Decay Rates of Solutions of Linear Stochastic Volterra Equations

The paper studies the exponential and non–exponential convergence rate to zero of solutions of scalar linear convolution Itˆo-Volterra equations in which the noise intensity depends lin-early on the current state. By exploiting the positivity of the solution, various upper and lower bounds in ﬁrst mean and almost sure sense are obtained, including Liapunov exponents .


Introduction
The stability and decay rate of linear deterministic Volterra integrodifferential equations to point equilibria has attracted much attention, in part because of applications to physics, biology and other sciences (cf., e.g., [13,Ch.1]). Since such real-world systems can be modelled to include random effects, it is natural to ask how the presence of such stochastic perturbations can influence the stability and convergence rate of solutions to these equilibria. Particular applications of Itô-Volterra equations arise in physics and mathematical finance. In material physics, for example, partial integro-differential equations of Itô type are used to model the temperature in heat conductors with memory (cf., e.g., [9]). The behaviour of viscoelastic materials under external stochastic loads has also been analysed using Itô-Volterra equations (cf., e.g, [12]). In financial mathematics, the presence of inefficiency in real markets can be modelled by using stochastic functional differential equations. Anh et al. [1; 2] have recently posited models of the evolution of the asset returns using stochastic Volterra equations with infinite memory. This paper is a mathematical contribution to the literature on the stability and rate of convergence to equilibrium. The non-exponential convergence reported in this paper is particularly interesting in economics, as it may be used in future work to represent the persistent memory of a shock in a financial system in the distant past.
The asymptotic behaviour of the solution of the deterministic linear convolution integrodifferential equation with x(0) = 0, is often analysed by considering the solutions of the characteristic equation wherek is the Laplace transform of k (cf., e.g., [13]). If all solutions λ of this equation obey Re λ < 0, then the solution decays to zero exponentially fast (cf., e.g., [16]). However it can happen that (2) has no solutions, in which case this equation gives no information about the rate of decay of solutions. Nonetheless, in this situation, it is possible for the rate of decay to be determined exactly when k is asymptotic to a positive weight function γ in the class U(µ), as introduced in [11]. In this case the solution of (1) is asymptotic as t → ∞ to γ(t) times a constant (cf., e.g., [7]). The class U(µ) has the property that for γ ∈ U(µ), γ(t) is asymptotic to the exponential function e µt times a slowly decaying function.
It is natural to ask how the asymptotic behaviour of solutions of (1) will change when it is perturbed by a simple state-dependent stochastic perturbation, which depends only on the present value and preserves the zero equilibrium of (1). The stochastic Itô-Volterra equation which results from such a linear perturbation is where σ = 0 and {B(t)} t≥0 is a one-dimensional standard Brownian motion. Assumptions are imposed on the data which that ensure solutions are non-negative.
In this paper we examine the asymptotic decay rate of solutions of (3) in both first mean and almost sure senses. By exploiting the scalar nature of (3) and positivity of solutions we obtain both lower and upper estimates on almost sure decay rates, and exact decay rates on the first mean. In fact we establish estimates closely related to those found for the deterministic equation, irrespective of whether (2) has solutions or not.
In particular, the following results are established. In the case when k has a Liapunov exponent i.e., when there exists µ ≤ 0 such that it is shown that lim in the cases when µ = 0 or when k ∈ U(µ) for µ < 0. In neither of these cases does (2) have a solution. When (2) has a solution, it can be shown that there is a constant η > 0 which depends only on k such that where λ < 0 is the unique positive solution of (2). A consequence of this result is that it is impossible to reduce arbitrarily the a.s. Liapunov exponent of the solution by increasing the noise intensity σ. This contrasts with the situation for scalar linear stochastic differential equations. Finally, in the case when k has a zero Liapunov exponent, and is in U(0), we can establish for each ε > 0 the estimates Other studies which consider the convergence rates of solutions of stochastic Volterra equations with state-dependent noise intensities include [3; 6; 18; 19]. [18] deals with exponential stability of solutions, while [3] examines the asymptotic stability of solutions, but not the rate of decay to equilibrium. The remaining papers look at non-exponetial decay rates. Further results on non-exponential decay of solutions of stochastic Volterra equations with state-dependent noise, as well as a review of literature, may be found in [4].

Notation
We first fix some standard notation. Denote by R + the set [0, ∞). The maximum of the real numbers x, y is denoted by x ∨ y; their minimum is denoted by x ∧ y.
If F : R + → R is in L 1 (R + ) and G : R + → R has bounded variation, we put Similarly, for f, g ∈ L 1 (R + ), the convolution of f with g is defined to be We denote by C the set of complex numbers, and the real part of z ∈ C by Re z. If f : We also define e α (t) = e −αt for all t ∈ R + .

Linear Itô-Volterra Equations
In this paper, we concentrate upon the asymptotic behaviour of the solution of the scalar linear convolution Itô-Volterra equation We have as a standing hypothesis (B(t)) t≥0 is an standard one-dimensional Brownian motion on a complete filtered probability space (Ω, F, (F B (t)) t≥0 , P), where the filtration is the natural one, namely F B (t) = σ{B(s) : 0 ≤ s ≤ t}. We introduce the initial condition where X 0 is a random variable, independent of (B(t)) t≥0 , with E[X 2 0 ] < ∞. Then there is a unique almost surely continuous solution to (4) (cf., e. g., Theorem 2E of [8]), satisfying (4) and (6). More precisely the solution is the unique adapted process satisfying (4) and (6) on the augmented filtered space (Ω, F, (F(t)) t≥0 , P), where (F(t)) t≥0 is the augmented filtration such that B is a (F(t)) t≥0 -standard Brownian motion, and X 0 is F(0)-measurable. We denote a realisation of the solution of (4) and (6) by t → X(t, ω; X 0 ). Usually this is abbreviated to t → X(t, ω), and the value of the process at time t by X(t).
We assume without loss of generality that The case that σ = 0 is the deterministic problem. For σ < 0 we may consider the Brownian motion B − (t) = −B(t) on (Ω, F, (F B (t)) t≥0 , P): for then, we can rewrite (4) as with σ − = −σ > 0, which is in exactly the form of (4). The augmentation of the filtration required to accommodate a random initial condition can then be achieved as indicated above.
We will impose additional hypotheses on the decay rate of k to obtain different decay rates on the solution X.

Stochastic Analysis Preliminaries
We collect in this subsection a few results concerning solutions of Itô-Volterra equations.
We say that the zero solution of (4) is almost surely globally asymptotically stable if, for each solution of (4) and (6), We say that the zero solution is almost surely exponentially asymptotically stable if there exists a postive random variable β 0 such that P ω ∈ Ω : lim sup We say that a process X = {X(t); t ≥ 0} with |X(t)| > 0 for all t ≥ 0 a.s., has an a.s. lower Liapunov exponent µ if there is a deterministic µ such that lim inf A scalar process X = {X(t); t ≥ 0} has an a.s. upper Liapunov exponent ν if there is a deterministic ν such that lim sup The positive process X has a.s. Liapunov exponent β if there is a deterministic β which is both a lower and upper Liapunov exponent.
The following result follows directly by using Problem 5.3.15 in [15].

Estimates on Lower Liapunov Exponents
In this section, we recall results which show that it is not destabilising to add the noise perturbation σX(t)dB(t) to (1), yielding (4). We also establish lower bounds on the decay rate of solutions. In each case, the results are established without any additional hypotheses being imposed on the kernel k. We will appeal to the following result frequently in the sequel.
Lemma 3.1. Suppose that X is the solution of (4) and (6).
A consequence of this is the following, which is Theorem 6 of [3].
Theorem 3.2. Suppose that k satisfies (5). If the solution r of (10) is in L 1 (R + ), then the solution of (4) and (6) satisfies To establish lower bounds on the decay rate of solutions of (4), we observe a result essentially contained in the proof of [6, Theorem 1].
Lemma 3.3. Suppose that k satisfies (5). If X is the solution of (4) and (6) with X 0 = 0 a.s., then The next result gives a deterministic lower estimate on the a.s. rate of decay of |X(t)| as t → ∞.
The lower estimate furnished by Theorem 3.4 can be simplified in the case when lim inf In order to see this, it is first necessary to prove the following preliminary lemma.
We may now combine Lemma 3.5 and Theorem 3.4 to obtain explicit and simple estimates on the Liapunov exponent and lower Liapunov exponent of |X|.

Decay Rate when Characteristic Equation has no Roots
Corollary 3.6 shows that rate of decay of X(t) to zero as t → ∞, cannot be exponential if the kernel obeys (20). However it does not precisely quantify the relationship between the decay rate of X(t) and that of the kernel k(t). In this section, we investigate this under additional assumptions on the decay of k. Moreover the corresponding problem for kernels obeying (18) can be reduced to one in which it obeys (20), by considering the equation for X µ (t) = e −µt X(t), which is Recall that k µ (t) = e −µt k(t), and that in our case µ ≤ 0.

k is in U(µ)
We recall the definition of a class of decaying functions employed in [7]: it was based on the hypotheses of Theorem 3 of [11].
The functions in U(0) are called subexponential. The reason for this nomenclature is that, if k is subexponential, then k(t)e ǫt → ∞ as t → ∞, for each ǫ > 0: consequently k obeys (20). The connection between U(µ) and U(0) is that, if k is in U(µ), then k µ is subexponential.
Many slowly decaying functions belong to the class U(0). For instance a positive function which is regularly varying at infinity with index α < −1 (such as k(t) = (t + 1) −α for α < −1) is a member of U(0). Functions such as k(t) = e −t α , for α ∈ (0, 1), which decay faster than any polynomial, but more slowly than any negative exponential, are also in U(0). We refer the reader to [5] for a further discussion regarding the properties of subexponential functions.
For k ∈ U(µ), applying [7,Theorem 15] to the scalar equation (10), we see that the solution r of (10) inherits precisely the decay rate of k.
Lemma 4.2. Suppose that k satisfies (5), is in U(µ) and Then the solution of (10) obeys We also state a corollary of Theorem 4.2 in [5], which is used later.
We start by determining the exact asymptotic decay rate of solutions in first mean. Its proof is a corollary of Lemmata 3.1 and 4.2.
Theorem 4.4. Suppose that k satisfies (5), is in U(µ) and (28) holds. Then the solution of (4) and (6) obeys The result of Theorem 4.4 is interesting in itself, and may be used to obtain an upper bound on the almost sure asymptotic behaviour of solutions of (4).
It suffices to suppose that k is in U(0), since we may consider (24) instead of (4). The idea of the proof of Theorem 4.5 is similar to that used to prove the main result in [3], which in turn uses the strategy employed to prove Theorem 4.3.1 in [17]. We use a carefully chosen partition P = ∪ ∞ n=1 [a n , a n+1 ) of the positive real half-line, where a n ր ∞, as n → ∞. By the appropriate choice of the partition, and using the properties of subexponential functions, together with Theorem 4.4 above, we can show that for sufficiently large n there is a finiteB > 0 such that E sup an≤t≤a n+1 |X(t)| ≤Bk(a n ).
A Borel-Cantelli argument along the lines of [17, Theorem 4.3.1] is then used to obtain an upper bound on the almost sure decay rate.
We proceed similarly for the third term on the right hand side of (39). First, we obtain a n+1 an (k * r)(s) ds ≤ k(a n ) sup Since r is integrable, by using Using (27) as before, and the above limit, we can infer from (41) the existence of n ′′ 0 ∈ N, and B 2 > 0 such that n > n ′′ 0 implies a n+1 an (k * r)(s) ds ≤ B 2 k(a n ).
Hence, by Markov's inequality, for every γ ′ > 0 and n > n 1 , we have P sup The summability of this sequence of probabilities enables us to conclude, by the first Borel-Cantelli lemma and (36), that lim n→∞ sup an≤t≤a n+1 |X(t)| k(a n ) n ηλ = 0 a.s.
The following result can be inferred from Theorem 4.5 and [6, Corollary 2] by considering the equation (24) for X µ .
Theorem 4.7. Suppose that k is U(µ) for some µ < 0, and obeys (28). If X is the solution of (4) and (6) with X 0 = 0 a.s.,then By Theorem 4.5, it follows that lim sup On the other hand, by Corollary 3.7 it follows that lim inf t→∞ 1 t log |X(t)| ≥ µ, a.s.
We close this section with a discussion on almost sure asymptotic behaviour for k ∈ U(0). Just as in the deterministic case, the decay rate of solutions of (4) is closely determined by the decay rate of the kernel. If the kernel decays polynomially in the sense that for some α > 1, then the solution inherits this polynomial decay, in the sense that For certain kernels whose decay rate is between exponential and polynomial, we can prove a very precise result about the asymptotic behaviour of solutions of (4). We will call these kernels superpolynomial kernels; more precisely, we give the following definition. We call the function λ a negative exponent of k (or say that k has negative exponent λ).
If k is a superpolynomial function, the solution of (4) inherits the superpolynomial character of the kernel to a significant degree. Indeed, by Theorem 4.6 we have and therefore lim sup It is instructive to compare this result with that which is obtained for (10) when k is superpolynomial, namely (29), which implies lim t→∞ log |r(t)| log k(t) = 1.
This serves to show the sharpness of (46).
When k is in U(µ), we observe an interesting result: although solutions of the stochastic Volterra equation (4) are asymptotically stable whenever those of the deterministic problem (1) are also stable, the decay rate of solutions of (4) is slower than that for solutions of (1) . Therefore, although the noise seems to be stabilising solutions of the original Volterra equation, it is doing so at the expense of the convergence rate (although that convergence rate cannot be worse by more than a factor of t −(1+ε) , for any ε > 0).

K is in S(µ)
We consider another class of kernels for which the characteristic equation (2) does not have a solution. but which is larger than U(µ). This enables us to study the decay rate of ∞ t e −µt |X(s)| ds under weaker assumptions on k than required for U(µ). We start by recalling the definition of a subexponential distribution introduced by Chistyakov [10]. The class of subexponential distribution functions is denoted by S.
Chistyakov [10] showed that if F is a subexponential distribution function, then uniformly for s in compact intervals of R + . Chistyakov [10] demonstrated that a consequence of (48) is that subexponential distribution functions have "heavy tails" which decay more slowly than any exponential functions. Indeed, he showed that for F ∈ S that lim t→∞ (1 − F (t))e γt = ∞, for all γ > 0.
We now extend this definition to include the possibility of exponential decay. We notice that S(0) = S. There is a fundamental connection between S and U(0), and between S(µ) and U(µ) in general: if k ∈ U(µ), then K defined by K(0+) = 0 and is in S(µ).
By considering the Volterra integrodifferential equation for r µ (t) = e −µt r(t), and applying Theorem 6.5 of [5], we can infer the following result.
Lemma 4.11. Suppose that k satisfies (5), that K defined by (50) is in S(µ) and (28) holds. Then the solution of (10) obeys We determine the decay rates of solutions of equations whose kernels satisfy the weaker hypothesis that K be in S(µ), where K is defined in terms of k by (50). The next result is an immediate corollary of Lemmata 3.1 and 4.11.
A similar-though less intricate-argument to that of Theorem 4.5 now enables us to obtain an upper bound on the a.s. decay rate of ∞ t |X(s)| ds to zero, as t → ∞. Note that this is meaningful because a conclusion of Theorem 3.2 applied to the solution X µ of (24) is that X µ is in L 1 (R + ) a.s.
For every t ∈ R + , let n(t) be the largest integer less than or equal to t. By defining gives the result, on taking the limit superior as t → ∞.

Decay Rates when Characteristic Roots exist
In the previous section, we saw that solutions of (4) decay more slowly that those of its deterministic counterpart (1), for k in a subclass of functions for which the characteristic equation (2) has no solutions. We now consider the asymptotic behaviour of solutions of (4) when (2) has solutions and the differential resolvent r satisfying (10) is in L 1 (R + ). By Grossman and Miller [14], this precludes solutions of (2) being in {z ∈ C : Re z ≥ 0}. Furthermore, there exists a γ > 0 such that ∞ 0 k(s)e γs ds < +∞.
In fact, because (2) has solutions, it follows that there is a λ > 0 such that We show in this case that the decay rate of solutions of (4) is at least as fast as that of solutions of (1). This is established by obtaining an upper bound on the a.s. Liapunov exponent of X.
Lower bounds on the decay rate are also given. For exponentially integrable kernels satisfying some additional mild conditions, these show that the a.s. Liapunov exponent cannot be less than −γ.
As at the beginning of the last section, we start with an estimate of the decay rate of E[|X|]. It is a consequence of Lemmata 3.1 and 5.1.
We remark that Lemma 5.1 implies Also, observe that Theorem 5.2 and Lemma 2.1 imply that there exists C > 0 such that We now show that X has an a.s. upper Liapunov exponent −λ, using the estimate on E[|X(t)|] given by Theorem 5.2. The proof has the same form as that of Theorem 4.5.

Lower Bounds
The final result of the paper involves obtaining lower bounds on the top a.s. Liapunov exponent in the case when (2) has a solution. For this problem, with γ > 0 defined by (56), it transpires that lim t→∞ 1 t log |r(t)| > −γ.
This indicates for the deterministic equation (1) that increasing a, the intensity of the negative instantaneous feedback, will not arbitrarily increase the rate of decay of the solution. This contrasts to the situation for the non-delay problem r ′ 0 (t) = −ar 0 (t), r 0 (0) = 1.
Moreover stabilisation occurs if an instantaneous multiplicative noise term is added to this deterministic non-delay equation, so that the dynamics of the resulting non-trivial process Z is governed by Z(0) = 0 and dZ(t) = −aZ(t) dt + σZ(t) dB(t).
Then the a.s. Liapunov exponent of Z is defined by Λ Z (a, σ) = lim t→∞ 1 t log |Z(t)|, a.s. and is given by the formula For fixed a, this is a decreasing function of |σ|. Furthermore, lim |σ|→∞ Λ Z (a, σ) = −∞, so that the decay rate can be made arbitrarily fast by increasing |σ|. Our final result shows that neither increasing the intensity of the negative instantaneous feedback a, nor that of the noise σ by an arbitrarily large amount, will decrease the a.s. top Liapunov exponent corresponding to (4) arbitrarily.
proving the left half of the inequality (65) in this case also.