Fuzzy Optim Decis MakingDOI 10.1007/s10700-014-9188-y

Almost sure stability for uncertain differential equation

Hongjian Liu · Hua Ke · Weiyin Fei

© Springer Science+Business Media New York 2014

Abstract Uncertain differential equation is a type of differential equation driven byLiu process. So far, concepts of stability and stability in mean for uncertain differentialequations have been proposed. This paper aims at providing a concept of almost surestability for uncertain differential equation. A sufficient condition is given for anuncertain differential equation being almost surely stable, and some examples aregiven to illustrate the effectiveness of the sufficient condition.

Keywords Uncertainty theory · Uncertain differential equation · Stability ·Almost sure stability

1 Introduction

Probability theory has been used to model indeterminacy phenomena for a long time.A premise of applying probability theory is that the obtained probability distributionis close enough to the frequency. However, we sometimes have no samples to estimatethe probability distribution. In this case, we invite some domain experts to evaluatetheir belief degree that each event happens. Since human tends to overweight unlikelyevents (Kahneman and Tversky 1979), the belief degree has a much larger variance

H. Liu (B) · W. FeiSchool of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, Chinae-mail: liu@ahpu.edu.cn

W. Feie-mail: wyfei@ahpu.edu.cn

H. KeSchool of Economics and Management, Tongji University, Shanghai 200092, Chinae-mail: hke@tongji.edu.cn

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than the frequency. In this case, probability theory is not applicable (Liu 2012). Inorder to deal with the belief degree, an uncertainty theory was founded by Liu (2007)and refined by Liu (2010).

Based on uncertainty theory, a concept of uncertain process was given by Liu(2008) as a sequence of uncertain variables indexed by time or space. Then Liu(2014) proposed a concept of uncertainty distribution of uncertain process. Liu (2009)designed a canonical Liu process, based on which an uncertain calculus theory wasfounded to deal with the integral and differential of an uncertain process. Recently,Chen and Ralescu (2013) defined a Liu process by the uncertain integral. In addi-tion, Yao (2012a) founded uncertain calculus with respect to an uncertain renewalprocess.

Uncertain differential equation, as a type of differential equation driven by a Liuprocess, was proposed by Liu (2008). Then Chen and Liu (2010) gave an analyticsolution to a linear uncertain differential equation. After that, Liu (2012) and Yao(2013b) gave two methods to solve some special types of nonlinear uncertain differ-ential equations. Besides, Yao and Chen (2013) and Yao (2013a) proposed a numericalmethod to solve an uncertain differential equation and calculate the extreme valuesof the solution. Uncertain differential equation was first introduced to finance by Liu(2009), in which he proposed an uncertain stock model. Then Chen (2011) gave aformula for pricing its American options. Besides, Peng and Yao (2011) proposed amean-reverting stock model to describe the stock price in a long term, and Yao (2012b)gave a sufficient condition for the mean-reverting stock model being no-arbitrage. Inaddition, uncertain currency model was proposed by Liu et al. (2013), and uncer-tain interest model was proposed by Chen and Gao (2013). A review about uncertainfinance was provided by Liu (2013).

With many applications of uncertain differential equation, its properties also drew alot of researches. Chen and Liu (2010) first gave a sufficient condition for an uncertaindifferential equation having a unique solution. Then Gao (2012) weakened the condi-tion. After that, Liu and Fei (2012) and (2013) addressed the existence and uniquenessconditions for uncertain functional differential equations and neutral uncertain delaydifferential equations, respectively. The concept of stability for uncertain differentialequation was first proposed by Liu (2009). Then Yao et al. (2013) gave a sufficientcondition for an uncertain differential equation being stable. After that, concepts ofstability in mean Yao and Sheng (2013) and stability in pth moment Sheng (2013)were proposed and studied. In this paper, we will propose a concept of almost surestability for an uncertain differential equation. The rest of this paper is organized asfollows. In Sect. 2, we will review some concepts and theorems in uncertainty theoryand uncertain differential equation. After that, a concept of almost sure stability willbe presented in Sect. 3, and a sufficient condition will be given in Sect. 4. At last, someconclusions will be made in Sect. 5.

2 Preliminary

In this section, we introduce some useful definitions about uncertain variable anduncertain differential equation.

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2.1 Uncertainty theory

Let � be a nonempty set, and L be a σ -algebra on �. Each element � ∈ L is calledan event.

Definition 1 (Liu 2007) Let L be a σ -algebra on a nonempty set �. A set functionM: L → [0, 1] is called an uncertain measure if it satisfies the following axioms:Axiom 1: (Normality Axiom) M{�} = 1 for the universal set �.Axiom 2: (Duality Axiom) M{�} + M{�c} = 1 for any event �.Axiom 3: (Subadditivity Axiom) For every countable sequence of events �1,�2, . . . ,

we have

M{ ∞⋃

i=1

�i

}≤

∞∑i=1

M {�i } .

Besides, the product uncertain measure on the product σ -algebra L was defined byLiu (2009) as follows,Axiom 4: (Product Axiom) Let (�k,Lk,Mk) be uncertainty spaces for k = 1, 2, . . .

Then the product uncertain measure M is an uncertain measure satisfying

M{ ∞∏

i=1

�k

}=

∞∧k=1

Mk{�k}

where �k are arbitrarily chosen events from Lk for k = 1, 2, . . ., respectively.

Definition 2 (Liu 2007) An uncertain variable ξ is a measurable function from anuncertainty space (�,L,M) to the set of real numbers �, i.e., for any Borel set B ofreal numbers, the set

{ξ ∈ B} = {γ ∈ � | ξ(γ ) ∈ B}is an event.

Definition 3 (Liu 2009) The uncertain variables ξ1, ξ2, . . . , ξn are said to be indepen-dent if

M{

n⋂i=1

(ξi ∈ Bi )

}=

n∧i=1

M {ξi ∈ Bi }

for any Borel sets B1, B2, . . . , Bn of real numbers.

In order to describe an uncertain variable, a concept of uncertainty distribution isdefined as follows.

Definition 4 (Liu 2007) The uncertainty distribution of an uncertain variable ξ isdefined by

Φ(x) = M{ξ ≤ x}for any x ∈ �.

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For an uncertain variable ξ with an uncertainty distribution Φ, if the inverse functionΦ−1 exists and is unique for each α ∈ (0, 1), then ξ is called a regular uncertainvariable, and Φ−1(α) is called its inverse uncertainty distribution. Inverse uncertaintydistribution plays a crucial role in the operational law of independent regular uncertainvariables.

Theorem 1 (Liu 2010) Let ξ1, ξ2, . . . , ξn be independent uncertain variables withuncertainty distributions Φ1, Φ2, . . . , Φn, respectively. If f (x1, x2, . . . , xn) is strictlyincreasing with respect to x1, x2, . . . , xm and strictly decreasing with respect toxm+1, xm+2, . . . , xn, then ξ = f (ξ1, ξ2, . . . , ξn) is an uncertain variable with aninverse uncertainty distribution

Φ−1(α) = f(Φ−1

1 (α), . . . , Φ−1m (α),Φ−1

m+1(1 − α), . . . , Φ−1n (1 − α)

).

Definition 5 (Liu 2007) The expected value of an uncertain variable ξ is defined by

E[ξ ] =+∞∫0

M{ξ ≥ x}dx −0∫

−∞M{ξ ≤ x}dx

provided that at least one of the two integrals exists.

For an uncertain variable ξ with an uncertainty distribution , Liu (2007) provedthat if E[ξ ] exists, then

E[ξ ] =+∞∫0

(1 − Φ(x))dx −0∫

−∞Φ(x)dx .

In addition, for independent uncertain variables ξ and η, Liu (2010) proved

E[aξ + bη] = aE[ξ ] + bE[η]

for any real numbers a and b.

2.2 Uncertain differential equation

Uncertain process was proposed by Liu (2008) to model the evolution of uncertain phe-nomena. Canonical Liu process, as a special type of uncertain process, was designedas follows.

Definition 6 (Liu 2009) An uncertain process Ct is said to be a canonical Liu processif

(i) C0 = 0 and almost all sample paths are Lipschitz continuous,(ii) Ct has stationary and independent increments,

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Almost sure stability for uncertain differential equation

(iii) every increment Cs+t − Cs is a normal uncertain variable with an uncertaintydistribution

Φt (x) =(

1 + exp

(− πx√

3t

))−1

, x ∈ �.

Based on canonical Liu process, a concept of Liu integral was defined, which maybe regarded as an uncertain counterpart of the Itô integral.

Definition 7 (Liu 2009) Let Xt be an uncertain process and Ct be a canonical Liuprocess. For any partition of closed interval [a, b] with a = t1 < t2 < · · · < tk+1 = b,

the mesh is written as

� = max1≤i≤k

|ti+1 − ti |.

Then the Liu integral of Xt is defined by

b∫a

Xt dCt = lim�→0

k∑i=1

Xti · (Cti+1 − Cti )

provided that the limit exists almost surely and is finite.

Liu (2010) proved that the Liu integral of an integrable function f (t) is a normaluncertain variable, i.e.,

s∫0

f (t)dCt ∼ N⎛⎝0,

s∫0

| f (t)|dt

⎞⎠ .

Definition 8 (Liu 2008) Suppose that Ct is a canonical Liu process, and f and g aretwo given functions. Then

dXt = f (t, Xt )dt + g(t, Xt )dCt

is called an uncertain differential equation.

In order to solve an uncertain differential equation numerically, Yao and Chen(2013) proposed a concept of α-path as follows.

Definition 9 (Yao and Chen 2013) Theα-path (0 < α < 1)of an uncertain differentialequation

dXt = f (t, Xt )dt + g(t, Xt )dCt

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is a deterministic function Xαt with respect to t that solves the corresponding ordinary

differential equation

dXαt = f (t, Xα

t )dt + |g(t, Xαt )|Φ−1(α)dt

where Φ−1(α) is the inverse uncertainty distribution of a standard normal uncertainvariable, i.e.,

Φ−1(α) =√

3

πln

α

1 − α, 0 < α < 1.

Theorem 2 (Yao and Chen 2013) Let Xt and Xαt be the solution and the α-path of

an uncertain differential equation

dXt = f (t, Xt )dt + g(t, Xt )dCt ,

respectively. Then

M{Xt ≤ Xαt ,∀t ≥ 0} = α,

M{Xt > Xαt ,∀t ≥ 0} = 1 − α.

Based on the above theorem, Yao and Chen (2013) proved that the α-paths providethe inverse uncertainty distribution of the solution, i.e., the solution Xt has an inverseuncertainty distribution

Ψ −1t (α) = Xα

t .

The concept of stability for uncertain differential equation was first proposed by Liu(2009), and was further developed by Yao and Sheng (2013) and Sheng (2013).

Definition 10 Let Xt and Yt be two solutions of the uncertain differential equation

dXt = f (t, Xt )dt + g(t, Xt )dCt (1)

with different initial values X0 and Y0. Then the uncertain differential Eq. (1) is saidto be stable (Liu 2009) if for any given number ε > 0, there exists a real number δ

such that

M{|Xt − Yt | > ε} ≤ ε

holds for any t ≥ 0 provided |X0 − Y0| ≤ δ; the uncertain differential Eq. (1) is saidto be stable in mean (Yao and Sheng 2013) if for any given number ε > 0, there existsa real number δ such that

E[|Xt − Yt |] ≤ ε

holds for any t ≥ 0 provided |X0 − Y0| ≤ δ.

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Almost sure stability for uncertain differential equation

3 Almost sure stability

In this section, we present a concept of almost sure stability for uncertain differentialequations.

Definition 11 Let Xt and Yt be two solutions of the uncertain differential equation

dXt = f (t, Xt )dt + g(t, Xt )dCt (2)

with different initial values X0 and Y0. Then the uncertain differential Eq. (2) is saidto be almost surely stable if

M{

lim|X0−Y0|→0|Xt − Yt | = 0

}= 1, ∀t > 0.

Example 1 Consider the uncertain differential equation

dXt = −Xt dt + σdCt . (3)

Clearly, its two solutions with initial values X0 and Y0 are

Xt = exp(−t)X0 + σ exp(−t)

t∫0

exp(s)dCs,

Yt = exp(−t)Y0 + σ exp(−t)

t∫0

exp(s)dCs,

respectively. Since

|Xt (γ ) − Yt (γ )| = exp(−t)|X0 − Y0| ≤ |X0 − Y0|, ∀t ≥ 0, γ ∈ �,

the uncertain differential (3) is almost surely stable.

Example 2 Consider the uncertain differential equation

dXt = Xt dt + σdCt . (4)

Clearly, its two solutions with initial values X0 and Y0 are

Xt = exp(t)X0 + σ exp(t)

t∫0

exp(−s)dCs,

Yt = exp(t)Y0 + σ exp(t)

t∫0

exp(−s)dCs,

respectively. Since

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|Xt (γ ) − Yt (γ )| = exp(t)|X0 − Y0| → ∞, ∀γ ∈ �

as t tends to infinity, the uncertain differential Eq. (4) is not almost surely stable.

4 Stability theorem

In this section, we give a sufficient condition for an uncertain differential equationbeing almost surely stable.

Theorem 3 The uncertain differential equation

dXt = f (t, Xt )dt + g(t, Xt )dCt (5)

is almost surely stable if the coefficients f (t, x) and g(t, x) satisfy the linear growthcondition

| f (t, x)| + |g(t, x)| ≤ R(1 + |x |), ∀x ∈ �, t ≥ 0

for some constant R, and the strong Lipschitz condition

| f (t, x) − f (t, y)| + |g(t, x) − g(t, y)| ≤ Lt |x − y|, ∀x, y ∈ �, t ≥ 0

for some integrable nonnegative function Lt on [0,+∞).

Proof It follows from Chen and Liu (2010) that for any given initial value, the uncertaindifferential Eq. (5) has a unique solution. Suppose that Xt and Yt are two solutions ofthe uncertain differential equation with different initial values X0 and Y0, respectively.Then for a Lipschitz continuous sample path Ct (γ ), we have

Xt (γ ) = X0 +t∫

0

f (t, Xt (γ ))dt +t∫

0

g(t, Xt (γ ))dCt (γ ),

Yt (γ ) = Y0 +t∫

0

f (t, Yt (γ ))dt +t∫

0

g(t, Yt (γ ))dCt (γ ).

Let K (γ ) denote the Lipstchitz constant of Ct (γ ). Then we have

|Xt (γ ) − Yt (γ )| ≤ |X0 − Y0| +∣∣∣∣∣∣

t∫0

( f (s, Xs(γ )) − f (s, Ys(γ )))ds

∣∣∣∣∣∣+

∣∣∣∣∣∣t∫

0

(g(s, Xs(γ )) − g(s, Ys(γ )))dCs(γ )

∣∣∣∣∣∣123

Almost sure stability for uncertain differential equation

≤ |X0 − Y0| +t∫

0

Ls |Xs(γ ) − Ys(γ )|ds

+K (γ )

t∫0

Ls |Xs(γ ) − Ys(γ )|ds

= |X0 − Y0| + (1 + K (γ ))

t∫0

Ls |Xs(γ ) − Ys(γ )|ds.

By the Grownwall inequality, we get

|Xt (γ ) − Yt (γ )| ≤ |X0 − Y0| exp

⎛⎝(1 + K (γ ))

t∫0

Lsds

⎞⎠

≤ |X0 − Y0| exp

⎛⎝(1 + K (γ ))

+∞∫0

Lsds

⎞⎠ .

Since Lt is integrable on [0,+∞) and K (γ ) is finite, we obtain that |Xt (γ )−Yt (γ )| →0 as long as |X0 − Y0| → 0. In other words,

M{

lim|X0−Y0|→0|Xt − Yt | = 0

}= 1, ∀t > 0.

So the uncertain differential Eq. (5) is almost surely stable. ��

Example 3 Consider the nonlinear uncertain differential equation

dXt = exp (−t) Xt dt + exp(−t2 − X2

t

)dCt . (6)

Since f (t, x) = exp (−t) x and g(t, x) = exp(−t2 − x2

)satisfy the linear growth

condition

| f (t, x)| + |g(t, x)| ≤ |x | + 1.

and the strong Lipschitz condition

| f (t, x) − f (t, y)| + |g(t, x) − g(t, y)| ≤(

exp (−t) + exp(−t2

))|x − y|

for all x, y ∈ � and t ≥ 0, the nonlinear uncertain differential Eq. (6) is almost surelystable.

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Remark 1 Theorem 3 gives a sufficient but not necessary condition for an uncertaindifferential equation being almost surely stable. Note that the uncertain differentialEq. (3) does not meet with the strong Lipstchitz condition, but it is still almost surelystable.

Corollary 1 The linear uncertain differential equation

dXt = (u1t Xt + u2t )dt + (v1t Xt + v2t )dCt (7)

is almost surely stable if u1t , u2t , v1t and v2t are bounded functions, and

+∞∫0

|u1t |dt < +∞,

+∞∫0

|v1t |dt < +∞.

Proof Take f (t, x) = u1t x + u2t and g(t, x) = v1t x + v2t in Theorem 3. Let Rdenote a common upper bound of |u1t |, |u2t |, |v1t | and |v2t |. Since

| f (t, x)| + |g(t, x)| = |u1t x + u2t | + |v1t x + v2t |≤ |u1t ||x | + |u2t | + |v1t ||x | + |v2t | ≤ 2R(1 + |x |),

the linear growth condition is satisfied. Besides, we have

rl| f (t, x) − f (t, y)| + |g(t, x) − g(t, y)| = |u1t ||x − y| + |v1t ||x − y|= (|u1t | + |v1t |)|x − y|.

Since

+∞∫0

Lt dt =+∞∫0

|u1t |dt ++∞∫0

|v1t |dt < +∞,

the strong Lipstchitz condition is also satisfied. Thus it follows from Theorem 3 thatthe linear uncertain differential Eq. (7) is almost surely stable. ��

5 Conclusions

This paper mainly proposed a concept of almost sure stability for an uncertain dif-ferential equation. A sufficient condition for an uncertain differential equation beingalmost surely stable was provided, and some examples were given to illustrate theeffectiveness of the condition.

Acknowledgments This work was supported by the National Natural Science Foundation of China GrantsNo.71171003, No.61203139, No.71371141, and No.71001080, and the Natural Science Foundation ofAnhui Universities Grant No.KJ2013B023.

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Almost sure stability for uncertain differential equation

References

Chen, X. (2011). American option pricing formula for uncertain financial market. International Journal ofOperations Research, 8(2), 32–37.

Chen, X., & Gao, J. (2013). Uncertain term structure model of interest rate. Soft Computing, 17(4), 597–604.Chen, X., & Liu, B. (2010). Existence and uniqueness theorem for uncertain differential equations. Fuzzy

Optimization and Decision Making, 9(1), 69–81.Chen, X., & Ralescu, D. A. (2013). Liu process and uncertain calculus. Journal of Uncertainty Analysis

and Applications, 1, Article 3.Gao, Y. (2012). Existence and uniqueness theorem on uncertain differential equations with local Lipschitz

condition. Journal of Uncertain Systems, 6(3), 223–232.Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica,

47(2), 263–292.Liu, B. (2007). Uncertainty Theory (2nd ed.). Berlin: Springer.Liu, B. (2008). Fuzzy process, hybrid process and uncertain process. Journal of Uncertain Systems, 2(1),

3–16.Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin:

Springer.Liu, B. (2012). Why is there a need for uncertainty theory? Journal of Uncertain Systems, 6(1), 3–10.Liu, B. (2013). Toward uncertain finance theory. Journal of Uncertainty Analysis and Applications, 1,

Article 1.Liu, B. (2014). Uncertainty distribution and independence of uncertain processes. Fuzzy Optimization and

Decision Making, to be published.Liu, Y. H. (2012). An analytic method for solving uncertain differential equations. Journal of Uncertain

Systems, 6(4), 244–249.Liu, H. J., & Fei, W. Y. (2012). Existence and uniqueness of solutions to uncertain functional differen-

tial equations. In: Proceedings of the ninth international conference on fuzzy systems and knowledgediscovery (pp. 63–66).

Liu, H. J., & Fei, W. Y. (2013). Neutral uncertain delay differential equations. Information. An InternationalInterdisciplinary Journal, 16(2(A)), 1225–1232.

Liu, Y. H., Chen, X., & Ralescu, D. A. (2013). Uncertain currency model and currency option pricing.International Journal of Intelligent Systems, to be published.

Peng, J., & Yao, K. (2011). A new option pricing model for stocks in uncertainty markets. InternationalJournal of Operations Research, 8(2), 18–26.

Sheng, Y. (2013). Stability in the pth moment for uncertain differential equation. Journal of Intelligent andFuzzy Systems, to be published.

Yao, K. (2012a). Uncertain calculus with renewal process. Fuzzy Optimization and Decision Making, 11(3),285–297.

Yao, K. (2012b). No-arbitrage determinant theorems on mean-reverting stock model in uncertain market.Knowledge-Based Systems, 35, 259–263.

Yao, K. (2013a). Extreme values and integral of solution of uncertain differential equation. Journal ofUncertainty Analysis and Applications, 1, Article 2.

Yao, K. (2013b). A type of uncertain differential equations with analytic solution. Journal of UncertaintyAnalysis and Applications, 1, Article 8.

Yao, K., & Chen, X. (2013). A numerical method for solving uncertain differential equations. Journal ofIntelligent and Fuzzy Systems, 25(3), 825–832.

Yao, K., & Sheng Y. (2013). Stability in mean for uncertain differential equation. http://orsc.edu.cn/online/120611

Yao, K., Gao, J., & Gao, Y. (2013). Some stability theorems of uncertain differential equation. FuzzyOptimization and Decision Making, 12(1), 3–13.

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