ON ASYMPTOTIC STABILITY OF LINEAR STOCHASTIC VOLTERRA ...

ON ASYMPTOTIC STABILITY OF LINEAR STOCHASTIC VOLTERRA DIFFERENCE EQUATIONS WITH RESPECT TO A FADING PERTURBATION JOHN A. D. APPLEBY, MARKUS RIEDLE, AND ALEXANDRA RODKINA Abstract. The paper concerns necessary and sufficient conditions on the fading intensity of a state–independent stochastic perturbation for the asymptotic stability of a linear stochastic Volterra difference equation. In broad terms, it is shown here that the results obtained in the deterministic case are robust to fading stochastic perturbations which are independent of the state, once it is known that these perturbations fade more rapidly than an identifiable critical rate.

1. Introduction This note can be considered as an extension of the continuous–time approaches and results of the paper [1] to the discrete case, in which necessary and sufficient conditions for the stability of linear Itô–Volterra equations with deterministically fading noise intensity is obtained. Similar conditions on the noise intensity are established in the continuous case in [2]. Almost sure asymptotic stability results for autonomous scalar non-linear difference equations without delays were obtained in [3] and in [4]. The outline of the note is as follows. Section 2 lists and comments upon the general results obtained. Section 3 shows how the general results can be applied to specific problems, and shows the connections between conditions required for the almost sure asymptotic stability of linear Volterra difference equations and corresponding continuous time Itô–Volterra equations driven by Brownian motion. Finally, Section 4 contains proofs of the results. 2. Results The following standard notation is employed. N denotes the natural numbers, Q the positive rational numbers, R the real numbers, and Rr r–dimensional real space, for r ∈ N. If E is a subset of R then the characteristic function χE : R → R is defined by 1, if x ∈ E, χE (x) = 0, if x 6∈ E. +

Date: 13 October 2006. Key words and phrases. Volterra difference equation; stochastic Volterra difference equation; resolvent; almost sure asymptotic stability. The first author was partially supported by an Albert College Fellowship awarded by Dublin City University’s Research Advisory Panel. The third author was partially supported by Programme Committee of the 11th International Conference of Difference Equations and by a New Initiative Fellowship awarded by University if the West Indies. 1

2

JOHN A. D. APPLEBY, MARKUS RIEDLE, AND ALEXANDRA RODKINA

The standard basis vectors in Rr are denoted by e1 , . . . , er . The standard innerproduct of x and y in Rr is denoted by hx, yi. The d × d real valued identity matrix is denoted P by Id . We say that a real sequence a = {a(n) : n ∈ N} obeys 1 a ∈ ` (N; R) if n∈N |a(n)| < ∞. We say that the d × r matrix–valued sequence a = {a(n) = (a(n))ij : n ∈ N} is in `1 (N; Rd × Rr ) if each entry aij ∈ `1 (N; R), i = 1, . . . , d, j = 1, . . . , r. The convolution of two sequences f = {f (n) : n ∈ N} and g = {g(n) : n ∈ N}, f ∗ g, is a sequence defined by (f ∗ g)(n) =

n X

f (n − k)g(k),

n ∈ N.

k=0

The ∗ notation is also used here for the convolution of distribution functions; however, the type of convolution being used will be clear from the context. We consider the stochastic Volterra difference equation (2.1) X(n + 1) = X(n) +

n X

A(n − k)X(k) + σ(n)ξ(n + 1),

n ∈ N;

X(0) = ζ.

k=0

Here A ∈ `1 (N; Rd ×Rd ), σ(n) ∈ Rd ×Rr for all n ∈ N. Each ξ(n) is a random vector in Rr . We make the following standing assumption about ξ through the paper: (2.2) ξ = {ξ(n) : n ∈ N} is a sequence of independent Rr –valued random vectors; (2.3)

ξj (n) := hξ(n), ej i have distribution function F , ∀n ∈ N, ∀j = 1, . . . , r;

(2.4)

E[ξj (n)] = 0 E[ξj (n)2 ] = 1,

(2.5)

{ξj (n)}rj=1

for fixed n,

∀n ∈ N, ∀j = 1, . . . , r;

are independent random variables.

Let R be the resolvent defined by (2.6)

R(n + 1) = R(n) +

n X

A(n − k)R(k),

n ∈ N;

R(0) = Id ,

k=0 d

d

so that each R(n) ∈ R × R . The significance of the resolvent in this context is that it allows X to be written purely in terms of the perturbation, according to the variation of parameters formula (2.7)

X(n) = R(n)ζ +

n X

R(n − k)σ(k − 1)ξ(k),

n = 1, 2, . . .

k=1

Everywhere in the paper we suppose that for A and R (defined by (2.6)) the following condition holds true: (2.8)

A ∈ `1 (N; Rd × Rr ),

R ∈ `1 (N; Rd × Rd ).

The importance of the process Σ defined by Σ(n + 1) := σ(n)ξ(n + 1) is that its almost sure asymptotic stability is equivalent to the almost sure asymptotic stability of X, provided the underlying deterministic resolvent R is summable. This result is proven in Proposition 2.1 below. Consequently, it is unnecessary to study the correlated unbounded delay equation for X directly; instead it is enough to determine the conditions under which an independent white noise process with zero lag tends to zero. Moreover, an explicit formula is not generally available for X, because it

LINEAR STOCHASTIC VOLTERRA DIFFERENCE EQUATIONS

3

depends on R, which is in general not known in closed form. On the other hand, however, the formula for Σ could hardly be simpler. Proposition 2.1. Suppose that the sequence of random variables ξ = {ξ(n) : n ∈ N} obeys (2.2)–(2.5). Let A and R obey (2.8) and let X be defined by (2.1). Then the following are equivalent: (a) limn→∞ σ(n)ξ(n + 1) = 0, a.s. (b) limn→∞ X(n) = 0, a.s. There is a simple sufficient condition which ensures σ(n)ξ(n + 1) → 0 as n → ∞ a.s. Proposition 2.2. Suppose that ξ = {ξ(n) : n ∈ N} is a Rr –valued sequence of random vectors (2.4), and σ = {σ(n) : n ∈ N} is a Rd × Rr –valued P∞which obeys 2 sequence. If n=1 |σ(n)| < ∞, then σ(n)ξ(n + 1) → 0 as n → ∞, a.s. The merit of this summability condition on σ is that it does not require independence of the stochastic sequence; however, it is not clear how close it comes to being optimal in ensuring almost sure stability of X. In the rest of the paper, we concern ourselves with establishing necessary and sufficient conditions on σ and the random sequence ξ which give stability. Next, we need to connect the condition on the stochastic equation (2.1) to the conditions on the data. This is achieved by the following result. Lemma 2.3. Let σ = {σ(n) : n ∈ N} be a Rd × Rr –valued sequence. Suppose that the sequence of random variables ξ = {ξ(n) : n ∈ N} obeys (2.2)–(2.5). Define the distribution function Fn,i,j by  F (x/|σij (n)|), σij (n) > 0,  1 − F (−x/|σij (n)|), σij (n) < 0, (2.9) Fn,i,j (x) =  χ[0,∞) (x), σij (n) = 0, and Fn,i as the convolution of the distributions Fn,i,j for j = 1, . . . , r. Then d X ∞ X 1 − Fn,i (ε) + Fn,i (−ε) < ∞

(2.10)

for all ε ∈ Q+

i=1 n=1

is equivalent to σ(n)ξ(n + 1) → 0 as n → ∞ a.s. In the scalar case (d = r = 1), this simplifies to give Lemma 2.4. Let σ = {σ(n) : n ∈ N} be a real sequence. Suppose that ξ = {ξ(n) : n ∈ N} is a sequence of independent random variables with the same distribution function F with E[ξ(n)] = 0 and E[ξ(n)2 ] = 1. (a) The following statements are equivalent: (2.11)

∞ X

1 − F (ε/|σ(n)|) + F (−ε/|σ(n)|) < ∞,

n=0

(2.12)

lim σ(n)ξ(n + 1) = 0,

n→∞

a.s.

for all ε ∈ Q+ ;

4


(b) The following statements are equivalent: (2.13)

∞ X

1 − F (ε/|σ(n)|) + F (−ε/|σ(n)|) = ∞,

for some ε ∈ Q+ ;

n=0

lim sup |σ(n)ξ(n + 1)| > ε,

(2.14)

a.s.

n→∞

In the general case, by combining Lemma 2.3 and Proposition 2.1, we get the following necessary and sufficient conditions on F and σ(n) such that X(n) → 0 a.s. as n → ∞. Theorem 2.5. Let σ = {σ(n) : n ∈ N} be a Rd × Rr –valued sequence. Suppose the sequence of random variables ξ = {ξ(n) ∈ Rr : n ∈ N} obeys (2.2)– (2.5). Let A and R obey (2.8) and let X be defined by (2.1). Then the following are equivalent: (a) (2.10) holds, where Fn,i and Fn,i,j are as defined in Lemma 2.3; (b) limn→∞ X(n) = 0, a.s. In the scalar case, by combining Lemma 2.4 and Proposition 2.1, we may establish a more easily–expressed result which characterises asymptotic stability. Theorem 2.6. Let σ = {σ(n) : n ∈ N} be a real sequence. Suppose ξ = {ξ(n) : n ≥ 1} is a sequence of independent and identically distributed random variables with the same distribution function F such that E[ξ(n)] = 0, E[ξ(n)2 ] = 1. Let A and R obey (2.8) and let X be defined by (2.1). (a) If F and σ obey (2.11), then P[limn→∞ X(n) = 0] = 1. (b) If F and σ do not obey (2.11), then P[limn→∞ X(n) = 0] = 0. 3. Applications and Extensions of Main Results If we additionally assume some monotonicity in n 7→ σ 2 (n), we can recover the critical rate of convergence in the case when the white noise process is Gaussian. We first state the result in the scalar case and then indicate how the result can be generalized to the finite dimensional case. Corollary 3.1. Let σ = {σ(n) : n ∈ N} be a real sequence. Let ξ = {ξ(n) : n ∈ N} be a sequence of independently and identically distributed standard normal random variables. Let A and R obey (2.8) and let X be defined by (2.1). (a) If (3.1)

∞ X

ε2 σ(n) exp − 2 2σ (n) n=1

< ∞,

for all ε ∈ Q+ ,

then P[limn→∞ X(n) = 0] = 1. (b) If (3.1) does not hold, then P[limn→∞ X(n) = 0] = 0. If, moreover, n 7→ σ 2 (n) is non–increasing, then the following are equivalent: (A) limn→∞ σ 2 (n) log n = 0; (B) P[limn→∞ X(n) = 0] > 0; (C) P[limn→∞ X(n) = 0] = 1.


5

A proof of the following finite–dimensional analogue of Corollary 3.1 is not given, but left to the interested reader as an exercise. The key observations in proving this result are that (i) the i–th component of σ(n)ξ(n + 1), which is Y (i) (n) = Pr Pr 2 j=1 σij (n)ξj (n + 1) is normally distributed N 0, j=1 σij (n) , and (ii) for each i = 1, . . . , d, Y (i) = {Y (i) (n) : n ∈ N} is a sequence of independent random variables. Corollary 3.2. Let σ = {σ(n) : n ∈ N} be a Rd × Rr –valued sequence. Let ξ = {ξ(n) ∈ Rr : n ∈ N} be a sequence of random vectors obeying (2.2)–(2.5) such that ξj (n) = hξ(n), ej iP are standard normal random variables, j = 1, . . . , r. Define 2 σi (n) ≥ 0 by σi (n) = rj=1 σij2 (n), i = 1, . . . , d. Let A and R obey (2.8) and let X be defined by (2.1). (a) If d X ∞ X ε2 (3.2) σi (n) exp − 2 < ∞, for all ε ∈ Q+ , 2σ (n) i i=1 n=1 then P[limn→∞ X(n) = 0] = 1. (b) If (3.2) does not hold, then P[limn→∞ X(n) = 0] = 0. If, moreover, each of the maps n 7→ σi2 (n), i = 1, . . . , d, is non–increasing, then the following are equivalent: (A) limn→∞ kσ(n)k2 log n = 0; (B) P[limn→∞ X(n) = 0] > 0; (C) P[limn→∞ X(n) = 0] = 1. These two results are analogous to those found in [1], which studies the asymptotic behaviour of linear continuous–time stochastic Volterra equations of Itô–type where the driving semimartingale is Brownian motion. The necessary and sufficient conditions on the rate of decay of the noise intensity σ given in (A) is the same as that required in [1]. The rate of decay σ 2 (n) log n → 0 has also been shown to be necessary and sufficient for almost sure asymptotic stability in a class of nonlinear stochastic delay–differential equations studied in [2]. This logarithmic decay condition was shown to be necessary for a class of non–delay gradient dynamical system in Chan and Williams [5]. Now give a result for a scalar equation in which the distribution of ξ has power–law decay in the tails, with the same rate of decay being present in each tail. Corollary 3.3. Let ξ = {ξ(n) : n ∈ N} be a sequence of independently and identically distributed random variables such that E[ξ(n)] = 0, E[ξ(n)2 ] = 1 and whose common distribution function F obeys lim F (x)|x|β = c1 ,

x→−∞

lim [1 − F (x)]xβ = c2 ,

x→∞

for some β > 2, and positive c1 and c2 . Let A and R obey (2.8) and let X be defined by (2.1). P β (a) If P∞ n=1 |σ(n)| < ∞, then P[limn→∞ X(n) = 0] = 1. ∞ β (b) If n=1 |σ(n)| = ∞, then P[limn→∞ X(n) = 0] = 0. To prove the results in this section, the following auxiliary results are needed.

6


Lemma 3.4. Let ξ = {ξ(n) : n ∈ N} be a sequence of independently and identically distributed random variables, with common distribution function F which does not have compact support. If (2.12) holds, then limn→∞ σ(n) = 0. Lemma 3.5. P Suppose that (a(n))n≥0 is a non–negative and non–increasing sequence such that ∞ n=0 a(n) < ∞. Then na(n) → 0 as n → ∞. 4. Proofs 4.1. Proof of Proposition 2.1. We show first that (a) implies (b). If Σ(n + 1) := σ(n)ξ(n + 1) → 0 as n → ∞, a.s., it follows immediately from the fact that R ∈ `1 (N; Rd × Rd ) that (R ∗ Σ)(n) → 0 as n → ∞, a.s. Moreover, because R ∈ `1 (N; Rd × Rd ), we have R(n) → 0 as n → ∞. Therefore by (2.7) we have that X(n) → 0 as n → ∞, a.s. To show (b) implies (a), suppose X(n) P → 0 as n → ∞ a.s. Rewriting (2.1) gives σ(n)ξ(n + 1) = X(n + 1) − X(n) − nk=0 A(n − k)X(k). Since A ∈ `1 (Rd×d ), we have (A ∗ X)(n) → 0 as n → ∞, a.s. Therefore, σ(n)ξ(n + 1) → 0 as n → ∞, a.s. Pr 4.2. Proof of Proposition 2.2. Consider Y (i) (n) = j=1 σij (n)ξj (n + 1), i = (i) 1, . . . , d. Then σ(n)ξ(n + 1) → 0 as n → ∞, a.s. once Y (n) → 0 a.s. as n → ∞. Evidently E[Y (i) (n)] = 0, while !2 r X r r X X |σij (n)||σil (n)| = |σij (n)| . E[Y (i) (n)2 ] ≤ j=1 l=1

j=1

This estimate, Chebyshev’s inequality, and |σ(n)ξ(n + 1)|1 = !2 ∞ d r ∞ X 1 XX X |σij (n)| , P[|σ(n)ξ(n + 1)|1 > ε] ≤ 2 ε n=1 i=1 n=1 j=1

Pd

i=1

|Y (i) (n)| give

for any ε > 0.

P 2 The righthand side can be bounded by an n–independent multiple of ∞ n=1 |σ(n)|F , which is finite by hypothesis. The result now follows from the Borel–Cantelli lemma. P 4.3. Proof of Lemma 2.3. Fix ε ∈ Q+ . Define Y (i) (n) = rj=1 σij (n)ξj (n + 1), i = 1, . . . , d. Proving σ(n)ξ(n + 1) → 0 as n → ∞ a.s. is equivalent to proving that (i) Y (i) (n) → 0 as n → ∞ a.s. for each i = 1, . . . , d. Define An,ε = {ω : |Y (i) (n)| > ε}. (i) Then P[A¯n,ε ev.] = 1 for every ε ∈ Q+ is equivalent to limn→∞ Y (i) (n) = 0, a.s. Indeed, since for each fixed i the sequence Y (i) = {Y (i) (n) : n ∈ N} is an independent (i) sequence, by the Borel–Cantelli lemma P[A¯n,ε ev.] = 1 for all ε ∈ Q+ is equivalent P∞ (i) to n=1 P[An,ε all ε ∈ Q+ . Hence σ(n)ξ(n + 1) → 0 as n → ∞ a.s. is P] ∞< ∞, for equivalent to n=1 P[|Y (i) (n)| > ε] < ∞ for all ε ∈ Q+ . Let Fn,i be the distribution function of Y (i) (n), F the distribution function of ξi , and Fn,i,j the distribution function of σij (n)ξj (n + 1). Then F (x/|σij (n)|), σij (n) > 0, Fn,i,j (x) = P[σij (n)ξj (n + 1) ≤ x] = 1 − F (−x/|σij (n)|), σij (n) < 0. If σij (n) = 0, Fn,i,j (x) = 1 if x ≥ 0 and Fn,i,j (x) = 0 if x < 0. Thus Fn,i,j (x) = χ[0,∞) (x). Since for each fixed n, (ξj (n))rj=1 are independent random variables, Fn,i = Fn,i,1 ∗ . . . ∗ Fn,i,r , so P[|Y (i) (n)| > ε] = 1 − Fn,i (ε) + Fn,i (−ε), completing the proof.


7

4.4. Proof of Lemma 2.4. Define Y (n) = σ(n)ξ(n + 1) and Aεn = {ω ∈ Ω : |Y (n, ω)| > ε}. Thus, if σ(n) 6= 0, we have P[Aεn ] = 1 − F (ε/|σ(n)|) + F (−ε/|σ(n)|), and P[Aεn ] = 0 if σ(n) = 0. To prove part (a), by theP Borel–Cantelli Lemma and ε ε ε ¯ independence of the An , P[An ev.] = 1 is equivalent to ∞ n=1 P[An ] < ∞. Therefore lim sup |Y (n)| ≤ ε on Ωε , with P[Ωε ] = 1 is equivalent to n→∞

∞ X

P[Aεn ] < ∞,

n=1 +

so by considering this equivalence for all ε ∈ Q , we conclude that (2.11) is equivalent to (2.12). To prove part (b), the independence of the Aεn and the Borel–Cantelli P∞ ε0 ¯ P[Aεn0 ] = ∞ for lemma imply that P[An ev.] = 0 for some ε > 0 is equivalent to n=1 P∞ ε0 some ε0 > 0. Thus P[An i.o.] = 1 for some ε0 > 0 is equivalent to n=1 P[Aεn0 ] = ∞ for some ε0 > 0. Therefore we conclude that (2.13) is equivalent to (2.14). 4.5. Proof of Corollary 3.1. Let the common distribution function be F . Then 1 2

1 2

lim F (x)|x|e 2 x = (2π)−1/2 , lim [1 − F (x)]|x|e 2 x = (2π)−1/2 .

x→−∞

x→−∞

To prove part (a), notice by Lemma 3.4 that we must have σ(n) → 0 as n → ∞. Now, let ε be a fixed positive rational number. Therefore, the estimates 1 1 − F (ε/|σ(n)|) 1 F (−ε/|σ(n)|) = √ , lim =√ , lim 2 2 1 ε 1 ε n→∞ |σ(n)| n→∞ |σ(n)| − − 2π 2π · e 2 σ2 (n) · e 2 σ2 (n) ε

ε

hold and so 1

(4.1)

ε2

lim [1 − F (ε/|σ(n)|) + F (−ε/|σ(n))]ε|σ(n)|−1 e 2 σ2 (n) = 2(2π)−1 .

n→∞

Thus (4.2)

∞ X

|σ(n)|ε−1 exp −2−1 ε2 σ −2 (n) < ∞

n=1

implies P[X(n) → 0 as n → ∞] = 1. To prove part (b), if the sum in (4.2) is infinite for some ε ∈ Q+ , then lim supn→∞ |σ(n)ξ(n + 1)| > 0 a.s., and so a brief perusal of the proof of Proposition 2.1 reveals that P[limn→∞ X(n) = 0] = 0. It is easily seen that the implication holds under the hypothesis (a) above. Now suppose that n 7→ σ 2 (n) is non–increasing. First, note that σ 2 (n) log n → 0 as n → ∞ implies (3.1). By part (a), this implies that X(n) → 0 a.s. as n → ∞. Thus (A) implies (C), and (C) obviously implies (B). Finally, we show that (B) implies (A). According to (B) and the proof of Proposition 2.1 it follows that P[limn→∞ σ(n)ξ(n+1) = 0] > 0. However, as (ξ(n))∞ n=1 is a sequence of independent random variables, this event is in the tail σ–field of the ξ and as it is of positive probability, by the Kolmogorov zero–one law it is almost sure. Thus by Lemma 2.4, and (4.1), (4.2) holds for every ε ∈ Q+ . Now, fix ε ∈ Q+ . Then the sequence (aε (n))n≥0 defined by aε (n) = ε−1 |σ(n)| exp (−ε2 /(2σ 2 (n))), n > N1 (ε) is non–increasing. By Lemma 3.5, limn→∞ naε (n) = 0, or limn→∞ n|σ(n)| exp (−ε2 /(2σ 2 (n))) = 0. Thus lim σ −2 (n) σ 2 (n) log n + 2−1 σ 2 (n) log σ 2 (n) − 2−1 ε2 = −∞. n→∞ Since limx→∞ x log x = 0, we have lim supn→∞ σ 2 (n) log n − 21 ε2 ≤ 0. However, this inequality holds for every ε ∈ Q+ , and so it follows that limn→∞ σ 2 (n) log n = 0.

8


P β 4.6. Proof of Corollary 3.3. To do this suppose first that ∞ n=1 |σ(n)| < ∞. Then limn→∞ σ(n) = 0. Let ε be any number in Q+ . Since ε/ limn→∞ σ(n) = ∞, (4.3) lim |σ(n)|−β F (−ε/|σ(n)|) = c1 ε−β , n→∞

lim |σ(n)|−β [1−F (ε/|σ(n)|)] = c2 ε−β ,

n→∞

and so (2.11) holds. ThenPby Theorem 2.6, limn→∞ X(n) = 0 a.s. On the other hand, β consider the case when ∞ n=1 |σ(n)| = ∞. Suppose that lim supn→∞ |σ(n)| > 0. Then lim supn→∞ |σ(n)ξ(n + 1)| = ∞ a.s., and so P[limn→∞ X(n) = 0] = 0. If lim supn→∞ |σ(n)| = 0, we can repeat theP calculations above to get (4.3). Since the denominator is not summable, we have ∞ n=0 [F (−ε/|σ(n)|) + 1 − F (ε/|σ(n)|)] = ∞. Thus by Theorem 2.6, we have P[limn→∞ X(n) = 0] = 0, whence the result. 4.7. Proof of Lemma 3.4. Suppose to that lim supn→∞ σ 2 (n) = c0 > 0. Then, for (c) (c) every c ∈ (0, c0 ) there is a sequence mn → ∞ such that σ 2 (mn ) > c for all n > (c) (c) (c) N (c). Since lim supn→∞ |ξ(mn +1)| = ∞, a.s., we have lim supn→∞ σ 2 (mn )ξ 2 (mn + 1) = ∞ a.s., which contradicts the assumption. Hence σ 2 (n) → 0 as n → ∞. 4.8. Proof of Lemma 3.5. By the monotonicity of a, the inequalities 2j−1 a(2j ) ≤ P2j−1 j−1 + l) hold for each j, so we get l=1 a(2 ∞ ∞ ∞ X X 1X j j a(j) ≥ a(1) + 2j−1 a(2j ) = a(1) + 2 a(2 ). 2 j=1 j=1 j=1 Therefore 2j a(2j ) → 0 as j → ∞. This means that for every ε > 0 there is J = J(ε) such that j > J implies 2j a(2j ) < ε. Consider now N = 2J+1 . Let n > N . Clearly, there is j > 0 such that 2j ≤ n < 2j+1 . Then 2j > n/2 > N/2 = 2J , so j > J. Hence na(n) < 2j+1 a(2j ) < 2ε, for all n > N . References [1] J. A. D. Appleby, M. Riedle. Almost sure asymptotic stability of stochastic Volterra integrodifferential equations with fading perturbations. Stoch. Anal. Appl., 24(4), 813–826, 2006. [2] J. A. D. Appleby, A. Rodkina. On the asymptotic stability of polynomial stochastic delay differential equations. Funct. Differ. Equ., 12(1–2), 35–66, 2005. [3] J. A. D. Appleby, D. Mackey, A. Rodkina. Almost sure polynomial asymptotic stability of stochastic difference equations. Contemporary Mathematics: Fundamental Directions, 17, 110– 128, 2006 (in Russian). [4] G. Berkolaiko, A. Rodkina. Almost Sure Convergence of Solutions to Non-Homogeneous Stochastic Difference Equation. Difference Equations and Applications, 12(6), 2006. [5] T. Chan, D. Williams. An “excursion” approach to an annealing problem. Math. Proc. Camb. Philos. Soc., 105 (1), 169–176, 1989. School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland E-mail address: [email protected] URL: http://webpages.dcu.ie/~applebyj Humboldt-University of Berlin, 10099 Berlin, Germany E-mail address: [email protected] Dept of Math/CSci, University of the West Indies, Kingston 7, Jamaica. E-mail address: [email protected]