of a scalar linear Itô-Volterra equation with state-independent diffusion coefficient. If the .... We will frequently use the following lemma, proved in [5]. Lemma 1.
FUNCTIONAL DIFFERENTIAL EQUATIONS
VOLUME ? 2003, NO ?-? PP. ??– ??
SUBEXPONENTIAL SOLUTIONS OF LINEAR ˆ ITO-VOLTERRA EQUATIONS WITH A DAMPED PERTURBATION J. A. D. APPLEBY
∗
Abstract. This paper studies the almost sure non-exponential decay rate of solutions of a scalar linear Itˆ o-Volterra equation with state-independent diffusion coefficient. If the kernel is subexponential, and the diffusion term decays sufficiently quickly, the decay rate is subexponential, and the same as in the deterministic case. If the diffusion coefficient is subexponential, there is a subexponential upper bound on the decay rate of solutions. Key Words. Volterra integro–differential equations, exponential asymptotic stability, subexponential functions. AMS(MOS) subject classification. 34K20, 34K50, 60H10
1. Introduction. In this paper we study the asymptotic behaviour of the scalar linear convolution Itˆo-Volterra equation (1)
dX(t) = (−aX(t) + (k ∗ X)(t)) dt + σ(t) dB(t),
where k is continuous, positive and integrable on [0, ∞), and σ is continuous on [0, ∞). Here, f ∗ g denotes the convolution of f, g ∈ C(0, ∞) (f ∗ g)(t) =
Z
t
f (t − s)g(s) ds.
0
We assume that the initial condition X(0) = ξ is deterministic, and, as conventional, that (1) is shorthand for the integral equation (2) X(t) = ξ +
Z 0
∗
t
{(−aX(s) + (k ∗ X)(s)} ds +
Z
t
σ(s) dB(s),
t ≥ 0.
0
Centre for Modelling with Differential Equations (CMDE), School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland 1
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J. A. D. APPLEBY
The probabilistic setting for this equation is a complete filtered probability space (Ω, F, (F (t))t≥0 , P ), where B is a scalar standard Brownian motion. According to [7], there is a unique continuous process, adapted to the filtration, which is a strong solution of (1). In the following, the abbreviation a.s. is used for “almost sure” or “almost surely”; in each case these refer to almost sure events relative to the objective probability measure P . Let z be the unique solution of z 0 (t) = −az(t) + (k ∗ z)(t),
(3)
t ≥ 0,
z(0) = 1.
In [4], the following necessary and sufficient conditions for the exponential stability of solutions of (1) were established: (a) The solution of (3) is integrable and
(5)
∞
Z
(4)
0 ∞
Z 0
k(s)eγ1 s ds < ∞ for some γ1 > 0,
e2γ2 s σ(s)2 ds < ∞ for some γ2 > 0.
(b) There is β0 > 0 such that all solutions of (1) obey lim sup t→∞
1 log |X(t)| ≤ −β0 , t
a.s.
Therefore, solutions cannot decay to zero exponentially if k or σ do not obey the exponential decay criteria (4), (5). Thus, when solutions of (1) are a.s. asymptotically stable (see, e.g., [1]), they obey lim sup t→∞
1 log |X(t)| = 0, t
a.s.
In this paper, we find precise estimates on the a.s. non-exponential decay of solutions, under non-exponential hypotheses on k and σ. Similar hypotheses are used to study non-exponential stability in Itˆo-Volterra equations in [6, 3]. 2. Subexponential functions and deterministic equations. The following class of functions, introduced in [5], derives from a definition in [8]. Definition 1. We say k ∈ L1 (R+ ) ∩ C(R+ ; R+ ) is subexponential if (6) (7)
Z ∞ (k ∗ k)(t) lim = 2 k(s) ds, t→∞ k(t) 0 k(t − s) lim sup − 1 = 0, ∀ τ > 0. t→∞ s∈[0,τ ] k(t)
SUBEXPONENTIAL STOCHASTIC VOLTERRA EQUATIONS
3
If k obeys these properties, we write k ∈ U. As pointed out in [5], condition (7) implies that k obeys (8)
lim k(t)eεt = ∞,
t→∞
for all ε > 0.
Thus, if k is subexponential, it obeys neither (4) nor (5). The class of subexponential functions is discussed in detail in [5]. It contains, for example, all positive functions which are regularly varying at infinity, with index α < −1. Denote by BC the space of bounded continuous functions on (0, ∞), and BCh the space of continuous functions f with f /h ∈ BC. For f ∈ BCh , let Λh f = lim sup t→∞
|f (t)| . h(t)
As in [5], we denote by BChl the space of continuous functions f ∈ BCh (t) where limt→∞ fh(t) exists. For f ∈ BChl , we write f (t) . t→∞ h(t)
Lh f = lim
We will frequently use the following lemma, proved in [5]. Lemma 1. Let h ∈ U. If f, g ∈ BChl , then f ∗ g ∈ BChl and Lh (f ∗ g) = Lh f
∞
Z
g(s) ds + Lh g
0
∞
Z
f (s) ds.
0
In this paper, we state an analogous result for functions in BCh . As it may be proved in a similar fashion to Lemma 1, its proof is omitted. Lemma 2. Let h ∈ U. If f, g ∈ BCh , then f ∗ g ∈ BCh and Λh (|f | ∗ |g|) ≤ Λh f
Z 0
∞
|g(s)| ds + Λh g
Z
∞
|f (s)| ds.
0
These results yield the following perturbation theorem, partly proven in [5]. Theorem 1. Let f ∈ C(0, ∞) ∩ L1 (0, ∞), k ∈ C([0, ∞); (0, ∞)) ∩ R L1 (0, ∞), a > 0∞ k(s) ds, and x be the solution of (9)
x0 (t) = −ax(t) + (k ∗ x)(t) + f (t),
t ≥ 0.
(a) If k ∈ U, f ∈ BCkl , then Lk f + 0∞ x(s) ds R . Lk x = a − 0∞ k(s) ds R
(b) If f obeys (6), (7), and Lf k = 0, then Lf x = (a − 0∞ k(s) ds)−1 . This result informs our study of the stochastically perturbed equation (1). It suggests that we find a deterministic proxy for the size of the random perturbation, to determine the critical decay rate of σ at which there is a transition from “small” to “large perturbation” asymptotics. R
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J. A. D. APPLEBY
3. Asymptotic stability of (1). We now study the a.s. asymptotic stability of solutions of (1) where σ ∈ L2 (0, ∞). This allows for direct comparison with equation (9) when f ∈ L1 (0, ∞), for when σ ∈ L2 (0, ∞), the Itˆo integral on the righthand side of (2) has a finite limit as t → ∞, a.s. Theorem 2. If the solution of (3) is in L1 (0, ∞), k ∈ C((0, ∞)) ∩ L1 (0, ∞), and σ ∈ C(0, ∞) ∩ L2 (0, ∞), then (a) limt→∞ E[X(t)2 ] = 0, E[X 2 ] ∈ L1 (0, ∞), and (b) limt→∞ X(t) = 0, X ∈ L2 (0, ∞), a.s. Proof. With z defined by (3), the solution of (1) is X(t) = X(0)z(t) + R Y (t), where Y (t) = 0t z(t − s)σ(s) dB(s) (see [1, 9]). In [1] it is noted that Y (t) is a normally distributed random variable with zero mean and variance v(t)2 , where v(t)2 = (z 2 ∗ σ 2 )(t). Since z ∈ L1 (0, ∞) implies z(t) → 0 as t → ∞, z 2 ∈ L1 (0, ∞). But σ 2 ∈ L1 (0, ∞) gives v(t) → 0 as t → ∞ and v ∈ L2 (0, ∞). Since E[X(t)2 ] ≤ 2z(t)2 X(0)2 + 2v(t)2 , part (a) holds. Part (b) follows by part (a) and the method of proof of Theorem 1 in [2]. R As k is positive here, the condition a > 0∞ k(s) ds implies z ∈ L1 (0, ∞). 4. Main Results. The proofs of the main results in this paper rely on rewriting the solution of (1) in terms of the solution of a perturbed Volterra integrodifferential equation whose solution, although random, is in C 1 (0, ∞). R Lemma 3. If k ∈ C([0, ∞); (0, ∞)) ∩ L1 (0, ∞), a > 0∞ k(s) ds, then for a.a. ω ∈ Ω, the path X(ω) obeys (10)
X(t, ω) = U (t, ω) + T (t, ω)
where T (ω) is the function defined by T (t, ω) = −
(11)
Z
∞
σ(s) dB(s) (ω), t
and f (ω), U (ω) obey (12) (13)
f (t, ω) = −aT (t, ω) + (k ∗ T (ω))(t), U 0 (t, ω) = −aU (t, ω) + (k ∗ U (ω))(t) + f (t, ω).
We refer the reader to a similar result, proven in [3], where a more general result on the asymptotic behaviour of the random variable T (t) also appears. Lemma 4. Let σ ∈ C([0, ∞); (0, ∞)) ∩ L2 (0, ∞), and (14)
2
Σ(t) =
Z t
∞
2
Z
∞
2
−1
σ(s) ds
σ(s) ds log log t
.
SUBEXPONENTIAL STOCHASTIC VOLTERRA EQUATIONS
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Then T defined by (11) obeys (15)
lim sup t→∞
|T (t)| √ = 2, Σ(t)
a.s.
This, along with Lemmas 1 and 2 suggests that the random perturbation f in (13) decays at the slower rate between k and Σ. Viewing Theorem 1 in the light of Lemma 3, the case in which Lk Σ = 0 seems to correspond to part (a) of the theorem, while that in which LΣ k = 0 roughly corresponds to part (b). The following result thus parallels Theorem 1, part (a). Theorem 3. If k ∈ C([0, ∞); (0, ∞)) ∩ L1 (0, ∞), σ ∈ L2 (0, ∞), σ ∈ R C([0, ∞); (0, ∞)), a > 0∞ k(s) ds, k is subexponential, and Σ obeys Lk Σ = 0, then the unique strong solution of (1) obeys R∞
(16)
Lk X =
X(s) =: G, a − 0 k(s) ds 0
R∞
a.s.
where G is a normally distributed F -measurable random variable, which is R R a.s. nonzero, and has mean ξ/(a− 0∞ k(s) ds), and variance 0∞ σ(s)2 ds/(a− R∞ 2 0 k(s) ds) . Moreover, X(ω) obeys (6), (7) for almost all ω ∈ Ω. The conclusion of the first part of the result is precisely that of Theorem 1, when Lk f = 0. So, although the sample paths of X are nowhere differentiable, a.s., they behave asymptotically like k, which can be in C ∞ (0, ∞). Proof. The second statement of the theorem follows from the first by arguments of [5] applied to each ω in the a.s. set on which (16) holds, and G is non-trivial. As to the proof of the first part, the hypotheses and (15) imply Lk T = 0 a.s. Since k is integrable, T ∈ L1 (0, ∞), a.s. Applying Lemma 1 to R f in (12) yields Lk f = 0∞ T (s) ds, a.s., so by Theorem 1, U obeys Lk f + 0∞ U (s) ds R Lk U = . a − 0∞ k(s) ds R
(10) now gives (16). To find the distribution of G, note that k ∈ L1 (0, ∞) implies X ∈ L1 (0, ∞). Therefore, letting t → ∞ on both sides of (2), and R noting that σ ∈ L2 (0, ∞) implies limt→∞ 0t σ(s) dB(s) exists a.s., gives Z 0
∞
ξ + 0∞ σ(s) dB(s) R X(s) ds = . a − 0∞ k(s) ds R
Since 0∞ σ(s) dB(s) is normally distributed with variance 0∞ σ(s)2 ds, G has the claimed distribution. Since σ 6≡ 0, G is not trivial, so Lk X 6= 0, a.s. We now supply the stochastic analogue of case (b) in Theorem 1. R
R
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J. A. D. APPLEBY
Theorem 4. If k ∈ C([0, ∞); (0, ∞)) ∩ L1 (0, ∞), σ ∈ L2 (0, ∞) ∩ R C([0, ∞); (0, ∞)), a > 0∞ k(s) ds, Σ defined by (14) is subexponential, and ΛΣ k = L ∈ [0, ∞), then the unique strong solution of (1) obeys ΛΣ X < ∞, a.s. Moreover, if LΣ k = 0 then √ 2 2a R ΛΣ X ≤ , a.s. a − 0∞ k(s) ds Proof. We study the case where LΣ k = 0 only: √ the case when L 6= 0 is similar and hence omitted. By (15), ΛΣ T = 2, a.s. Thus Lemma 2, LΣ k = 0, and (12) gives Z ∞ √ (17) ΛΣ f ≤ 2(a + k(s) ds), a.s. 0
−at
Define ea (t) = e , h = ea ∗k: then z = ea +ea ∗r, where r solves r = h+h∗r. Then LΣ k = 0, Lemma 2 imply ΛΣ h = 0. An argument involving Lemma 2 yields LΣ r = 0, so LΣ z = ΛΣ z = 0. As ΛΣ z = 0, ΛΣ U = ΛΣ (z ∗ f ). The result now follows, by Lemma 2, (13), (15), and (17). Acknowledgements The author is very grateful for discussions with Kieran Murphy, and also to the anonymous referee for their careful reading of the manuscript. REFERENCES [1] J. A. D. Appleby, Almost sure asymptotic stability of linear Itˆo-Volterra equations with damped stochastic perturbations, Electron. Comm. Probab., 7(2002), 223– 234. [2] J. A. D. Appleby, pth mean integrability and almost sure asymptotic stability of Itˆo-Volterra equations, J. Integral Equations Appl., (2003), to appear. [3] J. A. D. Appleby, Almost sure subexponential decay rates of scalar Itˆo-Volterra equations, Electron. J. Qual. Theory Differ. Equ., (2003), to appear. [4] J. A. D. Appleby and A. Freeman, Exponential asymptotic stability of linear ItˆoVolterra equations with damped stochastic perturbations, Electron. J. Probab., 2003, to appear. [5] J. A. D. Appleby and D. W. Reynolds, Subexponential solutions of linear Volterra integro-differential equations and transient renewal equations, Proc. Roy. Soc. Edinburgh. Sect. A, 132A(2002), 521–543. [6] J. A. D. Appleby and D. W. Reynolds, Non-exponential stability of scalar stochastic Volterra equations, Statist. Probab. Lett., 62(4)(2003), 335–343. [7] M. A. Berger and V. J. Mizel, Volterra equations with Itˆo integrals I, J. Integral Equations, 2(3)(1980), 187–245. [8] J. Chover, P. Ney, and S. Wainger, Functions of probability measures, J. Analyse. Math., 26(1972), 255–302. [9] U. K¨ uchler and S. Mensch, Langevin’s stochastic differential equation extended by a time-delay term, Stochastics Stochastics Rep., 40(1-2)(1992), 23–42.
