Research

# A note on the existence and uniqueness of the solutions to SFDES

Yeol JE Cho1, Sever S Dragomir2 and Young-Ho Kim3*

Author Affiliations

1 Department of Mathematics Education, Gyeongsang National University, Chinju 660-701, Republic of Korea

2 School of Computer Science and Mathematics, Victoria University of Technology, P.O. Box 14428, MCMC Melbourne, VIC 8001, Australia

3 Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea

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Journal of Inequalities and Applications 2012, 2012:126  doi:10.1186/1029-242X-2012-126

 Received: 26 March 2012 Accepted: 7 June 2012 Published: 7 June 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we establish a new proof of the existence and uniqueness of solution to stochastic functional differential equations with infinite delay at phase space BC((-∞, 0]: Rd) which denotes the family of bounded continuous Rd-valued functions φ defined on (-∞, 0] with norm ||φ|| = sup-∞ < θ≤0 |φ(θ)|.

Mathematics Subject Classification (2000): 60H05, 60H10.

##### Keywords:
existence; uniqueness; stochastic functional differential equations; infinite delay.

### 1. Introduction

Stochastic differential equations (SDEs) are well known to model problems from many areas of science and engineering. For instance, in 2006, Henderson and Plaschko [1] published the SDEs in Science and Engineering, in 2007, Mao [2] published the SDEs, in 2010, Li and Fu [3] considered the stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks.

In recent years, there is an increasing interest in stochastic functional differential equations(SFDEs) with finite and infinite delay under less restrictive conditions than Lipschitz condition. For instance, in 2007, Wei and Wang [4] discussed the existence and uniqueness of the solution for stochastic functional differential equations with infinite delay, in 2008, Mao et al. [5] discussed almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, in 2008, Ren et al. [6] considered the existence and uniqueness of the solutions to SFDEs with infinite delay, and in 2009, Ren and Xia [7], discussed the existence and uniqueness of the solution to neutral SFDEs with infinite delay. Furthermore, on this topic, one can see Halidias [8], Henderson and Plaschko [1], Kim [9,10], Ren [11], Ren and Xia [7], Taniguchi [12] and references therein for details.

On the other hand, Mao [2] discussed d-dimensional stochastic functional differential equations with finite delay

(1.1)

where xt = {x(t + θ): -τ θ ≤ 0} could be considered as a C([-τ, 0]; Rd)-value stochastic process. The initial value of (1.1) was proposed as follows:

Furthermore, Ren et al. [6] also considered the stochastic functional differential equations with infinite delay at phase space BC((-∞, 0]; Rd) to be described below.

(1.2)

where xt = {X(t + θ): -∞ ≤ θ ≤ 0} could be considered as a BC((-∞, 0]; Rd)-value stochastic process. The initial value of (1.2) was proposed as follows:

(1.3)

Following this way, now we recall the existence and uniqueness of the solutions to the Equation 1.2 with initial data (1.3) under the non-Lipschitz condition and the weakened linear growth condition. In this paper, we will give some new proof of the existence and uniqueness of the solutions to ISDEs under an alternative way.

### 2. Preliminary

Let | · | denote Euclidean norm in Rn. If A is a vector or a matrix, its transpose is denoted by AT; if A is a matrix, its trace norm is represented by . Let t0 be a positive constant and , throughout this paper unless otherwise specified, be a complete probability space with a filtration satisfying the usual conditions (i.e. it is right continuous and contains all P-null sets). Assume that B(t) is a m-dimensional Brownian motion defined on complete probability space, that is B(t) = (B1(t), B2(t), ..., Bm(t))T. Let BC((-∞, 0]; Rd) denote the family of bounded continuous Rd-value functions φ defined on (-∞, 0] with norm ||φ|| = sup-∞ < θ≤0 (θ)|. We denote by the family of all -measurable, ℝd-valued process ψ(t) = ψ(t, w), t ∈ (-∞, 0], such that . And let is the family of Rd-valued -adapted processes {f(t)}atb such that .

With all the above preparation, consider a d-dimensional stochastic functional differential equations:

(2.1)

where xt = {x(t + θ): -∞ < θ ≤ 0} can be considered as a BC((-∞, 0]; Rd)-value stochastic process, where

be Borel measurable. Next, we give the initial value of (2.1) as follows:

(2.2)

To find out the solution, we give the definition of the solution of the Equation 2.1 with initial data (2.2).

Definition 2.1. [6]Rd-value stochastic process x(t) defined on -∞ < t T is called the solution of (2.1) with initial data (2.2), if x(t) has the following properties:

(i) x(t) is continuous and is -adapted;

(ii) and ;

(iii) , for each t0 t T,

(2.3)

A solution x(t) is called as a unique if any other solution is indistinguishable with x(t), that is

The integral inequalities of Gronwall type have been applied in the theory of SDEs to prove the results on existence, uniqueness, and stability etc. [10,13-16]. Naturally, Gronwall's inequality will play an important role in next section.

Lemma 2.1. (Gronwall's inequality). Let u(t) and b(t) be non-negative continuous functions for t α, and let

where a ≥ 0 is a constant. Then

Lemma 2.2. (Bihari's inequality). Let u and b be non-negative continuous functions defined on R+. Let g(u) be a non-decreasing continuous function on R+ and g(u) > 0 on (0, ∞). If

for t R+, where k ≥ 0 is a constant. Then for 0 ≤ t t1,

where

and G-1 is the inverse function of G and t1 R+ is chosen so that

for all t R+ lying in the interval 0 ≤ t t1.

The following two lemmas are known as the moment inequality for stochastic integrals which will play an important role in next section.

Lemma 2.3. [2]. If p ≥ 2, such that

then

In particular, for p = 2, there is equality.

Lemma 2.4. [2]. If p ≥ 2, such that

then

### 3. Existence and Uniqueness of the Solutions

In order to obtain the existence and uniqueness of the solutions to (2.1) with initial data (2.2), we define and x0(t) = ξ(0), for t0 t T. Let , n = 1, 2, ... and define the Picard sequence

Now we begin to establish the theory of the existence and uniqueness of the solution. We first show that the non-Lipschitz condition and the weakened linea growth condition guarantee the existence and uniqueness.

Theorem 3.1. Assume that there exist a positive number K such that

(i) For any φ, ψ ∈ BC((-∞, 0]; Rd) and t ∈ [t0, T], it follows that

(3.1)

where κ(·) is a concave non-decreasing function from ℝ+ to ℝ+ such that κ(0) = 0, κ(u) > 0 for u > 0 and .

(ii) For any t ∈ [t0, T], it follows that f(0, t), g(0, t) ∈ L2 such that

(3.2)

Then the initial value problem (2.1) has a solution x(t). Moreover, . We prepare a lemma to prove this theorem.

Lemma 3.2. Let the assumption (3.1) and (3.2) of Theorem 3.1 hold. If x(t) is a solution of equation (2.1) with initial data (2.2), then

where c2 = c1 + Eǁ ξ ǁ2, c1 = 3E ǁξ ǁ2 + 6(T - t0 + 1)(T - t0)[K + a]. In particular, x(t) belong to .

Proof. For each number n ≥ 1, define the stopping time

Obviously, as n → ∞, τn T a.s. Let xn(t) = x(t τn), t ∈ (-∞, T]. Then, for t0 t T, xn(t) satisfy the following equation

Using the elementary inequality when p ≥ 1, we have

By the Hölder's inequality and the moment inequality, we have

Hence, by the condition (3.1) and (3.2) one can further show that

Given that κ(·) is concave and κ(0) = 0, we can find a positive constants a and b such that κ(u) ≤ a + bu for u ≥ 0. So, we obtains that

where c1 = 3E||ξ||2 + 6(T - t0 + 1)(T - t0)[K + a]. Noting the fact that , we obtain

where c2 = c1 + E||ξ||2. So, by the Gronwall inequality yields that

Letting t = T, it then follows that

Thus

Consequently the required result follows by letting n → ∞. □

Proof of Theorem 3.1. Let x(t) and be any tow solutions of (2.1). By Lemma 3.2, x(t), . Note that

By the elementary inequality (u + v)2 ≤ 2(u2 + v2), one then gets

By the Hölder's inequality, the moment inequality, and (3.1) we have

Since κ(·) is concave, by the Jensen inequality, we have

Consequently, for any ϵ > 0,

By the Bihari inequality, one deduces that, for all sufficiently small ϵ > 0,

(3.3)

where

on r > 0, and G-1(·) be the inverse function of G(·). By assumption and the definition of κ(·), one sees that limϵ↓0 G(ϵ) = -∞ and then

Therefore, by letting ϵ → 0 in (3.3), gives

This implies that for t0 t T, Therefore, for all -∞ < t T, a.s. The uniqueness has been proved.

Next to check the existence. Define and x0(t) = ξ(0) for t0 t T. For each n = 1, 2, ..., set and define, by the Picard iterations,

(3.4)

for t0 t T. Obviously, . Moreover, it is easy to see that , in fact

Taking the expectation on both sides and using the Hölder inequality and moment inequality, we have

Using the elementary inequality (u + v)2 ≤ 2u2 + 2v2, (3.1), and (3.2), we have

Given that κ(·) is concave and κ(0) = 0, we can find a positive constants a and b such that κ(u) ≤ a + bu for u ≥ 0. So, we have

where c1 = 3E||ξ||2 + 6(T - t0)(T - t0+ 1)[K + a]. It also follows from the inequality that for any k ≥ 1,

where c2 = c1 = 6b(T - t0)(T - t0+ 1) E||ξ||2. The Gronwall inequality implies

Since k is arbitrary, we must have

(3.5)

for all t0 t T, n ≥ 1.

Next, we that the sequence {xn(t)} is Cauchy sequence. For all n ≥ 0 and t0 t T, we have

Next, using an elementary inequality and the condition (H3), we derive that

On the other hand, by Hölder's inequality, Lemma (2.4), and the condition, one can show that

(3.6)

where β = (T - t0)/α + 4/(1 - α). Let

From (3.6), for any ϵ > 0, we get

By the Bihari inequality, one deduces that, for all sufficiently small ϵ > 0,

where

on r > 0, and G-1(·) be the inverse function of G(·). By assumption, we get Z(t) = 0. This shows the sequence {xn(t), n ≥ 0} is a Cauchy sequence in L2. Hence, as n → ∞, xn(t) - x(t), that is E |xn(t) - x(t)|2 → 0. Letting n → ∞ in (3.5) then yields that

for all t0 t T. Therefore, . It remains to show that x(t) satisfies Equation 2.3. Note that

Noting that sequence xn(t) is uniformly converge on (-∞, T ], it means that

as n → ∞, further

as n → ∞. Hence, taking limits on both sides in the Picard sequence, we obtain that

on t0 t T. The above expression demonstrates that x(t) is the solution of (2.3). So, the existence of theorem is complete.

Remark 3.1. In the proof of Theorem 3.1, the solution is constructed by the successive approximation. It shows that how to get the approximate solution of (2.1) and how to construct Picard sequence xn(t). For SFDEs, we know that a weakened linear growth condition imposed on Theorem 3.1 is rigorous for our discussion. In papers [6,7], the proofs of the assertions are based on some function inequalities. Recall that the procedures has become more and more complicated in those proofs. For this reason, although analogous problem is studied here, the proof of the assertion in the Theorem is completely different with respect to the ones from [7]. Moreover, our new proof in this paper is completed by Bihari's inequality and more simple than that reported in [7].

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

All authors read and approved the final manuscript.

### Acknowledgements

The authors wish to thank the anonymous referees for their endeavors and valuable comments. Also, this research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2011-0023547)

### References

1. Henderson, D, Plaschko, P: Stochastic Differential Equations in Science and Engineering. World Scientific Publishing Co., Singapore (2006)

2. Mao, X: Stochastic Differential Equations and Applications. Horwood Publication, Chichester (2007)

3. Li, X, Fu, X: Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks. J Comput Appl Math. 234, 407–417 (2010). Publisher Full Text

4. Wei, F, Wang, K: The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay. J Math Anal Appl. 331, 516–531 (2007). Publisher Full Text

5. Mao, X, Shen, Y, Yuan, C: Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching. Stoch Process Appl. 118, 1385–1406 (2008). Publisher Full Text

6. Ren, Y, Lu, S, Xia, N: Remarks on the existence and uniqueness of the solution to stochastic functional differential equations with infinite delay. J Comput Appl Math. 220, 364–372 (2008). Publisher Full Text

7. Ren, Y, Xia, N: Existence, uniqueness and stability of the solution to neutral stochastic functional differential equations with infinite delay. Appl Math Comput. 210, 72–79 (2009). Publisher Full Text

8. Halidias, N: Remarks and corrections on "An esistence theorem for stochastic functional differential equations with dealys under weak assumptions, Statistics and Probability Letters 78, 2008" by N. Halidias and Y. Ren. Stoch Probab Lett. 79, 2220–2222 (2009). Publisher Full Text

9. Kim, Y-H: An estimate on the solutions for stochastic functional differential equations. J Appl Math Informatics. 29(5-5), 1549–1556 (2011)

10. Kim, Y-H: Gronwall, Bellman and Pachpatte type integral inequalities with applications. Nonlinear Anal. 71(12), 2641–2656 (2009). Publisher Full Text

11. Ren, Y, Xia, N: A note on the existence and uniqueness of the solution to neutral stochastic functional differential equations with infinite delay. Appl Math Comput. 214, 457–461 (2009). Publisher Full Text

12. Taniguchi, T: Successive approximations to solutions of stochastic differential equations. J Differ Equ. 96, 152–169 (1992). Publisher Full Text

13. Agarwal, RP, Kim, Y-H, Sen, SK: Multidimensional Gronwall-Bellman-type integral inequalities with applications. Mem Diffe Equ Math Phys. 47, 19–122 (2009)

14. Cho, YJ, Dragomir, SS, Kim, Y-H: On some Gronwall type inequalities involving iterated integrals. Math Inequal Appl. 14(3), 605–620 (2011)

15. Cho, YJ, Dragomir, SS, Kim, Y-H: On some Gronwall type inequalities with iterated integrals. Math Commun. 12(1), 63–73 (2007)

16. Cho, YJ, Kim, Y-H, Pečarić, J: New Gronwall-Ou-Iang type integral inequalities and their applications. ANZIAM J. 50(1), 111–127 (2008). Publisher Full Text