Multifactor uncertain differential equation
 Shengguo Li^{1},
 Jin Peng^{1}Email author and
 Bo Zhang^{2}
https://doi.org/10.1186/s404670150031y
© Li et al.; licensee Springer. 2015
Received: 24 January 2015
Accepted: 5 March 2015
Published: 7 April 2015
Abstract
This paper proposes a type of multifactor uncertain differential equation within the framework of uncertainty theory. The analytic solutions of four special types of multifactor uncertain differential equations are first discussed. Then, a numerical method for solving general multifactor uncertain differential equation is presented. Finally, under the Lipschitz condition and linear growth condition, it is proved that the multifactor uncertain differential equation has a unique solution.
Keywords
Introduction
Uncertainty theory is a tool to study the indeterminacy phenomena in human systems, which was founded by Liu [1] in 2007. It was refined by Liu [2] and has become an axiomatic system via normality, duality, subadditivity, and product axioms of uncertain measure. Up to now, many branches of mathematics emerged based on uncertainty theory, such as mathematical programming [3], uncertain set and uncertain inference [4], uncertain logic [5], uncertain risk [6,7], and uncertain insurance [8].
Uncertain process is essentially a sequence of uncertain variables indexed by time which was first introduced by Liu [9]. After that, a significant uncertain process called canonical process was designed by [10]. The canonical process is a stationary independent increment process with Lipschitz continuous sample paths. Meanwhile, uncertain calculus with respect to canonical process called Liu calculus was developed by Liu [10]. In order to describe the evolution of uncertain phenomenon with some jumps, Liu [9] proposed the uncertain renewal process. Afterward, Yao [11] presented the uncertain calculus with respect to renewal process called the Yao calculus. Recently, Yao [12] proposed multidimensional uncertain calculus with Liu process, Chen [13] studied the uncertain calculus with finite variation processes. More research about uncertain process can be found in references [1416].
Uncertain differential equation was proposed by Liu [9], which is an important tool to deal with uncertain dynamic systems. Different from stochastic differential equation driven by a Wiener process [17], uncertain differential equation is a type of differential equation driven by uncertain process. In order to know well uncertain differential equation, many researchers did a lot of work. Chen and Liu [18] proved an existence and uniqueness theorem of solution under global Lipschitz condition and proposed an analytic solution for linear uncertain differential equation. Gao [19] gave an existence and uniqueness theorem with local Lipschitz condition. In 2009, Liu [10] gave a concept of stability of uncertain differential equation. After that, Yao et al. [20] proved some stability theorems of uncertain differential equation. In addition, Sheng and Wang [21] investigated the stability in pth moment for uncertain differential equation, Liu et al. [22] studied the almost sure stability, and Yao et al. [23] showed the stability in mean. In order to obtain the solution of uncertain differential equation, Liu [24] and Yao [25] provided the analytic solutions for some special nonlinear uncertain differential equations, respectively. Yao and Chen [26] presented a numerical method for solving uncertain differential equation when it is difficult to obtain analytic solution. Yao [27] also discussed the extreme values and integral of solution of uncertain differential equation.
Uncertain differential equation was first applied in finance by Liu [10] in 2009. Meanwhile, Liu [10] presented an uncertain stock model in uncertain financial market and proved the European option pricing formulas. After that, Chen [28] gave the America option pricing formulas. Besides, Peng and Yao [29] presented another uncertain stock model and corresponding option pricing formulas. Liu [30] discussed some possible applications of uncertain differential equations to financial markets. Li and Peng [31] proposed a stock model with uncertain stock diffusion. Liu et al. [32] built an uncertain currency model and proved the currency option pricing. Jiao and Yao [33] considered an interest rate model in uncertain environment. Yao [34] proved a noarbitrage theorem for uncertain stock model. In addition, uncertain differential equation was also applied in uncertain optimal control [35] and uncertain differential game [36].
The extensions of uncertain differential equation also attracted the attention of scholars. Several recent contributions in the extension literature have studied this question in many directions. Yao [11] suggested the uncertain differential equation with jumps. Ge and Zhu [37] discussed the backward uncertain differential equation. Barbacioru [38], Ge and Zhu [39], and Liu and Fei [40] focused on the uncertain delay differential equation. Yao [12] proposed the multidimentional uncertain differential equation via multidimensional uncertain calculus. Ji and Zhou [41] proved an existence and uniqueness theorem of solution for multidimensional uncertain differential equation. Yao [42] studied the higher order uncertain differential equation.
Usually, the uncertain factor influencing dynamic systems is not alone. In 2012, Liu and Yao [43] extended uncertain integral from single canonical process to multiple ones. This provides a motivation to consider the concept of uncertain differential equation driven by multiple uncertain processes. In this paper, we present a type of uncertain differential equation driven by multiple canonical processes which can be regarded as a generalization of the uncertain differential equation proposed by Liu [9].
The rest of the paper is organized as follows. Some preliminary concepts of uncertainty theory and uncertain calculus are recalled in the ‘Preliminary’ section. After that, the multifactor uncertain differential equation is presented. Following that, a numerical method is introduced. In addition, an existence and uniqueness theorem is proved. Finally, a brief summary is given.
Preliminary
In this section, uncertainty theory and uncertain calculus are introduced and some basic concepts are given.
Uncertainty theory

(Subadditivity axiom) for every countable sequence of events Λ _{1},Λ _{2},⋯, we have:
The triplet is called an uncertain space. In order to obtain an uncertain measure of compound event, Liu [10] defined a product uncertain measure which produces the fourth axiom of uncertainty theory:

(Product axiom) Let be uncertain spaces for k=1,2,⋯ The product uncertain measure is an uncertain measure on the product σalgebra \(\mathcal {L}_{1}\times \mathcal {L}_{2} \times \cdots \) satisfying:
where Λ _{ k } are arbitrarily chosen events from \({\mathcal L}_{k}\) for k=1,2,⋯, respectively.
The uncertainty distribution Φ:ℜ→[0,1] of an uncertain variable ξ is defined by Liu [1] as:
and the inverse function Φ ^{−1} is called the inverse uncertainty distribution of ξ.
The expected value of uncertain variable ξ is defined by Liu [1] as:
provided that at least one of the two integrals is finite. The variance of ξ is defined as V[ ξ]=E[ (ξ−E[ξ])^{2}].
Uncertain calculus
Definition 1.
Definition 2.
(Liu [10]) An uncertain process C _{ t } is said to be a canonical process if (i)C _{0}=0 and almost all sample paths are Lipschitz continuous; (i i)C _{ t } has stationary and independent increments; (i i i) every increment C _{ s+t }−C _{ t } is a normal uncertain variable with expected value 0 and variance t ^{2}.
Definition 3.
Example 1.
Definition 4.
is called an uncertain differential equation.
The uncertain differential with respect to canonical processes C _{1t },C _{2t },⋯,C _{ nt } is defined by Liu and Yao [43] as follows.
Definition 5.
In this case, Z _{ t } is called a differentiable uncertain process with drift μ _{ t } and diffusions σ _{1t },σ _{2t },⋯,σ _{ nt }.
Theorem 1.
Multifactor uncertain differential equation
Usually, the uncertain factor influencing dynamic systems is not alone. In order to model the dynamic systems with multiple factors, this section will extend the uncertain differential equation driven by single canonical process to one driven by multiple independent canonical processes.
Definition 6.
Theorem 2.
Definition 7.
is called an uncertain differential equation with respect to C _{1t },C _{2t },⋯,C _{ nt }. A solution is an uncertain process X _{ t } that satisfies Equation 4 identically in t.
Example 2.
Theorem 3.
Therefore, the uncertain differential Equation 8 has a solution (9).
Example 3.
Theorem 4.
Taking U _{0}=1 and V _{0}=X _{0}, we get the solutions (13) and (14). The theorem is proved.
Note that n=1, the uncertain differential Equation 12 degenerates to the linear uncertain differential equation in Chen and Liu [18].
Example 4.
provided that a≠0.
Example 5.
Theorem 5.
with initial value Z _{0}=X _{0}.
Defining Z _{ t }=X _{ t } Y _{ t }, we obtain \(X_{t}=Y_{t}^{1}Z_{t}\) and \({dZ}_{t}=Y_{t}f\left (t, Y_{t}^{1}Z_{t}\right)\). Furthermore, since Y _{0}=1, the initial value Z _{0} is just X _{0}. The theorem is proved.
Note that n=1, the uncertain differential Equation 19 degenerates to the nonlinear uncertain differential equation in Liu [24].
Example 6.
Theorem 6.
with initial value Z _{0}=X _{0}.
Define Z _{ t }=X _{ t } Y _{ t }, then \(X_{t}=Y_{t}^{1}Z_{t}\) and \({dZ}_{t}=Y_{t}\displaystyle \sum \limits _{i=1}^{n}g_{i}\left (t, Y_{t}^{1}Z_{t}\right){dC}_{\textit {it}}\). In addition, since Y _{0}=1, the initial value Z _{0} is just X _{0}. The theorem is proved.
Note that n=1, the uncertain differential Equation 24 degenerates to the nonlinear uncertain differential equation in Liu [24].
Example 7.
Numerical method
However, in many cases, it is difficult to find analytic solutions of uncertain differential equations. Yao and Chen [26] presented a numerical method called YaoChen method to obtain the inverse uncertainty distribution of solution.
YaoChen formula
Definition 8.
Theorem 7.
Theorem 8.
Generalization
In this subsection, we generalize the YaoChen formula to the multifactor uncertain differential equation.
Definition 9.
Example 8.
Lemma 9.

If k(t)g(t,x)≤K∣g(t,x)∣ for t∈[0,T], then ψ(T)≤ϕ(T),

If k(t)g(t,x)>K∣g(t,x)∣ for t∈[0,T], then ψ(T)>ϕ(T).
Theorem 10.
Let \(\Lambda _{1}^{+}\cap \Lambda _{1}^{}=\displaystyle \bigcap _{i=1}^{n}\left (\Lambda _{i1}^{+}\cap \Lambda _{i1}^{}\right)\). Because C _{1t },C _{2t },⋯,C _{ nt } are independent and , i=1,2,⋯,n, we have:
Let \(\Lambda _{2}^{+}\cap \Lambda _{2}^{}=\displaystyle \bigcap _{i=1}^{n}\left (\Lambda _{i2}^{+}\cap \Lambda _{i2}^{}\right)\). Because C _{1t },C _{2t },⋯,C _{ nt } are independent and , i=1,2,⋯,n, we have:
Since \(\left \{X_{t}\leq X_{t}^{\alpha },\forall t\right \}\) and \(\left \{X_{t} \not \leq X_{t}^{\alpha },\forall t\right \}\) are opposite events with each other. It follows from the duality axiom that:
Thus, the results follow from (33), (34), and (35).
Theorem 11.
Example 9.

Fix α on (0,1).

Solve the corresponding ordinary differential equation:and obtain \(X_{t}^{\alpha }\), for example, we can choose the recursion formula:$${dX}_{t}^{\alpha}=f\left(t, X_{t}^{\alpha}\right)dt+\sum\limits_{i=1}^{n} \mid g_{i}\left(t, X_{t}^{\alpha}\right)\mid \Phi^{1}(\alpha)dt $$where Φ ^{−1}(α) is the inverse standard normal uncertainty distribution and h is the step length.$$X_{i+1}^{\alpha}=X_{i}^{\alpha}+f\left(t_{i}, X_{i}^{\alpha}\right)h+\sum\limits_{j=1}^{n} \mid g_{j}\left(t_{i}, X_{i}^{\alpha}\right)\mid \Phi^{1}(\alpha)h $$

The inverse uncertainty distribution of X _{ t } is obtained.
Example 10.
Existence and uniqueness theorem
This section will give an existence and uniqueness theorem of solution for the multifactor uncertain differential equation under Lipschitz condition and linear growth condition.
Lemma 12.
Theorem 13.
for some constant L. Moreover, the solution is samplecontinuous.
Hence, \(X_{t}=X_{t}^{\star }\). The uniqueness is proved.
Note that n=1, the existence and uniqueness theorem degenerates to the one in Chen and Liu [18].
Conclusions
Uncertain differential equation is an important tool to deal with dynamic systems in uncertain environments. In this paper, the multifactor uncertain differential equation was proposed. Four special types of multifactor uncertain differential equations were studied and the corresponding analytic solutions were given. For general multifactor uncertain differential equation, a numerical method was provided for obtaining the solution. Also, an existence and uniqueness theorem that the multifactor uncertain differential equation has a unique solution was proved. The proposed multifactor uncertain differential equation can be used to describe the multifactor stock model in uncertain market.
Declarations
Acknowledgments
This work is supported by the Projects of the Humanity and Social Science Foundation of Ministry of Education of China (No.13YJA630065), the Key Project of Hubei Provincial Natural Science Foundation (No.2012FFA065).
Authors’ Affiliations
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