On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations

Abstract

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.

1. Introduction

Deterministic and stochastic delay differential equations are widely used to model systems in ecology, economics, engineering, and physics [1–10].

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 [14]. 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

𝑑𝑋(𝑡)=−^𝑡₀𝑘(𝑡−𝑠)𝑓(𝑋(𝑠))𝑑𝑠𝑑𝑡+𝜎𝑋(𝑡)𝑑𝐵(𝑡),𝑡≥0,𝑋(0)=𝛼>0, (1.1) remain positive for all time, or hit or cross zero in a finite time. In (1.1), 𝐵 is a standard one-dimensional Brownian motion or Wiener process. It is assumed that the kernel 𝑘 is a nonnegative, continuous, and integrable function and that the continuous function 𝑓 obeys 𝑥𝑓(𝑥)>0 for 𝑥≠0 and 𝑓(0)=0. Regularity assumptions on 𝑓 and 𝑘 are required to guarantee the existence of solutions. The sign conditions on 𝑓 and 𝑘 are motivated by the underlying deterministic Volterra integrodifferential equation

𝑥^′(𝑡)=−^𝑡₀𝑘(𝑡−𝑠)𝑓(𝑥(𝑠))𝑑𝑠,𝑡≥0,𝑥(0)=𝛼>0. (1.2) These conditions on 𝑓 and 𝑘 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 [15] and Györi and Ladas [16], 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 𝑓 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 𝑘 is not identically zero.

Second, we show that if 𝑓(𝑥) is of order 𝑥^𝛾 for 𝛾>1 as 𝑥→0⁺ (i.e., in the neighbourhood of the equilibrium), and 𝑓 also obeys a global superlinear upper bound on (0,∞), 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 𝑘 decays more quickly than some critical exponential rate (which depends on the noise intensity 𝜎). 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 𝑘 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 𝑓 acts on the solution. Therefore, we conclude that solutions of (1.1) will remain positive only if (i) 𝑘 decays more quickly than some critical noise-intensity dependent rate and (ii) 𝑓 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 𝐵, 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, [17].

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 𝑦 which has continuously differentiable paths and is of the form

𝑦^′(𝑡)=−^𝑡₀𝐾(𝑡,𝑠)𝐹(𝑠,𝑦(𝑠))𝑑𝑠,𝑡≥0,𝑦(0)=𝛼>0, (1.3) where 𝐾 and 𝐹 inherit positivity properties from 𝑘 and 𝑓. Therefore, the zero crossings of the solution 𝑋 of (1.1) correspond to zero crossings of the solution 𝑦 of (1.3). The proofs of the main results are given in the final three sections of the paper.

2. Preliminaries

2.1. Notation

In advance of stating and discussing our main results, we introduce some standard notation. We denote the maximum of the real numbers 𝑥 and 𝑦 by 𝑥∨𝑦 and the minimum of 𝑥 and 𝑦 by 𝑥⋀𝑦. Let 𝐶(𝐼;𝐽) denote the space of continuous functions 𝑓∶𝐼→𝐽 where 𝐼 and 𝐽 are intervals contained in ℝ. Similarly, we let 𝐶¹(𝐼;𝐽) denote the space of differentiable functions 𝑓∶𝐼→𝐽, where 𝑓^′∈𝐶(𝐼;𝐽). We denote by 𝐿¹(0,∞) the space of Lebesgue integrable functions 𝑓∶[0,∞)→ℝ such that

^∞₀|𝑓(𝑠)|𝑑𝑠<+∞. (2.1) If 𝐴 is an event we denote its complement by 𝐴. 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 (Ω,ℱ,ℙ) with a filtration (ℱ(𝑡))_𝑡≥0 satisfying the usual conditions and let (𝐵(𝑡)∶𝑡≥0) be a standard one-dimensional Brownian motion on this space. Let 𝜎 be a real positive constant. Suppose that

𝑘∈𝐶([0,∞);[0,∞)),𝑘∈𝐿¹(0,∞). (2.2) Suppose also that

𝑓∈𝐶(ℝ;ℝ),𝑥𝑓(𝑥)>0𝑥≠0,𝑓(0)=0. (2.3) Let 𝛼>0. We consider the stochastic Volterra equation

𝑑𝑋(𝑡)=−^𝑡₀𝑘(𝑡−𝑠)𝑓(𝑋(𝑠))𝑑𝑠𝑑𝑡+𝜎𝑋(𝑡)𝑑𝐵(𝑡),𝑡≥0,𝑋(0)=𝛼. (2.4) Let 𝑛∈ℕ. Suppose, in addition to (2.3), that 𝑓 is locally Lipschitz continuous. This means the following.

For every𝑛∈ℕthere exists𝐾_𝑛>0such that

|𝑓(𝑥)−𝑓(𝑦)|≤𝐾_𝑛|𝑥−𝑦|∀𝑥,𝑦∈ℝforwhich|𝑥|∨|𝑦|≤𝑛. (2.5) Then there is a unique continuous ℱ^𝐵-adapted process 𝑋 which obeys

𝑋(𝑡𝑡_𝑛)=𝛼−^{𝑡⋀𝑡𝑛}₀^𝑠₀𝑘(𝑠−𝑢)𝑓(𝑋(𝑢))𝑑𝑢𝑑𝑠+^{𝑡⋀𝑡𝑛}₀𝜎𝑋(𝑠)𝑑𝐵(𝑠),𝑡≥0,a.s, (2.6) where 𝑡_𝑛=inf{𝑡≥0∶|𝑋(𝑡)|=𝑛}. Suppose in addition that 𝑓 is globally linearly bounded. More precisely, this means that 𝑓 also obeys the following:

Thereexists𝐿₁≥0suchthat|𝑓(𝑥)|≤𝐿₁(1+|𝑥|),𝑥∈ℝ. (2.7) If 𝑘 obeys (2.2) and 𝑓 obeys (2.5) and (2.7), then there exists a unique continuous ℱ^𝐵-adapted process 𝑋 which obeys

𝑋(𝑡)=𝛼−^𝑡₀^𝑠₀𝑘(𝑠−𝑢)𝑓(𝑋(𝑢))𝑑𝑢𝑑𝑠+^𝑡₀𝜎𝑋(𝑠)𝑑𝐵(𝑠),𝑡≥0,a.s. (2.8) See, for example, Berger and Mizel [25, Theorem 2E]. 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 𝑓 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 Ω^∗. For each 𝜔∈Ω^∗ we denote by 𝑋(𝑡,𝜔) the value of 𝑋 at time 𝑡. We denote by 𝑋(𝜔) the realisation (or sample path) 𝑋(𝜔)={𝑋(𝑡,𝜔)∶𝑡≥0}.

2.3. Zero Crossing and Positivity of Solutions

Let 𝑋 be the solution of (2.4), where 𝛼>0. For each 𝜔∈Ω^∗, the stopping time 𝜏(𝛼) is defined by

𝜏(𝛼,𝜔)=inf{𝑡>0∶𝑋(𝑡,𝜔)=0}. (2.9) We interpret 𝜏(𝛼,𝜔)=+∞ in the case when {𝑡>0∶𝑋(𝑡,𝜔)=0} is the empty set. We say that the sample path 𝑋(𝜔) has a zero crossing if 𝜏(𝛼,𝜔)<+∞. Define

𝐴_𝛼={𝜔∈Ω^∗∶𝑋(𝜔)ispositiveon[0,∞)}={𝜔∈Ω^∗∶𝜏(𝛼,𝜔)=+∞}. (2.10)

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 𝑓 and 𝑘. We motivate these by considering the deterministic Volterra equation corresponding to (2.4). This deterministic equation can be constructed by setting 𝜎=0, resulting in

𝑥^′(𝑡)=−^𝑡₀𝑘(𝑡−𝑠)𝑓(𝑥(𝑠))𝑑𝑠,𝑡≥0,𝑥(0)=𝛼. (3.1) Conditions (2.2), (2.5), and (2.7) ensure that (3.1) possesses a unique continuous global solution. Clearly, in the case when 𝛼=0 the hypothesis (2.3) ensures that 𝑥(𝑡)=0 for all 𝑡≥0 is the unique steady-state solution. We also notice that there are no other steady-state solutions 𝐾 because (2.3) implies that 𝑓(𝐾)≠0 for 𝐾≠0. 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 𝑋(0)=0, then 𝑋(𝑡)=0 for all 𝑡≥0 a.s. Moreover, the stochastic perturbation does not produce more point equilibria.

The fact that 𝑘 is nonnegative and 𝑓(𝑥) is positive when 𝑥 is greater than the equilibrium solution of (3.1) means that the solution 𝑥 of (3.1) is initially attracted towards the equilibrium, because 𝑥^′(𝑡)≤0 for all 𝑡≥0, provided 𝑥(𝑡)>0 for all 𝑡≥0. 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 𝛼>0. Our first result demonstrates that the solutions of (2.4) has a zero crossing for a.a. sample paths in the case when 𝑓 has at least linear-order leading behaviour at the equilibrium, and when 𝑘 is not identically zero. More precisely we request that 𝑓 obeys the following:

Thereexists𝐿₃>0suchthatliminf_𝑥→0+𝑓(𝑥)𝑥=𝐿₃, (3.2) and that 𝑘(𝑡)≢0. Moreover, it transpires that 𝑋 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 𝑡(𝜔), provided that 𝑋(𝑡(𝜔),𝜔)≠0, it is nonetheless differentiable at 𝜏(𝛼), and the zero level is crossed because this first zero 𝜏(𝛼) of 𝑋 is a simple zero of 𝑋.

Theorem 3.1. Suppose that 𝑘 obeys (2.2), 𝑓 obeys (2.3) and (3.2), and that 𝑘(𝑡)≢0. Let 𝑋 be the unique strong solution of (2.4). If 𝜏(𝛼) is defined by (2.9), then for any 𝛼>0ℙ[𝜏(𝛼)<+∞]=1. (3.3) Moreover, 𝑋 is differentiable at 𝜏(𝛼) and 𝑋^′(𝜏(𝛼))<0.

See [20] 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

𝑑𝑋(𝑡)=−^𝑡₀𝑘(𝑡−𝑠)𝑋(𝑠)𝑑𝑠𝑑𝑡+𝜎𝑋(𝑡)𝑑𝐵(𝑡),𝑡≥0. (3.4) Under assumption (2.2), it follows that there is a unique strong solution of this equation (see, e.g., [25]).

Theorem 3.2. Suppose that 𝑘 obeys (2.2) and 𝑘(𝑡)≢0. Let 𝑋 be the unique strong solution of (3.4). If 𝜏(𝛼) is defined by (2.9), then for any 𝛼>0 one has ℙ[𝜏(𝛼)<+∞]=1. (3.5) Moreover, 𝑋 is differentiable at 𝜏(𝛼) and 𝑋^′(𝜏(𝛼))<0.

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 2], which applies to linear nonautonomous delay-differential equations. See also [13, Theorem 2.1.3]. 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 𝐵.

It is interesting to compare Theorem 3.2 with known results on the zero crossings of the corresponding deterministic linear Volterra integrodifferential equation

𝑥^′(𝑡)=−^𝑡₀𝑘(𝑡−𝑠)𝑥(𝑠)𝑑𝑠,𝑡≥0,𝑥(0)=1 (3.6) in the case when 𝑘 obeys (2.2) and 𝑘 is nontrivial. It has been shown (see, e.g., [16]) that (3.6) has zero crossings if and only if the characteristic equation of (3.6)

𝜆+^∞₀𝑘(𝑠)𝑒^−𝜆𝑠𝑑𝑠=0,𝜆∈ℂ (3.7) has no real solutions. However, solutions of (3.4) have zero crossings for a.a. sample paths provided that 𝑘 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 𝑓 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 𝑓 does not necessarily have linear-order leading behaviour at zero. We assume not only that 𝑓 is weakly nonlinear close to zero, but also that it obeys the following:

Thereexists𝛾>1and𝐿₂>0suchthat𝑓(𝑥)≤𝐿₂𝑥^𝛾∀𝑥≥0. (3.8) In addition, we suppose that 𝑘 decays more quickly than 𝑡↦𝑒^{−𝜎2𝑡/2} as 𝑡→∞ in the sense that

Thereexists𝜖>0suchthat^∞₀𝑒^{(𝜎2/2+𝜖)𝑡}𝑘(𝑡)𝑑𝑡<+∞. (3.9) 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 𝑘 obeys (2.2) and (3.9). Suppose also that 𝑓 obeys (2.3) and (3.8). Let 𝑋 be the unique strong solution of (2.4). If 𝐴_𝛼 is defined by (2.10), then there exists 𝛼_∗>0 such that ℙ[𝐴_𝛼]>0 for all 𝛼≤𝛼_∗. Moreover lim_𝛼→0+ℙ[𝐴_𝛼]=1. (3.10)

The result and proof are inspired by [19, Theorem 4.4], which applies to the stochastic delay differential equation

𝑑𝑋(𝑡)=−𝑓(𝑋(𝑡−𝜏))𝑑𝑡+𝜎𝑋(𝑡)𝑑𝐵(𝑡), (3.11) where 𝑓 obeys (2.3) and (3.8). Under these conditions, similar conclusions to those of Theorem 3.3 apply to the solutions of (3.11).

Theorems 3.1 and 3.3 show the importance of the linearity of 𝑓 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 𝑘 which is slightly stronger than the negation of condition (3.9). We assume that 𝑘 decays more slowly to zero than 𝑡↦𝑒^{−𝜎2𝑡/2} in the sense that Thereexists𝜖>0suchthat^∞₀𝑒^{(𝜎2/2−𝜖)𝑡}𝑘(𝑡)𝑑𝑡=+∞. (3.12) Under this condition, solutions of (2.4) cross zero on almost all sample paths, irrespective of how weakly the nonlinear restoring function 𝑓 acts on the solution.

Theorem 3.4. Suppose that 𝑘 obeys (2.2) and (3.12). Suppose also that 𝑓 obeys (2.3). Let 𝑋 be the unique continuous solution of (2.4). If 𝜏(𝛼) is defined by (2.9), then for any 𝛼>0 one has ℙ[𝜏(𝛼)<+∞]=1. (3.13) Moreover, 𝑋 is differentiable at 𝜏(𝛼) and 𝑋^′(𝜏(𝛼))<0.

This result is interesting because, in the case when 𝑓 is in 𝐶¹ and 𝑓^′(0)=0, the linearisation of (2.4) that is the linear SDE given by

𝑑𝑌(𝑡)=𝜎𝑌(𝑡)𝑑𝐵(𝑡),𝑡≥0,𝑌(0)=𝛼>0, (3.14) 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].

Define 𝜑={𝜑(𝑡)∶𝑡≥0} by

𝜑(𝑡)=𝑒^{𝜎𝐵(𝑡)−𝜎2𝑡/2},𝑡≥0. (4.1) Then 𝜑 is a strictly positive process (i.e., 𝜑(𝑡)>0 for all 𝑡≥0 a.s.) which obeys the stochastic differential equation

𝑑𝜑(𝑡)=𝜎𝜑(𝑡)𝑑𝐵(𝑡),𝑡≥0,𝜑(0)=1. (4.2) Then 𝜑⁻¹(𝑡)>0 for all 𝑡≥0 a.s. and by Itô's lemma, 𝜑⁻¹(𝑡) obeys the stochastic differential equation

𝑑𝜑⁻¹(𝑡)=𝜎²𝜑⁻¹(𝑡)𝑑𝑡−𝜎𝜑⁻¹(𝑡)𝑑𝐵(𝑡). (4.3) By (stochastic) integration by parts it follows that

𝑑(𝑋(𝑡)𝜑(𝑡))=𝑋(𝑡)𝑑𝜑⁻¹(𝑡)+𝜑(𝑡)⁻¹𝑑𝑋(𝑡)+𝜎𝑋(𝑡)(−𝜎𝜑⁻¹(𝑡))𝑑𝑡=−𝜑(𝑡)^−1𝑡₀𝑘(𝑡−𝑠)𝑓(𝑋(𝑠))𝑑𝑠𝑑𝑡. (4.4)

Therefore, as 𝑋(0)=𝛼, we have

𝑋(𝑡)𝜑(𝑡)=𝛼−^𝑡₀𝜑(𝑠)^−1𝑠₀𝑘(𝑠−𝑢)𝑓(𝑋(𝑢))𝑑𝑢𝑑𝑠,𝑡≥0. (4.5) Since 𝑋 has continuous sample paths, it follows from (2.3) and (2.2) that each realisation of

𝑡↦𝜑(𝑡)^−1𝑡₀𝑘(𝑡−𝑠)𝑓(𝑋(𝑠))𝑑𝑠 (4.6) is continuous. Therefore we have that each realisation of the process 𝑦={𝑦(𝑡)∶𝑡≥0} defined by

𝑦(𝑡)=𝑋(𝑡)𝜑(𝑡),𝑡≥0 (4.7) is in 𝐶¹((0,∞);ℝ) and by (4.5) we have

𝑦^′(𝑡)=−𝜑(𝑡)^−1𝑡₀𝑘(𝑡−𝑠)𝑓(𝜑(𝑠)𝑦(𝑠))𝑑𝑠,𝑡≥0,𝑦(0)=𝛼>0. (4.8) It is convenient here to record another fact concerning 𝜑: on account of the Strong Law of Large Numbers for standard Brownian motion, it follows that

lim_𝑡→∞1𝑡𝜎log𝜑(𝑡)=−²2,a.s. (4.9) and therefore we have that 𝜑(𝑡)→0 as 𝑡→∞ a.s.

5. Proof of Theorem 3.1

5.1. Supporting Lemmas

We start by developing a criterion independent of the solution of (2.4), but which depends on 𝜑 given by (4.1), which ensures that the solution of (2.4) exhibits a zero crossing a.s.

Lemma 5.1. Let 𝜎>0. Suppose that 𝑘 obeys (2.2) and that 𝑓 obeys (2.3) and (3.2). Suppose that 𝜑 is defined by (4.1), and 𝜅 by 𝜅(𝑡,𝑠)=𝜑(𝑡)⁻¹𝑘(𝑡−𝑠),0≤𝑠≤𝑡. (5.1) Suppose that there exists 𝜏>0 such that limsup_𝑡→∞^𝑡−𝜏_𝑡−2𝜏^𝑡_𝑡−𝜏1𝜅(𝑠,𝑢)𝜑(𝑢)𝑑𝑠𝑑𝑢>𝐿₃. (5.2) Let 𝛼>0. If 𝑋 is the unique strong solution of (2.4), and 𝜏(𝛼) is defined by (2.9), then ℙ[𝜏(𝛼)<+∞]=1. (5.3) Moreover, 𝑋 is differentiable at 𝜏(𝛼) and 𝑋^′(𝜏(𝛼))<0.

Proof. By (4.8) and the definition of 𝜅 in (5.1) we have 𝑦^′(𝑡)=−^𝑡₀𝜅(𝑡,𝑠)𝑓(𝜑(𝑠)𝑦(𝑠))𝑑𝑠,𝑡≥0. (5.4) Note that 𝐴_𝛼={𝜔∶𝑦(𝑡,𝜔)>0forall𝑡≥0}. Suppose that ℙ[𝐴_𝛼]>0. Let 𝜔∈𝐴_𝛼. Note that 𝑦^′(𝑡,𝜔)≤0 for all 𝑡≥0. Therefore 𝑦(𝑡,𝜔) tends to a nonnegative limit as 𝑡→∞. Since 𝜑(𝑡,𝜔)→0 as 𝑡→∞, we have that 𝑋(𝑡,𝜔)→0 as 𝑡→∞. We temporarily suppress the dependence on 𝜔. Since 𝑦(𝑡)>0, for any 𝑡≥2𝜏, by (5.4) we have 𝑦(𝑡−𝜏)=^𝑡_𝑡−𝜏^𝑠₀≥𝜅(𝑠,𝑢)𝑓(𝜑(𝑢)𝑦(𝑢))𝑑𝑢𝑑𝑠+𝑦(𝑡)_{𝑡𝑡−𝜏𝑠0}𝜅(𝑠,𝑢)𝜑(𝑢)𝑦(𝑢)𝑓(𝑋(𝑢))=𝑋(𝑢)𝑑𝑢𝑑𝑠_{𝑡0𝑡𝑢∨(𝑡−𝜏)}𝜅(𝑠,𝑢)𝜑(𝑢)𝑑𝑠⋅𝑓(𝑋(𝑢))≥𝑋(𝑢)𝑦(𝑢)𝑑𝑢_{𝑡−𝜏𝑡−2𝜏𝑡𝑢∨(𝑡−𝜏)}𝜅(𝑠,𝑢)𝜑(𝑢)𝑑𝑠⋅𝑓(𝑋(𝑢))𝑋(𝑢)𝑦(𝑢)𝑑𝑢≥inf_{𝑢∈[𝑡−2𝜏,𝑡−𝜏]}𝑓(𝑋(𝑢))𝑋(𝑢)^𝑡−𝜏_𝑡−2𝜏^𝑡_{𝑢∨(𝑡−𝜏)}𝜅(𝑠,𝑢)𝜑(𝑢)𝑑𝑠⋅𝑦(𝑢)𝑑𝑢. (5.5) Now, because 𝑦 is nonincreasing we have 𝑦(𝑡−𝜏)≥inf_{𝑢≥𝑡−2𝜏}𝑓(𝑋(𝑢))𝑋(𝑢)^𝑡−𝜏_𝑡−2𝜏^𝑡_{𝑢∨(𝑡−𝜏)}𝜅(𝑠,𝑢)𝜑(𝑢)𝑑𝑠𝑑𝑢⋅𝑦(𝑡−𝜏)=inf_{𝑢≥𝑡−2𝜏}𝑓(𝑋(𝑢))𝑋(𝑢)^𝑡−𝜏_𝑡−2𝜏^𝑡_𝑡−𝜏𝜅(𝑠,𝑢)𝜑(𝑢)𝑑𝑠𝑑𝑢⋅𝑦(𝑡−𝜏). (5.6) Since 𝑦(𝑡−𝜏)>0 for all 𝑡≥2𝜏 we have that inf_{𝑢≥𝑡−2𝜏}𝑓(𝑋(𝑢))𝑋(𝑢)^𝑡−𝜏_𝑡−2𝜏^𝑡_𝑡−𝜏𝜅(𝑠,𝑢)𝜑(𝑢)𝑑𝑠𝑑𝑢≤1,𝑡≥2𝜏. (5.7) Since 𝑋(𝑡)→0 as 𝑡→∞ and liminf_𝑥→0⁺𝑓(𝑥)/𝑥=𝐿₃ we have lim_𝑡→∞inf_{𝑢≥𝑡−2𝜏}𝑓(𝑋(𝑢))𝑋(𝑢)=𝐿₃. (5.8) Therefore limsup_𝑡→∞^𝑡−𝜏_𝑡−2𝜏^𝑡_𝑡−𝜏1𝜅(𝑠,𝑢)𝜑(𝑢)𝑑𝑠𝑑𝑢≤𝐿₃, (5.9) which contradicts (5.2). Therefore we have that ℙ[𝜏(𝛼)<+∞]=1, as required.
Now 𝐴_𝛼={𝜔∈Ω^∗∶thereexists𝑡^′=𝑡^′(𝜔)>0 such that 𝑋(𝑡^′(𝜔),𝜔)=0} = {𝜔∈Ω^∗∶𝜏(𝛼,𝜔)<+∞}, and this event is almost sure. Fix 𝜔∈𝐴_𝛼. Since 𝑦(𝑡)=0 if and only if 𝑋(𝑡)=0, by (4.7) we have from (2.9) that 𝑦(𝜏(𝛼,𝜔),𝜔)=0 and that 𝑦(𝑡,𝜔)>0 for all 𝑡∈[0,𝜏(𝛼,𝜔)). By (4.8) we have
𝑦^′(𝜏(𝛼))=−𝜑⁻¹(𝜏(𝛼))₀^𝜏(𝛼)𝑘(𝜏(𝛼)−𝑠)𝑓(𝜑(𝑠)𝑦(𝑠))𝑑𝑠. (5.10) Since (2.3) implies that 𝑓(𝜑(𝑠)𝑦(𝑠))>0 for all 𝑠∈[0,𝜏(𝛼)), and 𝑘 obeys (2.2) we have that 𝑦^′(𝜏(𝛼))≤0. Suppose that 𝑦^′(𝜏(𝛼))=0. Since 𝜑(𝜏(𝛼))>0, we must have that 𝑘(𝜏(𝛼)−𝑠)=0 for all 𝑠∈(0,𝜏(𝛼)) or 𝑘(𝑡)=0 for 𝑡∈(0,𝜏(𝛼)). Therefore we have that 𝑦^′(𝑡)=0 for all 𝑡∈(0,𝜏(𝛼)). Hence 𝑦(𝜏(𝛼))=𝑦(0)=𝛼>0, which contradicts the fact that 𝑦(𝜏(𝛼))=0. Therefore we must have 𝑦^′(𝜏(𝛼))<0. This implies that there exists 𝑡^′(𝜔)>𝜏(𝛼,𝜔) such that 𝑦(𝑡^′(𝜔),𝜔)<0 and therefore we have that 𝑋(𝑡^′(𝜔),𝜔)<0. Therefore for each 𝜔 in the a.s. event 𝐴_𝛼, there exists a 𝑡^′(𝜔)>0 such that 𝑋(𝑡^′(𝜔))<0.
We now show that 𝑋 is differentiable at 𝜏(𝛼) and that 𝑋^′(𝑡(𝛼))<0. Let 𝑡≠𝜏(𝛼). Then we have
𝑋(𝑡,𝜔)−𝑋(𝜏(𝛼,𝜔),𝜔)=𝑡−𝜏(𝛼,𝜔)𝑦(𝑡,𝜔)𝜑(𝑡,𝜔)−𝑦(𝜏(𝛼,𝜔),𝜔)𝜑(𝑡,𝜔)𝑡−𝜏(𝛼,𝜔)=𝜑(𝑡,𝜔)𝑦(𝑡,𝜔)−𝑦(𝜏(𝛼,𝜔),𝜔).𝑡−𝜏(𝛼,𝜔) (5.11) Now taking the limit as 𝑡→𝜏(𝛼) on the righthand side we have lim_{𝑡→𝜏(𝛼)}𝜑(𝑡,𝜔)𝑦(𝑡,𝜔)−𝑦(𝜏(𝛼,𝜔),𝜔)𝑡−𝜏(𝛼,𝜔)=𝜑(𝜏(𝛼,𝜔),𝜔)𝑦^′(𝜏(𝛼,𝜔),𝜔)<0. (5.12) Therefore we have lim_{𝑡→𝜏(𝛼)}𝑋(𝑡,𝜔)−𝑋(𝜏(𝛼,𝜔),𝜔)𝑡−𝜏(𝛼,𝜔)=𝜑(𝜏(𝛼,𝜔),𝜔)𝑦^′(𝜏(𝛼,𝜔),𝜔)<0, (5.13) so 𝑋^′(𝜏(𝛼,𝜔),𝜔) is well defined and indeed 𝑋^′(𝜏(𝛼,𝜔),𝜔)<0.

The next result develops a condition which depends only on the increments of 𝐵 and the kernel 𝑘 which implies condition (5.2).

Lemma 5.2. Let 𝜎>0. If 𝜅 is defined by (5.1) and 𝜑 by (4.1), and there exists 𝜏>0 such that limsup_𝑡→∞^𝜏₀^𝜏₀𝑒^{−𝜎(𝐵(𝑡+𝑢)−𝐵(𝑡−𝑤))}𝑘(𝑢+𝑤)𝑑𝑢𝑑𝑤=+∞,a.s. (5.14) then (5.2) holds.

Proof. Define for 𝑡≥2𝜏𝐴₁(𝑡)=^𝑡−𝜏_𝑡−2𝜏^𝑡_𝑡−𝜏𝜅(𝑠,𝑢)𝜑(𝑢)𝑑𝑠𝑑𝑢=^𝑡−𝜏_𝑡−2𝜏^𝑡_𝑡−𝜏𝜑(𝑠)⁻¹𝑘(𝑠−𝑢)𝜑(𝑢)𝑑𝑠𝑑𝑢. (5.15) Hence as 𝜑 is given by (4.1) we have 𝐴₁(𝑡)=^𝑡−𝜏_𝑡−2𝜏^𝑡_𝑡−𝜏𝑒^{−𝜎(𝐵(𝑠)−𝐵(𝑢))}𝑒^{𝜎2/2(𝑠−𝑢)}=𝑘(𝑠−𝑢)𝑑𝑠𝑑𝑢_{𝑡−𝜏𝑡−2𝜏𝑡−𝑢𝑡−𝑢−𝜏}𝑒^{−𝜎(𝐵(𝑣+𝑢)−𝐵(𝑢))}𝑒^𝜎2𝑣/2=𝑘(𝑣)𝑑𝑣𝑑𝑢_{𝜏0𝑤𝑤+𝜏}𝑒^{−𝜎(𝐵(𝑣+𝑡−𝜏−𝑤)−𝐵(𝑡−𝜏−𝑤))}𝑒^𝜎2𝑣/2≥𝑘(𝑣)𝑑𝑣𝑑𝑤_{𝜏0𝑤𝑤+𝜏}𝑒^{−𝜎(𝐵(𝑣+𝑡−𝜏−𝑤)−𝐵(𝑡−𝜏−𝑤))}𝑘(𝑣)𝑑𝑣𝑑𝑤. (5.16) Therefore if (5.14) holds, then limsup_𝑡→∞𝐴₁(𝑡)≥limsup_𝑡→∞^𝜏_0𝑤^𝑤+𝜏𝑒^{−𝜎(𝐵(𝑣+𝑡−𝜏−𝑤)−𝐵(𝑡−𝜏−𝑤))}𝑘(𝑣)𝑑𝑣𝑑𝑤=limsup_𝑡→∞^𝜏_0𝑤^𝑤+𝜏𝑒^{−𝜎(𝐵(𝑣+𝑡−𝑤)−𝐵(𝑡−𝑤))}𝑘(𝑣)𝑑𝑣𝑑𝑤=limsup_𝑡→∞^𝜏₀^𝜏₀𝑒^{−𝜎(𝐵(𝑢+𝑡)−𝐵(𝑡−𝑤))}𝑘(𝑢+𝑤)𝑑𝑢𝑑𝑤=∞, (5.17) which implies (5.2).

Lemma 5.3. Suppose that 𝜎>0. Suppose that 𝑘 obeys (2.2) and 𝑘(𝑡)≢0. Then there exists 𝜏>0 such that (5.14) holds.

Proof. If 𝑘(𝑡)≢0 and 𝑘(𝑡)≥0 it follows that there is a 𝑡₀≥0 such that 𝑘(𝑡₀)=∶2𝑘₀>0. Since 𝑘 is continuous on [0,∞) there exists 𝛿>0 such that |𝑘(𝑡)−𝑘(𝑡₀)|≤𝑘₀ for all 𝑡∈[𝑡₀,𝑡₀+𝛿]. Therefore we have 𝑘(𝑡)≥𝑘₀ for all 𝑡∈[𝑡₀+𝛿/2,𝑡₀+𝛿]=∶[𝜃₁,𝜃₂]. Hence Thereexists𝑘₀>0and0<𝜃₁<𝜃₂suchthat𝑘(𝑡)≥𝑘₀∀𝑡∈[𝜃₁,𝜃₂]. (5.18) Let 𝜏=𝜃₂. Then 𝜃₁≤𝜏, 𝜃₂≤𝜏. Define 𝐴₂(𝑡)=^𝜏₀^𝜏₀𝑒^{−𝜎(𝐵(𝑢+𝑡)−𝐵(𝑡−𝑤))}𝑘(𝑢+𝑤)𝑑𝑢𝑑𝑤,𝑡≥𝜏. (5.19) Equation (5.14) is equivalent to show that limsup_𝑡→∞𝐴₂(𝑡)=∞ a.s. Clearly by (5.18) we have 𝐴₂(𝑡)≥𝑘₀^𝜃2₀^𝜃2−𝑤_{(𝜃1−𝑤)∨0}𝑒^{−𝜎(𝐵(𝑢+𝑡)−𝐵(𝑡−𝑤))}𝑑𝑢𝑑𝑤. (5.20) Now [𝜃₁/2,𝜃₂/2]²⊂{(𝑤,𝑢)∶0≤𝑤≤𝜃₂,𝑢≥0,𝜃₁≤𝑤+𝑢≤𝜃₂}, so we have 𝐴₂(𝑡)≥𝑘₀^𝜃2_𝜃^/2_1/2^𝜃2_𝜃^/2_1/2𝑒^{−𝜎(𝐵(𝑢+𝑡)−𝐵(𝑡−𝑤))}𝑑𝑢𝑑𝑤=∶𝐴(𝑡). (5.21) Hence (5.14) follows if we can show that limsup_𝑡→∞𝐴(𝑡)=∞ a.s. Define 𝐴_𝑛=𝐴(𝑛𝜃₂)/𝑘₀ for 𝑛≥1 so that it suffices to prove that limsup_𝑛→∞𝐴_𝑛=∞ a.s. Note that 𝐴_𝑛=^𝜃2_𝜃^/2_1/2^𝜃2_𝜃^/2_1/2𝑒^{−𝜎(𝐵(𝑢+𝑛𝜃2)−𝐵(𝑛𝜃2−𝑤))}𝑑𝑢𝑑𝑤. (5.22) Now, we note that each 𝐴_𝑛 is a functional of increments of the standard Brownian motion 𝐵 over the interval [𝑛𝜃₂−𝜃₂/2,𝑛𝜃₂+𝜃₂/2]. Therefore as (𝑛+1)𝜃₂−𝜃₂/2=𝑛𝜃₂+𝜃₂/2, it follows that the intervals on which the increments of 𝐵 are considered for 𝐴_𝑛 and 𝐴_𝑛+1 are nonoverlapping. Since the increments of 𝐵 are independent, it follows that (𝐴_𝑛)_𝑛≥1 is a sequence of independent random variables. Hence by the Borel-Cantelli lemma, we are done if we can show that ^∞_𝑛=1ℙ[𝐴_𝑛>𝛽]=+∞,∀𝛽>0. (5.23) Note that 𝐵=−𝐵 is a standard Brownian motion. Then 𝐴_𝑛=^𝜃2_𝜃^/2_1/2^𝜃2_𝜃^/2_1/2𝑒^𝜎(𝐵_{(𝑢+𝑛𝜃2)−}𝐵_{(𝑛𝜃2−𝑤))}𝑑𝑢𝑑𝑤. (5.24) Define for 𝜂∈ℝ the event 𝐶_𝜂(𝑛)={𝜔∶min_{𝑢∈[𝜃1/2,𝜃2/2]}𝐵(𝑛𝜃₂+𝑢,𝜔)−max_{𝑤∈[𝜃1/2,𝜃2/2]}𝐵(𝑛𝜃₂−𝑤,𝜔)≥𝜂}. (5.25) If 𝜔∈𝐶_𝜂(𝑛), then 𝐵(𝑛𝜃₂+𝑢,𝜔)−𝐵(𝑛𝜃₂−𝑤,𝜔)≥𝜂 for all (𝑢,𝑤)∈[𝜃₁/2,𝜃₂/2]². Therefore 𝜔∈𝐶_𝜂(𝑛) implies 𝐴_𝑛(𝜔)=^𝜃2_𝜃^/2_1/2^𝜃2_𝜃^/2_1/2𝑒^𝜎(𝐵^(𝑛𝜃₂^{+𝑢,𝜔)−}𝐵^(𝑛𝜃₂^{−𝑤,𝜔))}𝜃𝑑𝑢𝑑𝑤≥(²2−𝜃₁2)²𝑒^𝜎𝜂. (5.26) Thus ℙ[𝐴_𝑛𝜃≥(²2−𝜃₁2)²𝑒^𝜎𝜂]≥ℙ[𝐶_𝜂(𝑛)],𝑛≥1,𝜂∈ℝ. (5.27) Now min_{𝑢∈[𝜃1/2,𝜃2/2]}𝐵(𝑛𝜃₂+𝑢)−max_{𝑤∈[𝜃1/2,𝜃2/2]}𝐵(𝑛𝜃₂−𝑤)=min_{𝑢∈[𝜃1/2,𝜃2/2]}{𝐵(𝑛𝜃₂+𝑢)−𝐵(𝑛𝜃₂+𝜃₁2)}+{𝐵(𝑛𝜃₂+𝜃₁2)−𝐵(𝑛𝜃₂−𝜃₁2)}−max_{𝑤∈[𝜃1/2,𝜃2/2]}{𝐵(𝑛𝜃₂−𝑤)−𝐵(𝑛𝜃₂−𝜃₁2)}=min_{𝑢∈[𝑛𝜃2+𝜃1/2,𝑛𝜃2+𝜃2/2]}{𝐵(𝑢)−𝐵(𝑛𝜃₂+𝜃₁2)}+{𝐵(𝑛𝜃₂+𝜃₁2)−𝐵(𝑛𝜃₂−𝜃₁2)}+min_{𝑤∈[𝑛𝜃2−𝜃2/2,𝑛𝜃2−𝜃1/2]}{𝐵(𝑛𝜃₂−𝜃₁2)−𝐵(𝑤)}=∶𝑊₁(𝑛)+𝑊₂(𝑛)+𝑊₃(𝑛). (5.28) Since 𝜃₁>0, each of 𝑊₁(𝑛), 𝑊₂(𝑛), and 𝑊₃(𝑛) is well defined and independent random variables. Hence ℙ[𝐶_𝜂(𝑛)]=ℙ[𝑊₁(𝑛)+𝑊₂(𝑛)+𝑊₃(𝑛)≥𝜂]≥ℙ[𝑊₁(𝑛)≥0,𝑊₂(𝑛)≥𝜂,𝑊₃(𝑛)≥0]=ℙ[𝑊₁(𝑛)≥0]⋅ℙ[𝑊₂(𝑛)≥𝜂]⋅ℙ[𝑊₃(𝑛)≥0]. (5.29) Now we note that 𝐵₁(𝑢)∶=𝐵(𝑢+𝑛𝜃₂+𝜃₁/2)−𝐵(𝑛𝜃₂+𝜃₁/2) for 𝑢∈[0,𝜃₂/2−𝜃₁/2] is a standard Brownian motion, so we have ℙ[𝑊₁(𝑛)≥0]=ℙ[inf_{𝑢∈[𝑛𝜃2+𝜃1/2,𝑛𝜃2+𝜃2/2]}𝐵(𝑢)−𝐵(𝑛𝜃₂+𝜃₁2)≥0]=ℙ[inf_{𝑢∈[0,𝜃2/2−𝜃1/2]}𝐵₁(𝑢)≥0]=1. (5.30) Next, if we define 𝐵₂(𝑢)∶=𝐵(𝑛𝜃₂−𝜃₁/2)−𝐵(𝑢) for 𝑢∈[0,𝜃₂/2−𝜃₁/2], then 𝐵₂ is another standard Brownian motion. Therefore we have ℙ[𝑊₃(𝑛)≥0]=ℙ[inf_{𝑤∈[𝑛𝜃2−𝜃2/2,𝑛𝜃2−𝜃1/2]}{𝐵(𝑛𝜃₂−𝜃₁2)−𝐵(𝑤)}≥0]=ℙ[inf_{𝑢∈[0,𝜃2/2−𝜃1/2]}{𝐵(𝑛𝜃₂−𝜃₁2)−𝐵(𝑛𝜃₂−𝜃₁2−𝑢)}≥0]=ℙ[inf_{𝑢∈[0,𝜃2/2−𝜃1/2]}𝐵₂(𝑢)≥0]=1. (5.31) Therefore if 𝑍 is a standard normal random variable and Φ is the distribution function of 𝑍, we have ℙ[𝐶_𝜂(𝑛)]=ℙ[𝑊₁(𝑛)+𝑊₂(𝑛)+𝑊₃(𝑛)≥𝜂]≥ℙ[𝑊₂(𝑛)≥𝜂]=ℙ[𝐵(𝑛𝜃₂+𝜃₁2)−𝐵(𝑛𝜃₂−𝜃₁2)≥𝜂]=ℙ[𝜃₁𝜂𝑍≥𝜂]=1−Φ(𝜃₁). (5.32) Hence by (5.27) we have ℙ[𝐴_𝑛𝜃≥(²2−𝜃₁2)²𝑒^𝜎𝜂𝜂]≥1−Φ(𝜃₁),∀𝜂∈ℝ,𝑛≥1. (5.33) Let 𝛽>0 and define 𝜂∈ℝ by 1𝜂=𝜎log(4𝛽(𝜃₂−𝜃₁)²). (5.34) Then 𝛽=(𝜃₂−𝜃₁)²𝑒^𝜎𝜂/4. Hence ℙ[𝐴_𝑛1≥𝛽]≥1−Φ(𝜎𝜃₁log(4𝛽(𝜃₂−𝜃₁)²))=∶𝑐(𝛽)>0,∀𝛽>0,𝑛≥1. (5.35) This implies that (5.23) holds, and therefore that limsup_𝑛→∞𝐴_𝑛=∞ 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 ℙ[𝜏(𝛼)<+∞]=1 for any 𝛼>0 and that 𝑋^′(𝜏(𝛼)) 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.

Lemma 6.1. Let 𝜎>0, 𝛾>1, and suppose that 𝜑 is given by (4.1). Suppose that 𝑘 obeys (2.2) and (3.9). Define 𝐼=^∞₀^𝑡₀𝜑(𝑡)⁻¹𝑘(𝑡−𝑠)𝜑^𝛾(𝑠)𝑑𝑠𝑑𝑡. (6.1) Then 𝐼<+∞ a.s.

Proof. By the definition of 𝜑 we have 𝐼=^∞₀^𝑡₀𝑘(𝑡−𝑠)𝑒^{𝜎𝛾𝐵(𝑠)−𝜎𝐵(𝑡)}𝑒^{𝜎2𝑡/2−𝜎2𝛾𝑠/2}𝑑𝑠𝑑𝑡. (6.2) By the Strong Law of Large Numbers for standard Brownian motion (see, e.g., Karatzas and Shreve [27]), there exists an almost sure event Ω₁ such that Ω₁={𝜔∶lim_𝑡→∞𝐵(𝑡,𝜔)𝑡=0}. (6.3) Therefore for each 𝜔∈Ω₁ and for every 𝜀>0 there exists a finite 𝑇(𝜔,𝜀)>0 such that |𝐵(𝑡,𝜔)|≤𝜀𝑡,𝑡≥𝑇(𝜔,𝜀). (6.4) Define 𝐼₁=₀^{𝑇(𝜔,𝜀)𝑡}₀𝑘(𝑡−𝑠)𝑒^{𝜎𝛾𝐵(𝑠)−𝜎𝐵(𝑡)}𝑒^{𝜎2𝑡/2−𝜎2𝛾𝑠/2}𝐼𝑑𝑠𝑑𝑡,₂=^∞_{𝑇(𝜔,𝜀)}^𝑡_{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{𝜎𝛾𝐵(𝑠)−𝜎𝐵(𝑡)}𝑒^{𝜎2𝑡/2−𝜎2𝛾𝑠/2}𝐼𝑑𝑠𝑑𝑡,₃=^∞_{𝑇(𝜔,𝜀)0}^{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{𝜎𝛾𝐵(𝑠)−𝜎𝐵(𝑡)}𝑒^{𝜎2𝑡/2−𝜎2𝛾𝑠/2}𝑑𝑠𝑑𝑡. (6.5) Then 𝐼=𝐼₁+𝐼₂+𝐼₃. The continuity of the integrand and finiteness of 𝑇(𝜔,𝜀)>0 ensures that 𝐼₁<+∞. Consider now 𝐼₃. Suppose that 𝜀>0 is so small that |𝜎|𝜀<𝜖 where 𝜖>0 is defined by (3.9) (note the distinction between the constant 𝜖 defined by (3.9) and the small parameter 𝜀>0). Define 𝛽₂=𝜎²/2+|𝜎|𝜀. By (3.9) we therefore have ^∞₀𝑘(𝑠)𝑒^{(𝜎2/2+|𝜎|𝜀)𝑠}𝑑𝑠=^∞₀𝑘(𝑠)𝑒^𝛽2𝑠𝑑𝑠<+∞. (6.6) Then by (6.4) we have 𝐼₃≤max_{0≤𝑠≤𝑇(𝜔,𝜀)}𝑒^{𝜎𝛾𝐵(𝑠)−𝜎2𝛾𝑠/2}⋅^∞_{𝑇(𝜔,𝜀)0}^{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{−𝜎𝐵(𝑡)}𝑒^𝜎2𝑡/2𝑑𝑠𝑑𝑡≤max_{0≤𝑠≤𝑇(𝜔,𝜀)}𝜑^𝛾(𝑠,𝜔)⋅^∞_{𝑇(𝜔,𝜀)0}^{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{𝛽2(𝑡−𝑠)}𝑒^𝛽2𝑠𝑑𝑠𝑑𝑡≤𝑒^{𝛽2𝑇(𝜔,𝜀)}max_{0≤𝑠≤𝑇(𝜔,𝜀)}𝜑^𝛾(𝑠,𝜔)⋅^∞_{𝑇(𝜔,𝜀)0}^{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{𝛽2(𝑡−𝑠)}𝑑𝑠𝑑𝑡. (6.7) Now by the nonnegativity of the integrand and Fubini's theorem we have ^∞_{𝑇(𝜔,𝜀)0}^{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{𝛽2(𝑡−𝑠)}=𝑑𝑠𝑑𝑡_{∞𝑇(𝜔,𝜀)𝑡𝑡−𝑇(𝜔,𝜀)}𝑘(𝑢)𝑒^𝛽2𝑢=𝑑𝑢𝑑𝑡_{∞0𝑢+𝑇(𝜀,𝜔)𝑢∨𝑇(𝜀,𝜔)}𝑑𝑡⋅𝑘(𝑢)𝑒^𝛽2𝑢=𝑑𝑢_{0𝑇(𝜀,𝜔)}𝑢𝑘(𝑢)𝑒^𝛽2𝑢𝑑𝑢+𝑇(𝜀,𝜔)^∞_{𝑇(𝜀,𝜔)}𝑘(𝑢)𝑒^𝛽2𝑢𝑑𝑢. (6.8) By (6.6) we have ^∞_{𝑇(𝜔,𝜀)0}^{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{𝛽2(𝑡−𝑠)}𝑑𝑠𝑑𝑡<+∞, (6.9) which implies that 𝐼₃<+∞.
Finally we show that 𝐼₂<+∞, a.s. Suppose that 𝜀>0 is so small that 𝜀<|𝜎|/2 and 𝜀<(𝛾−1)|𝜎|/[2(𝛾+1)]. Define 𝛽₁=𝛾(𝜎²/2−|𝜎|𝜀). Then 𝛽₁>0. Hence by (6.4) we have
𝐼₂=^∞_{𝑇(𝜔,𝜀)}^𝑡_{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{𝜎𝛾𝐵(𝑠)−𝜎𝐵(𝑡)}𝑒^{𝜎2𝑡/2−𝜎2𝛾𝑠/2}≤𝑑𝑠𝑑𝑡_{∞𝑇(𝜔,𝜀)𝑡𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{|𝜎|𝛾𝜀𝑠+|𝜎|𝜀𝑡}𝑒^{𝜎2𝑡/2−𝜎2𝛾𝑠/2}=𝑑𝑠𝑑𝑡_{∞𝑇(𝜔,𝜀)𝑡𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^−𝛽1𝑠𝑒^𝛽2𝑡𝑑𝑠𝑑𝑡. (6.10) Now 𝛽₁−𝛽₂=(𝛾−1)𝜎²/2−(𝛾+1)|𝜎|𝜀>0 since 𝜀<(𝛾−1)|𝜎|/[2(𝛾+1)]. Hence ^∞_{𝑇(𝜔,𝜀)}^𝑡_{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^−𝛽1𝑠𝑒^𝛽2𝑡=𝑑𝑠𝑑𝑡_{∞𝑇(𝜔,𝜀)𝑡𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^{𝛽1(𝑡−𝑠)}𝑑𝑠⋅𝑒^{(𝛽2−𝛽1)𝑡}=𝑑𝑡_{∞𝑇(𝜔,𝜀)0𝑡−𝑇(𝜔,𝜀)}𝑘(𝑢)𝑒^𝛽1𝑢𝑑𝑢⋅𝑒^{(𝛽2−𝛽1)𝑡}=𝑑𝑡_{∞0∞𝑇(𝜔,𝜀)+𝑢}𝑒^{−(𝛽1−𝛽2)𝑡}𝑑𝑡⋅𝑘(𝑢)𝑒^𝛽1𝑢=𝑒𝑑𝑢_{−(𝛽1−𝛽2)𝑇(𝜔,𝜀)}𝛽₁−𝛽₂^∞₀𝑒^𝛽2𝑢𝑘(𝑢)𝑑𝑢. (6.11) Therefore by (6.6) we have that ^∞_{𝑇(𝜔,𝜀)}^𝑡_{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^−𝛽1𝑠𝑒^𝛽2𝑡𝑑𝑠𝑑𝑡<+∞, (6.12) and so 𝐼₂≤^∞_{𝑇(𝜔,𝜀)}^𝑡_{𝑇(𝜔,𝜀)}𝑘(𝑡−𝑠)𝑒^−𝛽1𝑠𝑒^𝛽2𝑡𝑑𝑠𝑑𝑡<+∞. (6.13) Hence 𝐼=𝐼₁+𝐼₂+𝐼₃<+∞, as required.

6.1. Proof of Theorem 3.3

Let 𝜏(𝛼) be defined by (2.9). Let 𝜔∈𝐴_𝛼. Let 𝑦 be given by (4.7). Then for all 𝑡∈[0,𝜏(𝛼)) we have 0<𝑦(𝑡,𝜔)≤𝛼. Note also that 𝑦(𝜏(𝛼))=0. Therefore we have 0<𝑋(𝑡,𝜔)≤𝛼𝜑(𝑡) for all 𝑡∈[0,𝜏(𝛼)). Therefore by (3.8) it follows that

𝑓(𝑋(𝑡,𝜔))≤𝐿₂𝑋^𝛾(𝑡,𝜔)≤𝐿₂𝛼^𝛾𝜑^𝛾(𝑡),𝑡∈[0,𝜏(𝛼)]. (6.14) By (4.8) we have

0=𝑦(𝜏(𝛼))=𝛼−₀^{𝜏(𝛼)𝑡}₀𝜑(𝑡)⁻¹𝑘(𝑡−𝑠)𝑓(𝑋(𝑠))𝑑𝑠𝑑𝑡. (6.15) Hence by (6.14) and the nonnegativity of 𝑘 and 𝜑 we have

𝛼=₀^{𝜏(𝛼)𝑡}₀𝜑(𝑡)⁻¹𝑘(𝑡−𝑠)𝑓(𝑋(𝑠))𝑑𝑠𝑑𝑡≤𝐿₂𝛼^𝛾₀^{𝜏(𝛼)𝑡}₀𝜑(𝑡)⁻¹𝑘(𝑡−𝑠)𝜑^𝛾(𝑠)𝑑𝑠𝑑𝑡≤𝐿₂𝛼^𝛾∞₀^𝑡₀𝜑(𝑡)⁻¹𝑘(𝑡−𝑠)𝜑^𝛾(𝑠)𝑑𝑠𝑑𝑡. (6.16) Therefore as 𝐼 is defined by (6.1) we have

1𝐼(𝜔)≥𝐿₂𝛼^1−𝛾,foreach𝜔∈A_𝛼. (6.17) Therefore

1−ℙ[𝐴_𝛼]=ℙ[𝐴_𝛼𝛼]≤ℙ[𝐼≥^1−𝛾𝐿₂]. (6.18) By Lemma 6.1 it follows that ℙ[𝐼<+∞]=1. Therefore as 𝛾>1, by taking limits on both sides of (6.18), we obtain

1−liminf_𝛼→0+ℙ[𝐴_𝛼]=limsup_𝛼→0+{1−ℙ[𝐴_𝛼]}≤limsup_𝛼→0+𝛼ℙ[𝐼≥^1−𝛾𝐿₂]=0. (6.19) Therefore we have liminf_𝛼→0⁺ℙ[𝐴_𝛼]≥1. On the other hand, because we evidently have limsup_𝛼→0+ℙ[𝐴_𝛼]≤1, it follows that ℙ[𝐴_𝛼]→1 as 𝛼→0⁺.

On the other hand, by (6.18) we have ℙ[𝐴_𝛼]≥ℙ[𝐼<𝛼^1−𝛾/𝐿₂]. Therefore as 𝐼∈[0,∞) a.s. it follows that there is an 𝛼_∗>0 such that ℙ[𝐼<𝛼_∗^1−𝛾/𝐿₂]>0. Now suppose 𝛼≤𝛼_∗. Then as 𝛾>1 we have

𝛼ℙ[𝐼<^1−𝛾𝐿₂𝛼]≥ℙ[𝐼<^∗1−𝛾𝐿₂]>0, (6.20) which implies ℙ[𝐴_𝛼]>0 for all 𝛼≤𝛼_∗, proving the result.

7. Proof of Theorem 3.4

Let 𝐴_𝛼 be defined by (2.10). We suppose that ℙ[𝐴_𝛼]>0. Define also

ℎ(𝑡)=^𝑡₀𝑘(𝑡−𝑠)𝑓(𝑋(𝑠))𝑑𝑠,𝑡≥0. (7.1) Then by (4.5) we have

𝑋(𝑡)=𝜑(𝑡)(𝛼−^𝑡₀𝜑(𝑠)⁻¹ℎ(𝑠)𝑑𝑠),𝑡≥0. (7.2) Fix 𝜔∈𝐴_𝛼. Since ℎ(𝑡,𝜔)≥0 for all 𝑡≥0, we have that 𝑋(𝑡,𝜔)≤𝛼𝜑(𝑡,𝜔) for all 𝑡≥0. Also, as 𝜑(𝑡)>0 for all 𝑡≥0 and 𝑋(𝑡,𝜔)>0 we have

^𝑡₀𝜑(𝑠,𝜔)⁻¹ℎ(𝑠,𝜔)𝑑𝑠≤𝛼,∀𝑡≥0. (7.3) Hence

^∞₀𝜑(𝑠,𝜔)⁻¹ℎ(𝑠,𝜔)𝑑𝑠≤𝛼,foreach𝜔∈A_𝛼. (7.4) Let Ω₁ be the event defined in (6.3). Now define 𝐶_𝛼=𝐴_𝛼∩Ω₁. Then ℙ[𝐶_𝛼]=ℙ[𝐴_𝛼]>0. Let 𝜀>0 be so small that |𝜎|²/2−|𝜎|𝜀≥𝜎²/2−𝜖>0, where 𝜖>0 is defined by (3.12) (as in the proof of Lemma 6.1 above, note the distinction between the constant 𝜖 defined by (3.12) and the small parameter 𝜀>0). Then for every 𝜀>0 so chosen and 𝜔∈𝐶_𝛼 there exists 𝑇(𝜔,𝜀)>0 such that 𝐵 obeys (6.4). Then as 𝜔∈𝐶_𝛼 we have 𝑋(𝑡,𝜔)>0 for all 𝑡∈[0,𝑇(𝜔,𝜀)] and so by (2.3) we have that 𝑓(𝑋(𝑡,𝜔))>0 for all 𝑡∈[0,𝑇(𝜔,𝜀)]. Define

𝐹_𝜀(𝜔)=min_{𝑢∈[0,𝑇(𝜔,𝜀)]}𝑓(𝑋(𝑢,𝜔))>0. (7.5) Let 𝑠≥𝑇(𝜀,𝜔). Then

ℎ(𝑠,𝜔)≥₀^{𝑇(𝜀,𝜔)}𝑘(𝑠−𝑢)𝑓(𝑋(𝑢,𝜔))𝑑𝑢≥𝐹_𝜀(𝜔)₀^{𝑇(𝜀,𝜔)}𝑘(𝑠−𝑢)𝑑𝑢. (7.6) Therefore for all 𝜔∈𝐶_𝛼, by using (6.4) and (4.1), we obtain

^∞₀ℎ(𝑠,𝜔)𝜑(𝑠,𝜔)⁻¹≥𝑑𝑠_{∞𝑇(𝜀,𝜔)}ℎ(𝑠,𝜔)𝜑(𝑠,𝜔)⁻¹=𝑑𝑠_{∞𝑇(𝜀,𝜔)}ℎ(𝑠,𝜔)𝑒^{𝜎2/2𝑠−𝜎𝐵(𝑠,𝜔)}𝑑𝑠≥𝐹_𝜀(𝜔)^∞_{𝑇(𝜀,𝜔)0}^{𝑇(𝜀,𝜔)}𝑘(𝑠−𝑢)𝑑𝑢⋅𝑒^{𝜎2/2𝑠−𝜎𝐵(𝑠,𝜔)}𝑑𝑠≥𝐹_𝜀(𝜔)^∞_{𝑇(𝜀,𝜔)0}^{𝑇(𝜀,𝜔)}𝑘(𝑠−𝑢)𝑑𝑢⋅𝑒^{(𝜎2/2−|𝜎|𝜀)𝑠}𝑑𝑠. (7.7) Therefore

^∞₀ℎ(𝑠,𝜔)𝜑(𝑠,𝜔)⁻¹𝑑𝑠≥𝐹_𝜀(𝜔)^∞_{𝑇(𝜀,𝜔)0}^{𝑇(𝜀,𝜔)}𝑘(𝑠−𝑢)𝑑𝑢⋅𝑒^{(𝜎2/2−𝜖)𝑠}𝑑𝑠. (7.8) Define 𝛽₃=𝜎²/2−𝜖. Then 𝛽₃>0 and by (3.12) we have

^∞₀𝑘(𝑠)𝑒^𝛽3𝑠𝑑𝑠=+∞. (7.9) Then by the nonnegativity of the integrand and Fubini's theorem we have

^∞_{𝑇(𝜀,𝜔)0}^{𝑇(𝜀,𝜔)}𝑘(𝑠−𝑢)𝑑𝑢⋅𝑒^𝛽3𝑠=𝑑𝑠_{∞𝑇(𝜀,𝜔)0𝑇(𝜀,𝜔)}𝑘(𝑠−𝑢)𝑒^{𝛽3(𝑠−𝑢)}𝑒^𝛽3𝑢≥𝑑𝑢𝑑𝑠_{∞𝑇(𝜀,𝜔)0𝑇(𝜀,𝜔)}𝑘(𝑠−𝑢)𝑒^{𝛽3(𝑠−𝑢)}𝑑𝑢𝑑𝑠=^∞_{𝑇(𝜀,𝜔)}^𝑠_{𝑠−𝑇(𝜀,𝜔)}𝑘(𝑣)𝑒^𝛽3𝑣=𝑑𝑣𝑑𝑠_{∞0𝑣+𝑇(𝜀,𝜔)𝑣∨𝑇(𝜀,𝜔)}𝑑𝑠⋅𝑘(𝑣)𝑒^𝛽3𝑣=𝑑𝑣_{0𝑇(𝜀,𝜔)}𝑣𝑘(𝑣)𝑒^𝛽3𝑣𝑑𝑣+𝑇(𝜀,𝜔)^∞_{𝑇(𝜀,𝜔)}𝑘(𝑣)𝑒^𝛽3𝑣𝑑𝑣. (7.10) Therefore by (7.9), for each 𝜔∈𝐶_𝛼 we have

^∞_{𝑇(𝜀,𝜔)0}^{𝑇(𝜀,𝜔)}𝑘(𝑠−𝑢)𝑑𝑢⋅𝑒^𝛽3𝑠𝑑𝑠=+∞. (7.11) Therefore by (7.11) and (7.8) we have

^∞₀ℎ(𝑠,𝜔)𝜑(𝑠,𝜔)⁻¹𝑑𝑠=+∞,foreach𝜔∈𝐶_𝛼. (7.12) Recall that 𝐶_𝛼 is an a.s. subset of 𝐴_𝛼 so that (7.4) implies

^∞₀ℎ(𝑠,𝜔)𝜑(𝑠,𝜔)⁻¹𝑑𝑠≤𝛼,foreach𝜔∈𝐶_𝛼, (7.13) which contradicts (7.12). Therefore we have that ℙ[𝐴_𝛼]=0, or ℙ[𝐴_𝛼]=1. Therefore we have that ℙ[𝜏(𝛼)<+∞]=1. The proof that for each 𝜔 in the a.s. event {𝜏(𝛼)<+∞}, there exists a 𝑡^′(𝜔)>0 such that 𝑋(𝑡^′(𝜔))<0, that 𝑋 is differentiable at 𝜏(𝛼), and that 𝑋^′(𝑡(𝛼))<0, is identical to that in the proof of Lemma 5.1. The proof of the theorem is therefore complete.

Acknowledgments

The author wishes to thank the editors of this special issue on Recent Advances in Oscillation Theory for the opportunity to present his research in this volume. As well as acknowledging the reviewers for their careful and thorough reviews of the manuscript, he is also grateful to Huizhong Wu and David Reynolds for discussions concerning this paper. The author gratefully acknowledges Science Foundation Ireland for the support of this research under the Mathematics Initiative 2007 Grant 07/MI/008 “Edgeworth Centre for Financial Mathematics."