Dynamic Systems and Applications 15 (2006) 227-240

PREVENTION OF EXPLOSIONS IN SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS BY NOISE PERTURBATION

JOHN A. D. APPLEBY AND CÓNALL KELLY

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

ABSTRACT. The existence and uniqueness of global solutions of a class of scalar stochastic functional differential equations of Itô type is studied. It is not assumed, however, that the coefficients need to satisfy global linear bounds. For a subclass of these equations, it is known that the associated deterministic equation, which is not noise-perturbed, explodes in finite time. Therefore, a noise term may be added in such a way as to prevent the deterministic explosion. Finite dimensional analogues are also treated. AMS (MOS) Subject Classification. 34K50, 34K15, 60H10.

1. INTRODUCTION

There is a well-developed theory of the existence and uniqueness of strong solutions of stochastic functional differential equations of Itô type, due to Itô and Nisio [5], Berger and Mizel [4], among others. As indicated in Mao [10] and Kolmanovskii and Myskis [8], the existence of such global unique solutions is often proved under the assumption that the functional coefficients of the equation obey local Lipschitz continuity and global linear growth bounds. Protter [13] has shown that unique strong solutions can exist up to an explosion time under the assumption that the functionals are locally Lipschitz continuous.

It is natural to ask what minimal set of regularity and growth hypotheses on the coefficients will ensure that a stochastic functional differential equation has a unique and almost surely non-exploding solution. Mao [11] has given examples of stochastic delay differential equations which have unique global solutions despite the absence of a linear growth bound, but assuming the presence of a strongly nonlinear negative feedback in an instantaneous drift term. In these cases however, the solution of the corresponding deterministic equation has a global solution. It is therefore interesting to investigate the circumstances under which a solution will exist globally if the unperturbed equation experiences an explosion. Moreover, since it may be appropriate to study highly nonlinear systems (in mathematical biology, for example) in a noisy environment, it is important to determine whether noise perturbations are capable of suppressing explosions that are present in the underlying deterministic models.

The first author was partially supported by an Albert College Fellowship, awarded by Dublin City University’s Research Advisory Panel.

Received April 30, 2004          1056-2176 $15.00 \begin{matrix}c \\ ◯ \end{matrix}Dynamic Publishers, Inc.

This paper is principally concerned with an examination of the existence of stochastic functional differential equations, and more specifically with their non-explosion. A study of the simpler scalar case yields a minimal set of regularity and growth assumptions on the functional on the righthand side of the equation which guarantees a unique strong global solution. As it turns out, important features of the proof in the finite dimensional case are contained in the scalar analysis.

In our proof, we avail of some special structure imposed on the delay functional: roughly speaking, we consider equations in which the delay and instantaneous terms are separate, in the sense that feedback from the past never catches up with feedback from the present. Consequently, a version of the method of steps to be used to prove the existence of a non-exploding solution on a sequence of non-overlapping intervals whose union is the whole of the half-line. This is an adaptation of the method dealing with the existence of stochastic differential equations due to Mao [10], Mao, Marion & Renshaw [12], and Khasminskii [7], in which the boundedness of solutions on arbitrary compact intervals in established by constructing an appropriate Liapunov-like function.

The paper has the following structure. The delay functionals considered, and the structure of the parent deterministic functional differential equations in finite dimensions are outlined in Section 2. In Section 3, the global existence of a stochastic perturbation of these equations is established in the scalar case. Section 4 is concerned with showing the suppression effect of a state-dependent instantaneous noise perturbation on the exploding solution of a deterministic functional differential equation. It also contains an extension of the work in Section 3 to finite dimensions. Section 5 contains conclusions and maps out possible lines of enquiry arising out of the theory contained in this paper.

2. STRUCTURE OF THE DELAY

In this section, we concentrate on the structure of the delay and of the functionals which we study in the paper.

Let r ≥ 0. For each t ≥ 0 denote by C([−r, t]; ℝ^{d}) the family of continuous functions from [−r, t] to ℝ^{d} with the norm ∥φ∥ = sup_{−r≤s≤t,i=1,...,d} |φ_{i}(s)|. For φ ∈ C([−r, t]; ℝ^{d}), the history of φ, up to time t, will be defined by

φ_{t} = {φ(s) : −r ≤ s ≤ t}.

In this paper we study functional differential equations in which the influence of the past is distinct from that of the present. We do this by placing conditions on the structure of the functional, whose role is to determine the evolution of the system in terms of its past.

To make precise this idea that the past and present are distinct, define the functional g_{(τ,\overline{τ})} to be a mapping from C([−\overline{τ}, t]; ℝ^{d}) to ℝ^{d} where \overline{τ} is a real number which obeys

(2.1)  \overline{τ} ∈ [0, ∞),

and τ is a function satisfying

(2.2)  τ : [0, ∞) → (0, ∞) : t ↦ τ(t) is continuous,

such that

(2.3) for all t ≥ 0 and φ ∈ C([−\overline{τ}, t]; ℝ^{d})

(t, φ) ↦ g_{(τ,\overline{τ})}(t, φ_{t}) depends only on \{φ(s) : −\overline{τ} ≤ s ≤ t − τ(t)\}.

There is always a distinction between the dependence of solutions on the past and dependence on the present, because (2.2) implies that the function τ obeys

(2.4)  τ(t) > 0 for all t ≥ 0.

We impose (2.1) so as to exclude equations with infinite delay, although we anticipate that the method outlined here would enable us to prove existence results for functional differential equations with infinite delay. It transpires also that (2.1) does not preclude the study of equations with unbounded delay, in which t − τ(t) → ∞ as t → ∞.

Finally, we address the continuity requirements on the functional g_{(τ,\overline{τ})}. We suppose that

(2.5) for each continuous function φ ∈ C([−\overline{τ}, ∞); ℝ^{d})

t ↦ g_{(τ,\overline{τ})}(t, φ_{t}) is a continuous function from [0, ∞) to ℝ^{d}.

We say that a functional g_{(τ,\overline{τ})} is continuous with delay structure (τ, \overline{τ}) if there is a function τ and a number \overline{τ} which obey (2.1), (2.2) and (2.4), so that g_{(τ,\overline{τ})} obeys (2.3) and (2.5).

2.1. Deterministic functional differential equations. Suppose that g_{(τ,\overline{τ})} is continuous with delay structure τ, and consider the deterministic functional differential equation

(2.6a)  x′(t) = g_{(τ,\overline{τ})}(t, x_{t}), t > 0,
(2.6b)  x(t) = ψ(t), t ∈ [−\overline{τ}, 0],

where ψ ∈ C([−\overline{τ}, 0]; ℝ^{d}). It now transpires that (2.6) can be solved by the “method of steps”.

To see this, we avail of the properties (2.2) and (2.4) of the the delay function τ. Define the sequence {T_{n}}_{n≥0} with

(2.7)  T_{0} = 0, T_{n+1} = inf{t ≥ 0 : t − τ(t) = T_{n}}, n ≥ 0.

There are two possibilities; either there are finitely many members of this sequence, or there are infinitely many. Suppose first that there are infinitely many. Then we see that the sequence T diverges.

Lemma 2.1. Suppose that the function τ obeys (2.2), and (2.4). Then the sequence {T_{n}}_{n≥0} defined by (2.7) is strictly increasing and obeys lim_{n→∞} T_{n} = ∞.

Proof. By (2.7), T_{n+1} = T_{n} +τ(T_{n+1}), so by (2.4), T_{n+1} > T_{n}. Suppose that T_{n} ↑ L where 0 ≤ L < ∞ as n → ∞. Then, by (2.2) and (2.7)

0 = lim_{n→∞} T_{n+1} − τ(T_{n+1}) − T_{n} = L − τ(L) − L = τ(L).

But by (2.4), τ(t) > 0 for all t ≥ 0, so no such finite L can exist. Hence T_{n} → ∞.

On the other hand, if there are finitely many members of the sequence, there exists N ∈ ℕ and T_{N} ≥ 0 such that t − τ(t) ≤ T_{N} for all t ≥ 0.

In the first case, define I_{n} = [T_{n}, T_{n+1}], so ∪_{n∈ℕ}I_{n} = [0, ∞). Suppose a unique solution of (2.6) exists on [−\overline{τ}, T_{n}]. Then, for each t ∈ [T_{n}, T_{n+1}], (2.6b) gives

(2.8)  x(t) = x(T_{n}) + ∫_{T_{n}}^{t} g_{(τ,\overline{τ})}(s, x_{s}) ds.

We now wish to show that the quantity on the righthand side of (2.8) is finite and ensures that x(t) can be uniquely defined. This guarantees the uniqueness and existence of a solution on the subinterval [T_{n}, T_{n+1}].

Next, for each t ∈ [T_{n}, T_{n+1}], g_{(τ,\overline{τ})}(t, x_{t}) does not depend on \{x(s) : T_{n} ≤ s ≤ T_{n+1}}, by (2.3) and Lemma 2.1. Furthermore, by (2.5) t ↦ g_{(τ,\overline{τ})}(t, x_{t}) is a continuous function. Therefore (2.8) uniquely defines a C^{1} solution of (2.6a), (2.6b) on [T_{n}, T_{n+1}], and therefore on [−\overline{τ}, T_{n+1}]; therefore, by induction, a unique continuous solution exists on [−\overline{τ}, ∞).

In the case where there are only finitely many members of the sequence {T_{n}}_{1≤n≤N}, the same argument as above guarantees that (2.6a), (2.6b) has a unique continuous solution on [−\overline{τ}, T_{N}]. For t > T_{N}, notice that t−τ(t) ≤ T_{N}, so referring to (2.8), and using (2.3) and Lemma 2.1, and also (2.5), we see that a unique continuous solution is again guaranteed on [−\overline{τ}, ∞).

Hereinafter, we assume, without loss of generality, that the sequence {T_{n}} is increasing and contains infinitely many members.

3. GLOBAL EXISTENCE FOR STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

In this section we will set d = 1 for the duration of the argument. An example of how the theory can be generalised to finite dimensions can be found in section 4. We ask whether a stochastic perturbation of the scalar deterministic functional differential equation (2.6a), (2.6b) still has a unique global solution. To this end let B = {B(t); F^{B}(t); 0 ≤ t < ∞} be a standard one-dimensional motion defined on the probability triple (Ω, F, ℙ). Here (F^{B}(t))_{t≥0} is the natural filtration of B, viz., F^{B}(t) = σ({B(s) : 0 ≤ s ≤ t}). When we refer to an event as almost sure, this mean that it is ℙ-almost sure. We frequently abbreviate almost sure to a.s.

We study the scalar stochastic functional differential equation

(3.1a)  dX(t) = g_{(τ,\overline{τ})}(t, X_{t}) dt + h(X(t)) dB(t),
(3.1b)  X(t) = ψ(t), t ∈ [−\overline{τ}, 0].

Here, we request that h ∈ C(ℝ; ℝ) is a locally Lipschitz continuous function, ψ ∈ C([−\overline{τ}, 0]; ℝ) and that g_{(τ,\overline{τ})} is continuous with delay structure (τ, \overline{τ}), obeying (2.1), (2.2), and (2.4). In these circumstances, the existence of a unique strong solution is guaranteed up to a (possibly random) explosion time τ_{e}^{ψ}. We denote this solution at time 0 ≤ t < τ_{e}^{ψ} by X(t, [−\overline{τ}, 0], ψ), but we often choose to suppress the dependence on the initial function by writing the solution at time t as X(t). The explosion time is defined to be

(3.2)  τ_{e}^{ψ} = inf\{t > 0 : lim_{s↑t} |X(s, [−\overline{τ}, 0], ψ)| = ∞\}.

Observe that the dependence on the initial function ψ.

Theorem 3.1. Suppose that g_{(τ,\overline{τ})} is continuous with delay structure (τ, \overline{τ}) obeying (2.1), (2.2), and (2.4). Let h be a locally Lipschitz continuous function. If \overline{τ} is defined by (2.1), and ψ ∈ C([−\overline{τ}, 0]; ℝ), then there is a unique continuous adapted process X which is a strong solution of (3.1) on [0, ∞).

Remark 3.2. The result of Theorem 3.1 is equivalent to saying that τ_{e}^{ψ} defined by (3.2) obeys τ_{e}^{ψ} = ∞, almost surely.

Remark 3.3. Given the hypotheses (2.2) and (2.4), the sequence {T_{n}} defined in (2.7) is increasing, and, by the assumption at the end of Section 2, is infinite. It therefore obeys lim_{n→∞} T_{n} = ∞. Just as for the deterministic functional differential equation (2.6), we prove the existence of a unique strong solution of (3.1) on each of the subintervals I_{n} = [T_{n}, T_{n+1}] successively.

Remark 3.4. An application of Theorem 3.1 is needed in Appleby and Kelly [3], in which the oscillation and non–oscillation of solutions of nonlinear stochastic delay differential equations of the form

dX(t) = −f(X(t − τ)) dt + σh(X(t)) dB(t)

is considered. In order to exhibit both oscillation and non–oscillation of solutions, the usual global linear growth and local Lipschitz continuity conditions on f must be relaxed. Therefore, Theorem 3.1 is essential in guaranteeing the existence and uniqueness of a strong solution on [0, ∞); standard results which invoke such growth and continuity hypotheses are not always applicable.

Proof of Theorem 3.1. Define

(3.3)  τ_{k}^{ψ} = inf\{t ∈ [0, τ_{e}^{ψ}) : X(t, [−\overline{τ}, 0], ψ) ∉ (−k, k)\}

and, as {τ_{k}} is increasing, we may define τ_{∞}^{ψ} = lim_{k→∞} τ_{k}^{ψ}. Clearly τ_{∞}^{ψ} ≤ τ_{e}^{ψ}, a.s. Therefore, if τ_{∞}^{ψ} = ∞ a.s. for each ψ, then τ_{e}^{ψ} = ∞. We prove τ_{∞}^{ψ} = ∞ by contradiction, assuming that τ_{∞}^{ψ} = ∞, a.s. is false. First, we show that τ_{∞}^{ψ} > T_{1}, a.s. Consider the negation of this statement, namely that there exists some ψ such that ℙ[τ_{∞}^{ψ} > T_{1}] < 1. Hence there is k_{0} ∈ ℕ sufficiently large, and ε = ε(ψ) ∈ (0, 1) such that ℙ[τ_{k}^{ψ} ≤ T_{1}] ≥ ε, k ≥ k_{0}.

Consider a function V ∈ C^{2}(ℝ; ℝ) which obeys

V (x) > 0, for all x ∈ ℝ, (V1) V (x) = V (−x), x > 0, (V2) (V3) V′ is bounded on ℝ, (V4) There is x^{∗} > 0 such that V′′(x) < 0 for all |x| ≥ x^{∗},

lim_{|x|→∞} V (x) = ∞. (V5)

We note that functions obeying these properties exist; examples are furnished at the end of the proof.

Define F (x, μ) = V′(x)μ + \frac{1}{2}V′′(x)h(x)^{2} and G(x) = V′(x)h(x), where x, μ ∈ ℝ. Then Itô’s rule gives

(3.4) V (X(τ_{k}^{ψ} ∧ T_{1})) = V (ψ(0)) + ∫_{0}^{τ_{k}^{ψ}∧T_{1}} G(X(s)) dB(s)

+ ∫_{0}^{τ_{k}^{ψ}∧T_{1}} F (X(s), g_{(τ,\overline{τ})}(s, X_{s})) ds.

It can be shown by (V3) and (V4) that sup_{x∈ℝ} |F (x, μ)| ≤ \overline{F} (μ) := C_{1}|μ| + C_{2}, where C_{1}, C_{2} are μ-independent constants. Using this in (3.4) gives

V (X(τ_{k}^{ψ}∧T_{1})) ≤ V (ψ(0))+∫_{0}^{τ_{k}^{ψ}∧T_{1}} G(X(s)) dB(s)+∫_{0}^{T_{1}} \overline{F} (g_{(τ,\overline{τ})}(s, X_{s})) ds.

This is true as τ_{k}^{ψ}∧T_{1} ≤ T_{1}. The F(0)-measurability of F_{1}(t) = \overline{F} (g_{(τ,\overline{τ})}(t, X_{t})) for t ∈ [0, T_{1}] implies that

𝔼[V (X(τ_{k}^{ψ} ∧ T_{1}))] ≤ V (ψ(0)) + ∫_{0}^{T_{1}} F_{1}(s) ds < ∞.

Hence for k ≥ k_{0}

∞ > V (ψ(0)) + ∫_{0}^{T_{1}} F_{1}(s) ds ≥ 𝔼[V (X(τ_{k}^{ψ} ∧ T_{1}))] ≥ 𝔼[1_{{τ_{k}^{ψ}≤T_{1}}}V (X(τ_{k}^{ψ}))]

≥ ℙ[τ_{k}^{ψ} ≤ T_{1}]V (k) ≥ εV (k) → ∞ as k → ∞, by (V2) and (V5) yielding a contradiction. We therefore conclude that (3.1) has a unique adapted continuous solution on [0, T_{1}].

On [T_{1}, T_{2}] we need to show that

(3.5)  𝔼[1_{{τ_{∞}^{ψ}>T_{2}}}|F(T_{1})] = 1, a.s.,

for each ψ ∈ C([−\overline{τ}, 0]; ℝ), implying that τ_{∞}^{ψ} > T_{2}, a.s. If we can do this, it can be established, by induction and an identical argument, that there is a unique continuous adapted process which obeys (3.1) on [0, ∞). We suppose in contradiction to (3.5) that there exists ψ and A ⊂ Ω, with ℙ[A] > 0 such that

𝔼[1_{{τ_{∞}^{ψ}>T_{2}}}|F(T_{1})](ω) < 1 for all ω ∈ A.

So there exists ε ∈ (0, 1) and A^{ε} ⊆ A with ℙ[A^{ε}] > 0 such that 𝔼[1_{{τ_{∞}^{ψ}≤T_{2}}}|F(T_{1})] > ε on A^{ε}. Now define the sequence of random variables

(3.6)  P_{k} = 𝔼[1_{{τ_{k}^{ψ}≤T_{2}}}|F(T_{1})]

and the limiting random variable of (3.6) in k, namely P = 𝔼[1_{{τ_{∞}^{ψ}≤T_{2}}}|F(T_{1})]. So P_{k} ≥ P and P_{k} is decreasing in k. So for all k sufficiently large, P_{k} ≥ P > ε on A^{ε}. So we can say that there is ψ ∈ C([−\overline{τ}, 0]; ℝ) such that there is an ε ∈ (0, 1) and a set of positive probability A^{ε} such that 𝔼[1_{{τ_{k}^{ψ}≤T_{2}}}|F(T_{1})] > ε on A^{ε} for k ∈ ℕ. Now

V (X(τ_{k}^{ψ} ∧ T_{2})) = V (X(T_{1})) + ∫_{T_{1}}^{τ_{k}^{ψ}∧T_{2}} F (X(s), g_{(τ,\overline{τ})}(s, X_{s})) ds

+ ∫_{T_{1}}^{τ_{k}^{ψ}∧T_{2}} G(X(s)) dB(s).

If we consider in turn the three possibilities τ_{k}^{ψ} ≤ T_{1}, T_{1} ≤ τ_{k}^{ψ} ≤ T_{2}, and τ_{k}^{ψ} ≥ T_{2}, we get

(3.7) ∫_{T_{1}}^{τ_{k}^{ψ}∧T_{2}} F (X(s), g_{(τ,\overline{τ})}(s, X_{s})) ds

≤ max_{t∈[0,T_{1}]} ∫_{T_{1}}^{t} F (X(s), g_{(τ,\overline{τ})}(s, X_{s})) ds ⋁ ∫_{T_{1}}^{T_{2}} F_{1}(s) ds.

The right hand side of (3.7) is F(T_{1})-measurable. Because X does not explode on [−\overline{τ}, T_{1}], the conditional expectation relative to F(T_{1}) is bounded by an a.s. finite random variable. Therefore, if we can bound

𝔼[∫_{T_{1}}^{τ_{k}^{ψ}∧T_{2}} G(X(s)) dB(s)|F(T_{1})],

the proof will be complete.

Consider the process G(t) = G(X(t)) = V′(X(t))h(X(t)). Define a martingale M on [0, (τ_{k}^{ψ} ∧ T_{2}) ∨ T_{1}] by

(3.8)  M(t) = ∫_{0}^{t} G(s) dB(s).

By Lemma 3.8

𝔼[∫_{T_{1}}^{τ_{k}^{ψ}∧T_{2}} G(s) dB(s)|F(T_{1})] = ∫_{T_{1}}^{τ_{k}^{ψ}∧T_{1}} G(s) dB(s)

= 1_{{τ_{k}^{ψ}<T_{1}}} ∫_{T_{1}}^{τ_{k}^{ψ}} G(s) dB(s),

since ∫_{0}^{T_{1}} G(s) dB(s) is F(T_{1})-measurable. Thus

𝔼[∫_{T_{1}}^{τ_{k}^{ψ}∧T_{2}} G(s) dB(s)|F(T_{1})] ≤ max_{0≤t≤T_{1}} ∫_{T_{1}}^{t} G(s) dB(s).

So

(3.9) 𝔼[V (X(τ_{k}^{ψ} ∧ T_{2}))|F(T_{1})] ≤ V (X(T_{1})) + max_{0≤t≤T_{1}} ∫_{T_{1}}^{t} G(s) dB(s)

+ ∫_{T_{1}}^{T_{2}} |\overline{F} (g_{(τ,\overline{τ})}(s, X_{s}))| ds ⋁ max_{0≤t≤T_{1}} ∫_{T_{1}}^{t} F (X(s), g_{(τ,\overline{τ})}(s, X_{s})) ds.

The right hand side of (3.9) is F(T_{1})-measurable and independent of k. By (V1), we have

𝔼[V (X(τ_{k}^{ψ} ∧ T_{2}))|F(T_{1})] ≥ 𝔼 [1_{{τ_{k}^{ψ}≤T_{2}}}V (X(τ_{k}^{ψ} ∧ T_{2}))|F(T_{1})]

= 𝔼 [1_{{τ_{k}^{ψ}≤T_{2}}}V (X(τ_{k}^{ψ}))|F(T_{1})] ≥ V (k)𝔼[1_{{τ_{k}^{ψ}≤T_{2}}}|F(T_{1})] = V (k)ε on A^{ε}.

Since lim_{|k|→∞} V (k)ε = ∞, by (V5) and (V2), the induced contradiction implies that τ_{∞}^{ψ} > T_{2} a.s. for each ψ. We can proceed in this way, as in Section 2, to show that τ_{∞}^{ψ} = ∞ a.s. for each ψ.

The same method of proof as Theorem 3.1 ensures the existence of a global solution to a more general stochastic functional differential equation.

Theorem 3.5. Let h ∈ C(ℝ; ℝ) be locally Lipschitz continuous, and g_{(τ,\overline{τ})} and h_{(τ′,\overline{τ}′)} be continuous functionals with delay structures (τ, \overline{τ}) and (τ′, \overline{τ}′) respectively, obeying (2.1), (2.2), (2.4). Define τ̃ = \overline{τ} ∨ \overline{τ}′, and ψ ∈ C([−τ̃, 0]; ℝ). Then the stochastic functional differential equation

(3.10a) dX(t) = g_{(τ,\overline{τ})}(t, X_{t})dt + [h(X(t)) + h_{(τ′,\overline{τ}′)}(t, X_{t})] dB(t),
(3.10b) X(t) = ψ(t), t ∈ [−τ̃, 0]

has a unique adapted continuous strong solution on [0, ∞).

Remark 3.6. Theorem 3.1 applies to equations with drift coefficients of many forms. For example, we may consider very general equations with discrete delay. Suppose g̃ ∈ C(ℝ^{n}; ℝ), and τ_{1}, τ_{2}, … τ_{n} are n functions each of which obeys (2.2) and (2.4). Then, with τ(t) = min_{j=1...n} τ_{j}(t), −\overline{τ} = inf_{t≥0}\{t − max_{j=1...n} τ_{j}(t)\}, and φ ∈ C([−\overline{τ}, t]; ℝ), the functional

g_{(τ,\overline{τ})}(t, φ_{t}) = g̃(φ(t − τ_{1}(t)), φ(t − τ_{2}(t)), …, φ(t − τ_{n}(t)))

is continuous with delay structure (τ, \overline{τ}).

We can also consider equations with distributed delay. As an example, suppose that τ_{1}, τ_{2} are continuous nonnegative functions on [0, ∞) obeying τ_{1}(t) ≤ τ_{2}(t) for t ≥ 0, τ_{1}(t) > 0 for all t ≥ 0. Then τ_{1} and τ_{2} obey (2.2) and (2.4). Set τ = τ_{1}, and −\overline{τ} = inf_{t≥0} t − τ_{2}(t). Let g̃ ∈ C([−\overline{τ}, ∞) × [−\overline{τ}, ∞) × ℝ; ℝ). Then with φ ∈ C([−\overline{τ}, t]; ℝ), the functional

g_{(τ,\overline{τ})}(t, φ_{t}) = ∫_{t−τ_{2}(t)}^{t−τ_{1}(t)} g̃(s, t, φ(s)) ds

is continuous with delay structure (τ, \overline{τ}).

In neither case is the possibility of a vanishing delay (where lim_{t→∞} τ(t) = 0) precluded. This is notable as the ‘limiting’ equation (an equation which has only instantaneous contributions in the drift) need not in general have a unique global solution, in either the deterministic or stochastic case. We also can study equations with unbounded delay in each case, by letting t − τ_{2}(t) → ∞ and τ_{2}(t) → ∞ as t → ∞, for example.

Remark 3.7. Observe that the function V defined for x ≥ 0 by

V (x) = \{ \begin{matrix}log(x), & x ≥ e, \\ \frac{1}{4} + e^{−2}x^{2} − \frac{1}{4}e^{−4}x^{4}, & 0 ≤ x ≤ e, \end{matrix}

and which obeys V (−x) = V (x), for x ≤ 0, satisfies all the conditions (V1)-(V5) in Theorem 3.1.

Finally we prove a technical result required in Theorem 3.1.

Lemma 3.8. Let τ > 0, and ρ a bounded stopping time for the process M, which is well defined on [0, ρ∨τ]. If M is a martingale adapted to (F(t))_{t≥0}, then

𝔼[M(ρ)|F(τ)] = M(ρ ∧ τ).

Proof. Since M(τ) + M(ρ) = M(ρ ∧ τ) + M(ρ ∨ τ), we have

𝔼[M(ρ)|F(τ)] = 𝔼[M(ρ ∧ τ) + M(ρ ∨ τ) − M(τ)|F(τ)]

= 𝔼[M(ρ ∨ τ)|F(τ)] + 𝔼[M(ρ ∧ τ)|F(τ)] − 𝔼[M(τ)|F(τ)]

= M(τ) + M(ρ ∧ τ) − M(τ) = M(ρ ∧ τ),

since M(τ), M(ρ ∧ τ) are F(τ)-measurable, and M is a martingale.

4. PREVENTION OF EXPLOSION BY NOISE

Consider the scalar equation

(4.1a) dX(t) = (f(X(t)) + g_{(τ,\overline{τ})}(t, X_{t})) dt + h(X(t)) dB(t),
(4.1b)  X(t) = ψ(t), t ∈ [−\overline{τ}, 0].

We suppose here that f and h are locally Lipschitz continuous functions on ℝ. We assume that the functional g_{(τ,\overline{τ})} is nonnegative, so that g_{(τ,\overline{τ})}(t, φ_{t}) ≥ 0 for all t ≥ 0 and φ ∈ C([−\overline{τ}, t]; ℝ), and that it is locally Lipschitz continuous in the second argument.

By suppressing either the delay functional g_{(τ,\overline{τ})} or the diffusion term h (or both), we hope to understand the manner in which the interactions between noise and delay affect the explosion and non-explosion of the solutions of the various forms of (4.1).

4.1. The deterministic equation. We first consider (4.1) without a delay or noise term. Let f be a locally Lipschitz continuous function such that there is x^{∗} ∈ ℝ and f(x) > 0 for x > x^{∗}. Now consider the initial value problem

(4.2)  x′(t) = f(x(t)), t > 0; x(0) = x_{0},

where x(0) = x_{0} > x^{∗}. If

(4.3)  ∫_{x_{0}}^{∞} \frac{1}{f(x)} dx < ∞,

it is well known that the unique solution of (4.2) explodes in finite time.

A natural question that arises is how the addition of the delay component g_{(τ,\overline{τ})} to (4.2) affects the explosion of solutions of the perturbed equation, when f satisfies (4.3).

Consider the functional differential equation

(4.4) x′(t) = f(x(t)) + g_{(τ,\overline{τ})}(t, x_{t}), t > 0; x(t) = ψ(t), t ∈ [−\overline{τ}, 0],

where ψ ∈ C([−\overline{τ}, 0]; ℝ). The local Lipschitz continuity guarantees that there is a unique solution to this equation, up to an explosion time. A comparison argument establishes that the solution of (4.4) dominates that of (4.2) pointwise if the initial value of (4.4) dominates the initial value of the solution of (4.2). Therefore, as the solution of (4.2) explodes in finite time, the solution of (4.4) explodes in finite time. Moreover, (4.3), the condition which ensures the explosion of (4.4), does not involve the delay functional g_{(τ,\overline{τ})}. Therefore it does not appear that the explosion of (4.4) is influenced by the delay term.

4.2. The stochastic equation. If we perturb (4.4) stochastically to get (4.1) then the following theorem applies, provided that g_{(τ,\overline{τ})} is continuous with delay structure (τ, \overline{τ}) obeying the properties (2.1), (2.2), and (2.4).

Theorem 4.1. Suppose that g_{(τ,\overline{τ})} is continuous with delay structure (τ, \overline{τ}) obeying (2.1), (2.2), and (2.4). If ψ ∈ C([−\overline{τ}, 0]; ℝ), f and h are locally Lipschitz continuous, and

(4.5)  sup_{x∈ℝ} \frac{xf(x) − \frac{1}{2}h(x)^{2}}{1 + |x|^{2}} < ∞,

then there is a unique, continuous, adapted process which is a strong solution of (4.1) on [0, ∞).

This result can be proved in a manner similar to that of Theorem 3.1, using the function V mentioned in Remark 3.7 in the proof.

We observe that the condition (4.5) ensures that if the intensity of the diffusion term h is sufficiently large relative to f, the explosion exhibited by the solution of (4.4) can always be suppressed. Moreover, since the condition (4.5) does not involve the delay functional g_{(τ,\overline{τ})}, it appears that the delay term does not play a role in causing or preventing explosions of (4.1).

A similar result applies if there is no delay term in the drift. If the relative sizes of f and h in

(4.6)  dX(t) = f(X(t)) dt + h(X(t)) dB(t)

are as given in Feller’s test for explosion and non-explosion [6], then we can again suppress an explosion in the corresponding deterministic equation (4.2). In fact, the condition (4.5) is sufficient to guarantee the Feller test conditions, so the solution of (4.6) exists on [0, ∞) almost surely.

The example of this group of four equations (4.1), (4.2), (4.4), and (4.6) suggests that the presence of a sufficiently large state-dependent noise perturbation in a scalar system prevents the explosion that might be present in the unperturbed deterministic equation. Moreover, if the delay term is strictly in the past (in the sense that it is present as a functional g_{(τ,\overline{τ})} with continuous delay structure (τ, \overline{τ})), then the presence of an explosion does not depend on the structure of the delay term, but purely on the instantaneous terms.

4.3. Global existence in finite dimensions. Theorem 4.1 can be generalised to cover the global existence for a class of finite dimensional stochastic equations.

Theorem 4.2. Let h_{0} be a locally Lipschitz continuous function from ℝ^{d} to ℝ, and h(x) = h_{0}(x)x for all x ∈ ℝ^{d}. Suppose that g_{(τ,\overline{τ})} is continuous with delay structure (τ, \overline{τ}) obeying (2.1), (2.2), and (2.4). If ψ ∈ C([−\overline{τ}, 0]; ℝ^{d}), f and h are locally Lipschitz continuous, and

(4.7)  sup_{x∈ℝ^{d}} \frac{⟨x, f(x)⟩ − \frac{1}{2}∥h(x)∥^{2}}{1 + ∥x∥^{2}} < ∞,

then there is a unique, continuous, adapted ℝ^{d}-valued process which is a strong solution of

(4.8a) dX(t) = (f(X(t)) + g_{(τ,\overline{τ})}(t, X_{t})) dt + h(X(t)) dB(t),
(4.8b)  X(t) = ψ(t), t ∈ [−\overline{τ}, 0].

on [0, ∞).

Equations with an instantaneous multiplicative noise perturbation of this form can find application in modelling the population dynamics of several interacting species; we refer the reader to Mao, Marion and Renshaw [12] for further details. We note also that the condition (4.7) reduces to (4.5) in the scalar case (d = 1).

Proof of Theorem 4.2. Let V_{s} be the scalar function defined in Remark 3.7. Define V ∈ C(ℝ^{d}; ℝ) by V (x) = V_{s}(1 + ∥x∥^{2}), x ∈ ℝ^{d}. We redefine the explosion time given in (3.2) for finite dimensional processes as

τ_{e}^{ψ} = inf{t > 0 : lim_{s↑t} |X_{i}(s, [−τ̄, 0], ψ)| = ∞ for some i = 1, …, d},

where X_{i} = ⟨X, e_{i}⟩, and e_{i} is the vector with 1 in the i-th component, and zeros elsewhere. Let k^{∗} ∈ ℝ be sufficiently large that the initial data of each component of X lies within [−k^{∗}, k^{∗}]. Then for every integer k > k^{∗}, we can define the stopping time

τ_{k}^{ψ} = inf{t ∈ [0, τ_{e}^{ψ}) : X_{i}(t, [−τ̄, 0], ψ) ∉ (−k, k), for some i = 1, …, d}.

Clearly, τ_{k}^{ψ} is an increasing sequence. Set τ_{∞}^{ψ} = lim_{k→∞} τ_{k}^{ψ}, as before. We show that (4.7) is sufficient to ensure that τ_{∞}^{ψ} = ∞, and therefore that τ_{e}^{ψ} = ∞, by assuming the converse. That is, for some ψ ∈ C([−\overline{τ}, 0]; ℝ^{d}) there exists a pair of constants T > 0 and ε ∈ (0, 1) such that

ℙ[τ_{∞}^{ψ} ≤ T] > ε.

Hence there is an integer k_{1} > k^{∗} such that

ℙ[τ_{k}^{ψ} ≤ T] ≥ ε, for all k ≥ k_{1}.

We define a sequence of times (T_{n})_{n≥0} depending on the delay structure (τ, \overline{τ}) as in (2.7). Up to the truncated stopping time τ_{k}^{ψ} ∧ T_{1}, Itô’s rule shows that the semimartingale decomposition of V (X(t)) is given by

V (X(τ_{k}^{ψ} ∧ T_{1})) = V (X(0)) + ∫_{0}^{τ_{k}^{ψ} ∧T_{1}} F (X(s), g_{(τ,\overline{τ})}(s, X_{s})) ds

+ ∫_{0}^{τ_{k}^{ψ} ∧T_{1}} G(X(s)) dB(s) where F (x, μ) = L(x) + 2V′_{s}(∥x∥^{2})⟨x, μ⟩, G(x) = 2V′_{s}(∥x∥)^{2}⟨x, h(x)⟩, and

L(x) = 2(\frac{⟨x, f(x)⟩ − \frac{1}{2}∥h(x)∥^{2}}{1 + ∥x∥^{2}} ).

L is a globally bounded function by virtue of condition (4.7). Therefore (4.7) guarantees that sup_{x∈ℝ^{d}} |F (x, μ)| ≤ C_{1}∥μ∥ + C_{2} where C_{1}, C_{2} are positive and μ-independent constants. Letting |k| → ∞ delivers the contradiction required to show that τ_{∞}^{ψ} is not on (0, T_{1}]. We can continue this proof, as before, on successive intervals (T_{n−1}, T_{n}]. Thus, by induction, condition (4.7) ensures that a unique, strong solution to (4.8) exists on [−τ̄, ∞).

5. EPILOGUE

5.1. Conclusions. For the deterministic equation (2.6) with delay structure as described in (2.1), (2.2), (2.3), and (2.4) we merely require that g_{(τ,\overline{τ})} be continuous to ensure a unique global solution. The stochastic perturbation of (2.6) embodied in (3.1) can be shown to have a unique global solution if h is locally Lipschitz continuous. Thus, in this case, the addition of noise does not prevent the solution from existing globally. In fact if some of the feedback in the drift is instantaneous, it is possible to suppress an explosion in the solution of the deterministic equation with a carefully chosen noise term. To achieve this in either the scalar and finite dimensional cases, (4.5) and (4.7) respectively give sufficient conditions on h for a given drift coefficient. It is interesting to note that the delayed feedback component of the drift appears to play no role whatsoever in either causing or suppressing an explosion in such equations.

5.2. Further work - Removing the “gap” in the delay structure. It seems natural to ask whether similar methods will allow us to guarantee existence in equations of the form

(5.1a)  dX(t) = f(X(t))[g(t, X_{t}) dt + h(t, X_{t}) dB(t)],
(5.1b)  X(t) = ψ(t), t ∈ [−\overline{τ}, 0].

The delay structure in (5.1) has no characteristic ‘gap’ τ, as described in section 2, and feedback can originate from any point in the history right up to, and including, the present. We conjecture that, whatever the form of g, there will exist a functional h such that almost sure non-explosion of solutions is guaranteed. However, the absence of the delay structure which characterises the problems in this paper seems to preclude a straightforward extension of the methods developed here to the problems of the form (5.1).

5.3. Further work - Noise induced explosion. From work of Mao [9] it is known that a diagonal linear multiplicative noise perturbation can stabilise the equilibrium solution of a stochastic differential equation, and extensions of this result to stochastic functional differential equations are considered in Appleby [2]. It is notable that a more intense noise perturbation of the same form can suppress explosions in solutions of functional differential equations. Work of Mao [9] and Appleby [1] shows that a linear noise perturbation of a skew form destabilises equilibrium solutions of stochastic differential equations and stochastic functional differential equations. It is therefore tempting to conjecture that a more intense noise perturbation of this form may induce an explosion in the stochastic system which is not present in the underlying deterministic system.

Acknowledgement

The authors would like to thank the referee for their careful scrutiny of the manuscript, and for suggesting some amendments.

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