Convergence to Nonequilibrium Limits for Stochastic Volterra Equations

Volume 2008, Article ID 473156,27pages doi:10.1155/2008/473156

Research Article

Characterisation of Exponential

Convergence to Nonequilibrium Limits for Stochastic Volterra Equations

John A. D. Appleby,¹ Siobh ´an Devin,²and David W. Reynolds¹

1School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland

2School of Mathematical Sciences, University College Cork, Cork, Ireland

Correspondence should be addressed to John A. D. Appleby,john.appleby@dcu.ie Received 25 October 2007; Accepted 11 May 2008

Recommended by Jiongmin Yong

This paper considers necessary and sufficient conditions for the solution of a stochastically and deterministically perturbed Volterra equation to converge exponentially to a nonequilibrium and nontrivial limit. Convergence in an almost sure andpth mean sense is obtained.

Copyrightq2008 John A. D. Appleby et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, we study the exponential convergence of the solution of dXt

AXt _t

0

Kt−sXsdsft

dt ΣtdBt, t >0, 1.1a

X0 X0, 1.1b

to a nontrivial random variable. Here the solution X is an n-dimensional vector-valued function on 0,∞,A is a realn×n-dimensional matrix, K is a continuous and integrable n×n-dimensional matrix-valued function on0,∞,fis a continuousn-dimensional vector- valued function on0,∞,Σis a continuous n×d-dimensional matrix-valued function on 0,∞andBt B1t, B2t, . . . , Bdt,where each component of the Brownian motion is independent. The initial conditionX0is a deterministic constant vector.

The solution of1.1a-1.1bcan be written in terms of the solution of the resolvent equation

Rt ARt _t

0

Kt−sRsds, t >0, 1.2a

R0 I, 1.2b

where the matrix-valued function R is known as the resolvent or fundamental solution.

In 1 , the authors studied the asymptotic convergence of the solution R of 1.2a-1.2b to a nontrivial limit R∞. It was found that R −R∞ being integrable and the kernel being exponentially integrable were necessary and sufficient for exponential convergence. This built upon a result of Murakami 2 who considered the exponential convergence of the solution to a trivial limit and a result of Krisztin and Terj´eki3 who obtained necessary and sufficient conditions for the integrability ofR−R∞. A deterministically perturbed version of 1.2a-1.2b,

xt Axt _t

0

Kt−sxsdsft, t >0, 1.3a

x0 x0, 1.3b

was also studied in1 . It was shown that the exponential decay of the tail of the perturbation f combined with the integrability ofR−R∞and the exponential integrability of the kernel were necessary and sufficient conditions for convergence to a nontrivial limit.

The case where1.2a-1.2bis stochastically perturbed dXt

AXt _t

0

Kt−sXsds

dt ΣtdBt, t >0, 1.4a

X0 X0, 1.4b

has been considered. Various authors including Appleby and Freeman 4 , Appleby and Riedle 5 , Mao6 , and Mao and Riedle7 have studied convergence to equilibrium. In particular the paper by Appleby and Freeman4 considered the speed of convergence of solutions of 1.4a-1.4b to equilibrium. It was shown that under the condition that the kernel does not change sign on0,∞then ithe almost sure exponential convergence of the solution to zero,iithepth mean exponential convergence of the solution to zero, and iiithe exponential integrability of the kernel and the exponential square integrability of the noise are equivalent.

Two papers by Appleby et al.8,9 considered the convergence of solutions of1.4a- 1.4b to a nonequilibrium limit in the mean square and almost sure senses, respectively.

Conditions on the resolvent, kernel, and noise for the convergence of solutions to an explicit limiting random variable were found. A natural progression from this work is the analysis of the speed of convergence.

This paper examines1.1a-1.1band builds on the results in1,8,9 . The analysis of 1.1a-1.1b is complicated, particularly in the almost sure case, due to presence of both a deterministic and stochastic perturbation. Nonetheless, the set of conditions which characterise the exponential convergence of the solution of 1.1a-1.1b to a nontrivial random variable is found. It can be shown that the integrability ofR−R_∞, the exponential integrability of the kernel, the exponential square integrability of the noise combined with the exponential decay of the tail of the deterministic perturbation,t → _∞

t fsds, are necessary and sufficient conditions for exponential convergence of the solution to a nontrivial random limit.

2. Mathematical preliminaries

In this section, we introduce some standard notation as well as giving a precise definition of 1.1a-1.1band its solution.

LetRdenote the set of real numbers and letRⁿdenote the set ofn-dimensional vectors with entries inR. Denote byeitheith standard basis vector inRⁿ. Denote byAthe standard Euclidean norm for a vectorA a1, . . . , angiven by

A²ⁿ

i1

a²_i trAA^T, 2.1

where tr denotes the trace of a square matrix.

LetR^n×n be the space ofn×nmatrices with real entries whereIis the identity matrix.

Let diaga1, a2, . . . , andenote the n×nmatrix with the scalar entriesa1, a2, . . . , an on the diagonal and 0 elsewhere. ForA aij∈R^n×dthe norm denoted by·is defined by

A^{2 n}

i1

d j1

|aij|². 2.2

The set of complex numbers is denoted byC; the real part ofzinCbeing denoted by Rez. The Laplace transform of the functionA:0,∞→R^n×dis defined as

Az _∞

0

Ate^−ztdt. 2.3

If∈ Rand_∞

0Ate^−tdt <∞thenAz exists for Rez ≥andz →Az is analytic for Rez > .

IfJis an interval in RandV a finite-dimensional normed space with norm·then CJ, Vdenotes the family of continuous functionsφ:J→V. The space of Lebesgue integrable functionsφ :0,∞→V will be denoted byL¹0,∞, Vwhere_∞

0 φtdt < ∞. The space of Lebesgue square-integrable functionsφ:0,∞→V will be denoted byL²0,∞, Vwhere _∞

0φt²dt <∞. WhenV is clear from the context, it is omitted it from the notation.

We now make our problem precise. We assume that the functionK : 0,∞→R^n×n satisfies

K∈C0,∞,R^n×n∩L¹0,∞,R^n×n, 2.4 the functionf:0,∞→Rⁿsatisfies

f ∈C0,∞,Rⁿ∩L¹0,∞,Rⁿ, 2.5

and the functionΣ:0,∞→R^n×dsatisfies

Σ∈C0,∞,R^n×d. 2.6 Due to2.4we may defineK1to be a functionK1∈C0,∞,R^n×nwith

K1t _∞

t

Ksds, t≥0, 2.7

where this function defines the tail of the kernelK. Similarly, due to2.5, we may definef1

to be a functionf1∈C0,∞,Rⁿgiven by f1t

_∞

t

fsds, t≥0. 2.8

We let {Bt}_t≥0 denote d-dimensional Brownian motion on a complete probability space Ω,F,{F^Bt}_t≥0,Pwhere the filtration is the natural oneF^Bt σ{Bs: 0≤s≤t}.

Under the hypothesis2.4, it is well known that1.2a-1.2bhas a unique continuous solutionR, which is continuously differentiable. We define the functiont→Xt;X0, f,Σto be the unique solution of the initial value problem1.1a-1.1b. IfΣandfare continuous then for any deterministic initial conditionX0there exists an almost surely unique continuous and F^B-adapted solution to1.1a-1.1bgiven by

Xt;X0,Σ, f RtX0 _t

0

Rt−sfsds _t

0

Rt−sΣsdBs, t≥0. 2.9 WhenX0, f, andΣare clear from the context, we omit them from the notation.

The notion of convergence and integrability in pth mean and almost sure senses are now defined: theRⁿ-valued stochastic process{Xt}_t≥0 converges in pth mean toX_∞ if lim_{t→ ∞}EXt−X∞^p 0; the process ispth mean exponentially convergent toX∞ if there exists a deterministicβp>0 such that

lim sup

t→∞

1

t logEXt−X∞^p≤ −βp; 2.10

we say that the difference between the stochastic process{Xt}_t≥0andX∞is integrable in the pth mean sense if

_∞

0

EXt−X_∞^pdt <∞. 2.11

If there exists aP-null setΩ0such that for everyω /∈Ω0, the following holds: limt→ ∞ Xt, ω X∞ω, then X converges almost surely to X∞; we say X is almost surely exponentially convergent toX_∞if there exists a deterministicβ0>0 such that

lim sup

t→∞

1

t logXt, ω−X∞ω ≤ −β0, a.s. 2.12 Finally, the difference between the stochastic process{Xt}_t≥0andX_∞is square integrable in the almost sure sense if

_∞

0

Xt, ω−X∞ω²dt <∞. 2.13 Henceforth, EX^p will be denoted by EX^p except in cases where the meaning may be ambiguous. A number of inequalities are used repeatedly in the sequel; they are stated here for clarity. If, for p, q ∈ 0,∞, the finite-dimensional random variables X and Y satisfy EX^p < ∞ and EY^q < ∞, respectively, then the Lyapunov inequality is useful when considering the pth mean behaviour of random variables as any exponent p > 0 may be considered:

EX^{p 1/p}≤EX^{q 1/q}, 0< p≤q. 2.14

The following proves useful in manipulating norms:

_n

i1

|xi|

k

≤n^k−1 n

i1

|xi|^k, n, k∈N. 2.15

3. Discussion of results

We begin by stating the main result of this paper. That is, we state the necessary and sufficient conditions required on the resolvent, kernel, deterministic perturbation, and noise terms for the solution of1.1a-1.1bto converge exponentially to a limiting random variable. In this paper, we are particularly interested in the case when the limiting random variable is nontrivial, although the result is still true for the case when the limiting value is zero.

Theorem 3.1. LetKsatisfy2.4and _∞

0

t²Ktdt <∞, 3.1 letΣsatisfy2.6, and letfsatisfy2.5. IfKsatisfies

each entry ofK does not change sign on0,∞, 3.2 then the following are equivalent.

iThere exists a constant matrixR_∞such that the solutionRof1.2a-1.2bsatisfies

R−R∞∈L²0,∞,R^n×n, 3.3

and there exist constantsα >0, γ >0, ρ >0 , andc1 >0 such thatKsatisfies _∞

0

Kse^αsds <∞, 3.4

Σsatisfies

_∞

0

Σs²e^2γsds <∞, 3.5

andf1, the tail off, defined by2.8satisfies

f1t ≤c1e^−γt, t≥0. 3.6 iiFor all initial conditions X0 and constants p > 0 there exists an a.s. finite F^B∞- measurable random variable X∞X0,Σ, f with EX∞^p < ∞ such that the unique continuous adapted processX·;X0,Σ, fwhich obeys1.1a-1.1bsatisfies

EXt−X_∞^p ≤m^∗_pe^−β∗pt, t≥0, 3.7 whereβ_p^∗andm^∗_pm^∗_pX0are positive constants.

iiiFor all initial conditionsX0there exists an a.s. finiteF^B∞-measurable random variable X∞X0,Σ, fsuch that the unique continuous adapted processX·;X0,Σ, fwhich obeys 1.1a-1.1bsatisfies

lim sup

t→∞

1

tlogXt−X∞ ≤ −β₀^∗a.s., 3.8 whereβ₀^∗is a positive constant.

The proof ofTheorem 3.1is complicated by the presence of two perturbations so as an initial step the case whenf :0 is considered. That is we consider the conditions required for exponential convergence of1.4a-1.4bto a limiting random variable.

Theorem 3.2. LetK satisfy 2.4and 3.1 and let Σ satisfy 2.6. IfK satisfies 3.2 then the following are equivalent.

iThere exists a constant matrixR∞such that the solutionRof 1.2a-1.2bsatisfies3.3 and there exist constants α > 0 andγ > 0 such that K and Σ satisfy3.4and 3.5, respectively.

iiFor all initial conditions X0 and constants p > 0 there exists an a.s. finite F^B∞- measurable random variable X_∞X0,Σ with EX_∞^p < ∞ such that the unique continuous adapted processX·;X0,Σwhich obeys1.4a-1.4bsatisfies

EXt−X∞^p ≤mpe^−βpt, t≥0, 3.9 whereβpandmpmpX0are positive constants.

iiiFor all initial conditionsX0there exists an a.s. finiteF^B∞-measurable random variable X∞X0,Σ such that the unique continuous adapted process X·;X0,Σ which obeys 1.4a-1.4bsatisfies

lim sup

t→∞

1

tlogXt−X∞ ≤ −β0 a.s., 3.10 whereβ0is a positive constant.

This result is interesting in its own right as it generalises a result in4 where necessary and sufficient conditions for exponential convergence to zero are found. Theorem 3.2 collapses to this case ifR∞0.

It is interesting to note the relationship between the behaviour of the solutions of 1.1a-1.1b,1.2a-1.2b, 1.3a-1.3b, and1.4a-1.4band the behaviour of the inputs K, f, andΣ. It is seen in1 thatKbeing exponentially integrable is the crucial condition for exponential convergence when we consider the resolvent equation. Each perturbed equation then builds on this resolvent case: for the deterministically perturbed equation we require the exponential integrability ofKand the exponential decay of the tail of the perturbation f see1 ; for the stochastically perturbed case we require the exponential integrability of K and the exponential square integrability ofΣ. In the stochastically and deterministically perturbed case it is seen that the perturbations do not interact in a way that exacerbates or diminishes the influence of the perturbations on the system: we can isolate the behaviours of the perturbations and show that the same conditions on the perturbations are still necessary and sufficient.

Theorem 3.1has application in the analysis of initial history problems. In particular this theoretical result could be used to interpret the equation as an epidemiological model.

Conditions under which a disease becomes endemicwhich is the interpretation that is given when solutions settle down to a nontrivial limitwere studied in9 . The theoretical results obtained in this paper could be exploited to highlight the speed at which this can occur within a population.

The remainder of this paper deals with the proofs of Theorems3.1and3.2. InSection 4 we prove the sufficiency of conditions onR, K, and Σfor the exponential convergence of

the solution of1.4a-1.4b while inSection 5 we prove the necessity of these conditions.

InSection 6 we prove the sufficiency of conditions onR, K, Σ, and f for the exponential convergence of the solution of1.1a-1.1b, whileSection 7deals with the necessity of the condition on Σ. In Section 8we combine our results to prove the main theorems, namely, Theorems3.1and3.2.

4. Sufficient conditions for exponential convergence of solutions of1.4a-1.4b In this section, sufficient conditions for exponential convergence of solutions of1.4a-1.4b to a nontrivial limit are obtained.Proposition 4.1concerns convergence in thepth mean sense whileProposition 4.2deals with the almost sure case.

Proposition 4.1. LetKsatisfy2.4and3.1, letΣsatisfy2.6andR_∞be a constant matrix such that the solutionRof 1.2a-1.2bsatisfies3.3. If there exist constantsα >0 andγ >0 such that 3.4and3.5hold, then there exist constantsβp > 0, independent ofX0, andmp mpX0 >0, such that statement (ii) ofTheorem 3.2holds.

Proposition 4.2. LetKsatisfy2.4and3.1, letΣsatisfy2.6andR∞be a constant matrix such that the solutionRof 1.2a-1.2bsatisfies3.3. If there exist constantsα >0 andγ >0 such that 3.4and3.5hold, then there exists a constantβ0 >0, independent ofX0such that statement (iii) ofTheorem 3.2holds.

In 8 , the conditions which give mean square convergence to a nontrivial limit were considered. So a natural progression in this paper is the examination of the speed of convergence in the mean square case.Lemma 4.3examines the case whenp 2 in order to highlight this important case. This lemma may be then used when generalising the result to allp >0.

Lemma 4.3. LetKsatisfy2.4and3.1, letΣsatisfy2.6, and letR∞be a constant matrix such that the solutionRof 1.2a-1.2bsatisfies3.3. If there exist constantsα >0 andγ >0 such that 3.4and3.5hold, then there exist constantsλ >0, independent ofX0, andmmX0>0, such that

EXt−X∞²≤mX0e^−2λt, t≥0. 4.1 From8,9 it is evident thatR−R∞ ∈ L²0,∞,R^n×nis a more natural condition on the resolvent thanR−R∞ ∈L¹0,∞,R^n×nwhen studying convergence of solutions of 1.4a-1.4b. However, the deterministic results obtained in1 are based on the assumption thatR−R∞∈L¹0,∞,R^n×n.Lemma 4.4is required in order to make use of these results in this paper; this result isolates conditions that ensure the integrability ofR−R∞onceR−R∞

is square integrable.

Lemma 4.4. LetKsatisfy2.4and3.1and letR∞be a constant matrix such that the solutionR of 1.2a-1.2bsatisfies3.3. Then the solutionRof 1.2a-1.2bsatisfies

R−R∞∈L¹0,∞,R^n×n. 4.2

We now state some supporting results. It is well known that the behaviour of the resolvent Volterra equation influences the behaviour of the perturbed equation. It is unsurprising therefore that an earlier result found in1 concerning exponential convergence of the resolventRto a limitR_∞in needed in the proof of Theorems3.1and3.2.

Theorem 4.5. LetKsatisfy2.4and3.1. Suppose there exists a constant matrixR∞such that the solutionRof 1.2a-1.2bsatisfies4.2. If there exists a constantα >0 such thatKsatisfies3.4 then there exist constantsβ >0 andc >0 such that

Rt−R∞ ≤ce^−βt, t≥0. 4.3 In the proof of Propositions4.1and4.2, an explicit representation ofX∞is required. In 8,9 the asymptotic convergence of the solution of1.4a-1.4bwas considered. Sufficient conditions for convergence were obtained and an explicit representation ofX∞was found:

Theorem 4.6. LetKsatisfy2.4and _∞

0

tKtdt <∞, 4.4

and letΣsatisfy2.6and

_∞

0

Σt²dt <∞. 4.5

Suppose that the resolventR of 1.2a-1.2bsatisfies3.3. Then the solution X of 1.4a-1.4b satisfies lim_{t→ ∞}Xt X∞ almost surely, where X∞ is an almost surely finite and F^B∞- measurable random variable given by

X∞R∞

X0

_∞

0

ΣtdBt

a.s. 4.6

Lemma 4.7concerns the structure ofX_∞in the almost sure case. It was proved in9 . Lemma 4.7. LetKsatisfy2.4and4.4. Suppose that for all initial conditionsX0there is an almost surely finite random variable X∞X0,Σsuch that the solution t → Xt;X0,Σof 1.4a-1.4b satisfies

limt→∞Xt;X0,Σ X∞X0,Σa.s., 4.7

X·;X0,Σ−X_∞X0,Σ∈L²0,∞,Rⁿa.s. 4.8

Then

A _∞

0

Ksds

X∞0 a.s. 4.9

It is possible to apply this lemma using our a priori assumptions due toTheorem 4.8, which was proved in9 .

Theorem 4.8. LetKsatisfy2.4and4.4and letΣsatisfy2.6. IfΣsatisfies4.5and there exists a constant matrix R∞ such that the solution R of 1.2a-1.2b satisfies3.3, then for all initial conditionsX0there is an almost surely finiteF^B∞-measurable random variableX_∞X0,Σ, such that the unique continuous adapted processX·;X0,Σwhich obeys1.4a-1.4bsatisfies4.7.

Moreover, if the functionΣalso satisfies _∞

0

tR∞Σt²dt <∞, 4.10

then4.8holds.

Lemma 4.9below is required in the proof of Lemma 4.4. It is proved in 8 . Before citing this result some notation is introduced. LetMA_∞

0KtdtandT be an invertible matrix such thatJT⁻¹MThas Jordan canonical form. Letei1 if all the elements of theith row ofJare zero, andei0 otherwise. LetDpdiage1, e2, . . . , enand putP TDpT⁻¹and QI−P.

Lemma 4.9. LetKsatisfy2.4and4.4. If there exists a constant matrixR∞such that the resolvent Rof 1.2a-1.2bsatisfies3.3, then

detIFz /0, Rez≥0, 4.11

whereFis defined by

Ft −e^−tQQA−e∗QKt P _∞

t

Kudu, t≥0. 4.12

Lemma 4.10concerns the moments of a normally distributed random variable. It can be extracted from4, Theorem 3.3 and it is used inProposition 4.1.

Lemma 4.10. Suppose the functionσ∈C0,∞×0,∞,R^p×rthen

E _b

a

σs, tdBs

^2m≤dmp, r _b

a

σs, t²ds m

, 4.13

wheredmp, r p^m1r^2m12m!m!2^m−1c2p, r^m.

The following lemma is used inProposition 4.2. A similar result is proved in4 . Lemma 4.11. Suppose thatK ∈C0,∞,R^n×n∩L¹0,∞,R^n×nand

_∞

0

Kse ^αsds <∞. 4.14

Ifλ > 0 andη2λ∧αthen _t

0

e^−2λt−s e^−αsKsds ≤ce^−ηt, 4.15

wherecis a positive constant.

The proofs of Propositions4.1and4.2and Lemmas4.3and4.4are now given.

Proof ofLemma 4.3. FromTheorem 4.6we see thatXt→X∞almost surely whereX∞is given by4.6, so we see that

EX∞²EtrX∞X_∞^T R∞X0² _∞

0

R∞Σs²ds <∞. 4.16 Since

EXt−X_∞² EtrXt−X_∞Xt−X_∞^T , 4.17

we use2.9and4.6to expand the right hand side of4.17to obtain EXt−X∞² Rt−R∞X0²

_t

0

Rt−s−R∞Σs²ds _∞

t

R_∞Σs²ds.

4.18 In order to obtain an exponential upper bound on4.18each term is considered individually.

We begin by considering the first term on the right-hand side of 4.18. Using 3.1 and 3.3we can applyLemma 4.4to obtain4.2. Then using3.1,4.2, and3.4we see from Theorem 4.5that

Rt−R_∞X0² ≤c1X0²e^−2βt. 4.19 We provide an argument to show that the second term decays exponentially. Using3.5and the fact thatR decays exponentially quickly toR∞we can choose 0 < λ < minβ, γsuch thateλΣandeλR−R∞∈L²0,∞where the functioneλis defined byeλt e^λt. Since the convolution of anL²0,∞function with anL²0,∞function is itself anL²0,∞function we get

e^2λt _t

0

Rt−s−R∞Σs²ds≤ _t

0

e^2λt−sRt−s−R∞²e^2λsΣs²ds≤c2, 4.20 and so the second term of4.18decays exponentially quickly.

We can show that the third term on the right hand side of4.18decays exponentially using3.5and the following argument:

Σ: _∞

0

Σs²e^2γsds≥ _∞

t

Σs²e^2γsds≥e^2γt _∞

t

Σs²ds. 4.21

Combining these facts we see that

EXt−X_∞² ≤mX0e^−2λt, 4.22 wheremX0 c1X0²c2 ΣR_∞²andλ <minβ, γ.

Proof ofProposition 4.1. Consider the case where 0< p≤2 andp >2 separately. We begin with the case where 0 < p ≤ 2. The argument given by4.16shows thatEX∞² < ∞. Now applying Lyapunov’s inequality we see that

EX_∞^p≤EX_∞^{2 p/2}<∞. 4.23 We now show that3.9holds for 0 ≤ p ≤ 2. Lyapunov’s inequality andLemma 4.3can be applied as follows:

EXt−X∞^p ≤EXt−X∞^{2 p/2}≤mpX0e^−βpt, t≥0, 4.24 wherempX0 mX0^p/2andβpλp.

Now consider the case wherep > 2. In this case, there exists a constantm ∈Nsuch that 2m−1 < p≤ 2m. We now seek an upper bound onEX_∞^2mandEXt−X_∞^2m ,

which will in turn give an upper bound onEX∞^pandEXt−X∞^p by using Lyapunov’s inequality. By applyingLemma 4.10we see that

EX_∞^2m≤cR_∞X0^2mc

∞ 0

R_∞Σs²ds _m

<∞, 4.25

wherecis a positive constant, soEX_∞^p≤EX_∞^{2m p/2m}<∞.

Now consider EXt−X_∞^2m . Using the variation of parameters representation of the solution and the expression obtained forX∞, taking norms, raising both sides of the equation to the 2mth power, then taking expectations across the inequality, we arrive at

EXt−X∞^2m

≤3^2m−1

Rt−R_∞X0^2mE _t

0

Rt−s−R_∞ΣsdBs ^2m E

_∞

t

R_∞ΣsdBs ^2m

.

4.26

We consider each term on the right hand side of4.26. ByTheorem 4.5we have

Rt−R∞X0^2m≤c1X0^2me^−2mβt. 4.27 Now, consider the second term on the right-hand side of4.26. By4.20we see that_t

0Rt−

s−R∞Σs²ds≤c2e^−2λtwhereλ <minβ, γ. Using this andLemma 4.10we see that E

_t

0

Rt−s−R∞ΣsdBs ^2m

≤dmn, d t

0

Rt−s−R∞²Σs²ds m

≤dmn, dc^m₂e^−2mλt.

4.28

Using4.21combined withLemma 4.10and Fatou’s lemma, we show that the third term decays exponentially quickly:

E _∞

t

ΣsdBs ^2m

≤dmn, d _∞

t

Σs²ds m

≤dmn, dΣ^me^−2mγt.

4.29

Combining4.27,4.28, and4.29the inequality4.26becomes EXt−X∞^2m

≤3^2m−1c1X0^2me^−2mβtdmn, dc^m₂e^−2mλtdmn, dR∞^2mΣ^me^−2mγt. 4.30 Using Lyapunov’s inequality, the inequality4.30implies

EXt−X∞^p ≤mpX0e^−βpt, 4.31 where mpX0 3^p2m−1/2mc1X0^2mdmn, dc^m₂ dmn, dR∞^2mΣ^mp/2m and βp λp.

Proof ofProposition 4.2. In order to prove this proposition we show that E

sup

n−1≤t≤nXt−X_∞²

≤mX 0e^−2ηn−1, η >0. 4.32 For eacht >0 there existsn∈Nsuch thatn−1≤t < n. DefineΔt Xt−X_∞. Integrating 1.4a-1.4bovern−1, t , then adding and subtractingX∞on both sides we get

Xt−X_∞ Xn−1−X_{∞ t}

n−1AXs−X_∞ds

_t

n−1

_s

0

Ks−uXu−X_∞du ds _t

n−1ΣsdBs

_t

n−1

A

_∞

0

Kudu

X_∞ds− _t

n−1

_∞

s

KvX_∞dv ds.

4.33

By applyingTheorem 4.8, we see that4.7and4.8hold soLemma 4.7may be applied to obtain

Δt Δn−1 _t

n−1AΔs K∗Δsds _t

n−1ΣsdBs−

_t

n−1K1sdsX∞. 4.34 Taking norms on both sides of 4.34, squaring both sides, taking suprema, before finally taking expectations yields:

E

sup

n−1≤t≤nΔt²

≤4

EΔn−1²E _n

n−1AΔs K∗Δsds 2

E

sup

n−1≤t≤n

_t

n−1ΣsdBs

²

_n

n−1K1sds 2

EX_∞²

. 4.35 We now consider each term on the right hand side of4.35. FromLemma 4.3we see that the first term satisfies

EΔn−1² ≤mX0e^−2λn−1. 4.36 In order to obtain an exponential bound on the second term on the right hand side of4.26 we make use of the Cauchy-Schwarz inequality as follows:

_n

n−1AΔs K∗Δsds 2

≤2 _n

n−1

A²Δs² K∗Δ²s ds

≤2 _n

n−1

A²Δs² _s

0

e^αs−u/2Ks−u^1/2e^−αs−u/2Ks−u^1/2Δsdu 2

ds

≤2 _n

n−1

A²Δs²Kα

_s

0

e^−αs−uKs−uΔs²du

ds,

4.37

whereKα_∞

0 e^αtKtdt. Take expectations and examine the two terms within the integral.

UsingLemma 4.3we obtain E

n

n−1A²Δs²ds

≤ A²mX0 _n

n−1e^−2λsds≤c1X0e^−2λn−1. 4.38 In order to obtain an exponential upper bound for the second term within the integral we applyLemma 4.11withKK, αα, λ λandηη:

E _n

n−1Kα

_s

0

e^−αs−uKs−uΔs²du ds

≤mX0Kα

_n

n−1

_s

0

e^−αuKue^−2λs−udu ds

≤c2X0e^−ηn−1.

4.39 Next, we obtain an exponential upper bound on the third term. Using 4.21 and the Burkholder-Davis-Gundy inequality, there exists a constantc3>0 such that

E

sup

n−1≤t≤n

_t

n−1ΣsdBs ²

≤c3Σe^−2γn−1. 4.40

Now consider the last term on the right hand side of4.35. Using3.4we see that Kα:

_∞

0

Kse^αsds≥e^αt _∞

t

Ksds≥e^αtK1t. 4.41 Using this and the fact thatEX∞²<∞see4.16we obtain

_n

n−1K1sds 2

EX∞²≤EX∞² _n

n−1Kαe^−αsds 2

≤c4e^−2αn−1. 4.42 Combining4.36,4.38,4.39,4.40, and4.42we obtain

E

sup

n−1≤t≤nXt−X∞²

≤mX 0e^−2ηn−1, 4.43

wheremX 0 4mX0 c1X0 c2X0 c3Σ c4andη <min2λ, α.

We can now apply the line of reasoning used in10, Theorem 4.4.2 to obtain3.10.

Proof ofLemma 4.4. We use a reformulation of 1.2a-1.2b in the proof of this result. It is obtained as follows: multiply both sides ofRs ARs K∗Rsacross by the function Φt−s, whereΦt Pe^−tQ, integrate over0, t , use integration by parts, add and subtract R∞from both sides to obtain

Yt F∗Yt Gt, t≥0, 4.44 whereY R−R∞,Fis defined by4.12andGis defined by

Gt e^−tQ−e^−tQR∞QAR∞

_∞

t

_∞

s

P KuR∞du ds

− _∞

t

QKuR_∞du−e∗QKR∞t, t≥0.

4.45

Consider the reformulation of1.2a-1.2bgiven by4.44. It is well known thatYcan be expressed as

Yt Gt− _t

0

rt−sGsds, 4.46

where the functionr satisfiesrF∗r F andrr∗F F. We refer the reader to11 for details. Consider the first term on the right hand side of4.46. As3.1holds it is clear that the function Gis integrable. Now consider the second term. Since3.3and 4.4hold we may applyLemma 4.9to obtain4.11. Now we may apply a result of Paley and Wienersee 11 to see thatris integrable. The convolution of an integrable function with an integrable function is itself integrable. Now combining the arguments for the first and second terms we see that4.2must hold.

5. On the necessity of3.5for exponential convergence of solutions of1.4a-1.4b In this section, the necessity of condition3.5for exponential convergence in the almost sure andpth mean senses is shown.Proposition 5.1concerns the necessity of the condition in the almost sure case whileProposition 5.2deals with thepth mean case.

Proposition 5.1. LetKsatisfy2.4and4.4andΣsatisfy2.6. If there exists a constantα > 0 such that3.4 holds, and if for allX0 there is a constant vectorX∞X0,Σ such that the solution t→Xt;X0,Σof 1.4a-1.4bsatisfies statement (iii) ofTheorem 3.2, then there exists a constant γ >0, independent ofX0, such that3.5holds.

Proposition 5.2. LetKsatisfy2.4and4.4andΣsatisfy2.6. If there exists a constantα > 0 such that3.4 holds, and if for allX0 there is a constant vectorX_∞X0,Σ such that the solution t →Xt;X0,Σof 1.4a-1.4bsatisfies statement (ii) ofTheorem 3.2, then there exists a constant γ >0, independent ofX0, such that3.5holds.

In order to prove these propositions the integral version of1.4a-1.4bis considered.

By reformulating this version of the equation an expression for a term related to the exponential integrability of the perturbation is found. Using various arguments, including the Martingale Convergence Theorem in the almost sure case, this term is used to show that 3.5holds.

Some supporting results are now stated.Lemma 5.3is the analogue ofLemma 4.7in the mean square case. It was proved in8 .

Lemma 5.3. LetKsatisfy2.4and4.4. Suppose that for all initial conditionsX0there is aF^B∞- measurable and almost surely finite random variableX∞X0,Σwith EX∞² < ∞such that the solutiont→Xt;X0,Σof1.4a-1.4bsatisfies

t→∞limEXt;X0,Σ−X∞X0,Σ²0,

EX·;X0,Σ−X∞X0,Σ²∈L¹0,∞,R. 5.1 ThenX∞obeys

A

_∞

0

Ksds

X_∞0 a.s. 5.2

Lemma 5.4may be extracted from4 ; it is required in the proof ofProposition 5.2.

Lemma 5.4. LetN N1, . . . , NnwhereNi∼N0, v²_ifori1, . . . , n. Then there exists a{vi}ⁿ_i1- independent constantd1>0 such that

EN² ≤d1EN ². 5.3

Proof ofProposition 5.1. In order to prove this result we follow the argument used in 4, Theorem 4.1 . Let 0 < γ < α∧β0. By defining the process Zt e^γtXtand the matrix κt e^γtKtwe can rewrite1.4a-1.4bas

dZt

γIAZt _t

0

κt−sZsds

dte^γtΣtdBt, 5.4 the integral form of which is

Zt−Z0 γIA _t

0

Zsds _t

0

_s

0

κs−uZudu ds _t

0

e^γsΣsdBs. 5.5 UsingZt e^γtXtand rearranging this becomes

_t

0

e^γsΣsdBs e^γtXt−X0−γIA _t

0

e^γsXsds− _t

0

e^γs _s

0

Ks−uXudu ds.

5.6 Adding and subtractingX∞from the right hand side and applyingLemma 4.7we obtain:

_t

0

e^γsΣsdBs e^γtXt−X_∞−X0−X_∞−γIA _t

0

e^γsXs−X_∞ds

− _t

0

e^γs _s

0

Ks−uXu−X_∞du ds _t

0

e^γsK1sdsX_∞.

5.7

Consider each term on the right hand side of5.7. We see that the first term tends to zero as 3.10holds andγ < β0. The second term is finite by hypothesis. Again, using the fact that γ < β0and that assumption3.10holds we see thateγX−X∞∈L¹0,∞, so the third term tends to a limit ast→ ∞. Now consider the fourth term. Since 0< γ < α∧β0, we can choose γ1>0 such thatγ < γ1< α∧β0. So the functionst→e^γ1tKtandt→e^γ1tXt−X_∞are both integrable. The convolution of these two integrable functions is itself an integrable function,

so

_s

0

Ks−uXu−X_∞du

≤ce^−γ1s. 5.8 Thus, it is clear that the fourth term has a finite limit ast→ ∞. Finally, the fifth term on the right hand side of5.7has a finite limit at infinity, using4.41.

Each term on the right hand side of the inequality has a finite limit as t→ ∞, so therefore

limt→∞

_t

0

e^γsΣsdBsexists and is almost surely finite. 5.9 The Martingale Convergence Theorem 12, Proposition 5.1.8 may now be applied component by component to obtain3.5.

Proof ofProposition 5.2. By Lemma 5.3,5.7 still holds. Defineγ < α∧β1, take norms and expectations across5.7to obtain

E _t

0

e^γsΣsdBs

≤Ee^γtXt−X∞ EX0−X∞

γIA _t

0

Ee^γsXs−X∞ ds

_t

0

e^γs _s

0

KuEXs−u−X∞ du ds

_t

0

e^γsK1sdsEX_∞.

5.10

There existsm1such that

Ee^γtXt−X∞ ≤m1e^−β1−γt, 5.11 thus the first, second and third terms on the right hand side of5.10are uniformly bounded on0,∞. Now consider the fourth term. Since 0< γ < α∧β1, we can chooseγ1 >0 such that γ < γ1< α∧β1so that the functionst→e^γ1tKtandt→e^γ1tEXt−X∞are both integrable.

The convolution of two integrable functions is itself an integrable function, so _s

0

Ks−uEXu−X_∞du≤ce^−γ1s, 5.12

so it is clear that the fourth term is uniformly bounded on0,∞. Finally, we consider the final term on the right hand side of5.10. Using4.41we obtain

_t

0

e^γsK1sdsEX_∞ ≤KαEX_{∞ t}

0

e^−α−γsds <∞, 5.13

sinceγ < α. Thus there is a constantc >0 such that E

_t

0

e^γsΣsdBs

≤c. 5.14

The proof now follows the line of reasoning found in4, Theorem 4.3 : observe that _t

0

e^γsΣsdBs ²ⁿ

i1

Nit², 5.15

where

Nit ^d

j1

_t

0

e^γsΣijsdBjs. 5.16

It is clear thatNitis normally distributed with zero mean and variance given by

vit^2d

j1

_t

0

e^2γsΣijs²ds. 5.17

Lemma 5.4and5.14may now be applied to obtain:

_t

0

e^2γsΣs²dsⁿ

i1

d j1

_t

0

e^2γs|Σijs|²ds

ⁿ

i1

vit²

E _t

0

e^γsΣsdBs ²

≤d1E _t

0

e^γsΣsdBs 2

≤d1c².

5.18

Allowingt→ ∞on both sides of this inequality yields the desired result.

6. Sufficient conditions for exponential convergence of solutions of1.1a-1.1b In this section, sufficient conditions for exponential convergence of solutions of 1.1a- 1.1b to a nontrivial limit are found. Proposition 6.1 concerns the pth mean sense while Proposition 6.2deals with the almost sure case.

Proposition 6.1. LetKsatisfy2.4and3.1, letΣsatisfy2.6, letfsatisfy2.5, and letR∞be a constant matrix such that the solutionRof1.2a-1.2bsatisfies3.3. If there exist constantsα >0, ρ >0 andγ >0such that3.4,3.6and3.5hold, then there exist constantsβ^∗_p>0, independent ofX0, andm^∗_pm^∗_pX0>0, such that statement (ii) ofTheorem 3.1holds.

Proposition 6.2. LetK satisfy 2.4and 3.1, letΣsatisfy2.6, letf satisfy2.5, and let R_∞ be a constant matrix such that the solutionRof 1.2a-1.2bsatisfies3.3. If there exist constants α > 0, ρ > 0 and γ > 0such that3.4, 3.6and 3.5hold, then there exists constantβ^∗₀ > 0, independent ofX0such that statement (iii) ofTheorem 3.1holds.

As in the case wheref:0 we require an explicit formulation forX∞. The proof of this result follows the line of reasoning used in the proof ofTheorem 4.6and is therefore omitted.

Theorem 6.3. Let K satisfy 2.4 and 4.4, letΣ satisfy 2.6 and 4.5, and let f satisfy2.5.

Suppose that the resolventR of 1.2a-1.2b satisfies3.3, then the solutionX of 1.1a-1.1b satisfiesX→X∞X0,Σ, falmost surely, where

X∞X0,Σ, f X∞X0,Σ R∞

_∞

0

ftdt, a.s. 6.1

andX_∞X0,Σ, fis almost surely finite.

Proof ofProposition 6.1. We begin by showing thatEX∞X0,Σ, f^p is finite. Clearly, we see that

EX∞X0,Σ, f^p≤2^p−1

EX∞X0,Σ^p _∞

0

R∞fsds ^p

<∞. 6.2 Now, consider the difference between the solutionX·;X0,Σ, fof1.1a-1.1band its limit X∞X0,Σ, fgiven by6.1:

Xt;X0,Σ, f−X_∞X0,Σ, f Xt;X0,Σ−X∞X0,Σ

_t

0

Rt−s−R∞fsds− _∞

t

R∞fsds.

6.3

Using integration by parts this expression becomes Xt;X0,Σ, f−X∞X0,Σ, f

Xt;X0,Σ−X_∞X0,Σ−f1t Rt−R_∞f10− _t

0

Rt−sf1sds.

6.4

Taking norms on both sides of equation6.4, raising the power topon both sides, and taking expectations across we obtain

EXt;X0,Σ, f−X∞X0,Σ, f^p

≤4^p−1

EXt;X0,Σ−X∞X0,Σ^pf1t^p

Rt−R∞^pf10^p t

0

Rt−sf1sds p

.

6.5

Now consider the right hand side of6.5. The first term decays exponentially quickly due toTheorem 3.2. The second term decays exponentially quickly due to assumption3.6. By applyingLemma 4.4we see that4.2holds so we can applyTheorem 4.5to show that the third term must decay exponentially. In the sequel, an argument is provided to show that R decays exponentially; thus the final term must decay exponentially. Combining these arguments we see that3.7holds, whereβ^∗_p<minβp, β, ρ.

It is now shown thatR decays exponentially. It is clear from the resolvent equation 1.2a-1.2bthat

Rt ARt−R∞

_t

0

Kt−sRs−R∞ds−K1tR∞

A _∞

0

Ksds

R∞. 6.6 Consider each term on the right hand side of 6.6. We can apply Theorem 4.5 to obtain thatR decays exponentially quickly toR∞. In order to show that the second term decays exponentially we proceed as follows: sinceR−R_∞decays exponentially and3.4holds it is possible to chooseμsuch that the functionst→ e^μtKtandt→ e^μtRt−R∞are both in L¹0,∞,R^n×n. The convolution of two integrable functions is itself an integrable function, so

e^μt _t

0

Kt−sRs−R_∞ds

_t

0

e^μt−sKt−se^μsRs−R_∞ds

≤c. 6.7