Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 23, 1-14; http://www.math.u-szeged.hu/ejqtde/

EXPONENTIAL STABILITY OF LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS WITH BOUNDED DELAY AND THE W-TRANSFORM

RAMAZAN I. KADIEV AND ARCADY PONOSOV

Abstract. We demonstrate how the method of auxiliary (’reference’) equations, also known as N. V. Azbelev’s W-transform method, can be used to derive efficient conditions for the exponential Lyapunov stability of linear delay equations driven by a vector-valued Wiener process. For the sake of convenience the W-method is briefly outlined in the paper, its justification is however omitted. The paper contains a general stability result, which is specified in the last section in the form of seven corollaries providing sufficient stability conditions for some important classes of Itô equations with delay.

1. Introduction

Stability of solutions to delayed systems with random parameters is a popular topic in the mathematical literature. A rather complete, but definitely not exhaustive, list of publications can e. g. be found in [7], [8], [9], [11]. Most authors apply the method of auxiliary functionals (the Lyapunov-Krasovskii-Razumikhin method). On the other hand, an alternative method (Azbelev’s W -transform method) to study asymptotic properties of linear functional-differential equations, based on auxiliary equations and developed in [2], [4], [5], [6], proved to be efficient as well, especially in the case of non-diffusion equations driven by semimartingales.

While in the case of the Lyapunov-Krasovskii-Razumikhin method a successful stability analysis depends on a skillful choice of an auxiliary functional with certain properties, the W -transform method is very much dependent on a suitable auxiliary equation that has the required asymptotic properties. The auxiliary equation gives rise to a certain integral transform which is applied to the equation in question, and the challenge is to find conditions under which the latter equation would inherit the asymptotic properties of the auxiliary equation. By this reason the auxiliary equation may be called a reference equation. Some hints on what kind of reference equations can be suitable for certain classes of linear stochastic functional differential equations, are discussed in [2], [4], [5], [6].

In the present paper we use the W -transform method to study the exponential Lyapunov 2p–stability (1 ≤ p < ∞) of the zero solution of linear and scalar differential equations with bounded delay, which are driven by a vector-valued Wiener process. We apply the general framework developed in our previous papers [2], [4], [5], [6]. The framework provides sufficient stability conditions in terms of the parameters of the given equation.

Note that all the results of the paper are new (the only exception is the lemma in Section 4, which was proved by the authors in [6]). We found it difficult to compare our stability results with those proved by the Lyapunov-Krasovskii-Razumikhin method (see e.g. [8] and references therein), as in most cases the results are almost independent of each other. We feel that a combination of these two methods will give better results, but this was beyond the scope of this particular paper.

2. Notation and definitions

In this paper we always assume that the real number p satisfies 1 ≤ p < ∞.

Below we introduce some basic notation to be used in the sequel. Let (Ω, F, (F_{t})_{t≥0}, ℙ) be a given stochastic basis consisting of a probability space (Ω, F, ℙ) and an increasing family (a filtration) (F_{t})_{t≥0} of complete σ-algebras on it, satisfying the usual assumptions (see [1]). By 𝔼 we denote the expectation on this probability space. The stochastic (m − 1)-vector process B consists of jointly independent Wiener processes

EJQTDE, 2008 No. 23, p. 1 B_{i}, i = 2, ..., m w.r.t. (F_{t})_{t≥0}. The linear space K consists of functions η : Ω → R ≡ (−∞, ∞) (random variables) which are F_{0}-measurable (in our previous papers we used the notation k for this space).

1991 Mathematics Subject Classification. 34K50, 34D20.

Key words and phrases. Stochastic differential equations, aftereffect, exponential stability, integral transforms.

In the sequel we will also use the universal constants c_{p} (1 ≤ p < ∞), which come from the celebrated Burkholder-Davis-Gandy inequalities and help to estimate stochastic integrals. These constants are involved in the stability conditions of Sections 4 and 5. From [10] it is known that c_{p} = 2\sqrt{12}p. However, other sources give other values (see e.g. [1]). All these values are not optimal. For instance, in the results presented below we may assume that c_{1} = 1. Indeed, we estimate sup_{t} 𝔼|x(t)|^{2} and not 𝔼 sup_{t} |x(t)|^{2} as in the

Burkholder-Davis-Gandy inequalities.

The following scalar Itô-type equation with delay and the ’prehistory’ condition is considered in this paper:

dx(t) = ∑_{j=0}^{m_{1}} a_{1j}(t)x(h_{1j}(t))dt + ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} a_{ij}(t)x(h_{ij}(t))dB_{i}(t) (t ≥ 0),  (1)
x(ν) = φ(ν) (ν < 0),  (2)

where φ is a measurable stochastic process, which is independent of the vector-valued process B = (B_{2}, ..., B_{m}) and which almost surely (a. s.) has trajectories that are essentially bounded, i. e. belong to L^{∞}. The functions a_{ij}, h_{ij}, i = 1, ..., m, j = 0, ..., m_{i} are Lebesgue-measurable on [0, ∞) and satisfy the following assumptions:

1) a_{1j}, j = 0, ..., m_{1} are locally integrable functions;

2) a_{ij}, i = 2, ..., m, j = 0, ..., m_{i} are locally square-integrable functions;

3) h_{ij}(t) ≤ t for t ∈ [0, ∞) a. s., vraisup_{t≥0} (t − h_{ij}(t)) < ∞ for i = 1, ..., m, j = 0, ..., m_{i}.

In what follows these hypotheses are always assumed to be fulfilled.

A solution to the equation (1), (2) is a stochastic process x(t) (t ∈ (−∞, ∞)) satisfying (1) and (2). This means, in particular, that x(t) must be (F_{t})-adapted and have continuous trajectories for t ≥ 0. We observe also that varying x(0) (which is not fixed in (1), (2)) we obtain different solutions.

It can be proved (see [3]) that the assumptions 1)-3) above guarantee that for any given x_{0} ∈ k there exists a unique (up to the ℙ–equivalence) solution of (1), (2). We will denote this solution by x(t, x_{0}, φ).

Remark 1. Unlike most works on stochastic functional differential equations, we do not require here that the trajectories x(t) (t ≥ 0) should be continuous extensions of the trajectories of the ’prehistory’ process φ. This allows for use discontinuous φ. But the solution may even make a jump at t = 0, when the initial function φ is continuous. This property seems to be more natural in the case of discontinuous stochastic processes. But even in the continuous case it may be convenient to allow jumps at t = 0. For instance, in the conventional framework it is sometimes desirable to extend the set of initial functions from the space of continuous functions C to the space of square-integrable functions L_{2} (see e.g. [9]), which necessarily requires a new, extended phase space L_{2} × R (where φ ∈ L^{2} and x_{0} ∈ R = (−∞, ∞) can be chosen arbitrarily).

The stochastic process

\{ \begin{matrix}φ(ν), if ν < 0, \\ x_{0}, if ν = 0, \end{matrix}

will be conventionally addressed as the initial function for the delay equation (1).

The equation (1), (2) is called homogeneous if φ(ν) ≡ 0 (ν < 0):

dx(t) = ∑_{j=0}^{m_{1}} a_{1j}(t)x(h_{1j}(t))dt + ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} a_{ij}(t)x(h_{ij}(t))dB_{i}(t) (t ≥ 0),

x(ν) = 0 (ν < 0).

Remark 2. It is important to stress that our interpretation of homogeneous delay equations is different from the conventional one (see e.g. [5]). Usually, one calls the equation (1) homogeneous, while (2) is interpreted as the initial condition. Our definition (see e. g. [5]) is more restrictive, as we in addition require that φ(ν) = 0 a. s. for all ν < 0, so that the only initial assumption left is x(0) = x_{0}. The reason

EJQTDE, 2008 No. 23, p. 2 for that is technical: it gives us an opportunity to treat the homogeneous equation as finite dimensional, and thus exploit the most essential features of the W -transform. The ”prehistory function” φ(ν) (ν < 0) does not disappear, but becomes a part of the (inhomogeneous) equation (1), (2). This will be demonstrated in the beginning of Section 4.

In any case, the final output in our approach - the exponential estimate (EXP) - is the same as in the conventional theory. This is summarized in the definition below.

Definition. The zero solution x(t, 0, 0) ≡ 0 of the homogeneous equation corresponding to the equation (1), (2) is called exponentially (Lyapunov) 2p–stable w.r.t. the initial function, if there exist positive numbers c̄, β such that

𝔼|x(t, x_{0}, φ)|^{2p} ≤ c̄(𝔼|x_{0}|^{2p}+ vraisup_{ν<0} 𝔼|φ(ν)|^{2p}) exp{−βt} (t ≥ 0)  (EXP)

for any x_{0} ∈ k and any measurable stochastic process φ, which is independent of the vector-valued process B = (B_{2}, ..., B_{m}) and which almost surely (a. s.) has trajectories belonging to L^{∞}.

Remark 3. Usually, the exponential Lyapunov stability w.r.t. the initial function is simply called the exponential Lyapunov stability. However, in this paper we would like to use the terminology which is more consistent with that used in our previous papers [2]-[6], where we also study (Lyapunov) stability w.r.t. the initial value x_{0} (keeping φ fixed).

Of course, 2p can be replaced by another letter (say q) giving the usual definition of the stochastic exponential q-stability. However, in this paper we only intend to study stability of order 2 and higher. That is why the notation 2p-stability with p ≥ 1 is more convenient for our purposes.

3. Two auxiliary results

The first result provides a uniform estimate on the solutions of the equation (1), (2). The second result gives a convenient technical tool for deriving exponential estimates in the forthcoming sections.

Theorem 1. For any given s ∈ [0, ∞) and any admissible φ, x_{0} such that vraisup_{ν<0} 𝔼|φ(ν)|^{2p} < ∞,

𝔼|x_{0}|^{2p} < ∞ the solution x(t, x_{0}, φ) of (1), (2) satisfies sup_{0≤t≤s} 𝔼|x(t, x_{0}, φ)|^{2p} < ∞ .

Proof. Let s be an arbitrary number from [0, ∞) and φ, x_{0} be such that vraisup_{ν<0} 𝔼|φ(ν)|^{2p} < ∞, 𝔼|x_{0}|^{2p} <

∞. Given a natural number k we put s_{k} = inf{ζ : ∑_{j=0}^{m_{1}} ∫_{0}^{ζ} |a_{1j}(τ)|dτ + ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} c_{p}(∫_{0}^{ζ}(a_{ij}(τ))^{2}dτ)^{0.5} ≥ k/2}. From

this definition we immediately obtain that s_{l} ≥ s for l = 2[ ∑_{j=0}^{m_{1}} ∫_{0}^{s} |a_{1j}(τ)|dτ + ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} c_{p} (∫_{0}^{s}(a_{ij}(τ))^{2}dτ)^{0.5}] + 1.

Here [r] stands for the integer part of r.

Put

(S_{h}x)(t) = \{ \begin{matrix}x(h(t)), if h(t) ≥ 0, \\ 0, if h(t) < 0, \end{matrix}

φ_{h}(t) = \{ \begin{matrix}0, if h(t) ≥ 0, \\ φ(h(t)), if h(t) < 0. \end{matrix}

To simplify calculations we introduce the following notation:

Z(t) = (t, B_{2}(t), ..., B_{m}(t))^{T} , (V x)(t) = ((V_{1}x)(t), ..., (V_{m}x)(t)) , (V_{i}x)(t) = ∑_{j=0}^{m_{i}} a_{ij}(t)(S_{h_{ij}}x)(t) (i = 1, ...m).

Clearly, (1)-(2) is equivalent to the integral equation

x(t) = x_{0}(t) + ∫_{0}^{t}(V x)(τ)dZ(τ) (t > 0),

where

x_{0}(t) = x_{0} + ∑_{j=0}^{m_{1}} ∫_{0}^{t} a_{1j}(τ)φ_{h_{1j}}(τ)dτ + ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} ∫_{0}^{t} a_{ij}(τ)φ_{h_{ij}}(τ)dB_{i}(τ).

Consider the finite sequence of equations

x_{k}(t) = x_{k−1}(t) + ∫_{0}^{t} I_{k}(τ)(V x_{k})(τ)dZ(τ) (t ≥ 0, k = 1, 2, ...),  (3)

where I_{k}(τ) is the indicator of the interval [s_{k−1}, s_{k}], x_{0}(t) is the same as above, and x_{k−1}(t) is a solution of the (k − 1)-th equation in the sequence (3).

Let us show by induction that the solution of the k-th equation in the sequence (3) coincides with the solution of (1), (2) on the interval [0, s_{k}], i. e. x_{k}(t) = x(t, x_{0}, φ) ≡ x(t) for 0 ≤ t ≤ s_{k}, k ≥ 1.

If k = 1 and t ∈ [0, s_{1}], then the definition of I_{1}(τ) gives

x_{1}(t) = x_{0}(t) + ∫_{0}^{t}(V x_{1})(τ)dZ(τ).

The integral representation of x(t) and the uniqueness of solutions to (1)-(2) imply x_{1}(t) = x(t) for t ∈ [0, s_{1}].

Assume now that x_{n}(t) = x(t) for t ∈ [0, s_{n}], n = 1, ..., k − 1 and divide the interval [0, s_{k}] in two subintervals: [0, s_{k−1}] and [s_{k−1}, s_{k}].

For t ∈ [0, s_{k−1}] we immediately obtain that x_{k}(t) = x_{k−1}(t) = x(t), as the integral in (3) is zero by the definition of I_{k}(τ).

Assume that t ∈ [s_{k−1}, s_{k}]. Then

x_{k}(t) = x_{k−1}(t) + ∫_{0}^{t} I_{k}(τ)(V x_{k})(τ)dZ(τ) = x_{k−1}(t) + ∫_{s_{k−1}}^{t} (V x_{k})(τ)dZ(τ)

= x_{k−2}(t) + ∫_{0}^{t} I_{k−1}(τ)(V x_{k−1})(τ)dZ(τ) + ∫_{s_{k−1}}^{t} (V x_{k})(τ)dZ(τ)

= x_{k−2}(t) + ∫_{s_{k−2}}^{s_{k−1}}(V x_{k−1})(τ)dZ(τ) + ∫_{s_{k−1}}^{t} (V x_{k})(τ)dZ(τ) = ...

... = x_{0}(t) + ∫_{0}^{s_{1}}(V x_{1})(τ)dZ(τ) + ... + ∫_{s_{k−2}}^{s_{k−1}}(V x_{k−1})(τ)dZ(τ) + ∫_{s_{k−1}}^{t} (V x_{k})(τ)dZ(τ).

We have already proved that x_{k}(t) = x(t) if t ∈ [0, s_{k−1}]. On the other hand, x_{n}(t) = x(t) for t ∈ [0, s_{n}], n = 1, ..., k − 1. Hence x_{k}(t) = x_{n}(t) for t ∈ [0, s_{n}], n = 1, ..., k − 1 and we obtain

\begin{matrix}x_{k}(t) = x_{0}(t) + ∫_{0}^{s_{1}}(V x_{k})(τ)dZ(τ) + ... + ∫_{s_{k−2}}^{s_{k−1}}(V x_{k})(τ)dZ(τ) + ∫_{s_{k−1}}^{t} (V x_{k})(τ)dZ(τ) \\ = x_{0}(t) + ∫_{0}^{t}(V x_{k})(τ)dZ(τ) (t ∈ [s_{k−1}, s_{k}]). \end{matrix}

Again uniqueness of solutions gives the equality x_{k}(t) = x(t) for t ∈ [s_{k−1}, s_{k}], and the induction is completed.

In particular, x_{l}(t) = x(t) = x(t, x_{0}, φ) for 0 ≤ t ≤ s_{l} and hence for 0 ≤ t ≤ s_{l}, as s ≤ s_{l}. If we now show

that sup_{t≥0}(𝔼|x_{k}(t)|^{2p})^{1/2p} < ∞ for k = 1, ..., l and any φ, x_{0} such that vraisup_{ν<0} 𝔼|φ(ν)|^{2p} < ∞, 𝔼|x_{0}|^{2p} < ∞,

then it would give us sup_{0≤t≤s} 𝔼|x(t, x_{0}, φ)|^{2p} < ∞ for any φ, x_{0} such that vraisup_{ν<0} 𝔼|φ(ν)|^{2p} < ∞, 𝔼|x_{0}|^{2p} < ∞.

To see it, we first of all prove that sup_{t≥0}(𝔼|x_{0}(t)|^{2p})^{1/2p} < ∞. Let χ_{h}(t) be a function on [0, ∞) defined

by χ_{h}(t) = \{ \begin{matrix}0, if h(t) ≥ 0, \\ 1, if h(t) < 0. \end{matrix} Put ā_{ij}(t) = χ_{h_{ij}}(t)a_{ij}(t) for i = 1, ..., m, j = 0, ...m_{i}. Evidently, x_{0}(t) =

x_{0} + ∑_{j=0}^{m_{1}} ∫_{0}^{t} ā_{1j}(τ)φ_{h_{1j}}(τ)dτ + ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} ∫_{0}^{t} ā_{ij}(τ)φ_{h_{ij}}(τ)dB_{i}(τ).

Thus we obtain

sup_{t≥0}(𝔼|x_{0}(t)|^{2p})^{1/2p} ≤ (𝔼|x_{0}|^{2p})^{1/2p} + ∑_{j=0}^{m_{1}} sup_{t≥0}(𝔼| ∫_{0}^{t} ā_{1j}(τ)φ_{h_{1j}}(τ)dτ|^{2p})^{1/2p}+

∑_{i=2}^{m} ∑_{j=0}^{m_{i}} sup_{t≥0}(𝔼| ∫_{0}^{t} ā_{ij}(τ)φ_{h_{ij}}(τ)dB_{i}(τ)|^{2p})^{1/2p} ≤

(𝔼|x_{0}|^{2p})^{1/2p} + ∑_{j=0}^{m_{1}} sup_{t≥0}((∫_{0}^{t} |ā_{1j}(τ)|dτ)^{(2p−1)/2p}(𝔼 ∫_{0}^{t} |ā_{1j}(τ)||φ_{h_{1j}}(τ)|^{2p}dτ)^{1/2p})+

∑_{i=2}^{m} ∑_{j=0}^{m_{i}} c_{p} sup_{t≥0}(𝔼(∫_{0}^{t}(ā_{ij}(τ)φ_{h_{ij}}(τ))^{2}dτ)^{p})^{1/2p} ≤ (𝔼|x_{0}|^{2p})^{1/2p}+

∑_{j=0}^{m_{1}} sup_{t≥0} ∫_{0}^{t} |ā_{1j}(τ)|dτ vraisup_{t≥0} (𝔼|φ_{h_{1j}}(t)|^{2p})^{1/2p}+

∑_{i=2}^{m} ∑_{j=0}^{m_{i}} c_{p} sup_{t≥0}((∫_{0}^{t}(ā_{ij}(τ))^{2}dτ)^{(p−1)/2p}(𝔼 ∫_{0}^{t}(ā_{ij}(τ))^{2}|φ_{h_{ij}}(τ)|^{2p}dτ)^{1/2p}) ≤

(𝔼|x_{0}|^{2p})^{1/2p} + (∑_{j=0}^{m_{1}} sup_{t≥0} ∫_{0}^{t} |ā_{1j}(τ)|dτ + ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} c_{p} sup_{t≥0}(∫_{0}^{t}(ā_{ij}(τ))^{2}dτ)^{0.5}) vraisup_{ν<0} (𝔼|φ_{(}ν)|^{2p})^{1/2p}.

From this and from the assumptions 1), 2), 3) on the parameters in (1), (2) we conclude that

sup_{t≥0}(𝔼|x_{0}(t)|^{2p})^{1/2p} < ∞ for any φ, x_{0} such that vraisup_{ν<0} 𝔼|φ(ν)|^{2p} < ∞, 𝔼|x_{0}|^{2p} < ∞.

From (3) we obtain

sup_{t≥0}(𝔼|x_{k}(t)|^{2p})^{1/2p} ≤ sup_{t≥0}(𝔼|x_{k−1}(t)|^{2p})^{1/2p} + ∑_{j=0}^{m_{1}} sup_{t≥0}(𝔼| ∫_{0}^{t} I_{k}(τ)a_{1j}(τ)(S_{h_{1j}}x_{k})(τ)dτ|^{2p})^{1/2p}+

∑_{i=2}^{m} ∑_{j=0}^{m_{i}} sup_{t≥0}(𝔼| ∫_{0}^{t} I_{k}(τ)a_{ij}(τ)(S_{h_{ij}}x_{k})(τ)dB_{i}(τ)|^{2p})^{1/2p} ≤

sup_{t≥0}(𝔼|x_{k−1}(t)|^{2p})^{1/2p} + ∑_{j=0}^{m_{1}} sup_{t≥0}((∫_{0}^{t} |I_{k}(τ)a_{1j}(τ)|dτ)^{(2p−1)/2p}(𝔼 ∫_{0}^{t} |I_{k}(τ)a_{1j}(τ)||(S_{h_{1j}}x_{k})(τ)|^{2p}dτ)^{1/2p})+

∑_{i=2}^{m} ∑_{j=0}^{m_{i}} c_{p} sup_{t≥0}(𝔼(∫_{0}^{t}(I_{k}(τ)a_{ij}(τ)(S_{h_{ij}}x_{k})(τ))^{2}dτ)^{p})^{1/2p} ≤

EJQTDE, 2008 No. 23, p. 5

sup_{t≥0}(𝔼|x_{k−1}(t)|^{2p})^{1/2p} + ∑_{j=0}^{m_{1}} sup_{t≥0} ∫_{0}^{t} |I_{k}(τ)a_{1j}(τ)|dτ sup_{t≥0}(𝔼|(S_{h_{1j}}x_{k})(t)|^{2p})^{1/2p}+

∑_{i=2}^{m} ∑_{j=0}^{m_{i}} c_{p} sup_{t≥0}((∫_{0}^{t}(I_{k}(τ)a_{ij}(τ))^{2}dτ)^{(p−1)/2p})(𝔼 ∫_{0}^{t}(I_{k}(τ)a_{ij}(τ))^{2}|(S_{h_{ij}}x_{k})(τ)|^{2p}dτ)^{1/2p}) ≤

sup_{t≥0}(𝔼|x_{k−1}(t)|^{2p})^{1/2p} + (∑_{j=0}^{m_{1}} sup_{t≥0} ∫_{0}^{t} |I_{k}(τ)a_{1j}(τ)|dτ+

∑_{i=2}^{m} ∑_{j=0}^{m_{i}} c_{p} sup_{t≥0}(∫_{0}^{t}(I_{k}(τ)a_{ij}(τ))^{2}dτ)^{0.5}) sup_{t≥0}(𝔼|x_{k}(t)|^{2p})^{1/2p} ≤ sup_{t≥0}(𝔼|x_{k−1}(t)|^{2p})^{1/2p} + ρ sup_{t≥0}(𝔼|x_{k}(t)|^{2p})^{1/2p},

where ρ < 1 in accordance with the choice of s_{k} (k = 1, ..., l).

This implies that for any φ, x_{0} such that vraisup_{ν<0} 𝔼|φ(ν)|^{2p} < ∞, 𝔼|x_{0}|^{2p} < ∞ one has

sup_{t≥0}(𝔼|x_{k}(t))|^{2p})^{1/2p} < ∞ for k = 1, ..., l. This is exactly what we needed. By this, the theorem is proved.

Let T be an arbitrary number from the interval [0, ∞). In addition to (1), (2) let us consider the following equation

dy(t) = ∑_{j=0}^{m_{1}} a_{1j}(t)y(h_{1j}(t))dt + ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} a_{ij}(t)y(h_{ij}(t))dB_{i}(t) (t ≥ T),  (4)
y(ν) = ψ(ν) (ν < T),  (5)

where ψ(ν) (ν < T) is a measurable stochastic process which is independent of the Wiener process B(t) = (B_{2}(t), ..., B_{m}(t)), (t ≥ T) and which a. s. has trajectories from L^{∞}. Note that the other parameters are the same as in (1), (2). The following result will be used in the next section.

Theorem 2. The zero solution of the homogeneous equation corresponding to (1), (2) is exponentially 2p–stable w.r.t. the initial function (i.e. exponentially Lyapunov 2p–stable) if for some T ∈ [0, ∞) the zero solution of the homogeneous equation corresponding to (4), (5) is exponentially 2p–stable w.r.t. the initial function.

Proof. Let y(t, y_{0}, ψ) be solution of (4), (5) satisfying y(T, y_{0}, ψ) = y_{0}.

Assume that the zero solution y(t, 0, 0) of the homogeneous equation corresponding to (4), (5) is exponentially 2p–stable w.r.t. the initial function. This means that there exist positive numbers ĉ, β_{1}, for which 𝔼|y(t, y_{0}, ψ)|^{2p} ≤ ĉ(𝔼|y_{0}|^{2p}+ vraisup_{ν<T} 𝔼|ψ(ν)|^{2p}) exp{−β_{1}t} (t ≥ T). We are to show that

𝔼|x(t, x_{0}, φ)|^{2p} ≤ c̄(𝔼|x_{0}|^{2p}+ vraisup_{ν<0} 𝔼|φ(ν)|^{2p}) exp{−βt} (t ≥ 0) for some positive constants c̄, β. In the

equation (4), (5) we define ψ(ν) to be φ(ν) if ν < 0 and x(ν, x_{0}, φ) if 0 ≤ ν < T. Then taking y_{0} = x(T, x_{0}, φ) implies x(t, x_{0}, φ) = y(t, y_{0}, ψ) for t ≥ T. From Theorem 1 we obtain 𝔼|y_{0}|^{2p} < ∞, vraisup_{ν<T} 𝔼|ψ(ν)|^{2p} < ∞

for any φ, x_{0} such that vraisup_{ν<0} 𝔼|φ(ν)|^{2p} < ∞, 𝔼|x_{0}|^{2p} < ∞. This and the exponential 2p–stability of

the zero solution y(t, 0, 0) w.r.t. the initial function yields the exponential 2p–stability of the zero solution x(t, 0, 0) of the equation (1), (2) w.r.t. the initial function.

The theorem is proved.

4. The main theorem

We start with a brief description of the W-transform method referring the reader to the paper [6] for further details. However, one of the results from [6] will be formulated explicitly (in a simplified form adjusted to our objectives).

We have already mentioned that any W -transform comes from an auxiliary equation, which is called a reference equation. Like in the Lyapunov function(al) approach, there is no precise algorithm describing how

EJQTDE, 2008 No. 23, p. 6 to choose the reference equation. Roughly speaking, this is an equation, which is similar to the equation in question, but ”simpler”. In addition, the reference equation must a priori have the asymptotic properties which we want the original equation to have as well.

Below we again use the notation Z(t) = (t, B_{2}(t), ..., B_{m}(t))^{T} . The stochastic process Z(t) is a continuous m-dimensional semimartingale (see e. g. [1]). The equation (1), (2) can then be rewritten as follows:

dx(t) = [(V x)(t) + f(t)]dZ(t) (t ≥ 0),

To do it, we use the notation from Section 3:

\begin{matrix}(V x)(t) = ((V_{1}x)(t), ..., (V_{m}x)(t)) , (V_{i}x)(t) = ∑_{j=0}^{m_{i}} a_{ij}(t)(S_{h_{ij}}x)(t) (i = 1, ...m), \\ f(t) = (f_{1}(t), ..., f_{m}(t)) , f_{i}(t) = ∑_{j=0}^{m_{i}} a_{ij}(t)φ_{h_{ij}}(t) (i = 1, ...m). \end{matrix}

Let the reference equation have the form

dx(t) = [(Qx)(t) + g(t)]dZ(t) (t ≥ 0),

where Q is a K-linear Volterra operator (see e. g. [6] where the notation k was used instead of K).

Assume further that the reference equation admits ”the Cauchy representation” x(t) = U(t)x(0) + (W g)(t) (t ≥ 0), where U(t) is the fundamental matrix of the associated homogeneous equation, and W is the corresponding Cauchy (also called Green’s) operator. This representation gives rise to the W-transform, which is applied to the original equation in the following manner:

dx(t) = [(Qx)(t) + ((V − Q)x)(t) + f(t)]dZ(t) (t ≥ 0),

or, alternatively,

x(t) = U(t)x(0) + (W (V − Q)x)(t) + (W f)(t) (t ≥ 0).

Denoting W (V − Q) = Θ, we obtain the operator equation

((I − Θ)x)(t) = U(t)x(0) + (W f)(t) (t ≥ 0).

Put

M_{2p,T} ≡ {x : x ∈ C, ||x||_{M_{2p,T}} := (sup_{t≥T} 𝔼|x(t)|^{p})^{1/2p} < ∞},

where C denotes the set of all (F_{t})_{t≥T} -adapted stochastic processes with a. s. continuous trajectories. The norm in this linear space is defined by sup_{t≥T}(𝔼|x(t)|^{2p})^{1/2p}. The main idea of the W-transform approach is to

check invertibility of the operator I − Θ in the space M_{2p} ≡ M_{2p,0} with the norm ||x||_{M_{2p,0}} ≡ ||x||_{M_{2p}}, as the theory [6] says that this would imply exponential 2p-stability.

To justify this reduction, we need some additional hypotheses.

R1. The fundamental solution U(t) satisfies |U(t)| ≤ c̄, where c̄ ∈ R_{+}.

R2. The W-operator is integral:

(W g)(t) = ∫_{0}^{t} C(t, s)g(s)dZ(s) (t ≥ 0),

where the function C(t, s) is defined on G := {(t, s) : t ∈ [0, ∞), 0 ≤ s ≤ t}, and satisfies

|C(t, s)| ≤ ¯ c̄ exp{−α(t − s)}

for some α > 0, ¯ c̄ > 0.

Remark 4. In our paper [5] we showed that asymptotic properties of linear stochastic equations can be deduced, as in the deterministic case, from the property of admissibility of certain pairs of functional spaces. We exploit this idea, however somewhat implicitly, in the present paper as well. This partly explains why we claim that the reference equation should have the same asymptotic properties as the equation to be studied,

EJQTDE, 2008 No. 23, p. 7 while obviously the estimate analogous to (EXP) is not valid for an arbitrary g(t) in the reference equation. In fact, we mean that as soon as perturbations f(t) and g(t) are of the same kind, then the asymptotic properties of the two equations will be the same, too. For instance, if we assume that g(t) is similar to f(t) above, i.e. g(t) has a compact support, then the solutions of the reference equation would satisfy a similar exponential estimate. Likewise, if f(t) is as general as g(t) is assumed to be, then the solutions of neither equation would satisfy the estimate (EXP) or simliar, in general.

Lemma Let Z(t) = (t, B_{2}(t), ..., B_{m}(t))^{T}, the conditions 1)-3) be fulfilled and the reference equation satisfy the hypotheses R1-R2. Assume that the operator (I − Θ) : M_{2p} → M_{2p} has a bounded inverse. Then the solutions to the equation (1), (2) satisfies the estimate (EXP) for some β > 0.

Proof. The lemma is just a particular case of our previous result [6, Corollary 4.2], where we should put ξ = 1 and observe that Condition D2 follows from our assumption 3).

Remark 5. The above lemma holds true if we replace the interval [0, ∞) with the interval [T, ∞) where T > 0. Accordingly, we should also replace M_{2p} = M_{2p,0} with M_{2p,T}. This observation will be used in the course of the proof of Theorem 3 below.

We let h^{T} (t) be a function on [T, ∞), which is defined for a given function h(t)(t ∈ [T, ∞)) in the following way

h^{T} (t) = \{ \begin{matrix}h(t), if h(t) ≥ T, \\ T, if h(t) < T. \end{matrix}

Theorem 3. Let for some subset I ⊂ {0, ..., m_{1}} and some numbers a_{0} > 0, γ_{i} > 0, i = 1, 2, T ∈ [0, ∞)

the estimates ∑_{k∈I} a_{1k}(t) ≤ −a_{0}, ∑_{k∈I} |a_{1k}(t)|[ ∑_{j=0}^{m_{1}} ∫_{h_{1k}^{T}(t)}^{t} |a_{1j}(s)|ds+c_{p} ∑_{i=2}^{m} ∑_{j=0}^{m_{i}}( ∫_{h_{1k}^{T}(t)}^{t}(a_{ij}(s))^{2}ds)^{0.5}]+ ∑_{k¯∈I} |a_{1k}(t)| ≤

−γ_{1} ∑_{k∈I} a_{1k}(t), ∑_{i=2}^{m} ∑_{j=0}^{m_{i}}(a_{ij}(t))^{2} ≤ −2γ_{2} ∑_{k∈I} a_{1k}(t) be valid for t ≥ T a.s. If, in addition, γ_{1} + c_{p}\sqrt{γ_{2}} < 1,

where c_{p} are specified in Section 2, then the solutions to the equation (1), (2) satisfy the estimate (EXP) for some β > 0.

Proof. First of all, we notice that without loss of generality we may assume that the ’prehistory’ process φ(ν) (ν < 0) is F_{0}-measurable. Indeed, it is independent of the Wiener process in (1), so that adding the σ-algebra generated by φ(ν) (ν < 0) to the filtration (F_{t})_{t≥0}, results in an extended filtration, under which the Wiener process preserves its properties, while φ(ν) (ν < 0) becomes F_{0}-measurable.

Due to Theorem 2 it suffices to show that under the assumptions of the present theorem the zero solution of the homogeneous equation corresponding to the equation (4), (5) is exponentially 2p–stable w.r.t. the initial function for some T ∈ [0, ∞).

To do this, we make use of Lemma above as well as Remark 4.

We introduce the operator S_{h}^{T} putting

(S_{h}^{T} x)(t) = \{ \begin{matrix}x(h(t)), if h(t) ≥ T, \\ 0, if h(t) < T \end{matrix}

(so that S_{h}^{0} = S_{h}).

The reference equation is chosen as follows:

dy(t) = [∑_{k∈I} a_{1k}(t)y(t) + g_{0}(t)]dt + ∑_{i=1}^{m} g_{i}(t)dB_{i}(t) (t ≥ T),

where g_{0} is an (F_{t})_{t≥T} -adapted stochastic process with a.s. locally integrable trajectories, g_{i} (i = 2, ..., m) is an (F_{t})_{t≥T} -adapted stochastic process with a.s. locally square-integrable trajectories, and the other parameters are defined in (1), (2). This is an ordinary differential equation with stochastic perturbations. Thus, it satisfies the assumptions R1–R2 (with α = a_{0}), which means that the theorem will be proved if the operator (I − Θ) : M_{2p,T} → M_{2p,T} is shown to be invertible.

EJQTDE, 2008 No. 23, p. 8 To see this, let us write down the operator Θ explicitly:

(Θy)(t) = ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}{[∑_{k∈I} a_{1k}(s) ∫_{h_{1k}^{T}(s)}^{s} (∑_{j=0}^{m_{1}} a_{1j}(τ)(S_{h_{1j}}^{T}y)(τ)dτ+

∑_{i=2}^{m} ∑_{j=0}^{m_{i}} a_{ij}(τ)(S_{h_{ij}}^{T}y)(τ)dB_{i}(τ)) + ∑_{k¯∈I} a_{1k}(s)(S_{h_{1k}}^{T}y)(s)]ds + ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} a_{ij}(s)(S_{h_{ij}}^{T}y)(s)dB_{i}(s)},

and M_{2p,T} is described right before the lemma.

Now, to prove the property of invertibility of the linear operator I − Θ we estimate the norm of the operator Θ in the space M_{2p,T}. We have

||Θy||_{M_{2p,T}} ≤ sup_{t≥T}(𝔼| ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}[∑_{k∈I} a_{1k}(s)(∑_{j=0}^{m_{1}} ∫_{h_{1k}^{T}(s)}^{s} a_{1j}(τ)(S_{h_{1j}}^{T}y)(τ)dτ+

∑_{i=2}^{m} ∑_{j=0}^{m_{i}} ∫_{h_{1k}^{T}(s)}^{s} a_{ij}(τ)(S_{h_{ij}}^{T}y)(τ)dB_{i}(τ)) + ∑_{k¯∈I} a_{1k}(s)(S_{h_{1k}}^{T}y)(s)]ds|^{2p})^{1/2p}+

c_{p} ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} sup_{t≥T}(𝔼(∫_{T}^{t} exp{2 ∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}(a_{ij}(s)(S_{h_{ij}^{T}}y)(s))^{2}ds)^{p})^{1/2p} ≤

sup_{t≥T}(𝔼| ∑_{k∈I} ∑_{j=0}^{m_{1}} ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}a_{1k}(s) ∫_{h_{1k}^{T}(s)}^{s} a_{1j}(τ)(S_{h_{1j}}^{T}y)(τ)dτds+

∑_{k∈I} ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}a_{1k}(s) ∫_{h_{1k}^{T}(s)}^{s} a_{ij}(τ)(S_{h_{ij}}^{T}y)(τ)dB_{i}(τ)ds+

∑_{k¯∈I} ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}a_{1k}(s)(S_{h_{1k}}^{T}y)(s)ds|^{2p})^{1/2p}+

c_{p} ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} sup_{t≥T}((∫_{T}^{t} exp{2 ∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}(a_{ij}(s))^{2}ds)^{(p−1)/2p}×

(𝔼 ∫_{T}^{t} exp{2 ∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}(a_{ij}(s))^{2}|(S_{h_{ij}}^{T}y)(s)|^{2p}ds)^{1/2p}) ≤

∑_{k∈I} ∑_{j=0}^{m_{1}} sup_{t≥T}((∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}|a_{1k}(s)| ∫_{h_{1k}^{T}(s)}^{s} |a_{1j}(τ)|dτds)^{(2p−1)/2p}×

(𝔼 ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}|a_{1k}(s)|( ∫_{h_{1k}^{T}(s)}^{s} |a_{1j}(τ)|dτ)^{1−2p}| ∫_{h_{1k}^{T}(s)}^{s} a_{1j}(τ)(S_{h_{1j}}y)(τ)dτ|^{2p}ds)^{1/2p})+

∑_{k∈I} ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} sup_{t≥T}((∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}|a_{1k}(s)|( ∫_{h_{1k}^{T}(s)}^{s} |a_{ij}(τ)|^{2}dτ)^{0.5}ds)^{(2p−1)/2p}×

(𝔼 ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}|a_{1k}(s)|( ∫_{h_{1k}^{T}(s)}^{s} |a_{ij}(τ)|^{2}dτ)^{0.5−p}| ∫_{h_{1k}^{T}(s)}^{s} a_{ij}(τ)(S_{h_{ij}}^{T}y)(τ)dB_{i}(τ)|^{2p}ds)^{1/2p})+

∑_{k¯∈I} sup_{t≥T}((∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}|a_{1k}(s)|ds)^{(2p−1)/2p}×

(𝔼 ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}|a_{1k}(s)| |(S_{h_{1k}}^{T}y)(s)|^{2p}ds)^{1/2p})+

c_{p} ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} sup_{t≥T}((∫_{T}^{t} exp{2 ∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}(a_{ij}(s))^{2}ds)^{(p−1)/2p}×

(𝔼 ∫_{T}^{t} exp{2 ∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}(a_{ik}(s))^{2}|(S_{h_{ij}}^{T}y)(s)|^{2p}ds)^{1/2p}).

Taking into account that

𝔼| ∫_{h_{1k}^{T}(s)}^{s} a_{1j}(τ)(S_{h_{1j}}^{T}y)(τ)dτ|^{2p} ≤ ( ∫_{h_{1k}^{T}(s)}^{s} |a_{1j}(τ)|dτ)^{2p−1}𝔼 ∫_{h_{1k}^{T}(s)}^{s} |a_{1j}(τ)| |(S_{h_{1j}}^{T}y)(τ)|^{2p}dτ ≤

( ∫_{h_{1k}^{T}(s)}^{s} |a_{1j}(τ)|dτ)^{2p}(||y||_{M_{2p,T}})^{2p}

for k ∈ I, j = 0, ..., m_{1} and

𝔼| ∫_{h_{1k}^{T}(s)}^{s} a_{ij}(τ)(S_{h_{ij}}^{T}y)(τ)dB_{i}(τ)|^{2p} ≤ c_{p}𝔼( ∫_{h_{1k}^{T}(s)}^{s} (a_{ij}(τ)(S_{h_{ij}}^{T}y)(τ))^{2}dτ)^{p} ≤

c_{p}( ∫_{h_{1k}^{T}(s)}^{s} (a_{ij}(τ))^{2}dτ)^{p−1}𝔼 ∫_{h_{1k}^{T}(s)}^{s} (a_{ij}(τ))^{2}|(S_{h_{ij}}^{T}y)(τ)|^{2p}dτ ≤ ( ∫_{h_{1k}^{T}(s)}^{s} (a_{ij}(τ))^{2}dτ)^{p}(||y||_{M_{2p,T}})^{2p}

for k ∈ I, i = 2, ..., m, j = 0, ..., m_{i} we obtain

||Θy||_{M_{2p,T}} ≤ [sup_{t≥T} ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}(∑_{k∈I} |a_{1k}(s)|[∑_{j=0}^{m_{1}} ∫_{h_{1k}^{T}(s)}^{s} |a_{1j}(τ)|dτ+

c_{p} ∑_{i=2}^{m} ∑_{j=0}^{m_{i}}( ∫_{h_{1k}^{T}(s)}^{s} (a_{ij}(τ))^{2}dτ)^{0.5}] + ∑_{k¯∈I} |a_{1k}(s)|)ds+

c_{p} sup_{t≥T}(∫_{T}^{t} exp{2 ∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ} ∑_{i=2}^{m} ∑_{k=0}^{m_{i}}(a_{ik}(s))^{2}ds)^{0.5}]||y||_{M_{2p,T}} ≤

[sup_{t≥T} ∫_{T}^{t} exp{∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ}(−γ_{1} ∑_{k∈I} a_{1k}(s))ds+

c_{p} sup_{t≥T}(∫_{T}^{t} exp\{2 ∫_{s}^{t}(∑_{k∈I} a_{1k}(τ))dτ\}(−2γ_{2} ∑_{k∈I} a_{1k}(s))ds)^{0.5}]||y||_{M_{2p,T}} ≤ [γ_{1} + c_{p}\sqrt{γ_{2}}]||y||_{M_{2p,T}} .

As we assumed that γ_{1} + c_{p}\sqrt{γ_{2}} < 1, the estimate ||Θ||_{M_{2p,T}} < 1 is proved, and so is the theorem.

In the last section we show how Theorem 3 provides sufficient conditions for exponential 2p–stability of the zero solution to specific stochastic equations.

5. Some corollaries

In the corollaries below we again use the universal constants c_{p} (1 ≤ p < ∞) which are described in [1, p. 65].

Corollary 1. Let h_{10}(t) ≡ t for t ∈ [0, ∞). Assume that there exist numbers a_{0} > 0, γ_{i} > 0, i = 1, 2,

T ∈ [0, ∞) such that a_{10}(t) ≤ −a_{0}, ∑_{j=1}^{m_{1}} |a_{1j}(t)| ≤ −γ_{1}a_{10}(t), ∑_{i=2}^{m} ∑_{j=0}^{m_{i}}(a_{ij}(t))^{2} ≤ −2γ_{2}a_{10}(t) for t ≥ T a.s. If,

in addition, γ_{1} + c_{p}\sqrt{γ_{2}} < 1, then the solutions to the equation (1), (2) satisfy the estimate (EXP) for some β > 0.

Corollary 1 follows directly from Theorem 3 if we put I = {0}.

Corollary 2. Assume that there exist numbers a_{0} > 0, γ_{i} > 0, i = 1, 2, T ∈ [0, ∞) such

that ∑_{k=0}^{m_{1}} a_{1k}(t) ≤ −a_{0}, ∑_{k=0}^{m_{1}} |a_{1k}(t)|[ ∑_{j=0}^{m_{1}} ∫_{h_{1k}^{T}(t)}^{t} |a_{1j}(s)|ds + c_{p} ∑_{i=2}^{m} ∑_{j=0}^{m_{i}}( ∫_{h_{1k}^{T}(t)}^{t}(a_{ij}(s))^{2}ds)^{0.5}] ≤ −γ_{1} ∑_{k=0}^{m_{1}} a_{1k}(t),

∑_{i=2}^{m} ∑_{j=0}^{m_{i}}(a_{ij}(t))^{2} ≤ −2γ_{2} ∑_{k=0}^{m_{1}} a_{1k}(t) for t ≥ T a.s. If now γ_{1} + c_{p}\sqrt{γ_{2}} < 1. Then the solutions to the equation

(1), (2) satisfy the estimate (EXP) for some β > 0.

Corollary 2 follows directly from Theorem 3 if we put I = {0, ..., m_{1}}.

Corollary 3. Assume that there exists a number T ∈ [0, ∞) such that the coefficients in (1), (2) satisfy

a_{1j}(t) = A_{1j}r(t), k = 0, ..., m_{1}, a_{ij}(t) = A_{ij}\sqrt{r(t)}, i = 2, ..., m, j = 0, ..., m_{i},
r(t) ≥ r_{0} > 0 (t ∈ [T, ∞)) a.s.

Assume also that for some I ⊂ \{0, ..., m_{1}\} the estimates ∑_{k∈I} A_{1k} < 0 and γ_{1} + c_{p}\sqrt{γ_{2}} < 1 are valid, where

γ_{1} = lim_{t→∞} sup_{T ≤τ≤t}[ ∑_{k∈I} |A_{1k}[ ∫_{h_{1k}^{T}(τ)}^{τ} r(s)ds ∑_{j=0}^{m_{1}} |A_{1j}|+c_{p}( ∫_{h_{1k}^{T}(τ)}^{τ} r(s)ds)^{0.5} ∑_{i=2}^{m} ∑_{j=0}^{m_{i}} |A_{ij}|]+ ∑_{k¯∈I} |A_{1k}]/(− ∑_{k∈I} A_{1k}), γ_{2} =

(∑_{i=2}^{m} ∑_{j=0}^{m_{i}} A_{ij}^{2})/(−2 ∑_{k∈I} A_{1k}). Then the solutions to the equation (1), (2) satisfy the estimate (EXP) for some

β > 0.

The next two corollaries provide sufficient stability conditions for a particular case of (1), (2) given by the following scalar stochastic differential equation

dx(t) = (−a(t)x(t) − b(t)x(h(t)))dt + c(t)x(g(t))dB(t) (t ≥ 0),  (6)
x(ν) = φ(ν) (ν < 0),  (7)

where φ is a stochastic process which is independent of the standard scalar Wiener process B and which a. s. has trajectories from L^{∞}. The functions a, b, c, g, h in (6) are all Lebesgue-measurable, a, b are, in addition, locally integrable, c is locally square-integrable, h(t) ≤ t, g(t) ≤ t for t ∈ [0, ∞) a.s., vraisup_{t≥0} (t − h(t)) < ∞,

vraisup_{t≥0} (t − h(t)) < ∞.

Corollary 4. Assume that there exist numbers a_{0} > 0, γ_{i} > 0, i = 1, 2, T ∈ [0, ∞) such that for the equation (6), (7) one of the following conditions holds:

1) a(t) ≥ a_{0}, |b(t)| ≤ γ_{1}a(t), (c(t))^{2} ≤ 2γ_{2}a(t) (t ≥ T) a.s.,

EJQTDE, 2008 No. 23, p. 11

2) a(t) + b(t) ≥ a_{0}, |b(t)|[ ∫_{h^{T} (t)}^{t} (|a(s)| + |b(s)|)ds + c_{p}( ∫_{h^{T} (t)}^{t} (c(s))^{2}ds)^{0.5}] ≤

γ_{1}(a(t) + b(t)), (c(t))^{2} ≤ 2γ_{2}(a(t) + b(t)) (t ≥ T) a.s.

If, in addition, γ_{1} + c_{p}\sqrt{γ_{2}} < 1, then the solutions to the equation (6), (7) satisfy the estimate (EXP) for some β > 0.

Corollary 5. Assume that there exists a number T ∈ [0, ∞) such that the coefficients in (6), (7) satisfy

a(t) = Ar(t), b(t) = Br(t), c(t) = C\sqrt{r(t)}, r(t) ≥ r_{0} > 0 (t ∈ [T, ∞)) a.s.

Assume also that one of the following conditions holds:

1) A > 0, |B|/A + c_{p}|C|/\sqrt{2A} < 1,

2) A+B > 0, lim_{t→∞} sup_{T ≤τ≤t} |B|[ ∫_{h^{T} (τ)}^{τ} r(s)ds(|A|+|B|)+c_{p}( ∫_{h^{T} (τ)}^{τ} r(s)ds)^{0.5}|C|]/(A+B)+c_{p}|C|/\sqrt{2(A + B)} < 1,

3) A > 0, B > 0, lim_{t→∞} sup_{T ≤τ≤t} B[ ∫_{h^{T} (τ)}^{τ} r(s)ds + c_{p}( ∫_{h^{T} (τ)}^{τ} r(s)ds)^{0.5}|C|/(A + B)] + c_{p}|C|/\sqrt{2(A + B)} < 1.

Then the solutions to the equation (6), (7) satisfy the estimate (EXP) for some β > 0.

Finally, we consider another particular case of the equation (1), (2) given by

dx(t) = (−a(t)x(t) − b(t)x(h(t)) − d(t)x(l(t)))dt + c(t)x(g(t))dB(t) (t ≥ 0),  (8)
x(ν) = φ(ν) (ν < 0),  (9)

where φ is a stochastic process which is independent of the standard scalar Wiener process B and which a. s. has trajectories from L^{∞}. The functions a, b, c, d, g, h, l in (8) are all assumed to be Lebesgue-measurable, where a, b, d are, in addition, locally integrable, c is locally square-integrable, h(t) ≤ t, l(t) ≤ t, g(t) ≤ t for t ∈ [0, ∞) a. s., vraisup_{t≥0} (t − h(t)) < ∞, vraisup_{t≥0} (t − l(t)) < ∞, vraisup_{t≥0} (t − g(t)) < ∞.

Corollary 6. Assume that there exist numbers a_{0} > 0, γ_{i} > 0, i = 1, 2, T ∈ [0, ∞) such that for the equation (8), (9) one of the following conditions holds:

1) a(t) ≥ a_{0}, |b(t)| + |d(t)| ≤ γ_{1}a(t), (c(t))^{2} ≤ 2γ_{2}a(t) (t ≥ T) a.s.,

2) a(t) + b(t) ≥ a_{0}, |b(t)|[ ∫_{h^{T} (t)}^{t} (|a(s)| + |b(s)| + |d(s)|)ds + c_{p}( ∫_{h^{T} (t)}^{t} (c(s))^{2}ds)^{0.5}] ≤

γ_{1}(a(t) + b(t)), (c(t))^{2} ≤ 2γ_{2}(a(t) + b(t)) (t ≥ T) a.s.,

3) a(t) + d(t) ≥ a_{0}, |d(t)|[ ∫_{l^{T} (t)}^{t} (|a(s)| + |b(s)| + |d(s)|)ds + c_{p}( ∫_{l^{T} (t)}^{t} (c(s))^{2}ds)^{0.5}] ≤

γ_{1}(a(t) + d(t)), (c(t))^{2} ≤ 2γ_{2}(a(t) + d(t)) (t ≥ T) a.s.,

4) a(t) + b(t) + d(t) ≥ a_{0}, |b(t)|[ ∫_{h^{T} (t)}^{t} (|a(s)| + |b(s)| + |d(s)|)ds + c_{p}( ∫_{h^{T} (t)}^{t} (c(s))^{2}ds)^{0.5}]+

|d(t)|[ ∫_{l^{T} (t)}^{t} (|a(s)| + |b(s)| + |d(s)|)ds + c_{p}( ∫_{l^{T} (t)}^{t} (c(s))^{2}ds)^{0.5}] ≤

γ_{1}(a(t) + b(t) + d(t)), (c(t))^{2} ≤ 2γ_{2}(a(t) + b(t) + d(t)) (t ≥ T) a.s. EJQTDE, 2008 No. 23, p. 12 If, in addition, γ_{1} + c_{p}\sqrt{γ_{2}} < 1, then the solutions to the equation (8), (9) satisfy the estimate (EXP) for some β > 0.

Corollary 7. Assume that there exists a number T ∈ [0, ∞) such that the coefficients in (8), (9) satisfy

a(t) = Ar(t), b(t) = Br(t), d(t) = Dr(t), c(t) = C\sqrt{r(t)}, r(t) ≥ r_{0} > 0 (t ∈ [T, ∞)) a.s.

Assume also that one of the following conditions holds:

1) A > 0, (|B| + |D|)/A + c_{p}|C|/\sqrt{2A} < 1,

2) A+B >0, lim_{t→∞} sup_{T ≤τ≤t}|B|[ ∫_{h^{T} (τ)}^{τ} r(s)ds(|A|+|B|+|D|)+c_{p}( ∫_{h^{T} (τ)}^{τ} r(s)ds)^{0.5}|C|]/(A+B)+c_{p}|C|/\sqrt{2(A + B)}<1,

3) A+D >0, lim_{t→∞} sup_{T ≤τ≤t}|D|[ ∫_{l^{T} (τ)}^{τ} r(s)ds(|A|+|B|+|D|)+c_{p}( ∫_{l^{T} (τ)}^{τ} r(s)ds)^{0.5}|C|]/(A+D)+c_{p}|C|/)\sqrt{2(A + D)}<1,

4) A + B + D > 0, lim_{t→∞} sup_{T ≤τ≤t} |B|[ ∫_{h^{T} (τ)}^{τ} r(s)ds(|A| + |B| + |D|) + c_{p}( ∫_{h^{T} (τ)}^{τ} r(s)ds)^{0.5}|C|]/(A + B + D)+

lim_{t→∞} sup_{T ≤τ≤t} |D|[ ∫_{l^{T} (τ)}^{τ} r(s)ds(|A|+|B|+|D|)+c_{p}( ∫_{l^{T} (τ)}^{τ} r(s)ds)^{0.5}|C|]/(A+B +D)+c_{p}|C|/\sqrt{2(A + B + D)} < 1,

5) A > 0, B > 0, D > 0, lim_{t→∞} sup_{T ≤τ≤t} B[ ∫_{h^{T} (τ)}^{τ} r(s)ds + c_{p}( ∫_{h^{T} (τ)}^{τ} r(s)ds)^{0.5}|C|/(A + B + D)]+

lim_{t→∞} sup_{T ≤τ≤t} D[ ∫_{l^{T} (τ)}^{τ} r(s)ds + c_{p}( ∫_{l^{T} (τ)}^{τ} r(s)ds)^{0.5}|C|/(A + B + D)] + c_{p}|C|/\sqrt{2(A + B + D)} < 1.

Then the solutions to the equation (8), (9) satisfy the estimate (EXP) for some β > 0.

Acknowledgments. The authors thank an anonymous referee for the valuable comments and suggestions which helped us to improve the paper.

References

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EJQTDE, 2008 No. 23, p. 13

(Received January 30, 2007)

Dagestan Scientific Center, Russian Academy of Sciences, Makhachkala 367005 (Russia) E-mail address: dgu@dgu.ru

CIGENE & Department of Mathematical Sciences and Technology, Norwegian University of Life Sciences, NO-1432 Ås (Norway)

E-mail address: arkadi@umb.no