doi:10.1155/2010/508217
Research Article
On the Positivity and Zero Crossings of Solutions of Stochastic Volterra Integrodifferential Equations
John A. D. Appleby
Edgeworth Centre for Financial Mathematics, School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland
Correspondence should be addressed to John A. D. Appleby,john.appleby@dcu.ie Received 1 November 2009; Accepted 14 January 2010
Academic Editor: Elena Braverman
Copyrightq2010 John A. D. Appleby. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the zero crossings and positive solutions of scalar nonlinear stochastic Volterra integrodifferential equations of It ˆo 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 physics1–10.
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 research4,11–13; further, we would like to draw attention to the recent survey paper 14. However, the effect that random perturbations of It ˆo 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
dXt − t
0
kt−sfXsds dt σXtdBt, t≥0, X0 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 functionfobeys xfx>0 forx /0 andf0 0. Regularity assumptions onfandkare required to guarantee the existence of solutions. The sign conditions onf andk are motivated by the underlying deterministic Volterra integrodifferential equation
xt − t
0
kt−sfxsds, t≥0, x0 α >0. 1.2
These conditions onfandkensure that zero is the unique steady-state solution of1.2and 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 of1.2include work by Gopalsamy and Lalli15 and Gy ¨ori and Ladas16, 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 iff is of linear or lower order in the neighbourhood of the zero equilibrium, then the solution of1.1has a zero crossing almost surely, provided that the kernelkis not identically zero.
Second, we show that if fx is of order xγ for γ > 1 as x → 0 i.e., in the neighbourhood of the equilibrium, andf also obeys a global superlinear upper bound on 0,∞, then any solution of1.1which 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 kdecays more slowly than the critical exponential rate, then all solutions of1.1will have zero crossings, regardless of how weakly the restoring functionf acts on the solution. Therefore, we conclude that solutions of1.1will remain positive only ifik decays more quickly than some critical noise-intensity dependent rate andiifis 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 of1.1is 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 motionB, 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 singleand finitedelay in18–21.
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 include5,22–24.
The paper is organised as follows. Mathematical preliminaries, including remarks on the existence and uniqueness of solutions of1.1, are presented inSection 2. The main results of the paper are stated and discussed inSection 3. InSection 4, we show that solutions of 1.1can 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 solutionywhich has continuously differentiable paths and is of the form
yt − t
0
Kt, sF s, ys
ds, t≥0, y0 α >0, 1.3
whereKandFinherit positivity properties fromkandf. Therefore, the zero crossings of the solutionX of1.1correspond to zero crossings of the solutionyof1.3. The proofs of the main results are given in the final three sections of the paper.
2. Preliminaries
2.1. NotationIn advance of stating and discussing our main results, we introduce some standard notation.
We denote the maximum of the real numbersxandybyx∨yand the minimum ofxandy byx∧y. LetCI;Jdenote the space of continuous functionsf :I → J whereIandJare intervals contained inR. Similarly, we letC1I;Jdenote the space of differentiable functions
f : I → J, wheref ∈ CI;J. We denote byL10,∞ the space of Lebesgue integrable functionsf :0,∞ → Rsuch that
∞
0
fsds < ∞. 2.1
IfAis an event we denote its complement byA. 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 spaceΩ,F,Pwith a filtrationFtt≥0satisfying the usual conditions and letBt : t ≥ 0be a standard one-dimensional Brownian motion on this space. Letσbe a real positive constant. Suppose that
k∈C0,∞;0,∞, k∈L10,∞. 2.2
Suppose also that
f∈CR;R, xfx>0 x /0, f0 0. 2.3
Letα >0. We consider the stochastic Volterra equation
dXt − t
0
kt−sfXsds dt σXtdBt, t≥0, X0 α. 2.4
Letn∈N. Suppose, in addition to2.3, thatfis locally Lipschitz continuous. This means the following.
For everyn∈Nthere existsKn>0 such that fx−f
y≤Knx−y ∀x, y∈Rfor which|x| ∨y≤n. 2.5 Then there is a unique continuousFB-adapted processXwhich obeys
Xt∧tn α− t∧tn
0
s
0
ks−ufXudu ds t∧tn
0
σXsdBs, t≥0, a.s, 2.6
wheretn inf{t≥ 0 : |Xt| n}. Suppose in addition thatf is globally linearly bounded.
More precisely, this means thatfalso obeys the following:
There existsL1≥0 such thatfx≤L11 |x|, x∈R. 2.7
Ifkobeys2.2andfobeys2.5and2.7, then there exists a unique continuousFB-adapted processXwhich obeys
Xt α− t
0
s
0
ks−ufXudu ds t
0
σXsdBs, t≥0, a.s. 2.8
See, for example, Berger and Mizel25, Theorem 2E. In this situation, we say that2.4has a unique strong solution. Throughout the paper, we will assume that2.4has a unique strong solution but will not necessarily impose conditions2.5or2.7onfin order to guarantee this. Hereinafter we will often refer to the solution rather than the strong solution of2.4.
We denote the almost sure event on which2.8holds byΩ∗. For eachω ∈Ω∗we denote by Xt, ωthe value ofX at timet. We denote byXωthe realisationor sample pathXω {Xt, ω:t≥0}.
2.3. Zero Crossing and Positivity of Solutions
LetXbe the solution of2.4, whereα >0. For eachω∈Ω∗, the stopping timeταis defined by
τα, ω inf{t >0 :Xt, ω 0}. 2.9
We interpretτα, ω ∞in the case when{t >0 :Xt, ω 0}is the empty set. We say that the sample pathXωhas a zero crossing ifτα, ω< ∞. Define
Aα
ω∈Ω∗:Xωis positive on 0,∞
{ω∈Ω∗:τα, 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 onfandk. 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
xt − t
0
kt−sfxsds, t≥0, x0 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 hypothesis2.3ensures that xt 0 for all t ≥ 0 is the unique steady-state solution. We also notice that there are no other steady-state solutions K because2.3implies thatfK/0 forK /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 3.1means that this stochastic perturbation preserves the unique equilibrium solution of the deterministic equation3.1indeed ifX0 0, thenXt 0 for allt≥0 a.s. Moreover, the stochastic perturbation does not produce more point equilibria.
The fact that k is nonnegative and fx is positive when x is greater than the equilibrium solution of3.1means that the solutionxof3.1is initially attracted towards
the equilibrium, becausext≤ 0 for allt≥ 0, providedxt >0 for allt ≥ 0. The question then arises: does the solution ever reach the zero equilibrium solution in finite time? If so, does it overshoot and become negativethis 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 equation2.4.
Letα > 0. Our first result demonstrates that the solutions of2.4has a zero crossing for a.a. sample paths in the case whenf has at least linear-order leading behaviour at the equilibrium, and whenkis not identically zero. More precisely we request thatfobeys the following:
There existsL3>0 such that lim inf
x→0
fx
x L3, 3.2
and that kt/≡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 tω, provided thatXtω, ω/0, it is nonetheless differentiable atτα, and the zero level is crossed because this first zeroταofXis a simple zero ofX.
Theorem 3.1. Suppose thatkobeys2.2,f obeys2.3and 3.2, and thatkt/≡0. LetX be the unique strong solution of2.4. Ifταis defined by2.9, then for anyα >0
Pτα< ∞ 1. 3.3
Moreover,Xis differentiable atταandXτα<0.
See20for 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 ofTheorem 3.1concerns the linear stochastic Volterra equation
dXt − t
0
kt−sXsds dt σXtdBt, t≥0. 3.4
Under assumption2.2, it follows that there is a unique strong solution of this equationsee, e.g.,25.
Theorem 3.2. Suppose thatkobeys2.2andkt/≡0. LetXbe the unique strong solution of 3.4.
Ifταis defined by2.9, then for anyα >0 one has
Pτα< ∞ 1. 3.5
Moreover,Xis differentiable atταandXτα<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 of18–20the 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 motionB.
It is interesting to compareTheorem 3.2with known results on the zero crossings of the corresponding deterministic linear Volterra integrodifferential equation
xt − t
0
kt−sxsds, t≥0, x0 1 3.6
in the case whenkobeys2.2andkis nontrivial. It has been shownsee, e.g.,16that3.6 has zero crossings if and only if the characteristic equation of3.6
λ ∞
0
kse−λsds0, 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 of3.6possesses zero crossings, then so does that of3.4. Therefore, the presence of a stochastic term tends to induce oscillatory behaviour in the solution.
Theorems3.1and3.2show that positive solutions are impossible iff 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 whenfdoes not necessarily have linear-order leading behaviour at zero. We assume not only thatfis weakly nonlinear close to zero, but also that it obeys the following:
There existsγ >1 andL2>0 such thatfx≤L2xγ ∀x≥0. 3.8 In addition, we suppose thatkdecays more quickly thant→ e−σ2t/2ast → ∞in the sense that
There exists >0 such that ∞
0
eσ2/2 tktdt < ∞. 3.9
Under these conditions, the next result states that2.4can 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 thatkobeys2.2and3.9. Suppose also thatf obeys2.3and3.8. Let Xbe the unique strong solution of 2.4. IfAαis defined by2.10, then there existsα∗>0 such that PAα>0 for allα≤α∗. Moreover
α→lim0 PAα 1. 3.10
The result and proof are inspired by19, Theorem 4.4, which applies to the stochastic delay differential equation
dXt −fXt−τdt σXtdBt, 3.11
where f obeys 2.3 and 3.8. Under these conditions, similar conclusions to those of Theorem 3.3apply to the solutions of3.11.
Theorems 3.1and 3.3show the importance of the linearity off local to zero in the presence or absence of zero crossings. However, it is natural to ask whether condition3.9 inTheorem 3.3is essential in allowing for positive solutions of2.4, or whether it is merely a convenient condition which enables us to establish positivity in some cases. The next result shows that condition3.9is 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 ofkwhich is slightly stronger than the negation of condition3.9. We assume thatkdecays more slowly to zero thant→e−σ2t/2in the sense that
There exists >0 such that ∞
0
eσ2/2−tktdt ∞. 3.12
Under this condition, solutions of2.4cross zero on almost all sample paths, irrespective of how weakly the nonlinear restoring functionfacts on the solution.
Theorem 3.4. Suppose thatkobeys2.2and3.12. Suppose also thatf obeys2.3. LetXbe the unique continuous solution of 2.4. Ifταis defined by2.9, then for anyα >0 one has
Pτα< ∞ 1. 3.13
Moreover,Xis differentiable atταandXτα<0.
This result is interesting because, in the case when f is in C1 and f0 0, the linearisation of2.4that is the linear SDE given by
dYt σYtdBt, t≥0, Y0 α >0, 3.14
has positive solutions with probability one. In a complete contrast however,2.4has 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 in18–20.
Defineϕ{ϕt:t≥0}by
ϕt eσBt−σ2t/2, t≥0. 4.1
Thenϕis a strictly positive processi.e.,ϕt>0 for allt≥0 a.s.which obeys the stochastic differential equation
dϕt σϕtdBt, t≥0, ϕ0 1. 4.2
Thenϕ−1t > 0 for allt ≥ 0 a.s. and by It ˆo’s lemma,ϕ−1tobeys the stochastic differential equation
dϕ−1t σ2ϕ−1tdt−σϕ−1tdBt. 4.3
Bystochasticintegration by parts it follows that
d Xt
ϕt Xtdϕ−1t ϕt−1 dXt σXt
−σϕ−1t dt −ϕt−1
t
0
kt−sfXsds dt.
4.4
Therefore, asX0 α, we have Xt
ϕt α− t
0
ϕs−1 s
0
ks−ufXudu ds, t≥0. 4.5
SinceXhas continuous sample paths, it follows from2.3and2.2that each realisation of
t−→ϕt−1 t
0
kt−sfXsds 4.6
is continuous. Therefore we have that each realisation of the processy{yt:t≥0}defined by
yt Xt
ϕt, t≥0 4.7
is inC10,∞;Rand by4.5we have
yt −ϕt−1 t
0
kt−sf
ϕsys
ds, t≥0, y0 α >0. 4.8
It is convenient here to record another fact concerningϕ: on account of the Strong Law of Large Numbers for standard Brownian motion, it follows that
t→ ∞lim 1
tlogϕt −σ2
2 , a.s. 4.9
and therefore we have thatϕt → 0 ast → ∞a.s.
5. Proof of Theorem 3.1
5.1. Supporting LemmasWe start by developing a criterion independent of the solution of2.4, but which depends onϕgiven by4.1, which ensures that the solution of2.4exhibits a zero crossing a.s.
Lemma 5.1. Letσ >0. Suppose thatkobeys2.2and thatfobeys2.3and3.2. Suppose thatϕ is defined by4.1, andκby
κt, s ϕt−1kt−s, 0≤s≤t. 5.1
Suppose that there existsτ >0 such that
lim sup
t→ ∞
t−τ
t−2τ
t
t−τκs, uϕuds du > 1
L3. 5.2
Letα >0. IfXis the unique strong solution of 2.4, andταis defined by2.9, then
Pτα< ∞ 1. 5.3
Moreover,Xis differentiable atταandXτα<0.
Proof. By4.8and the definition ofκin5.1we have
yt − t
0
κt, sf
ϕsys
ds, t≥0. 5.4
Note thatAα {ω : yt, ω > 0 for allt ≥ 0}. Suppose thatPAα > 0. Letω ∈ Aα. Note thatyt, ω≤0 for allt≥0. Thereforeyt, ωtends to a nonnegative limit ast → ∞. Since
ϕt, ω → 0 ast → ∞, we have thatXt, ω → 0 ast → ∞. We temporarily suppress the dependence onω. Sinceyt>0, for anyt≥2τ, by5.4we have
yt−τ t
t−τ
s
0
κs, uf
ϕuyu
du ds yt
≥ t
t−τ
s
0
κs, uϕuyufXu
Xu du ds
t
0
t
u∨t−τκs, uϕuds·fXu
Xu yudu
≥ t−τ
t−2τ
t
u∨t−τκs, uϕuds·fXu
Xu yudu
≥ inf
u∈t−2τ,t−τ
fXu Xu
t−τ
t−2τ
t
u∨t−τκs, uϕuds·yudu.
5.5
Now, becauseyis nonincreasing we have
yt−τ≥ inf
u≥t−2τ
fXu Xu
t−τ
t−2τ
t
u∨t−τκs, uϕuds du·yt−τ inf
u≥t−2τ
fXu Xu
t−τ
t−2τ
t
t−τκs, uϕuds du·yt−τ.
5.6
Sinceyt−τ>0 for allt≥2τwe have that
u≥t−2τinf
fXu Xu
t−τ
t−2τ
t
t−τκs, uϕuds du≤1, t≥2τ. 5.7
SinceXt → 0 ast → ∞and lim infx→0 fx/xL3we have
tlim→ ∞ inf
u≥t−2τ
fXu
Xu L3. 5.8
Therefore
lim sup
t→ ∞
t−τ
t−2τ
t
t−τκs, uϕuds du≤ 1
L3, 5.9
which contradicts5.2. Therefore we have thatPτα< ∞ 1, as required.
NowAα {ω∈Ω∗: there existsttω>0 such thatXtω, ω 0}{ω ∈Ω∗ : τα, ω< ∞}, and this event is almost sure. Fixω∈Aα. Sinceyt 0 if and only ifXt 0,
by4.7we have from2.9thatyτα, ω, ω 0 and thatyt, ω>0 for allt∈0, τα, ω.
By4.8we have
yτα −ϕ−1τα τα
0
kτα−sf
ϕsys
ds. 5.10
Since2.3implies thatfϕsys >0 for alls ∈ 0, τα, andkobeys2.2we have that yτα≤0. Suppose thatyτα 0. Sinceϕτα>0, we must have thatkτα−s 0 for alls ∈ 0, τα or kt 0 fort ∈ 0, τα. Therefore we have that yt 0 for all t ∈ 0, τα. Henceyτα y0 α > 0, which contradicts the fact that yτα 0.
Therefore we must haveyτα<0. This implies that there existstω> τα, ωsuch that ytω, ω < 0 and therefore we have that Xtω, ω < 0. Therefore for eachωin the a.s.
eventAα, there exists atω>0 such thatXtω<0.
We now show thatX is differentiable atταand thatXtα<0. Lett /τα. Then we have
Xt, ω−Xτα, ω, ω
t−τα, ω yt, ωϕt, ω−yτα, ω, ωϕt, ω t−τα, ω
ϕt, ωyt, ω−yτα, ω, ω t−τα, ω .
5.11
Now taking the limit ast → ταon the righthand side we have
t→limταϕt, ωyt, ω−yτα, ω, ω
t−τα, ω ϕτα, ω, ωyτα, ω, ω<0. 5.12 Therefore we have
t→ταlim
Xt, ω−Xτα, ω, ω
t−τα, ω ϕτα, ω, ωyτα, ω, ω<0, 5.13 soXτα, ω, ωis well defined and indeedXτα, ω, ω<0.
The next result develops a condition which depends only on the increments ofBand the kernelkwhich implies condition5.2.
Lemma 5.2. Letσ >0. Ifκis defined by5.1andϕby4.1, and there existsτ >0 such that
lim sup
t→ ∞
τ
0
τ
0
e−σBt u−Bt−wku wdu dw ∞, a.s. 5.14
then5.2holds.
Proof. Define fort≥2τ A1t
t−τ
t−2τ
t
t−τκs, uϕuds du t−τ
t−2τ
t
t−τϕs−1ks−uϕuds du. 5.15
Hence asϕis given by4.1we have
A1t t−τ
t−2τ
t
t−τe−σBs−Bueσ2/2s−uks−uds du
t−τ
t−2τ
t−u
t−u−τe−σBv u−Bueσ2v/2kvdv du
τ
0
w τ
w
e−σBv t−τ−w−Bt−τ−weσ2v/2kvdv dw
≥ τ
0
w τ
w
e−σBv t−τ−w−Bt−τ−wkvdv dw.
5.16
Therefore if5.14holds, then lim sup
t→ ∞ A1t≥lim sup
t→ ∞
τ
0
w τ
w
e−σBv t−τ−w−Bt−τ−wkvdv dw lim sup
t→ ∞
τ
0
w τ
w
e−σBv t−w−Bt−wkvdv dw lim sup
t→ ∞
τ
0
τ
0
e−σBu t−Bt−wku wdu dw ∞,
5.17
which implies5.2.
Lemma 5.3. Suppose thatσ > 0. Suppose thatkobeys2.2andkt/≡0. Then there existsτ > 0 such that5.14holds.
Proof. Ifkt/≡0 andkt≥0 it follows that there is at0 ≥0 such thatkt0 : 2k0 >0. Since kis continuous on0,∞there existsδ >0 such that|kt−kt0| ≤k0for allt∈t0, t0 δ.
Therefore we havekt≥k0for allt∈t0 δ/2, t0 δ :θ1, θ2. Hence
There existsk0>0 and 0< θ1< θ2 such thatkt≥k0 ∀t∈θ1, θ2. 5.18 Letτθ2. Thenθ1 ≤τ,θ2≤τ. Define
A2t τ
0
τ
0
e−σBu t−Bt−wku wdu dw, t≥τ. 5.19
Equation5.14is equivalent to show that lim supt→ ∞A2t ∞ a.s. Clearly by5.18we have
A2t≥k0
θ2
0
θ2−w
θ1−w∨0e−σBu t−Bt−wdu dw. 5.20
Nowθ1/2, θ2/22⊂ {w, u: 0≤w≤θ2, u≥0, θ1≤w u≤θ2}, so we have
A2t≥k0
θ2/2
θ1/2
θ2/2
θ1/2
e−σBu t−Bt−wdu dw:At. 5.21
Hence5.14follows if we can show that lim supt→ ∞At ∞a.s. DefineAnAnθ2/k0for n≥1 so that it suffices to prove that lim supn→ ∞An∞a.s. Note that
An θ2/2
θ1/2
θ2/2
θ1/2
e−σBu nθ2−Bnθ2−wdu dw. 5.22
Now, we note that eachAn is a functional of increments of the standard Brownian motion B over the interval nθ2 −θ2/2, nθ2 θ2/2. Therefore as n 1θ2 −θ2/2 nθ2 θ2/2, it follows that the intervals on which the increments of Bare considered forAn and An 1
are nonoverlapping. Since the increments ofBare independent, it follows thatAnn≥1 is a sequence of independent random variables. Hence by the Borel-Cantelli lemma, we are done if we can show that
∞ n1
P
An> β
∞, ∀β >0. 5.23
Note thatB−Bis a standard Brownian motion. Then
An θ2/2
θ1/2
θ2/2
θ1/2
eσBu nθ 2−Bnθ 2−wdu dw. 5.24
Define forη∈Rthe event
Cηn
ω: min
u∈θ1/2,θ2/2Bnθ 2 u, ω− max
w∈θ1/2,θ2/2Bnθ 2−w, ω≥η
. 5.25
Ifω ∈Cηn, thenBnθ 2 u, ω−Bnθ 2−w, ω≥ηfor allu, w∈θ1/2, θ2/22. Therefore ω∈Cηnimplies
Anω θ2/2
θ1/2
θ2/2
θ1/2
eσBnθ 2 u,ω−Bnθ 2−w,ωdu dw≥ θ2
2 −θ1
2
2
eση. 5.26
Thus
P
An≥ θ2
2 −θ1
2
2
eση
≥P Cηn
, n≥1, η∈R. 5.27
Now
u∈θmin1/2,θ2/2Bnθ 2 u− max
w∈θ1/2,θ2/2Bnθ 2−w min
u∈θ1/2,θ2/2
Bnθ 2 u−B
nθ2 θ1
2 B
nθ2 θ1
2 −B
nθ2−θ1
2
− max
w∈θ1/2,θ2/2
Bnθ 2−w−B
nθ2−θ1
2 min
u∈nθ2 θ1/2,nθ2 θ2/2
Bu −B
nθ2 θ1
2 B
nθ2 θ1
2 −B
nθ2−θ1
2
w∈nθ2−θmin2/2,nθ2−θ1/2
B
nθ2−θ1
2 −Bw
:W1n W2n W3n.
5.28
Sinceθ1 > 0, each ofW1n,W2n, andW3nis well defined and independent random variables. Hence
P Cηn
P
W1n W2n W3n≥η
≥P
W1n≥0, W2n≥η, W3n≥0 PW1n≥0·P
W2n≥η
·PW3n≥0.
5.29
Now we note thatB1u : Bu nθ2 θ1/2−Bnθ 2 θ1/2foru ∈ 0, θ2/2−θ1/2is a standard Brownian motion, so we have
PW1n≥0 P
u∈nθ2 θ1inf/2,nθ2 θ2/2Bu −B
nθ2 θ1
2 ≥0
P
u∈0,θinf2/2−θ1/2B1u≥0
1.
5.30
Next, if we defineB2u:Bnθ 2−θ1/2−Bu foru ∈0, θ2/2−θ1/2, thenB2 is another standard Brownian motion. Therefore we have
PW3n≥0 P
w∈nθ2−θ2inf/2,nθ2−θ1/2
B
nθ2−θ1
2 −Bw
≥0
P
u∈0,θ2inf/2−θ1/2
B
nθ2−θ1
2 −B
nθ2− θ1
2 −u ≥0
P
u∈0,θ2inf/2−θ1/2B2u≥0
1.
5.31
Therefore ifZis a standard normal random variable andΦis the distribution function ofZ, we have
P Cηn
P
W1n W2n W3n≥η
≥P
W2n≥η P
B
nθ2 θ1
2 −B
nθ2− θ1
2 ≥η
P
θ1Z≥η
1−Φ η
θ1 .
5.32
Hence by5.27we have P
An≥
θ2
2 −θ1
2
2
eση
≥1−Φ η
θ1 , ∀η∈R, n≥1. 5.33
Letβ >0 and defineη∈Rby
η 1 σlog
4β θ2−θ12
. 5.34
Thenβ θ2−θ12eση/4. Hence P
An≥β
≥1−Φ 1
σθ1
log
4β θ2−θ12
:c
β
>0, ∀β >0, n≥1. 5.35
This implies that5.23holds, and therefore that lim supn→ ∞An ∞a.s., from which it has already been shown that the lemma follows.
5.2. Proof ofTheorem 3.1
The proof of Theorem 3.1is now an immediate consequence of the last three lemmas. By Lemma 5.3it follows that5.14 holds. ByLemma 5.2it therefore follows that5.2holds.
Hence byLemma 5.1it follows thatPτα< ∞ 1 for anyα >0 and thatXταexists and is negative. Since these are the desired conclusions ofTheorem 3.1, the proof is complete.
6. Proof of Theorem 3.3
We start by proving a technical lemma.
Lemma 6.1. Letσ >0,γ >1, and suppose thatϕis given by4.1. Suppose thatkobeys2.2and 3.9. Define
I ∞
0
t
0
ϕt−1kt−sϕγsds dt. 6.1
ThenI < ∞a.s.
Proof. By the definition ofϕwe have
I ∞
0
t
0
kt−seσγBs−σBteσ2t/2−σ2γs/2ds dt. 6.2
By the Strong Law of Large Numbers for standard Brownian motionsee, e.g., Karatzas and Shreve27, there exists an almost sure eventΩ1such that
Ω1
ω: limt→ ∞Bt, ω
t 0
. 6.3
Therefore for eachω∈Ω1and for everyε >0 there exists a finiteTω, ε>0 such that
|Bt, ω| ≤εt, t≥Tω, ε. 6.4
Define
I1 Tω,ε
0
t
0
kt−seσγBs−σBteσ2t/2−σ2γs/2ds dt, I2
∞
Tω,ε
t
Tω,εkt−seσγBs−σBteσ2t/2−σ2γs/2ds dt, I3
∞
Tω,ε
Tω,ε
0
kt−seσγBs−σBteσ2t/2−σ2γs/2ds dt.
6.5
ThenI I1 I2 I3. The continuity of the integrand and finiteness ofTω, ε>0 ensures that I1 < ∞. Consider nowI3. Suppose thatε >0 is so small that|σ|ε < where >0 is defined
by3.9 note the distinction between the constantdefined by3.9and the small parameter ε >0. Defineβ2σ2/2 |σ|ε. By3.9we therefore have
∞
0
kseσ2/2 |σ|εsds ∞
0
kseβ2sds < ∞. 6.6
Then by6.4we have
I3≤ max
0≤s≤Tω,εeσγBs−σ2γs/2· ∞
Tω,ε
Tω,ε
0
kt−se−σBteσ2t/2ds dt
≤ max
0≤s≤Tω,εϕγs, ω· ∞
Tω,ε
Tω,ε
0
kt−seβ2t−seβ2sds dt
≤eβ2Tω,ε max
0≤s≤Tω,εϕγs, ω· ∞
Tω,ε
Tω,ε
0
kt−seβ2t−sds dt.
6.7
Now by the nonnegativity of the integrand and Fubini’s theorem we have ∞
Tω,ε
Tω,ε
0
kt−seβ2t−sds dt
∞
Tω,ε
t
t−Tω,εkueβ2udu dt
∞
0
u Tε,ω
u∨Tε,ωdt·kueβ2udu
Tε,ω
0
ukueβ2udu Tε, ω ∞
Tε,ωkueβ2udu.
6.8
By6.6we have
∞
Tω,ε
Tω,ε
0
kt−seβ2t−sds dt < ∞, 6.9
which implies thatI3< ∞.
Finally we show thatI2 < ∞, a.s. Suppose thatε > 0 is so small thatε < |σ|/2 and ε <γ−1|σ|/2γ 1. Defineβ1γσ2/2− |σ|ε. Thenβ1>0. Hence by6.4we have
I2 ∞
Tω,ε
t
Tω,εkt−seσγBs−σBteσ2t/2−σ2γs/2ds dt
≤ ∞
Tω,ε
t
Tω,εkt−se|σ|γεs |σ|εteσ2t/2−σ2γs/2ds dt
∞
Tω,ε
t
Tω,εkt−se−β1seβ2tds dt.
6.10
Nowβ1−β2 γ−1σ2/2−γ 1|σ|ε >0 sinceε <γ−1|σ|/2γ 1. Hence ∞
Tω,ε
t
Tω,εkt−se−β1seβ2tds dt
∞
Tω,ε
t
Tω,εkt−seβ1t−sds·eβ2−β1tdt
∞
Tω,ε
t−Tω,ε
0
kueβ1udu·eβ2−β1tdt
∞
0
∞
Tω,ε ue−β1−β2tdt·kueβ1udu e−β1−β2Tω,ε
β1−β2
∞
0
eβ2ukudu.
6.11
Therefore by6.6we have that ∞
Tω,ε
t
Tω,εkt−se−β1seβ2tds dt < ∞, 6.12
and so
I2≤ ∞
Tω,ε
t
Tω,εkt−se−β1seβ2tds dt < ∞. 6.13
HenceII1 I2 I3< ∞, as required.
6.1. Proof ofTheorem 3.3
Letταbe defined by2.9. Letω∈Aα. Letybe given by4.7. Then for allt∈0, ταwe have 0< yt, ω≤α. Note also thatyτα 0. Therefore we have 0< Xt, ω≤αϕtfor all t∈0, τα. Therefore by3.8it follows that
fXt, ω≤L2Xγt, ω≤L2αγϕγt, t∈0, τα. 6.14
By4.8we have
0yτα α− τα
0
t
0
ϕt−1kt−sfXsds dt. 6.15
Hence by6.14and the nonnegativity ofkandϕwe have
α τα
0
t
0
ϕt−1kt−sfXsds dt
≤L2αγ τα
0
t
0
ϕt−1kt−sϕγsds dt
≤L2αγ ∞
0
t
0
ϕt−1kt−sϕγsds dt.
6.16
Therefore asIis defined by6.1we have
Iω≥ 1
L2α1−γ, for eachω∈Aα. 6.17
Therefore
1−PAα P Aα
≤P
I ≥ α1−γ L2
. 6.18
ByLemma 6.1it follows thatPI < ∞ 1. Therefore asγ >1, by taking limits on both sides of6.18, we obtain
1−lim inf
α→0 PAα lim sup
α→0 {1−PAα} ≤lim sup
α→0
P
I≥ α1−γ L2
0. 6.19
Therefore we have lim infα→0 PAα ≥ 1. On the other hand, because we evidently have lim supα→0 PAα≤1, it follows thatPAα → 1 asα → 0 .
On the other hand, by6.18we havePAα≥PI < α1−γ/L2. Therefore asI ∈0,∞ a.s. it follows that there is anα∗>0 such thatPI < α1−γ∗ /L2>0. Now supposeα≤α∗. Then asγ >1 we have
P
I < α1−γ L2
≥P
I < α1−γ∗ L2
>0, 6.20
which impliesPAα>0 for allα≤α∗, proving the result.
7. Proof of Theorem 3.4
LetAαbe defined by2.10. We suppose thatPAα>0. Define also
ht t
0
kt−sfXsds, t≥0. 7.1
Then by4.5we have
Xt ϕt
α−
t
0
ϕs−1hsds
, t≥0. 7.2
Fixω∈Aα. Sinceht, ω≥0 for allt≥0, we have thatXt, ω≤αϕt, ωfor allt≥0. Also, as ϕt>0 for allt≥0 andXt, ω>0 we have
t
0
ϕs, ω−1hs, ωds≤α, ∀t≥0. 7.3
Hence
∞
0
ϕs, ω−1hs, ωds≤α, for eachω∈Aα. 7.4
LetΩ1be the event defined in6.3. Now defineCα Aα∩Ω1. ThenPCα PAα>0. Let ε >0 be so small that|σ|2/2− |σ|ε≥σ2/2− >0, where >0 is defined by3.12 as in the proof ofLemma 6.1above, note the distinction between the constantdefined by3.12and the small parameterε >0. Then for everyε >0 so chosen andω∈Cαthere existsTω, ε>0 such thatBobeys6.4. Then asω∈Cαwe haveXt, ω>0 for allt∈0, Tω, εand so by 2.3we have thatfXt, ω>0 for allt∈0, Tω, ε. Define
Fεω min
u∈0,Tω,εfXu, ω>0. 7.5
Lets≥Tε, ω. Then
hs, ω≥ Tε,ω
0
ks−ufXu, ωdu≥Fεω Tε,ω
0
ks−udu. 7.6
Therefore for allω∈Cα, by using6.4and4.1, we obtain ∞
0
hs, ωϕs, ω−1ds
≥ ∞
Tε,ωhs, ωϕs, ω−1ds
∞
Tε,ωhs, ωeσ2/2s−σBs,ωds
≥Fεω ∞
Tε,ω
Tε,ω
0
ks−udu·eσ2/2s−σBs,ωds
≥Fεω ∞
Tε,ω
Tε,ω
0
ks−udu·eσ2/2−|σ|εsds.
7.7
Therefore ∞
0
hs, ωϕs, ω−1ds≥Fεω ∞
Tε,ω
Tε,ω
0
ks−udu·eσ2/2−sds. 7.8
Defineβ3σ2/2−. Thenβ3>0 and by3.12we have ∞
0
kseβ3sds ∞. 7.9
Then by the nonnegativity of the integrand and Fubini’s theorem we have ∞
Tε,ω
Tε,ω
0
ks−udu·eβ3sds
∞
Tε,ω
Tε,ω
0
ks−ueβ3s−ueβ3udu ds
