Second Moment Boundedness of Linear Stochastic Delay Differential Equations

This paper studies the second moment boundedness of solutions of linear stochastic delay differential equations. First, we give a framework, for general $\mathrm{N}$-dimensional linear stochastic differential equations with a single discrete delay, of calculating the characteristic function for the second moment boundedness. Next, we apply the proposed framework to a special case of a type of 2-dimensional equation that the stochastic terms are decoupled. For the 2-dimensional equation, we obtain the characteristic function explicitly given by equation coefficients, the characteristic function gives sufficient conditions for the second moment to be bounded or unbounded.


Introduction
Stochastic delay differential equations have been extensively studied in the last several decades from different points of view (see [3, 4, 6-13, 15, 16] and the references therein). However, many basic issues remain unsolved even for linear equations with constant coefficients. In this paper, we study general N-dimensional linear stochastic delay differential equations with a single discrete delay (here we assume the delay τ = 1) (i, j = 1, · · · , N ; k = 1, · · · , K) (1.1) Here the Einstein summation convention has been used so that repeated indices are implicitly summed over, all coefficients a j i , b j i , µ k i , σ jk i , η jk i are constants, and W k are independent 1-dimensional Wiener Processes. The initial functions are assumed to be x i = φ i ∈ C([−1, 0], R) (i = 1, · · · , N). This paper considers the second moment boundedness of solutions of (1.1). We always assume Itô interpretation for the stochastic integral, and results for Stratonovich interpretation can be obtained similarly.
To the best of our knowledge, there are very few results for the stability and second moment boundedness for equation (1.1), and most of known results are obtained through the method of Lyapunov functional [9][10][11]14]. However, it remains unclear how can we define the characteristic equation for the boundedness of the solution moments of (1.1). In 2007, Lei and Mackey [8] introduced the method of Laplace transform to study the second moment boundedness of 1-dimensional equations with a single discrete delay, and proposed a characteristic equation. In [17], the authors extended the method to the 1-dimensional equation with distributed delay.
Based on previous studies [8,17], this paper aims at proposing a general framework to calculate the characteristic function for high dimensional linear stochastic delay differential equations with a single delay. The obtained characteristic function (as we can see below) is complicate, that shows the elaborate correlations when delay and stochastic effects are coupled into an dynamical system. As an example, we apply the framework to study a special situation of a 2-dimensional equation in which the stochastic terms are decoupled (N = K = 2, and µ k i = σ jk i = η jk i = 0 when i = k). Rest of this paper is organized as follows. In Section 2, we briefly introduce basic results for linear delay differential equations. In Section 3, a general framework for defining the characteristic function of a N-dimensional stochastic delay differential equation is given. In Section 4, we discuss the boundedness of the second moment of (1.1) for a simple case: N = K = 2 and the stochastic terms are decoupled. Theorem 4.1 establishes the unbounded condition for the second moment if the trivial solution of the unperturbed equation is unstable. When the trivial solution of the unperturbed equation is stable, we obtain a characteristic function given explicitly by the equation parameters. Boundedness of the second moments depends on the supremum of the real parts of all roots of the characteristic equation (Theorem 4.3). An explicit condition for the boundedness of the second moment is also proved following framework of calculation given here (Theorem 4.7).

Preliminaries
In this paper, we always use the L 1 norm for a tensor. For example, the L 1 norm of a second order tensor , the norm is defined as In this section we give some basic results for the N-dimensional linear delay differential equation The linear autonomous functional differential equation (2.3) has been studied extensively and details can be referred to [1,2,5]. The fundamental matrix of (2.3), denoted by is the solution of (2.3) with the initial condition (hereinafter δ means the Kronecker delta) Using the fundamental matrix X(t), the solution of (2.3) with initial function φ ∈ C([−1, 0], R N ) can be represented as Denote the matrices Taking the Laplace transform on both sides of (2.3), we obtain Thus, h(λ) = det(∆(λ)) is the characteristic function for the linear stability of (2.3).
The following results are straightforward from the above discussions.

Then
(i) for any α > α 0 there exists a constantK =K(α) ≥ 1 such that the fundamental matrix X(t) satisfies (iii) for any α 1 < α 0 , there existsᾱ ∈ (α 1 , α 0 ) and a subset U ⊂ R + with measure m(U) = +∞ such that for any i, j = 1, · · · , N, Here, the number α 0 is also termed as the Lyapunov exponent (Definition 1.19 in [1]). The proofs of (i) and (ii) in Theorem 2.1 is referred to the proofs in [1,Theorem 1.21 in Chapter 3]. The proof of (iii) is similar to that of [17,Theorem 2.3,2] and is detailed at Appendix A.
From Theorem 2.1, the trivial solution of (2.3) is locally asymptotically stable if and only if α 0 < 0.

Moments of linear stochastic delay differential equations-General cases
Now, we discuss the solution moments and a framework for calculating the characteristic function of the second moment of solutions for general cases.
The existence and uniqueness results for stochastic delay differential equations have been established in [6,11,14]. Using the fundamental matrix X(t), the solution of (1.1) with initial function x = φ ∈ C([−1, 0], R N ) is a N-dimensional stochastic process given by Itô integral as follows We denote by E the mathematical expectation. Now we give the definitions of the p th moment exponential stability and the p th moment boundedness.
Definition 3.1. A solution x(t; φ) of (1.1) is said to be the first moment exponentially stable if there exist two positive constants γ and R such that for all φ ∈ C((−1, 0], R N ). When p ≥ 2, a solution of (1.1) is said to be the p th moment exponentially stable if there exist two positive constants γ and R such that Otherwise, the p th moment is said to be unbounded.

First moment
From (3.1) and applying the properties of Itô integral, it is easy to have Thus, the following result is straightforward from Theorem 2.1.
In particular, the first moment of (1.1) is exponentially stable when α 0 < 0.

Second moment
Now we study the second moment. First, we give some notations. Let x(t; φ) (x i (t)) N ×1 be a solution of (1.1) andx i (t) = x i (t) − E(x i (t)) (i = 1, · · · , N), and define Then M ii (t) is the second moment of x i (t). It is easy to havex i (t) = M ij (t) = N ij (t) = 0 when t ∈ [−1, 0], and E(x i (t)) = 0 for all t ≥ 0.

Additive noise
We have additive noise when σ jk l = η jk l = 0 for any i, j, k and µ = 0 (µ is defined at (2.5)). In this case, we have Thus the upper bound of M(t) is determined by that of the fundamental matrix X(t). Hence, we have the following sufficient conditions for the second moments M(t) to be bounded or unbounded.
The proof is similar to that in [17,Theorem 3.4] and is omitted here. The critical case α 0 = 0 is not discussed here and the boundedness issue remains open.
Then P lp (s) = P pl (s), Q lp (s) = Q pl (s) and We note that Therefore, Similarly, we have From (3.9), we have |M ii (t)| ≥ |F ii (t)| (∀i). If α 0 > 0 (α 0 is defined by (2.8)), similar to discussions in [17], |F ii (t)| approaches to infinity exponentially, and therefore the second moment is unbounded (a proof for the case of a 2-dimensional equation is given in the next section). Thus, we only need to study the situation when α 0 < 0.
From (3.9) and (3.10), and take Laplacians to both sides of them (existence of the Laplacians are proved in Lemmas 3.5 and 3.6 below), we obtain and Here S ij hp is the inverse tensor of L(X h i X p j ). Next, substitute the obtained L(P hp ) + L(Q hp ) into (3.12) to linearly express L(N ij ) through L(M ij ): Then, put the resulting L(N ij ) into (3.13) so that L(Q hp ) linearly depends on L(M ij ) in the form L(Q hp ) = T kl hp L(M kl ). Finally, substituting L(Q hp ) back to (3.11) to obtain an equation for Then equation (3.14) is a linear equation of L(M kl ). Thus, the determinant of the coefficients, denoted by is the desired characteristic function. We note that (3.14) contains N 2 linear equations. Nevertheless, we can simplify the calculation due to symmetry. For example, since L(M kl ) = L(M lk ) and L(P kl ) = L(P lk ) in (3.14), we only need to solve equations for L(M kl ) with k ≤ l, and therefore have N(N + 1)/2 equations.
The above procedure gives a general framework to obtain the characteristic function. However, it is too complicate to obtain an explicit expression for general cases. In the next section, we study a 2-dimensional equation with a specific form.
Denote the matrices Before introducing the results for the 2-dimensional equation, we give some estimates for F (t), M(t) and N(t) for general situation. These estimations ensure the existence of Laplace transforms of F (t), M(t) and N(t).

Application to 2-dimensional equations
In this section, we apply the general framework established in the above to a special case of a 2-dimensional equation that N = K = 2 and µ k i = σ jk i = η jk i = 0 when i = k. Hereafter, we do not use the Einstein summation convention, and introduce following notations for simplicity: In this particular case, the expressions of P ij (t), Q ij (t) and F ij (t), M ij (t), N ij (t) (i, j = 1, 2) in the previous section are as follows: Before we state and prove the main results, we introduce some preliminaries below.
When N = 2, we consider the delay differential equation The characteristic function of (4.2) is given by Here we give some properties of the fundamental solution X(t) that are useful for the estimate of the second moment below. Recall When α 0 < 0, from (4.4), we have Thus, from (4.4), we obtain Obviously, X k i (t)X k j (t) and X k i (t)X k j (t−1) (i, j, k = 1, 2) have Laplace transforms. When α 0 < 0, explicit expressions and estimates for the Laplace

Unboundedness of the second moment
Here we give a result for the unboundedness of the second moment of (4.1) when α 0 > 0. First, we note a rare situation when the coefficients µ i , σ j i , η j i (i, j = 1, 2) satisfy the following assumption.
Assumption H: µ 1 = µ 2 = 0, and there is a root λ ∈ R of When the assumption H is satisfied, the stochastic equation (4.1) has a deterministic solution x(t) = e λt c (t ≥ 0) with initial function φ = e λt c (−1 ≤ t < 0), and the corresponding second moment M(t) = 0. This is a very rare situation and is excluded in discussions below.
The following result shows that in general the second moment of (4.1) is unbounded when the trivial solution of (2.3) is unstable.
Theorem 4.1. Let α 0 be defined as (4.5) and α 0 > 0. If the assumption H is not satisfied, the second moment of (4.1) is unbounded.
Proof. We only need to show that there is a special solution x(t; φ) of (4.1) such that the corresponding second moment is unbounded. Note that P ii , Q ii ≥ 0 (i = 1, 2), and hence Let λ be a solution of h(λ) = 0 with 0 < Re(λ) ≤ α 0 , and c ∈ R 2 the corresponding eigenvector. Then x φ 1 (t) = Re(e λt c) is the solution of (4.2) with initial function φ 1 = Re(e λt c) (−1 ≤ t < 0). Since the assumption H is not satisfied, x φ 1 (t) is not a solution of (4.1). Following the proof of Theorem 2.1 (3), there is a subset U ⊂ R + with m(U) = +∞ and positive constants C 1 , C 2 such that for any t ∈ U, Thus, from (4.6), the second moment M(t) is unbounded.
(ii). Now we assume β 0 > 0. We only need to show that there is a special solution x(t; φ) such that the corresponding second moment is unbounded. Similar to the proof of Theorem 4.1, let λ = α+iω (α ≤ α 0 < 0) be a solution of h(λ) = 0, and c = c 1 c 2 ∈ R 2 is an eigenvector corresponding to the eigenvalue λ, then x φ 2 (t) = Re(e λt c) (t ≥ −1) is a solution of (2.3) with initial function φ 2 = Re(e λt c) ∈ C([−1, 0], R 2 ). Hence, for this particular initial function φ 2 , since the assumption H is not satisfied, x φ 2 (t) is not a solution of (1.1) and therefore P 11 (t) or P 22 (t) is nonzero. Thus the Laplacian L(P 11 ) or L(P 22 ) is nonzero.
Since M(t) ≥ M (t) ≥ M 11 (t), in the following, we only need to show that M 11 (t) is unbounded for the initial function φ 2 . From (4.18), we have and are analytic functions for Re(s) >ᾱ 0 and are nonzeros. It is easy to see that L(P 11 )(s), L(P 22 )(s) are analytic for Re(s) =c > 0. Thus, similar to the proof of Theorem 2.1 (3), there is a sequence {t k } k≥1 such that t k → +∞ and M 11 (t k ) → +∞ as k → +∞, which implies that the second moment is unbounded. Thus, the theorem is proved. The critical case ofᾱ 0 < 0, β 0 = 0 is not considered here, and the issue of boundedness criteria remains open.
The characteristic function H(λ) depends not only on the coefficients of equation (4.1), but also on the Laplace transforms of X k i (t)X k j (t) and X k i (t)X k j (t − 1) (i, j, k = 1, 2). One can calculate these functions numerically according to Lemma Appendix B.1 and Appendix B.2, however, it is difficult to obtain β 0 = sup{Re(λ) : H(λ) = 0} for a given equation. Hence the sufficient conditions for the second moment boundedness of (4.1) established in Theorem 4.3 are not practical. In applications, one need to derive useful criteria in terms of equation coefficients, either from the proposed characteristic function or following the procedure in the above discussions. Here, for applications, we give a practical condition for the boundedness of the second moment.
Theorem 4.7. Assume α 0 < 0. If there exists α ∈ (α 0 , 0) and the a positive constantK =K(α) so that where Let Then By (4.33), we obtain In this paper, we have established framework procedure to calculate the characteristic function for the second moment boundedness of linear delay differential equations with a single discrete delay, we also applied the procedure to study a special case of 2-dimensional equations. However, as we have seen, the resulting function has a very complicate form. These complicate results is in fact show the elaborate correlations of non-Markov processes when both delay and stochastic effects are taken into involved. In spite of the complicate form of final formulations, the procedure of calculating is simple and easy to follow. Thus, in applications, one can develop the characteristic function, for particular equation of studied, following the scheme given here. We leave these further applications to future works.
Similar to Lemma Appendix B.1, we have the following expressions and estimates.