# A type of uncertain differential equations with analytic solution

## Abstract

Uncertain differential equation is a type of differential equation driven by the Liu process. So far, an analytic solution of linear uncertain differential equation has been obtained. This paper aims at proposing a method to solve a type of nonlinear uncertain differential equation.

## Introduction

Uncertainty theory, as a branch of axiomatic mathematics dealing with human’s belief degree, was founded by Liu  in 2007 and refined by Liu  in 2010. During the past 6 years, many researchers have contributed in this area. For example, Peng and Iwamura  gave a sufficient condition for the uncertainty distribution of an uncertain variable. Liu and Ha  gave a formula to calculate the expected value of a function of multiple uncertain variables. Chen and Dai  showed that a normal uncertain variable has the maximum entropy given the expected value and variance. Especially, Liu  proposed uncertain programming as a type of mathematical programming involving uncertain variables.

In order to model the evolution of an uncertain phenomenon, Liu  proposed a concept of uncertain process. Meanwhile, Liu  gave an uncertain renewal process as an example. After that, Liu  proposed an uncertain renewal reward process, and Yao and Li  proposed an uncertain alternating renewal process. In addition, Zhang et al.  proposed an uncertain delayed renewal process. In 2009, Liu  mathematically defined a type of uncertain process, named canonical Liu process, which has independent and stationary uncertain normal increments and of which almost all the sample paths are Lipschitz continuous. In addition, Liu  proved the extreme value theorems for an independent increment uncertain process.

In 2009, Liu  founded an uncertain calculus to deal with the integral and differential of an uncertain process with respect to Liu process, which are called Liu integral and Liu differential afterwards. Then Liu and Yao  studied an uncertain integral with respect to multiple Liu processes. After that, Chen and Ralescu  proposed an uncertain integral with respect to the general Liu process. Inspired by the Liu integral, Yao  proposed an uncertain calculus with respect to an uncertain renewal process.

Uncertain differential equation was first proposed by Liu  as a type of differential equation driven by Liu process. Then Chen and Liu  gave a sufficient condition for an uncertain differential equation having a unique solution. After that, Gao  provided an existence and unique theorem under weaker conditions. In 2013, Yao et al.  gave a sufficient condition for it being stable. After that, Sheng  studied the stability in p th moment.

Nowadays, uncertain differential equation has been applied to many areas especially in finance. In 2009, Liu  assumed that the stock price follows a geometric Liu process in the short run and proposed a stock model in an uncertain environment. After that, Chen  derived its American option formulae. In 2011, Peng and Yao  proposed another model to describe the stock price displaying a mean-reverting property in the long run via uncertain differential equation. In 2013, Chen et al.  proposed a stock model with periodic dividends. Assuming that the interest rate follows an uncertain differential equation, Chen and Gao  presented an uncertain interest rate model and calculated its zero-coupon bond. Besides, Liu et al.  proposed an uncertain currency model via uncertain differential equation. For more recent developments in uncertain finance, please refer to Liu .

In 2010, Chen and Liu  gave an analytic solution for the linear uncertain differential equation. Then Liu  proposed a method to solve a special type of nonlinear uncertain differential equation. In addition, some numerical methods were designed to calculate the uncertainty distributions of the solution and the extreme values of the solution by Yao and Chen  and Yao . In this paper, we will give an analytic method to solve another special type of uncertain differential equation. The rest of this paper is organized as follows. Some concepts and theorems about uncertainty theory, uncertain calculus, and uncertain differential equation will be introduced in the 'Uncertainty theory’ section, 'Uncertain calculus’ section, and 'Uncertain differential equation’ section, respectively. Then a special type of uncertain differential equation is solved in the 'An analytic method’ section. Finally, some remarks are made in the 'Conclusions’ section.

## Uncertainty theory

This section will briefly introduce some basic concepts in uncertainty theory, including uncertainty space, uncertain variable, independence, and the operational law.

### Definition 1

(Liu ) Let be a σ-algebra on a nonempty set Γ. A set function $ℳ:ℒ→[0,1]$ is called an uncertain measure if it satisfies the following axioms:Axiom 1. (Normality Axiom) $ℳ{Γ}=1$ for the universal set Γ.Axiom 2. (Duality Axiom) $ℳ{Λ}+ℳ{ Λ c }=1$ for any event Λ.Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ 1,Λ 2,…, we have

$ℳ ⋃ i = 1 ∞ Λ i ≤ ∑ i = 1 ∞ ℳ Λ i .$

Besides, the product uncertain measure on the product σ-algebre is defined by Liu  as follows:Axiom 4. (Product Axiom) Let $( Γ k , ℒ k , ℳ k )$ be uncertainty spaces for k=1,2,… Then the product uncertain measure on the product σ-algebra satisfies

$ℳ ∏ i = 1 ∞ Λ k = ∧ k = 1 ∞ ℳ k { Λ k }$

where Λ k are arbitrarily chosen events from $ℒ k$ for k=1,2,…, respectively.

In order to represent the quantities with uncertainty, an uncertain variable was proposed as a real valued function on an uncertainty space.

### Definition 2

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

The uncertainty distribution Φ of an uncertain variable is defined by $Φ(x)=ℳ{ξ≤x}$ for any real number x. The expected value of an uncertain variable ξ is

$E [ ξ ] = ∫ 0 + ∞ ℳ { ξ ≥ r } d r - ∫ - ∞ 0 ℳ { ξ ≤ r } d r$

provided that at least one of the two integrals is finite, and the variance of ξ is

$V [ ξ ] = E ξ - E [ ξ ] 2 .$

An uncertain variable ξ is said to be normal if it has an uncertainty distribution

$Φ ( x ) = 1 + exp π ( e - x ) 3 σ - 1 , x ∈ ℜ$

where e and σ2 are the expected value and variance of ξ, respectively.

### Definition 3

(Liu ) The uncertain variables ξ 1,ξ 2,…,ξ n are said to be independent if

$ℳ ⋂ i = 1 n ( ξ i ∈ B i ) = ∧ i = 1 n ℳ { ξ i ∈ B i }$

for any Borel sets B 1,B 2,…,B n .

Since the definition of independence for uncertain variables is quite different from that for random variables, the operational law of uncertain variables is also different.

### Theorem 1

(Liu ) Let ξ 1,ξ 2,…,ξ n be independent uncertain variables with uncertainty distributions Φ 1,Φ 2,…,Φ n , respectively. If f(x 1,x 2,…,x n ) is strictly increasing with respect to x 1,x 2,…,x m and strictly decreasing with respect to x m+1,x m+2,…,x n , then ξ=f(ξ 1,ξ 2,…,ξ n ) is an uncertain variable with an inverse uncertainty distribution

$Φ - 1 r = f Φ 1 - 1 r , … , Φ m - 1 r , Φ m + 1 - 1 1 - r , … , Φ n - 1 1 - r .$

## Uncertain calculus

An uncertain process is essentially a sequence of uncertain variables indexed by time or space. As one of the most important types of uncertain processes, the canonical Liu process is defined as follows:

### Definition 4

(Liu ) An uncertain process C t is called a canonical Liu process if:

1. 1.

C 0=0 and almost all sample paths are Lipschitz continuous.

2. 2.

C t has stationary and independent increments.

3. 3.

Every increment C s+t -C s is a normal uncertain variable $N(0,t)$ with expected value 0 and variance t 2 whose uncertainty distribution is

$Φ t x = 1 + exp - πx 3 t - 1 , x ∈ R.$

Based on the canonical Liu process, Liu  proposed an uncertain integral of an uncertain process with respect to the canonical Liu process and thus founded a theory of uncertain calculus.

### Definition 5

(Liu ) Let X t be an uncertain process and C t be a canonical Liu process. For any partition of closed interval [ a,b] with a=t 1<t 2<<t k+1=b, the mesh is written as

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

Then the Liu integral of X t is defined by

$∫ a b X t d C t = lim Δ → 0 ∑ i = 1 k X t i · C t i + 1 - C t i$

provided that the limit exists almost surely and is finite.

For example, the Liu integral of an integrable real function f(t) is

$∫ 0 t f s d C s ∼ 0 , ∫ 0 t | f s | d s$

at each time t.

### Definition 6

(Liu ) Let X t be an uncertain process and N t be an uncertain renewal process. Then the Yao integral of X t is defined by

$∫ a b X t d N t = ∑ a < t ≤ b X t - ( N t - N t - )$

provided that the sum exists almost surely and is finite.

For a continuously differentiable function h(t,c,n), the uncertain process Z t =h(t,C t ,N t ) has an uncertain differential

$d Z t = ∂h ∂t ( t , C t , N t ) d t + ∂h ∂c ( t , C t , N t ) d C t + h ( t , C t , N t ) - h ( t , C t , N t - ) .$

For example, the uncertain differential of an uncertain process X t =μ t+σ C t +γ N t is

$d X t = μ d Z t + σ d C t + γ ( N t - N t - ) ,$

and the uncertain differential of an uncertain process Y t =tC t N t is

$d Y t = C t N t d t + t N t d C t + tC t ( N t - N t - ) .$

## Uncertain differential equation

### Definition 7

(Liu ) Suppose C t is a canonical Liu process, and f and g are two given functions. Then

$d X t =f(t, X t )dt+g(t, X t )d C t$
(1)

is called an uncertain differential equation.

In 2010, Chen and Liu  solved the linear uncertain differential equation

$d X t = ( u 1 t X t + u 2 t ) d t + ( v 1 t X t + v 2 t ) d C t$

and obtained a solution

$X t = Y t X 0 + ∫ 0 t u 2 s Y s d s + ∫ 0 t v 2 s Y s d C s$

where

$Y t = exp ∫ 0 t u 1 s d s + ∫ 0 t v 1 s d C s .$

After that, Liu  provided a method to solve the following types of uncertain differential equations:

$d X t = f ( t , X t ) d t + σ t X t d C t ,$
(2)
$d X t = μ t X t d t + g ( t , X t ) d C t ,$
(3)

and showed that uncertain differential equation (2) has a solution $X t = Y t - 1 Z t$ where

$Y t = exp - ∫ 0 t σ s d C s$

and Z t solves uncertain differential equation $d Z t = Y t f(t, Y t - 1 Z t )dt$ with an initial value Z 0=X 0, and uncertain differential equation (3) has a solution $X t = Y t - 1 Z t$ where

$Y t = exp - ∫ 0 t α s d s$

and Z t solves uncertain differential equation $d Z t = Y t g(t, Y t - 1 Z t )d C t$ with an initial value Z 0=X 0.

## An analytic method

In this section, we will propose an analytic method to solve uncertain differential equations

$d X t = f ( t , X t ) d t + σ t d C t$

and

$d X t = μ t d t + g ( t , X t ) d C t .$

### Model I

#### Theorem 2

Let f be a function of two variables, and σ t be an integrable function on +. Then the uncertain differential equation

$d X t = f ( t , X t ) d t + σ t d C t$

has a solution

$X t = Y t + Z t$

where

$Y t = ∫ 0 t σ s d C s$

and Z t is the solution of uncertain differential equation

$d Z t = f ( t , Y t + Z t ) d t , Z 0 = X 0 .$

#### Proof

The uncertain process Y t has an uncertain differential

$d Y t = μ t d t.$

Then we have

$d ( X t - Y t ) = d X t - d Y t = f ( t , X t ) d t + σ t d C t - σ t d C t .$

That is,

$d ( X t - Y t ) = f ( t , X t ) d t.$

Defining Z t =X t -Y t , we obtain X t =Y t +Z t and dZ t =f(t,Y t +Z t )dt. Furthermore, since Y 0=0, the initial value Z 0 is just X 0. The theorem is verified. □

#### Remark 1

If σ t degenerates to a constant σ, then Y t =σ C t . The uncertain differential equation

$d X t = f ( t , X t ) d t + σ d C t$

has a solution

$X t = Z t + σ C t$

where Z t solves the uncertain differential equation

$d Z t = f ( t , Z t + σ C t ) d t , Z 0 = X 0 .$

#### Example 1

Let m t ,μ t , and σ t be some real functions on . Consider the uncertain differential equation

$d X t = ( m t - μ t X t ) d t + σ t d C t .$

At first, we have

$Y t = ∫ 0 t σ s d C s$

and Z t solves the uncertain differential equation

$d Z t = m t - μ t ∫ 0 t σ s d s + Z t d t ,$

so

$d Z t + μ t Z t d t = m t - μ t ∫ 0 t σ s d C s d t.$

It follows from the fundamental theorem of uncertain calculus that

$d exp ∫ 0 t μ s d s Z t = exp ∫ 0 t μ s d s m t - μ t ∫ 0 t σ s d C s d t.$

That is,

$exp ∫ 0 t μ s d s Z t - Z 0 = ∫ 0 t exp ∫ 0 s μ v d v m s - μ s ∫ 0 s σ v d C v d s.$

As a result,

$Z t = exp - ∫ 0 t μ s d s Z 0 + ∫ 0 t exp - ∫ s t μ v d v m s - μ s ∫ 0 s σ v d C v d s = exp - ∫ 0 t μ s d s X 0 + ∫ 0 t exp - ∫ s t μ v d v m s - μ s ∫ 0 s σ v d C v d s.$

By Theorem 2, we have

$X t = Y t + Z t = exp - ∫ 0 t μ s d s X 0 + ∫ 0 t σ s d C s + ∫ 0 t exp - ∫ s t μ v d v m s - μ s ∫ 0 s σ v d C v d s.$

#### Example 2

Let μ and σ be real numbers with μ≠0. Consider the uncertain differential equation

$d X t = μ exp ( X t ) d t + σ d C t .$

At first, we have Y t =σ C t and Z t satisfies the uncertain differential equation

$d Z t = μ exp ( σ C t + Z t ) d t ,$

so

$exp ( - Z t ) d Z t = μ exp ( σ C t ) d t.$

It follows from the fundamental theorem of uncertain calculus that

$d exp ( - Z t ) = - μ exp ( σ C t ) d t.$

That is,

$exp ( - Z t ) - exp ( - Z 0 ) = - μ ∫ 0 t exp ( σ C s ) d s.$

As a result,

$Z t = Z 0 - ln 1 - μ ∫ 0 t exp Z 0 + σ C s d s = X 0 - ln 1 - μ ∫ 0 t exp X 0 + σ C s d s .$

By Theorem 2, we have

$X t = Y t + Z t = X 0 + σ C t - ln 1 - μ ∫ 0 t exp X 0 + σ C s d s .$

### Model II

#### Theorem 3

Let μ t be an integrable function on +, and g be a function of two variables. Then the uncertain differential equation

$d X t = μ t d t + g ( t , X t ) d C t$

has a solution

$X t = Y t + Z t$

where

$Y t = ∫ 0 t μ s d s$

and Z t is the solution of uncertain differential equation

$d Z t = g ( t , Y t + Z t ) d C t , Z 0 = X 0 .$

#### Proof

The uncertain process Y t has an uncertain differential

$d Y t = μ t d t.$

Then we have

$d ( X t - Y t ) = d X t - d Y t = μ t d t + g ( t , X t ) d C t - μ t d C t .$

That is,

$d ( X t - Y t ) = g ( t , X t ) d C t .$

Defining Z t =X t -Y t , we obtain X t =Y t +Z t and dZ t =g(t,Y t +Z t )dC t . Furthermore, since Y 0=0, the initial value Z 0 is just X 0. The theorem is verified. □

#### Remark 2

If μ t degenerates to a constant μ, then Y t =μ t, and the uncertain differential equation

$d X t = μ d t + g ( t , X t ) d C t$

has a solution

$X t = Z t + μt$

where Z t solves the uncertain differential equation

$d Z t = g ( t , Z t + μt ) d t , Z 0 = X 0 .$

#### Example 3

Let m t ,μ t , and σ t be some real functions on . Consider the uncertain differential equation

$d X t = μ t d t + ( m t - σ t X t ) d C t .$

At first, we have

$Y t = ∫ 0 t μ s d s$

and Z t solves the uncertain differential equation

$d Z t = m t - σ t ∫ 0 t μ s d s + Z t d C t ,$

so

$d Z t + σ t Z t d C t = m t - σ t ∫ 0 t μ s d s d C t .$

It follows from the fundamental theorem of uncertain calculus that

$d exp ∫ 0 t σ s d C s Z t = exp ∫ 0 t σ s d C s m t - σ t ∫ 0 t μ s d s d C t .$

That is,

$exp ∫ 0 t σ s d C s Z t - Z 0 = ∫ 0 t exp ∫ 0 s σ v d C v m s - σ s ∫ 0 s μ v d v d C s .$

As a result,

$Z t = exp - ∫ 0 t σ s d C s Z 0 + ∫ 0 t exp - ∫ s t σ v d C v m s - σ s ∫ 0 s μ v d v d C s = exp - ∫ 0 t σ s d C s X 0 + ∫ 0 t exp - ∫ s t σ v d C v m s - σ s ∫ 0 s μ v d v d C s .$

By Theorem 3, we have

$X t = Y t + Z t = exp - ∫ 0 t σ s d C s X 0 + ∫ 0 t μ s d s + ∫ 0 t exp - ∫ s t σ v d C v m s - σ s ∫ 0 s μ v d v d C s .$

#### Example 4

Let μ and σ be real numbers with σ≠0. Consider the uncertain differential equation

$d X t = μ d t + σ exp ( X t ) d C t .$

At first, we have Y t =μ t and Z t satisfies the uncertain differential equation

$d Z t = σ exp ( μt + Z t ) d C t ,$

so

$exp ( - Z t ) d Z t = σ exp ( μt ) d C t .$

It follows from the fundamental theorem of uncertain calculus that

$d exp ( - Z t ) = - σ exp ( μt ) d C t .$

That is,

$exp ( - Z t ) - exp ( - X 0 ) = - σ ∫ 0 t exp ( μs ) d C s .$

As a result,

$Z t = Z 0 - ln 1 - σ ∫ 0 t exp Z 0 + μs d C s = X 0 - ln 1 - σ ∫ 0 t exp X 0 + μs d C s .$

By Theorem 3, we have

$X t = Y t + Z t = X 0 + μt - ln 1 - σ ∫ 0 t exp X 0 + μs d C s .$

## Conclusions

This paper proposed a method to solve a special type of uncertain differential equation and employed some examples to illustrate the method.

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## Acknowledgements

This work was supported by National Natural Science Foundation of China Grant Nos. 61273044 and 91224008.

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Correspondence to Kai Yao.

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Yao, K. A type of uncertain differential equations with analytic solution. J. Uncertain. Anal. Appl. 1, 8 (2013). https://doi.org/10.1186/2195-5468-1-8 