α} .$ The proof is completed. □

To calculate the inverse uncertainty distribution of the solution, we design a numerical algorithm as follow.

Table 1

Inverse uncertainty distribution (IUD) of the solution

Algorithm 3.1	Let $Z_{0}^{α} = Z_{0}$ be an initial guess.
Step 1	Set α = 0, step length Δα.
Step 2	α = α + Δα
Step 3	Fix l as the step length. Set j = 0,and $N = \frac{T}{l}$
	for a fixed α in (0,1)
Step 4	Use the recursion formula
	$Z_{j + 1}^{α} = Z_{j}^{α} + f (t_{j}, Z_{j}^{α}) l$
	$+ \| g (t_{j}, Z_{j}^{α}) \| Φ^{- 1} (α) l$
	$+ i \| g (t_{j}, Z_{j}^{α}) \| Φ^{- 1} (α) l$
	and calculate $Z_{j + 1}^{α}$
Step 5	Set j ← j + 1 .
Step 6	Repeat Step 4 and Step 5 for N times.
Step 7	Obtain the IUD Z_t for α.
Step 8	Return to step 2, while α + Δα < 1 .
Step 9	Output the iterative results. Then obtain
	the IUD $Z_{t}^{- 1} (α)$ at each α in (0,1).
Step 10	The expected value at time T is
	$E_{α} = \frac{1}{N - 1} \sum_{j = 1}^{N - 1} Re (Z_{T}^{\frac{j}{N}})$
	$+ i \frac{1}{N - 1} \sum_{j = 1}^{N - 1} Im (Z_{T}^{\frac{j}{N}})$

Remark 3.1. For the outer loop, for each α in (0,1), break up α in (0,1), if we suppose the number of shares is M. While the internal time complexity is O (N). So the time complexity of this algorithm becomes $O (M \times \frac{T}{l})$ .

Example 3.7. Let a, b be real numbers and let C_t = C_1t + iC_2t be complex Liu process. The uncertain differential equation:

${dZ}_{t} = {aZ}_{t} dt + {bZ}_{t} {dC}_{1 t} + {ibZ}_{t} {dC}_{2 t}, Z_{0} = 1$ (16) has a solution: $Z_{t} = exp (at + {bC}_{1 t} + {ibC}_{2 t})$ with an inverse uncertainty distribution: $ϒ_{t}^{- 1} (α) = exp (at + (∣ b ∣ + i ∣ b ∣) Φ^{- 1} (α) t) .$

In order to illustrate the numerical method, let a = 1, b = 1,then $d Z_{t} = Z_{t} d t + Z_{t} d C_{1 t} + {iZ}_{t} d C_{2 t}, Z_{0} = 1$ whose solution is Z_t = exp(t + C_1t + iC_2t) .

The α-path of Equation (1)5 is $\begin{matrix} {dZ}_{t}^{α} = (Z_{t}^{α} + ((| Z_{t}^{α} | + i | Z_{t}^{α} |) Φ^{- 1} (α))) dt \end{matrix}$

Based on the Algorithm 3.1 we design (see Table 1), we obtain the running time of this example is 8.23088806s. The inverse uncertainty distribution of Z_t is shown in Fig. 1. At the same time, we give the analytic solution as a comparison in Fig. 1. The error of the real part is 2.8802, the error of the imaginary part is 3.5553. And we can get $\begin{matrix} E_{α} [Z_{t}^{α}] = 4.4519 + 1.7740 i \end{matrix}$

Fig. 1

The IUD simulatin of Z_t about CUDE.

4 Applications

In this section, we will give CUDEs applied in time integral, that is the inverse uncertainty distribution of time integral.

4.1 Inverse uncertainty distribution of time integral

Theorem 4.1. Let Z_t and $Z_{t}^{α}$ be the solution and α-path of the CUDE,respectively. For C (z) is a strictly increasing function. Then the integral $\int_{0}^{T} C (Z_{t}) dt$ have the IUD $ϒ_{T}^{- 1} (α) = \int_{0}^{T} C (Z_{t}^{α}) dt .$ for C (z) is a strictly decreasing function. Then the integral $\int_{0}^{T} C (Z_{t}) dt$ have the IUD $ϒ_{T}^{- 1} (α) = \int_{0}^{T} C (Z_{t}^{1 - α}) dt .$

Proof. When C (z) is a strictly increasing function with respect to z, it is always true that $\begin{matrix} {Re (\int_{0}^{T} C (Z_{t}) dt) \leq Re (\int_{0}^{T} C (Z_{t}^{α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) \leq Im (\int_{0}^{T} C (Z_{t}^{α}) dt)} \\ \supset {Re (C (Z_{t}) dt) \leq Re (C (Z_{t}^{α}) dt), \\ Im (C (Z_{t}) dt) \leq Im (C (Z_{t}^{α}) dt), \forall t} \\ = {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t}, \end{matrix}$ and $\begin{matrix} {Re (\int_{0}^{T} C (Z_{t}) dt) > Re (\int_{0}^{T} C (Z_{t}^{α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) > Im (\int_{0}^{T} C (Z_{t}^{α}) dt)} \\ \supset {Re (C (Z_{t}) dt) > Re (C (Z_{t}^{α}) dt), \\ Im (C (Z_{t}) dt) > Im (C (Z_{t}^{α}) dt), \forall t} \\ = {Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α}), \forall t}, \end{matrix}$

By Lemma 3.2, we obtain $\begin{matrix} M {Re (\int_{0}^{T} C (Z_{t}) dt) \leq Re (\int_{0}^{T} C (Z_{t}^{α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) \leq Im (\int_{0}^{T} C (Z_{t}^{α}) dt)} \geq \\ M {Re (Z_{t}) \leq Re (Z_{t}^{α}), Im (Z_{t}) \leq Im (Z_{t}^{α}), \forall t} \\ = α . \end{matrix}$

Similarly, we have $\begin{matrix} M {Re (\int_{0}^{T} C (Z_{t}) dt) > Re (\int_{0}^{T} C (Z_{t}^{α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) > Im (\int_{0}^{T} C (Z_{t}^{α}) dt)} \geq \\ M {Re (Z_{t}) > Re (Z_{t}^{α}), Im (Z_{t}) > Im (Z_{t}^{α}), \forall t} \\ = 1 - α . \end{matrix}$

It follows that $\begin{matrix} M {Re (\int_{0}^{T} C (Z_{t}) dt) \leq Re (\int_{0}^{T} C (Z_{t}^{α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) \leq Im (\int_{0}^{T} C (Z_{t}^{α}) dt)} \\ = α . \end{matrix}$

In other words, $\int_{0}^{T} C (Z_{t}) dt$ has an inverse uncertainty distribution $ϒ_{T}^{- 1} (α) = \int_{0}^{T} C (Z_{t}^{α}) dt$

Similarly, for a strictly decreasing function C (z), it is always true that $\begin{matrix} {Re (\int_{0}^{T} C (Z_{t}) dt) \leq Re (\int_{0}^{T} C (Z_{t}^{1 - α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) \leq Im (\int_{0}^{T} C (Z_{t}^{1 - α}) dt)} \\ \supset {Re (C (Z_{t}) dt) \leq Re (C (Z_{t}^{1 - α}) dt), \\ Im (C (Z_{t}) dt) \leq Im (C (Z_{t}^{1 - α}) dt), \forall t} \\ = {Re (Z_{t}) \geq Re (Z_{t}^{1 - α}), \\ Im (Z_{t}) \geq Im (Z_{t}^{1 - α}), \forall t}, \end{matrix}$ and $\begin{matrix} {Re (\int_{0}^{T} C (Z_{t}) dt) > Re (\int_{0}^{T} C (Z_{t}^{1 - α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) > Im (\int_{0}^{T} C (Z_{t}^{1 - α}) dt)} \\ \supset {Re (C (Z_{t}) dt) > Re (C (Z_{t}^{1 - α}) dt), \\ Im (C (Z_{t}) dt) > Im (C (Z_{t}^{1 - α}) dt), \forall t} \\ = {Re (Z_{t}) < Re (Z_{t}^{1 - α}), \\ Im (Z_{t}) < Im (Z_{t}^{1 - α}), \forall t}, \end{matrix}$

By Lemma 3.2, we obtain $\begin{matrix} M {Re (\int_{0}^{T} C (Z_{t}) dt) \leq Re (\int_{0}^{T} C (Z_{t}^{1 - α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) \leq Im (\int_{0}^{T} C (Z_{t}^{1 - α}) dt)} \geq \\ M {Re (Z_{t}) \leq Re (Z_{t}^{1 - α}), Im (Z_{t}) \leq Im (Z_{t}^{1 - α}), \\ \forall t} = α . \end{matrix}$

Similarly, we have $\begin{matrix} M {Re (\int_{0}^{T} C (Z_{t}) dt) > Re (\int_{0}^{T} C (Z_{t}^{1 - α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) > Im (\int_{0}^{T} C (Z_{t}^{1 - α}) dt)} \geq \\ M {Re (Z_{t}) > Re (Z_{t}^{1 - α}), Im (Z_{t}) > Im (Z_{t}^{1 - α}), \\ \forall t} = 1 - α . \end{matrix}$

It follows that $\begin{matrix} M {Re (\int_{0}^{T} C (Z_{t}) dt) \leq Re (\int_{0}^{T} C (Z_{t}^{1 - α}) dt), \\ Im (\int_{0}^{T} C (Z_{t}) dt) \leq Im (\int_{0}^{T} C (Z_{t}^{1 - α}) dt)} \\ = α . \end{matrix}$

In other words, $\int_{0}^{T} C (Z_{t}) dt$ has an inverse uncertainty distribution $ϒ_{T}^{- 1} (α) = \int_{0}^{T} C (Z_{t}^{1 - α}) dt,$ we can get the conclusion. □

To calculate the inverse uncertainty distribution of the integral of C (Z_t), we design a numerical algorithm for two situations as follow.

Table 2
The situation about a strictly increasing function C (z)

Algorithm 4.1 Let $Z_{0}^{α} = Z_{0}$ be an initial guess.

Step 1 Set α = 0, step length Δα.

Step 2 α = α + Δα

Step 3 Fix l as the step length. Set j = 0, $N = \frac{T}{l}$

for a fixed α in (0,1)

Step 4 Use the recursion formula

$Z_{j + 1}^{α} = Z_{j}^{α} + f (t_{j}, Z_{j}^{α}) l$

$+ | g (t_{j}, Z_{j}^{α}) | Φ^{- 1} (α) l + i | g (t_{j}, Z_{j}^{α}) | Φ^{- 1} (α) l$

and calculate $Z_{j + 1}^{α}$ and $C (Z_{j + 1}^{α})$

Step 5 Set j ← j + 1 .

Step 6 Repeat Step 4 and Step 5 for N times.

Step 7 Obtain the IUD $\int_{0}^{T} C (Z_{t}) dt$

$= \sum_{j = 1}^{N} Re (C (Z_{j}^{α})) l$

$+ i \sum_{j = 1}^{N} Im (C (Z_{j}^{α})) l$ for α.

Step 8 Return to step 2, while α + Δα < 1 .

Step 9 Output the iterative results.

Then the IUD $ϒ_{T}^{- 1} (α) = \sum_{j = 1}^{N} Re (C (Z_{j}^{α})) l$

$+ i \sum_{j = 1}^{N} Im (C (Z_{j}^{α})) l$ at each α in (0,1).

Algorithm 4.1	Let $Z_{0}^{α} = Z_{0}$ be an initial guess.
Step 1	Set α = 0, step length Δα.
Step 2	α = α + Δα
Step 3	Fix l as the step length. Set j = 0, $N = \frac{T}{l}$
	for a fixed α in (0,1)
Step 4	Use the recursion formula
	$Z_{j + 1}^{α} = Z_{j}^{α} + f (t_{j}, Z_{j}^{α}) l$
	$+ \| g (t_{j}, Z_{j}^{α}) \| Φ^{- 1} (α) l + i \| g (t_{j}, Z_{j}^{α}) \| Φ^{- 1} (α) l$
	and calculate $Z_{j + 1}^{α}$ and $C (Z_{j + 1}^{α})$
Step 5	Set j ← j + 1 .
Step 6	Repeat Step 4 and Step 5 for N times.
Step 7	Obtain the IUD $\int_{0}^{T} C (Z_{t}) dt$
	$= \sum_{j = 1}^{N} Re (C (Z_{j}^{α})) l$
	$+ i \sum_{j = 1}^{N} Im (C (Z_{j}^{α})) l$ for α.
Step 8	Return to step 2, while α + Δα < 1 .
Step 9	Output the iterative results.
	Then the IUD $ϒ_{T}^{- 1} (α) = \sum_{j = 1}^{N} Re (C (Z_{j}^{α})) l$
	$+ i \sum_{j = 1}^{N} Im (C (Z_{j}^{α})) l$ at each α in (0,1).

Table 3

The situation about a strictly decreasing function C (z)

Algorithm 4.2	Let $Z_{0}^{α} = Z_{0}$ be an initial guess.
Step 1	Set α = 0, step length Δα.
Step 2	α = α + Δα
Step 3	Fix l as the step length. Set j = 0, $N = \frac{T}{l}$
	for a fixed α in (0,1)
Step 4	Use the recursion formula
	$Z_{j + 1}^{1 - α} = Z_{j}^{1 - α} + f (t_{j}, Z_{j}^{1 - α}) l$
	$+ \| g (t_{j}, Z_{j}^{1 - α}) \| Φ^{- 1} (α) l + i \| g (t_{j}, Z_{j}^{1 - α}) \| Φ^{- 1} (α) l$
	and calculate $Z_{j + 1}^{1 - α}$ and $C (Z_{j + 1}^{1 - α})$
Step 5	Set j ← j + 1 .
Step 6	Repeat Step 4 and Step 5 for N times.
Step 7	Obtain the IUD $\int_{0}^{T} C (Z_{t}) dt$
	$= \sum_{j = 1}^{N} Re (C (Z_{j}^{1 - α})) l$
	$+ i \sum_{j = 1}^{N} Im (C (Z_{j}^{1 - α})) l$ for α.
Step 8	Return to step 2, while α + Δα < 1 .
Step 9	Output the iterative results.
	Then the IUD $ϒ_{T}^{- 1} (α) = \sum_{j = 1}^{N} Re (C (Z_{j}^{1 - α})) l$
	$+ i \sum_{j = 1}^{N} Im (C (Z_{j}^{1 - α})) l$ at each α in (0,1).

Remark 4.1. Similar to algorithm 1, algorithm 2 and 3 have similar time complexity. For the outer loop, for each α in (0,1), break up α in (0,1), if we suppose the number of shares is M. While the internal time complexity is O (N). So the time complexity of this algorithm becomes $O (M \times \frac{T}{l})$ .

4.2 Numerical examples

Example 4.1. Let η, a, b be real number. For the following nonlinear CUDE,

$\begin{matrix} {dZ}_{t} = & (η - {aZ}_{t}) dt + {bdC}_{1 t} + {ibdC}_{2 t}, \\ 0 \leq t < 1 \end{matrix}$ (17) with given initial value Z₀ = 1 . The α-path equivalent equation of Equation (1)6 is $\begin{matrix} {dZ}_{t}^{α} = (η - {aZ}_{t}^{α} + ((| b | + i | b |) Φ^{- 1} (α))) dt \end{matrix}$

(i) Consider the time integral

$\int_{0}^{T} exp (- δ t) (Z_{t} - H) dt$ (18) where δ and H are real numbers. The α-path of Equation (1)7 is

$\int_{0}^{T} exp (- δ t) (Z_{t}^{α} - H) dt$ (19) for given times T > 0 . We set η = 5, a = 1, b = 2, t = 0.9 and N = 1, 000 . in Equation (1)6 and choose the parameters δ = 0.02 and H = 1 in Equation (1)8. Based on the Algorithm 4.1 we design (see Table 2), the IUD of time integral at T = 0.9 is shown in Fig. 2. The running time of this example is 8.68379906s.

Fig. 2

The time integral simulatin of CUDE when C (z) is a strictly increasing function.

(ii) Consider the time integral

$\int_{0}^{T} exp (δ t) (Z_{t} - H) dt$ (20) where δ and H are real numbers. The α-path of Equation (1)9 is

$\int_{0}^{T} exp (δ t) (Z_{t}^{1 - α} - H) dt$ (21) for given times T > 0 . We set η = 5, a = 1, b = 2, t = 0.9 and N = 1, 000 . in Equation (1)6 and choose the parameters δ = 0.02 and H = 1 in Equation (19). Based on the Algorithm 4.2 we design (see Table 3), the inverse uncertainty distribution of time integral at T = 0.9 is shown in Fig. 3. The running time of this example is 8.72653301s.

Fig. 3

The time integral simulatin of CUDE when C (z) is a strictly decreasing function.

5 Conclusions

This paper mainly discussed existence and uniqueness theorem, numerical algorithm and time integral of CUDEs. First of all, we proposed the concepts of some classes of CUDEs. Furthermore, from the definition of CUDE, deduced existence, uniqueness, α-path, inverse uncertainty distributions successively, thus designing the numerical algorithm. Some analytic examples and numerical examples were provided to illustrate existence, uniqueness, inverse uncertainty distributions. Future researches may consider stability analysis and some applications in control fields.

Footnotes

Acknowledgments

The research was supported by Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_-0170).

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