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On a Class of Nonlocal Stochastic FunctionalDifferential Equations with Infinite DelayFuke Wu a & Shigeng Hu aa School of Mathematics and Statistics, Huazhong University of Science and Technology,Wuhan, Hubei, P. R. China

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To cite this article: Fuke Wu & Shigeng Hu (2011): On a Class of Nonlocal Stochastic Functional Differential Equations withInfinite Delay, Stochastic Analysis and Applications, 29:4, 713-721

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Stochastic Analysis and Applications, 29: 713–721, 2011Copyright © Taylor & Francis Group, LLCISSN 0736-2994 print/1532-9356 onlineDOI: 10.1080/07362994.2011.581112

On a Class of Nonlocal Stochastic FunctionalDifferential Equations with Infinite Delay

FUKE WU AND SHIGENG HU

School of Mathematics and Statistics, Huazhong University of Scienceand Technology, Wuhan, Hubei, P. R. China

This article considers a class of nonlocal stochastic functional differential equationswith infinite delay whose coefficients are dependent the pth moment and establishesthe existence-and-uniqueness theorem under the conditions that are similar to theclassical linear growth condition and the Lipschitz condition. Compared with theexisting results, the conditions of this article are easier to test.

Keywords Existence and uniqueness; Moment dependence; Nonlocal stochasticfunctional differential equations.

Mathematics Subject Classification 60H10; 34K50.

1. Statement of the Problem

Since Itô established his stochastic calculus, the theory of stochastic differentialequations has been developed very quickly and widely applied in populationbiology, engineering, neural networks, mathematical finance and almost all appliedsciences. The classical Itô-type stochastic differential equations may be written asthe form

dx�t� = f�t� x�t��dt + g�t� x�t��dw�t�� (1.1)

whose solution (the strong solution) is a sample path which only depends on its owninitial value and not those of other sample paths. This class of equations is called asthe local stochastic differential equation in [5, 6] in term of the point � in the samplespace �.

Received October 7, 2010; Accepted October 12, 2010The authors would like to thank the referees for their detailed comments and

helpful suggestions. They also wish to thank the National Natural Science Foundationof China (Grant No. 11001091) and Chinese University Research Foundation (Grant No.2010MS129) for their financial supports.

Address correspondence to Fuke Wu, School of Mathematics and Statistics, HuazhongUniversity of Science and Technology, Wuhan, Hubei 430074, P. R. China; E-mail:wufuke@mail.hust.edu.cn

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However, the stock market shows that the stock price depends not only on itsown price, but also on its expectation and variance, namely, the first moment andthe second moment. It is more reasonable to replace Equation (1.1) by the stochasticdifferential equation of the form

dx�t� = f�t� x�t���x�t����x�t��2�dt + g�t� x�t���x�t����x�t��2�dw�t�� (1.2)

where �x�t� and ��x�t��2 represent the mean and the second moment, respectively.Since the first and second moments are determined by all the sample points inthe sample space �, Equation (1.2) is therefore a nonlocal stochastic differentialequation.

There exist some, but not many papers to concern the systems similar toEquation (1.2). Hernandez et al. [3] examined a class of stochastic evolutionequations dependent a family of probability distributed measure of the solutionsand established the existence and uniqueness of the solutions. [5, 6] considered thestochastic differential equation of the form

dx�t� = f�t� x�t�� x�t� ·��dt + g�t� x�t�� x�t� ·��dw�t�� (1.3)

where x�t� ·� represents all the sample points in the sample space �. These equationsdepend on all the sample and are named as the nonlocal stochastic differentialequation. It is obvious that both Equation (1.3) and the equation in [3] containEquation (1.2) as a special case.

Although the so-called Efficient Market Hypothesis implies that all informationavailable is already reflected in the present price of the stock and past stockperformance gives no information, some statistical data of stock prices (see [1, 9])indicated the dependence on past behaviors. It is, therefore, interesting to considera class of stochastic differential equations dependent upon not only their momentsbut also their history. This article will examine the nonlocal stochastic functionaldifferential equation with infinite delay

dx�t� = f�t� xt� �xt�p�dt + g�t� xt� �xt�p�dw�t�� (1.4)

where xt = xt��� =� �x�t + �� � � ∈ �−�� 0, f � �+ × BC��−�� 0��n�×�+ →�n and g � �+ × BC��−�� 0��n�×�+ → �n×m are Borel measurable, w�t� isan m-dimensional Brownian motion, p ≥ 2 and �xt�p is a norm in the spaceLp��−�� 0×���n�, namely,

�xt�p =[ ∫ 0

−���x�t + ���pd����

] 1p

� (1.5)

in which � is a probability measure.It is clear that Equation (1.3) and the equation in [3] are more general than

Equation (1.4), but it is difficult to test their conditions on the existence anduniqueness of solutions, for example, in [5, 6], the conditions of the existenceand uniqueness of solutions for Equation (1.3) contain the computation of theexpectation which is difficult since the distributed function is generally unknown.In [3], the conditions of the existence and uniqueness are dependent on a metricfunction, which is also difficult to be computed. For Equation (1.4), as a special

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Nonlocal SFDEs with Infinite Delay 715

case of the Equation (1.3) and the equation in [3], this article will present someconditions under which the existence-and-uniqueness theorems are established.These conditions are similar to the classical Lipschitz condition and the lineargrowth condition in [2, 7] and hence it is easy for these conditions to be tested.

2. Nonlocal Stochastic Functional Differential Equations

Throughout this article, unless otherwise specified, we use the following notation.Let ���� ��� be a complete probability space with a filtration ��tt≥0 satisfyingthe usual conditions, that is, it is right continuous and increasing while �0 containsall �-null sets. Let w�t� be an m-dimensional Brownian motion defined on thisprobability space. If x�t� is an �n-valued stochastic process, define xt = xt��� �=�x�t + �� � −� < � ≤ 0 for t ≥ 0.

Let � · � be the Euclidean norm in �n. Let �+ = 0���. Denote byC��−�� 0��n� the family of continuous functions from �−�� 0 to �n. Similarly,denote by BC��−�� 0��n� the family of bounded continuous functions from�−�� 0 to �n with the norm ��� = sup−�<�≤0 ������ < �, which forms aBanach space. Let L

p�0��−�� 0��n� denote the family of all �0-measurable

BC��−�� 0��n�-valued random variables � such that ����p < �. If a� b ∈ �, a ∨b represents the maximum of a and b.

The main aim of this article is to establish the existence-and-uniqueness theoremfor Equation (1.4). Assume that the initial data x0 = � satisfies

x0 = � ∈ Lp�0��−�� 0��n�� (2.1)

Then we present the definition of the solution for Equation (1.4):

Definition 2.1. An �n-valued stochastic process x�t� on −� < t ≤ T is called asolution to Equation (1.4) with initial data (2.1) if it has the following properties:

(i) x�t� is continuous and �t-measurable and �xt � 0 ≤ t ≤ T is an �0-measurableBC��−�� 0��n�-valued stochastic process;

(ii) x�t� satisfies

∫ T

0�[

sup−�<s≤t

�x�s��p]dt < �� (2.2a)

∫ T

0�f�t� xt� �xt�p��dt ∨

∫ T

0�g�t� xt� �xt�p��2dt < �� a�s� (2.2b)

(iii) x0 = � and

x�t� = ��0�+∫ t

0f�s� xs� �xs�p�ds +

∫ t

0g�s� xs� �xs�p�dw�s� (2.3)

for all t ∈ 0� T.

A solution x�t� is said to be unique if any other solution x̄�t� is indistinguishablefrom it, that is

��x�t� = x̄�t� for all −� < t ≤ T = 1� (2.4)

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Remark. Compared with the existing definition of the solution in [7, Definition2.1, P. 149], Definition 2.1 requires condition (2.2a). This is reasonable sinceEquation (1.4) is dependent on the pth moment.

To establish the existence and uniqueness of the solution to Equation (1.4), letus impose the following assumptions on the coefficients f and g:

Assumption 2.1 (Uniform Lipschitz Condition). For all �� �̄ ∈ BC��−�� 0��n�and x� x̄ ∈ �n, there exists constant K such that for all t ∈ 0� T,

�f�t� �� x�− f�t� �̄� x̄�� ∨ �g�t� �� x�− g�t� �̄� x̄�� ≤ K���− �̄� + �x − x̄��� (2.5)

Assumption 2.2 (Linear Growth Condition). For all � ∈ BC��−�� 0��n� and x ∈�n, there exists constant K such that for all t ∈ 0� T,

�f�t� �� x�� ∨ �g�t� �� x�� ≤ K�1+ ��� + �x��� (2.6)

Then the existence-and-uniqueness of the solutions to Equation (1.4) follows.

Theorem 2.1. Under Assumptions 2.1 and 2.2, there exists a unique solution x�t� toEquation (1.4) with the initial data (2.1) on all t ∈ 0� T.

Proof. We divide this proof into the following four steps.

Step 1. For any � > 0, let us establish the function space

X = �x�t� � x�t� is �t-measurable �n-vaule stochastic process with �x�� < ��(2.7)

where

�x�� = sup0≤t≤T

e−�t

[ ∫ t

0�(

sup−�<r≤s

�x�r��p)ds

] 1p

� (2.8)

It is obvious that for any x ∈ X and t ∈ 0� T,

∫ t

0�(

sup−�<r≤s

�x�r��p)ds ≤ e�pt�x�p� � (2.9)

It is easy to test that � · �� is a norm defined in X. Let

D =� �x ∈ X � x0 = �� (2.10)

We claim that D ⊂ X is not empty. Choose x�t� ≡ ��0� for all t ∈ 0� T. It is clearthat x�t� ∈ D, which shows that D is not empty. We then claim that D is completewith respect to the norm � · ��. Choose the sequence �xkk≥0 ∈ D satisfying �xk −xl�� → 0 as k� l → �. By (2.9),

∫ T

−���xk�t�− xl�t��pdt → 0 as k� l → �� (2.11)

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Nonlocal SFDEs with Infinite Delay 717

Note that the space Lp��−�� 0��n� is complete. By (2.11), there exists asubsequence �xki �t�i≥0 such that xki → x a.s. where x�t� �� � �−�� 0×� → �n

is �t-measurable with x0 = �. For any � > 0, choose k0 > 0 such that for anyk� ki ≥ k0,

�xk − xki�� < ��

Applying the Fatou theorem (cf. [4]) yields that �xk − x�� ≤ � for any k ≥ k0. Thisshows that �xk − x�� → 0 as k → � and hence

�x�� ≤ �+ �xk�� < ��

which implies that x ∈ D and D is complete.

Step 2. Introduce the mapping � � x → x̂, wherex̂�t� = ��t� for t ∈ �−�� 0�

x̂�t� = ��0�+∫ t

0f�s� xs� �xs�p�ds +

∫ t

0g�s� xs� �xs�p�dw�s� for t ∈ 0� T�

(2.12)

Choose x ∈ D. It is obvious that x̂�t� is �t-measurable. Recalling the elementaryinequality: for any c1� � � � � ch ≥ 0 and any integer h ≥ 1,

( h∑i=1

ci

)p

≤ hp−1h∑

i=1

cpi � (2.13)

For any t ∈ 0� T, by (2.9) and (2.12), applying the Hölder inequality, theBurkholder-Davis-Gundy inequality (cf. [7, Theorem 7.3, p. 40]) and Assumption 2.2yields

�[

sup−�<s≤t

�x̂�s��p]

≤ ����p +�[sup0≤s≤t

�x̂�s��p]

= ����p +�[sup0≤s≤t

∣∣∣��0�+∫ s

0f�r� xr� �xr�p�dr +

∫ t

0g�r� xr� �xr�p�dw�r�

∣∣∣p]

≤ ����p + 3p−1����p + 3p−1�[ ∫ t

0�f�s� xs� �xs�p��ds

]p

+ C0p�[ ∫ t

0�g�s� xs� �xs�p��2ds

] p2

≤ �1+ 3p−1�����p + CT�1p�∫ t

0 �f�s� xs� �xs�p��p + �g�s� xs� �xs�p��pds

≤ �1+ 3p−1�����p + 2 · 3p−1CT�1pKp�

∫ t

0 1+ �xs�p + �xs�ppds

≤ CT�p + CT�2p�∫ t

0

[�xs�p +

∫ 0

−���x�s + ���pd����

]ds

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≤ CT�p + 2CT�2p

∫ t

0�[

sup−�<r≤s

�x�r��p]ds

≤ CT�p + 2CT�2pe�pt�x�p�� (2.14)

where C0p� CT�1p� CT�2p and CT�p are constants dependent on T� p. This, together with(2.8), gives

�x̂�p� ≤ sup0≤t≤T

e−�pt∫ t

0 CT�p + 2CT�2pe

�ps�x�p�ds < ��

which implies that x̂ ∈ D. This shows that � maps D into D.For any x� x̄ ∈ D, define z = x − x̄ and Z = �x −�x̄. By Assumption 2.1, (2.8),

(2.12), and (2.13), applying the Hölder inequality, the Burkholder-Davis-Gundyinequality, and the triangle inequality of the norm � · �p gives

�[

sup−�<s≤t

�Z�s��p]= �

[sup0≤s≤t

∣∣∣∫ s

0 f�r� xr� �xr�p�− f�r� x̄r � �x̄r�p�dr

+∫ s

0 g�r� xr� �xr�p�− g�r� x̄r � �x̄r�p�dw�r�

∣∣∣p]

≤ 2p−1�[ ∫ t

0�f�s� xs� �xs�p�− f�s� x̄s� �x̄s�p��dr

]p

+ c0p�[ ∫ t

0�g�s� xs� �xs�p�− g�s� x̄s� �x̄s�p��2dr

] p2

≤ cT�1p

[�∫ t

0 �f�s� xs� �xs�p�− f�s� x̄s� �x̄s�p��p

+ �g�s� xs� �xs�p�− g�s� x̄s� �x̄s�p��p]

≤ 2cT�1pKp[�∫ t

0 �zs� + ��xs�p − �x̄s�p�pds

]

≤ cT�2p�∫ t

0

[�zs�p +

∫ 0

−���z�s + ���pd����

]ds

≤ 2cT�2p∫ t

0�[

sup−�<r≤s

�z�r��p]ds

≤ 2cT�2pe�pt�z�p�� (2.15)

where c0p� cT�1p and cT�2p are constants dependent on T and p. By (2.8), we have

�Z�p� ≤ sup0≤t≤T

e−�pt∫ t

02cT�2pe

�ps�z�p�ds

= sup0≤t≤T

2cT�2p�p

�1− e−�pt��z�p�

= 2cT�2p�p

�1− e−�pT ��z�p�=� �r�z���p�

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Nonlocal SFDEs with Infinite Delay 719

where

r =[2cT�2p�p

�1− e−�pT �

] 1p

�

Choose � sufficiently large such that r ∈ �0� 1�. This shows that the mapping � is acontraction mapping, namely,

��x −�x̄�� ≤ r�x − x̄��for any x� x̄ ∈ D.

Step 3. Since the mapping � is a contraction mapping, there exists x∗ ∈ D suchthat x∗ = �x∗. By the definition of �, x∗�t� is continuous and satisfies (2.3). Notingthat x∗ ∈ X, by (2.9),

∫ T

0�[

sup−�<s≤t

�x∗�s��p]dt < �� (2.16)

This shows that

∫ T

0

[sup

−�<s≤t

�x∗�s��p]dt < �� a�s�

By Assumption 2.2,

∫ T

0�f�t� x∗t � �x∗t �p��dt ≤ KT + K

∫ T

0�x∗t �dt + K

∫ T

0�x∗t �pdt� (2.17)

Applying the Hölder inequality gives

∫ T

0�x∗t �dt =

∫ T

0

[sup

−�<s≤t

�x∗�s��]dt

≤ Tp

p−1

∫ T

0

[sup

−�<s≤t

�x∗�s��p]dt < ��

Noting that (2.16) implies that �x∗t �p < �, we therefore have

∫ T

0�f�t� x∗t � �x∗t �p��dt < �� (2.18)

By Assumption 2.2, applying the inequality (2.13) gives

∫ T

0�g�t� x∗t � �x∗t �p��2dt ≤ 3KT + 3K

∫ T

0�x∗t �2dt + K

∫ T

0�x∗t �2pdt�

Applying the similar technique gives that

∫ T

0�g�t� x∗t � �x∗t �p��2dt < �� (2.19)

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(2.16) and (2.18) as well as (2.19) show x∗�t� is the solution of Equation (1.4) withthe initial data (2.1).

Step 4. Let x∗ be defined by Step 3. Assume that x̄∗ is another solution ofEquation (1.4) on �−�� 0 with the same initial data � as x∗. By (2.8), noting that∫ T

0 � sup−�<s≤t �x̄∗�s��pdt < �, we have �x̄∗�� < �, which implies that x̄∗ ∈ D. Bythe definition of the mapping � and (2.3), �x̄∗ = x̄∗. Note that � is a contractionmapping and �x∗ = x∗. The contraction mapping principle shows that

�x∗ − x̄∗�� = 0�

It follows from (2.9) that

�∫ T

0

[sup0≤s≤t

�x∗�s�− x̄∗�s��p]dt = 0�

which implies that for almost all � ∈ �,

∫ T

0

[sup0≤s≤t

�x∗�s� ��− x̄∗�s� ���p]dt = 0�

This shows that for almost all � ∈ �, x∗�s� �� = x̄∗�s� �� on all t ∈ 0� T, that is,(2.4) holds. Hence, x∗�t� is the unique solution of Equation (1.4) with the initial data(2.1). This completes this proof. �

In Theorem 2.1, Assumptions 2.1 and 2.2 are similar to the classical Lipschitzcondition and the linear growth condition (cf. [2, 7]). This shows that, comparedwith some existing conditions (e.g., [3, 5, 6]), it is easier for these conditions tobe tested and Equation (1.4) also looks “like” the common Itô-type stochasticfunctional differential equations with infinite delay.

It is well-known that for the existence and uniqueness of the solution tocommon Itô-type stochastic functional differential equations that the uniformLipschitz condition may be replaced by the local Lipschitz condition. Here we givethe local Lipschitz condition for the coefficients of Equation (1.4):

Assumption 2.3 (Local Lipschitz Condition). For every integer n ≥ 1, there existsconstant Kn such that for all t ∈ 0� T and those �� �̄ ∈ BC��−�� 0��n� and x� x̄ ∈�n with ��� ∨ ��̄� ∨ �x� ∨ �x̄� ≤ n,

�f�t� �� x�− f�t� �̄� x̄�� ∨ �g�t� �� x�− g�t� �̄� x̄�� ≤ Kn���− �̄� + �x − x̄��� (2.20)

Then the following theorem follows:

Theorem 2.2. Under Assumptions 2.2 and 2.3, there exists a unique solution x�t� toEquation (1.4) with the initial data (2.1) on all t ∈ 0� T.

The proof of this theorem follows from the truncation technique and thestopping time similar to [7, Theorem 3.4, p. 56], so we omit it.

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Nonlocal SFDEs with Infinite Delay 721

In Equation (1.4), this article only considers that the coefficients are dependentthe pth moment, but the technique of this article can be extended to examine thenonlocal stochastic differential equations with infinite delay of the form

dx�t� = f�t� xt� �xt�p1� � � � � �xt�pl �dt + g�t� xt� �xt�p1� � � � � �xt�pl �dw�t��

where l ≥ 1 is an arbitrary integer. Noting that the Lyapunov inequality shows thatfor any random variable Y , and � ≥ � > 0,

��Y �� 1� ≤ ��Y �� 1

� �

The analysis of this article can also be extended to the case p ∈ �0� 2�.

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