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On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations
We consider the zero crossings and positive solutions of scalar nonlinear stochastic Volterra integrodifferential equations of Itô type. In the equations considered, the diffusion coefficient is linear and depends on the current state, and the drift term is a convolution integral which is in some sense mean reverting towards the zero equilibrium. The state dependent restoring force in the integral can be nonlinear. In broad terms, we show that when the restoring force is of linear or lower order in the neighbourhood of the equilibrium, or if the kernel decays more slowly than a critical noise-dependent rate, then there is a zero crossing almost surely. On the other hand, if the kernel decays more rapidly than this critical rate, and the restoring force is globally superlinear, then there is a positive probability that the solution remains of one sign for all time, given a sufficiently small initial condition. Moreover, the probability that the solution remains of one sign tends to unity as the initial condition tends to zero.
Very often in deterministic systems, interest focusses on solutions of such equations which are oscillatory, as these could plausibly reflect cyclic motion of a system around an equilibrium. Over the last thirty years, an extensive theory of oscillatory solutions of deterministic equations has developed. Numerous papers and several monographs illustrate the extent of research [4, 11–13]; further, we would like to draw attention to the recent survey paper . However, the effect that random perturbations of Itô type might have on the existence—creation or destruction—of oscillatory solutions of delay differential equations seems, at present, to have received comparatively little attention.
In this paper we consider whether solutions of the stochastic Volterra convolution integrodifferential equation
remain positive for all time, or hit or cross zero in a finite time. In (1.1), is a standard one-dimensional Brownian motion or Wiener process. It is assumed that the kernel is a nonnegative, continuous, and integrable function and that the continuous function obeys for and . Regularity assumptions on and are required to guarantee the existence of solutions. The sign conditions on and are motivated by the underlying deterministic Volterra integrodifferential equation
These conditions on and ensure that zero is the unique steady-state solution of (1.2) and that solutions tend to revert towards the equilibrium (at least ab initio). If the strength of the mean reversion is sufficiently strong, or the kernel fades sufficiently slowly, solutions of (1.2) can hit zero in finite time. This phenomenon is referred to as a zero crossing. Results on the zero crossing of solutions of (1.2) include work by Gopalsamy and Lalli  and Györi and Ladas , and a significant literature exists for the zero crossings of such deterministic equations. However, less seems to be known in the stochastic case. Therefore, the question addressed in this paper is: how does a linear state-dependent, instantaneous and equilibrium preserving stochastic perturbation effect the zero crossing and positivity properties of solutions of (1.2)? We answer this question by proving three interrelated results.
First, we show that if is of linear or lower order in the neighbourhood of the zero equilibrium, then the solution of (1.1) has a zero crossing almost surely, provided that the kernel is not identically zero.
Second, we show that if is of order for as (i.e., in the neighbourhood of the equilibrium), and also obeys a global superlinear upper bound on , then any solution of (1.1) which starts sufficiently close to the equilibrium will remain strictly positive with a probability arbitrarily close to unity. This result holds if the kernel decays more quickly than some critical exponential rate (which depends on the noise intensity ). Therefore, if the restoring force is sufficiently weak close to the equilibrium (relative to the linear stochastic intensity), solutions will never change sign. Indeed, it is a fortiori shown that solutions can remain positive with arbitrarily high probability once the initial value is small enough.
Finally, if decays more slowly than the critical exponential rate, then all solutions of (1.1) will have zero crossings, regardless of how weakly the restoring function acts on the solution. Therefore, we conclude that solutions of (1.1) will remain positive only if (i) decays more quickly than some critical noise-intensity dependent rate and (ii) is superlinear (at least in the neighbourhood of the equilibrium).
It is interesting to observe that the change in sign of solutions is similar to that seen for the corresponding deterministic equations: at the first zero, the sample path of the solution is differentiable and the derivative is negative. This is notable because the sample path of the solution of (1.1) is not differentiable at any other point. Therefore “oscillation” is not a result of the lack of regularity in the sample path of the nondifferentiable Brownian motion , but rather results from the fluctuation properties of its increments. The presence of delay is important as well: for a stochastic ordinary differential equation, the presence of noise does not induce an oscillation about the equilibrium, if it is a strong solution, see, for example, .
Although results in this paper are established for convolution equations, the elegant theory of zero crossings and oscillation for deterministic Volterra equations, which hinge on the existence of real zeros of the characteristic equation, is not employed here. See, for example, [15, 16]. This is largely because the effect of the stochastic perturbation dominates. Instead we employ ideas developed for stochastic functional differential equations with a single (and finite) delay in [18–21].
One motivation for this work is to establish that in the presence of uncertainty, mean reverting systems with delay tend to overshoot equilibrium levels, rather than to approach them monotonically, as appears more likely in the absence of stochastic shocks. This is postulated as a mechanism by which economic systems overshoot an equilibrium, in which the system experiences external stochastic shocks whose intensity depends on the state of the system. Therefore, overconfidence among economic agents, and their feedback behaviour based on the past history of the system, is likely to have a significant impact on the adjustment of the system towards, and overshooting of, its equilibrium, when the system is truly random. Examples of stochastic functional differential and difference equation models of financial markets in which agents use the past information of the system to determine their trading behaviour include [5, 22–24].
The paper is organised as follows. Mathematical preliminaries, including remarks on the existence and uniqueness of solutions of (1.1), are presented in Section 2. The main results of the paper are stated and discussed in Section 3. In Section 4, we show that solutions of (1.1) can be written as the product of the positive solution of a linear stochastic differential equation and the solution of a random Volterra integrodifferential equation. This Volterra equation has solution which has continuously differentiable paths and is of the form
where and inherit positivity properties from and . Therefore, the zero crossings of the solution of (1.1) correspond to zero crossings of the solution of (1.3). The proofs of the main results are given in the final three sections of the paper.
In advance of stating and discussing our main results, we introduce some standard notation. We denote the maximum of the real numbers and by and the minimum of and by . Let denote the space of continuous functions where and are intervals contained in . Similarly, we let denote the space of differentiable functions , where . We denote by the space of Lebesgue integrable functions such that
If is an event we denote its complement by . We frequently use the standard abbreviations a.s. to stand for almost sure, and a.a. to stand for almost all.
2.2. Existence of Solutions of the Stochastic Equation
Let us fix a complete probability space with a filtration satisfying the usual conditions and let be a standard one-dimensional Brownian motion on this space. Let be a real positive constant. Suppose that
Suppose also that
Let . We consider the stochastic Volterra equation
Let . Suppose, in addition to (2.3), that is locally Lipschitz continuous. This means the following.
For everythere existssuch that
Then there is a unique continuous -adapted process which obeys
where . Suppose in addition that is globally linearly bounded. More precisely, this means that also obeys the following:
See, for example, Berger and Mizel [25, Theorem ]. In this situation, we say that (2.4) has a unique strong solution. Throughout the paper, we will assume that (2.4) has a unique strong solution but will not necessarily impose conditions (2.5) or (2.7) on in order to guarantee this. Hereinafter we will often refer to the solution rather than the strong solution of (2.4). We denote the almost sure event on which (2.8) holds by . For each we denote by the value of at time . We denote by the realisation (or sample path) .
2.3. Zero Crossing and Positivity of Solutions
Let be the solution of (2.4), where . For each , the stopping time is defined by
We interpret in the case when is the empty set. We say that the sample path has a zero crossing if . Define
3. Statement and Discussion of Main Results
Before stating our main results on the solutions of (2.4), we discuss the significance of the hypotheses on and . We motivate these by considering the deterministic Volterra equation corresponding to (2.4). This deterministic equation can be constructed by setting , resulting in
Conditions (2.2), (2.5), and (2.7) ensure that (3.1) possesses a unique continuous global solution. Clearly, in the case when the hypothesis (2.3) ensures that for all is the unique steady-state solution. We also notice that there are no other steady-state solutions because (2.3) implies that for . The fact that the intensity of the stochastic perturbation is zero if and only if the solution is at the steady-state solution of (3.1) means that this stochastic perturbation preserves the unique equilibrium solution of the deterministic equation (3.1)∶ indeed if , then for all a.s. Moreover, the stochastic perturbation does not produce more point equilibria.
The fact that is nonnegative and is positive when is greater than the equilibrium solution of (3.1) means that the solution of (3.1) is initially attracted towards the equilibrium, because for all , provided for all . The question then arises: does the solution ever reach the zero equilibrium solution in finite time? If so, does it overshoot and become negative (this is referred to as a zero crossing), or hit zero and remain there indefinitely thereafter. The paper addresses these questions for the solutions of the stochastic equation (2.4).
Let . Our first result demonstrates that the solutions of (2.4) has a zero crossing for a.a. sample paths in the case when has at least linear-order leading behaviour at the equilibrium, and when is not identically zero. More precisely we request that obeys the following:
and that . Moreover, it transpires that not only hits the zero level, but even assumes negative values. Furthermore, although the sample path of solutions of (2.4) is not differentiable at time , provided that , it is nonetheless differentiable at , and the zero level is crossed because this first zero of is a simple zero of .
See  for related comments concerning the zero set of the solution of a stochastic delay differential equation with a single fixed delay. An immediate and interesting corollary of Theorem 3.1 concerns the linear stochastic Volterra equation
The proof of these results is a consequence of Lemma 5.1 below. This lemma is inspired by a result of Staikos and Stavroulakis [26, Theorem ], which applies to linear nonautonomous delay-differential equations. See also [13, Theorem ]. This theorem has been employed in [18–20] to demonstrate the existence of a.s. oscillatory solutions of stochastic delay differential equations with a single delay. In each of [18–20] the analysis of the large fluctuations of integral functionals of increments of standard Brownian motion plays an important role in verifying the deterministic oscillation criterion. Similarly, the proofs of Theorems 3.1 and 3.2 in this work hinge on an analysis of increments of the standard Brownian motion .
It is interesting to compare Theorem 3.2 with known results on the zero crossings of the corresponding deterministic linear Volterra integrodifferential equation
has no real solutions. However, solutions of (3.4) have zero crossings for a.a. sample paths provided that is nontrivial. Therefore, the presence of the noise term tends to induce crossing of the equilibrium, even when this is absent in the underlying deterministic equation. On the other hand, if the solution of (3.6) possesses zero crossings, then so does that of (3.4). Therefore, the presence of a stochastic term tends to induce oscillatory behaviour in the solution.
Theorems 3.1 and 3.2 show that positive solutions are impossible if is of linear, or lower order, leading behaviour at zero. It is reasonable therefore to ask whether positive solutions can ever persist in the presence of a stochastic perturbation. To this end, we now consider the case when does not necessarily have linear-order leading behaviour at zero. We assume not only that is weakly nonlinear close to zero, but also that it obeys the following:
In addition, we suppose that decays more quickly than as in the sense that
Under these conditions, the next result states that (2.4) can possess positive solutions with positive probability, provided that the initial condition is sufficiently small. Moreover, the probability that the solution remains positive for all time approaches unity as the positive initial condition tends to zero.
Theorem 3.3. Suppose that obeys (2.2) and (3.9). Suppose also that obeys (2.3) and (3.8). Let be the unique strong solution of (2.4). If is defined by (2.10), then there exists such that for all . Moreover
The result and proof are inspired by [19, Theorem ], which applies to the stochastic delay differential equation
Theorems 3.1 and 3.3 show the importance of the linearity of local to zero in the presence or absence of zero crossings. However, it is natural to ask whether condition (3.9) in Theorem 3.3 is essential in allowing for positive solutions of (2.4), or whether it is merely a convenient condition which enables us to establish positivity in some cases. The next result shows that condition (3.9) is more or less essential if zero crossings are to be precluded with positive probability.
In order to show this, we consider a condition on the rate of decay of which is slightly stronger than the negation of condition (3.9). We assume that decays more slowly to zero than in the sense that Under this condition, solutions of (2.4) cross zero on almost all sample paths, irrespective of how weakly the nonlinear restoring function acts on the solution.
Theorem 3.4. Suppose that obeys (2.2) and (3.12). Suppose also that obeys (2.3). Let be the unique continuous solution of (2.4). If is defined by (2.9), then for any one has Moreover, is differentiable at and .
This result is interesting because, in the case when is in and , the linearisation of (2.4) that is the linear SDE given by
has positive solutions with probability one. In a complete contrast however, (2.4) has zero crossings with probability one.
4. Reformulation in Terms of a Random Differential Equation
The results in the paper are often a consequence of a reformulation of (2.4) as a random differential equation with continuously differentiable sample paths. This approach has proved successful for studying the oscillation and positivity of solutions of stochastic delay differential equations in [18–20].
Then is a strictly positive process (i.e., for all a.s.) which obeys the stochastic differential equation
Then for all a.s. and by Itô's lemma, obeys the stochastic differential equation
By (stochastic) integration by parts it follows that
Therefore, as , we have
is continuous. Therefore we have that each realisation of the process defined by
is in and by (4.5) we have
It is convenient here to record another fact concerning : on account of the Strong Law of Large Numbers for standard Brownian motion, it follows that
and therefore we have that as a.s.
5. Proof of Theorem 3.1
5.1. Supporting Lemmas
Lemma 5.1. Let . Suppose that obeys (2.2) and that obeys (2.3) and (3.2). Suppose that is defined by (4.1), and by Suppose that there exists such that Let . If is the unique strong solution of (2.4), and is defined by (2.9), then Moreover, is differentiable at and .
Proof. By (4.8) and the definition of in (5.1) we have
Note that . Suppose that . Let . Note that for all . Therefore tends to a nonnegative limit as . Since as , we have that as . We temporarily suppress the dependence on . Since , for any , by (5.4) we have
Now, because is nonincreasing we have
Since for all we have that
Since as and we have
which contradicts (5.2). Therefore we have that , as required.
Now such that = , and this event is almost sure. Fix . Since if and only if , by (4.7) we have from (2.9) that and that for all . By (4.8) we have
Since (2.3) implies that for all , and obeys (2.2) we have that . Suppose that . Since , we must have that for all or for . Therefore we have that for all . Hence , which contradicts the fact that . Therefore we must have . This implies that there exists such that and therefore we have that . Therefore for each in the a.s. event , there exists a such that .
We now show that is differentiable at and that . Let . Then we have
Now taking the limit as on the righthand side we have Therefore we have so is well defined and indeed .
The next result develops a condition which depends only on the increments of and the kernel which implies condition (5.2).
Proof. If and it follows that there is a such that . Since is continuous on there exists such that for all . Therefore we have for all . Hence Let . Then , . Define Equation (5.14) is equivalent to show that a.s. Clearly by (5.18) we have Now , so we have Hence (5.14) follows if we can show that a.s. Define for so that it suffices to prove that a.s. Note that Now, we note that each is a functional of increments of the standard Brownian motion over the interval . Therefore as , it follows that the intervals on which the increments of are considered for and are nonoverlapping. Since the increments of are independent, it follows that is a sequence of independent random variables. Hence by the Borel-Cantelli lemma, we are done if we can show that Note that is a standard Brownian motion. Then Define for the event If , then for all . Therefore implies Thus Now Since , each of , , and is well defined and independent random variables. Hence Now we note that for is a standard Brownian motion, so we have Next, if we define for , then is another standard Brownian motion. Therefore we have Therefore if is a standard normal random variable and is the distribution function of , we have Hence by (5.27) we have Let and define by Then . Hence This implies that (5.23) holds, and therefore that a.s., from which it has already been shown that the lemma follows.
5.2. Proof of Theorem 3.1
The proof of Theorem 3.1 is now an immediate consequence of the last three lemmas. By Lemma 5.3 it follows that (5.14) holds. By Lemma 5.2 it therefore follows that (5.2) holds. Hence by Lemma 5.1 it follows that for any and that exists and is negative. Since these are the desired conclusions of Theorem 3.1, the proof is complete.
6. Proof of Theorem 3.3
We start by proving a technical lemma.
Proof. By the definition of we have
By the Strong Law of Large Numbers for standard Brownian motion (see, e.g., Karatzas and Shreve ), there exists an almost sure event such that
Therefore for each and for every there exists a finite such that
Then . The continuity of the integrand and finiteness of ensures that . Consider now . Suppose that is so small that where is defined by (3.9) (note the distinction between the constant defined by (3.9) and the small parameter ). Define . By (3.9) we therefore have
Then by (6.4) we have
Now by the nonnegativity of the integrand and Fubini's theorem we have
By (6.6) we have
which implies that .
Finally we show that , a.s. Suppose that is so small that and . Define . Then . Hence by (6.4) we have
Now since . Hence Therefore by (6.6) we have that and so Hence , as required.
6.1. Proof of Theorem 3.3
By (4.8) we have
Hence by (6.14) and the nonnegativity of and we have
Therefore as is defined by (6.1) we have
Therefore we have . On the other hand, because we evidently have , it follows that as .
On the other hand, by (6.18) we have . Therefore as a.s. it follows that there is an such that . Now suppose . Then as we have
which implies for all , proving the result.
7. Proof of Theorem 3.4
Let be defined by (2.10). We suppose that . Define also
Then by (4.5) we have
Fix . Since for all , we have that for all . Also, as for all and