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

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]: R^d) which denotes the family of bounded continuous R^d-valued functions φ defined on (-∞, 0] with norm ||φ|| = sup_{-∞ < θ≤0} |φ(θ)|.

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

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

where x_t = {x(t + θ): -τ ≤ θ ≤ 0} could be considered as a C([-τ, 0]; R^d)-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]; R^d) to be described below.

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

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.

Let | · | denote Euclidean norm in Rⁿ. If A is a vector or a matrix, its transpose is denoted by A^T; 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), ..., B_m(t))^T. Let BC((-∞, 0]; R^d) denote the family of bounded continuous R^d-value functions φ defined on (-∞, 0] with norm ||φ|| = sup_{-∞ < θ≤0} |φ(θ)|. We denote by the family of all -measurable, ℝ^d-valued process ψ(t) = ψ(t, w), t ∈ (-∞, 0], such that . And let is the family of R^d-valued -adapted processes {f(t)}_a≤t≤b such that .

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

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

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

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]R^d-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,

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

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]; R^d) and t ∈ [t₀, T], it follows that

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

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 → ∞, τ_n ↑ T a.s. Let xⁿ(t) = x(t ∧ τ_n), 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,

where

on r > 0, and G^-1(·) be the inverse function of G(·). By assumption and the definition of κ(·), one sees that lim_ϵ↓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,

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

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

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].

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

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)

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