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A Ray-Knight Theorem for Spectrally Positive Stable Processes
Wei Xu
To cite this version:
Wei Xu. A Ray-Knight Theorem for Spectrally Positive Stable Processes. 2021. �hal-03219926v3�
BY WEIXU1
1Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany.xuwei@math.hu-berlin.de
We generalize a classical second Ray-Knight theorem to spectrally positive stable pro- cesses. It is shown that their local time processes are solutions of certain stochastic Volterra equations driven by Poisson random measure and they belong to a class of fully novel non- Markov branching processes, named asrough continuous-state branching processes. Also, we prove the weak uniqueness of solutions to the stochastic Volterra equations by provid- ing explicit exponential representations of their characteristic functionals in terms of unique solutions to some associated nonlinear Volterra equations.
1. Introduction. This paper is preoccupied with generalizing the classical second Ray-Knight the- orem to spectrally positive stable processes. The classical second Ray-Knight theorem was originally proved by Ray [35] and Knight [25] independently to understand the law of the Brownian local time processes at the first time that the amount of local time accumulated at some level exceeds a given value. It is shown that for a constantζ >0the Brownian local time process{LB(t, τBL(ζ));t≥0}with τBL(ζ) := inf{s≥0 :LB(0, s)≥ζ}turns to be a0-dimensional Bessel process. It is also a critical Feller branching diffusion and solves the stochastic differential equation
Yζ(t) =ζ+ Z t
0
Z Yζ(s) 0
2WB(ds, dz), (1.1)
whereWB(ds, dz)is a Gaussian white noise onR2+with densitydsdz; see Theorem 3.1 in [9].
For a general Markov process, its local time process has the Markov property if and only if it has continuous paths; see Theorem 1.1 in [11]. Therefore, it is usually a challenge to generalize the classical Ray-Knight theorems to Markov processes with jumps and limited work has been done in understanding the law of their local time processes. In a considerable important work, Eisenbaum et al. [12] provided a generalization of the classical second Ray-Knight theorem to any strongly symmetric recurrent Markov process {S(t) :t≥0} with state spaceV being a locally compact separable metric space, i.e., there exists a mean-zero Gaussian process{G(x) :x∈V}such that for anya∈Randζ >0,
LS(x, τSL(ζ)) + (G(x) +a)2/2 :x∈V =
G(x) +p
2ζ+a22
/2 :x∈V (1.2)
in distribution. It is recently proved again in [36] using a martingale related to the reversed vertex- reinforced jump process. WhenSis a real-valued symmetric stable process with index1+α∈(1,2], the mean-zero Gaussian process turns to be a fractional Brownian motion with Hurst indexα/2. Specially, it is a standard Brownian motion whenα= 2and the equivalence (1.2) is an alternate formulation of the second Ray-Knight theorem for Brownian motion.
Our main interest is in a one-dimensional spectrally positive stable process{ξ(t) :t≥0}with index 1 +α∈(1,2). It is a nonsymmetric, discontinuous Markov process defined on a complete probabil- ity space (Ω,F,P) endowed with filtration{Ft}t≥0 satisfying the usual hypotheses. Its distribution is usually characterized by the Laplace transform E[exp{−λξ(t)}] = exp{tΦ(λ)} forλ≥0and the Laplace exponentΦ(λ)is of the form
Φ(λ) =bλ+cλ1+α=bλ+ Z ∞
0
e−λy−1 +λy
να(dy), (1.3)
MSC2020 subject classifications:Primary 60G51, 60J55; secondary 60G22, 60J80, 60F05.
Keywords and phrases: Ray-Knight theorem, spectrally positive stable process, stochastic Volterra equation, rough continuous-state branching process, marked Hawkes point process.
1
whereb≥0,c >0and theLévy measureνα(dy)is given by να(dy) :=cα(α+ 1)
Γ(1−α) y−α−2dy, y >0.
(1.4)
Let {W(x) :x∈R} denote the scale function of ξ, which is identically zero on x∈(−∞,0) and characterized on[0,∞)as a strictly increasing function whose Laplace transform is given by
Z ∞
0
e−λxW(x)dx= 1
Φ(λ), λ >0.
(1.5)
The scale functionW is infinitely differentiable on(0,∞)with derivative denoted asW0.
Let {Lξ(y, t) :y∈R} be the local time processof ξ, which is a joint continuous two-parameter process satisfying the well-known occupation density formula
Z t 0
f(ξ(s))ds= Z
R
f(y)Lξ(y, t)dy
for every nonnegative measurable function f onR; readers may refer to Chapter V in [4] for details.
Specially, the process{Lξ(0, t) :t≥0}is non-decreasing andLξ(0,∞) =∞a.s. if and only ifb= 0.
This allows us to define its right inverse{τξL(ζ) :ζ≥0}byτξL(ζ) =∞ifζ > Lξ(0,∞)and τξL(ζ) := inf{t≥0 :Lξ(0, t)≥ζ}, ifζ∈[0, Lξ(0,∞)].
It is usual to interpretτξL(ζ)as the first time that the amount of local time accumulated at level0exceeds ζ. Our first main result in this paper is the following generalization of the classical second Ray-Knight theorem (1.1) toξ.
THEOREM1.1. For anyζ >0, conditioned onτξL(ζ)<∞the local time process {Lξ(t, τξL(ζ)) : t≥0}is the unique weak solution to thestochastic Volterra equation
Xζ(t) =ζ(1−bW(t)) + Z t
0
Z ∞
0
Z Xζ(s) 0
W(t−s)−W(t−s−y)N˜α(ds, dy, dz), (1.6)
where N˜α(ds, dy, dz) is a compensated Poisson random measure on (0,∞) ×R2+ with intensity dsνα(dy)dz. Moreover, it is Hölder-continuous of any order strictly less than α/2 on any bounded interval and the Hölder coefficient has finite moments of all orders.
The stochastic Volterra integral in (1.6) is well defined as an Itô integral. Indeed, there exists a constantC >0such that |W(t)−W(t−y)| ≤Ctα−1·(t∧y)for anyt, y≥0. A simple calculation shows that for anyt≥0,
Z t 0
ds Z ∞
0
W(t−s)−W(t−s−y)
2να(dy)< Ctα
and hence the stochastic Volterra integral has finite quadratic variation. The Hölder regularity and the exact modulus of continuity of local times processes have already provided in [7] and [3] respectively.
For the caseb= 0, the finiteness of moments of the Hölder coefficient has been proved in [14]. Different to the methods developed in the previous literature, we prove the Hölder regularity with the help of (1.6) and the Kolmogorov continuity theorem. The finiteness of moments of the Hölder coefficient is proved by using the Garsia-Rodemich-Rumsey inequality.
Conditioned on the event τξL(ζ)<∞, the local time process Lξ(·, τξL(ζ))will fall into the trap0 in finite time, i.e. τ0 := inf{t≥0 :Lξ(t, τξL(ζ)) = 0}<∞ a.s. and Lξ(τ0 +t, τξL(ζ)) = 0 for any t≥0. However, this asymptotic result is difficult to be obtained from the solutionXζto (1.6) because the Markov property fails to hold for Xζ. Indeed, by interpreting the stochastic Volterra integral as a
convolution in intuition, we observe that (1.6) is path-dependent and its solutions may move continually into the negative half line after hitting0. Fortunately, thanks to the weak uniqueness, we can assert that 0is an absorbing state for any solution to (1.6).
Whenb= 0, the Lévy processξis recurrent withLξ(0,∞) =∞a.s. and hence the conditional law in Theorem1.1reduces to an unconditional law. Whenb >0, the processξis transient withLξ(0,∞)<∞ a.s., then with positive probabilityτξL(ζ)<∞for anyζ≥0. We now generalize the classical second Ray-Knight theorem with unconditional law to the transient spectrally positive stable process ξ with b >0. Forλ≥0, let{Uλ(x, dy) :x∈R}be thepotential measuresofξ, also known as theresolvent kernel, with
Uλ(x, A) :=
Z ∞
0
e−λtP{ξ(t)∈A}dt
for any setAin the Borelσ-algebraB(R). Specially, the measureUλ(0, dy)is absolutely continuous with respect to the Lebesgue measure and its density, denoted as{uλ(y) :y∈R}, is bounded, positive and continuous; see Theorem 16 and 19 in [4, p.61-65]. The process{τξL(ζ) :ζ≥0}is a subordinator killed at an independent exponential time with meanu0(0)and its Laplace transform is of the form
E[exp{−λτξL(ζ)}] = exp{−ζ/uλ(0)}, λ≥0.
This yields thatLξ(0,∞)is an exponential random variable with meanu0(0). In the next theorem we define %to be an exponential random variable with mean u0(0), which is independent of the Poisson random measureNα(ds, dy, dz).
THEOREM1.2. When b >0, the local process{Lξ(t, τξL(Lξ(0,∞))) :t≥0}is the unique weak solution to (1.6) withζ=%.
In the second part of this paper we understand the law of the solutionXζ by providing explicit rep- resentations for its characteristic functionals in terms of solutions to some associated nonlinear Volterra equations. In addition, we should indicate that the weak uniqueness of solutions to (1.6) is proved by verifying the uniqueness of solutions to the associated nonlinear Volterra equations. As we mentioned before, because of the stochastic Volterra integral, solutions to (1.6) are neither Markov nor semi- martingale, which makes it not possible to prove the pathwise uniqueness using the method developed in [9,15]. In the next theorem we define
K(t) := c·t−α
Γ(1−α), t >0 (1.7)
andVα to be an operator acting on anC-valued functionf by Vαf(t) :=
Z ∞
0
expnZ t (t−y)+
f(r)dro
−1− Z t
(t−y)+
f(r)dr
να(dy), t≥0.
(1.8)
LetC−be the space of all complex numbers with non-positive real part. LetB(R+;C−)be the space of all boundedC−-valued functions onR+. Denote byf∗gthe convolution of two functionsf andg onR+, i.e.,f∗g(t) =Rt
0f(t−s)g(s)dsfor anyt≥0.
THEOREM1.3. For anyλ∈C−andg∈B(R+;C−), we have E
exp
λXζ(T) +g∗Xζ(T) = exp
ζ·K∗vλg(T) , T≥0, (1.9)
where{vλg(t) :t≥0}is the unique global solution to the nonlinear Volterra equation vλg(t) =λW0(t) + g+Vαvgλ
∗W0(t).
(1.10)
Noting that the derivative function W0 is singular at0 and so isvgλ, we should considerK∗vλg(0) as the right limit ofK∗vλg at0. Indeed, we show that K∗W0(t)∼1ast→0+andVαvgλ(t)can be uniformly bounded byCtα−1in the neighbor of0. Thus
K∗vλg(0)∼K∗vλg(t) =λ+o(tα)
as t→0+ and hence the equality (1.9) still holds true for T = 0. Actually, the representation (1.9)- (1.10) indicates that the solutionXζ is a fully novel non-Markov branching system. We refer it as a rough continuous-state branching processbecause of the following two properties:
• It has the branching property and0is an absorbing state. Indeed, for anyζ1, ζ2>0, letXζ1,Xζ2 and Xζ1+ζ2 be three mutually independent unique weak solutions to (1.6) with initial state ζ1, ζ2 and ζ1+ζ2respectively. Then we haveXζ1+Xζ2=Xζ1+ζ2 in distribution;
• Compared to the Feller branching diffusion defined by (1.1) that is locally Hölder-continuous of any order strictly less than1/2, the processXζhas lesser degree of Hölder regularity and hence its simple paths are much rougher.
As the last result in this paper, we provide in the next theorem alternate representations for the stochastic Volterra equation (1.6) and the nonlinear Volterra equation (1.10) in terms of fractional in- tegration and differential equation. Let Iaρ and Daρ be the Riemann-Liouville fractional integral and derivative operator of orderρ∈(0,1]modified by a constanta >0. They act on a measurable function f onR+according to
Iaρf(t) := 1 aΓ(ρ)
Z t 0
(t−s)ρ−1f(s)ds and Dρaf(t) := a Γ(1−ρ)
d dt
Z t 0
(t−s)−ρf(s)ds.
For simplicity, we writeIρ=I1ρandDρ=Dρ1.
THEOREM1.4. The stochastic Volterra equation (1.6) is equivalent to Xζ(t) =ζ−b
Z t 0
(t−s)α−1
cΓ(α) Xζ(s)ds+ Z t
0
Z ∞
0
Z Xζ(s) 0
Z t−s (t−s−y)+
rα−1
cΓ(α)drN˜α(ds, dy, dz) (1.11)
and the nonlinear Volterra equation (1.10) is equivalent to
Dαcvλg(t) =−bvλg(t) +g(t) +Vαvλg(t), Ic1−αvλg(0) =λ.
(1.12)
Our main idea of the generalization of the second classic Ray-Knight theorem to spectrally positive stable processes is characterizing the cluster points of local times processes of a sequence of nearly recurrent compound Poisson processes with negative drift and Pareto-distributed jump distribution. An important tool used in the characterizations is a new representation, named asHawkes representation, for the local time processes of compound Poisson processes in terms of stochastic Volterra-Fredholm integral equation driven by Poisson random measure. It is established based on the following two properties:
• The local time process of a compound Poisson process with drift −1 stopped at hitting0 is equal in distribution to a homogeneous, binaryCrump-Mode-Jagers process(CMJ-process) starting from one ancestor; see [28];
• A homogeneous, binary CMJ-process can be reconstructed as the density process of a marked Hawkes point measurewith arrivals and marks of events representing the birth times and life-lengths of offsprings; see [20,37].
The main difficulty we encountered is to characterize the limit of the stochastic Volterra integrals in the new representation. Indeed, because of the absence of martingale property, the instruments provided by modern probability theory, e.g., the martingale representation theorem and martingale problems, are
out of work. To overcome this difficulty, we first find a good approximation for the stochastic Volterra integral by exploring the genealogy and the dynamic evolution of CMJ-processes. The tightness of approximation processes is proved by using Kolmogorov tightness criterion. For the convergence in finite-dimensional distributions, we first construct a sequence of martingales that are equal to the ap- proximation processes in the finite-dimensional distributions at the given time points. Then we show that these martingales converge to a limit process whose finite-dimensional distributions at the given time points equal to those of the desired stochastic Volterra integral in (1.6).
Different types of limit results have been established for self-exciting system in many literature, but none of them contains our case. A functional central limit theorem has been established for multivariate Hawkes processes in [2] and for Marked Hawkes point measures in [20] respectively. For the nearly unstable Hawkes process with light-tailed kernel, Jaisson and Rosenbaum [23] proved the weak con- vergence of the rescaled intensity to a Feller diffusion, which was generalized to multivariate marked Hawkes point processes and their shot noise processes in [37]. For the heavy-tailed case, they also proved that the rescaled point process converges weakly to the integral of a rough fractional diffusion;
see [13,24]. However, they left the weak convergence of the rescaled intensity as an open problem.
Our exponential representations (1.9) of characteristic functionals are established by extending the method developed in the proof of Theorem 4.3 in [1]. The key step in the proof is to write the Doob mar- tingale related toλXζ(T) + (g+Vαvgλ)∗Xζ(T)as a stochastic integral with respect to a compensated Poisson random measure. Furthermore, because of the impact of the nonlinear operatorV defined by (1.8), the existence and uniqueness of solutions to (1.10) turn to be a challenge. Indeed, different to the nonlinear Volterra equations that have been widely studied in many literature, e.g., [1,17], we shall see that the operatorV not only fails to be Lipschitz continuous but also turns (1.10) into path-dependent.
To overcome this difficulty, we first give a crucial pre-estimate about the asymptotics of solutions to (1.10). It not only reveals the convex sets in which solutions are but also guarantees that any local solu- tion can be successfully extended on the whole half real line. We then improve the standard proofs well developed in the previous literature and apply them together with the prior-estimate to solve (1.10).
Organization of this paper.In Section2, we provide some properties of the scale function and then introduce a sequence of compound Poisson processes whose local time processes converge weakly to that of a spectrally positive stable process. In Section3we give the Hawkes representations for the local times processes of compound Poisson processes. The local time processes of spectrally positive stable processes is characterized in Section4and the representation (1.9) of their characteristic functionals is proved in Section5. The existence and uniqueness of solution to (1.10) are proved in Section6. Finally, we prove Theorem1.4in Section7.
Notation.For the background and notation of Lévy processes we refer to [4]. For notational conve- nience, we writeR+:= [0,∞),R−:= (−∞,0],C−:={z∈C:Rez∈R−}andiR:={z∈C:Rez= 0}. For anyx∈R, letx+=x∨0,x−=x∧0andbxcbe the floor ofx. Given a topology spaceE, let D([0,∞),E)be the space of cádlágE-valued functions endowed with Skorokhod topology.
Let∆h and∇h be the forward and backward difference operator with step sizeh >0respectively, i.e., ∆hf(x) :=f(x+h)−f(x)and∇hf(x) :=f(x)−f(x−h). Throughout this paper, we make the conventions
Z y x
=− Z x
y
= Z
(x,y]
,
Z y−
x−
= Z
[x,y)
and Z ∞
x
= Z
(x,∞)
forx, y∈Rsatisfying thaty≥x. For any0≤a≤b≤ ∞andp∈(0,∞], letLp((a, b];C)be the space ofC-valuedp-integrable functions on(a, b], i.e.,
Lp((a, b];C) :=
n
f: (a, b]7→C:kfkpLp (a,b]:=
Z b a
|f(x)|pdx o
. For simplicity, we writekfkLp
b =kfkLp
(0,b] andkfkLp=kfkLp
(0,∞).
We useCto denote a positive constant whose value might change from line to line.
2. Preliminaries.
2.1. The scale function. As shown in the SDE-representation (1.6) of local time processes, the scale functionW will play a crucial role in our following proofs and analysis. We here provide some elements about the regularity and asymptotic behavior of this function. Define theMittag-Leffler function Eα,α
onR+by
Eα,α(x) :=
∞
X
k=0
xn Γ(α(n+ 1)).
It is locally Hölder continuous with indexα; see [18] for a precise definition of it and a survey of some of its properties, e.g. for anya≥0we have the well-known Laplace transform
Z ∞
0
e−λxaxα−1Eα,α(−a·yα)dx= a
a+λα, λ≥0.
(2.1)
Applying integration by parts to (1.5), we have Z ∞
0
e−λxW0(x)dx= Z ∞
0
λe−λxW(x)dx= 1
b+cλα, λ >0.
From this and (2.1), we see that the derivative functionW0 is of the form W0(x) =c−1xα−1·Eα,α(−b/c·xα), x >0.
Specially, whenb= 0we haveEα,α(0) = 1/Γ(α)and W(x) = xα
cΓ(1 +α), W0(x) = xα−1
cΓ(α), W00(x) = (α−1)xα−2 cΓ(α) . (2.2)
When b >0, according to (2.1) we see thatbW0 is a density function on R+, well known as Mittag- Leffler function, and henceW(x)→1/basx→ ∞. Moreover, the following properties ofW andW0 are direct consequences of properties of Mittag-Leffler density function; see [18,34,32]. The function W is locally Hölder continuous with indexα. Moreover, we have asx→0+,
W(x)∼ xα
cΓ(α+ 1), W0(x)∼ xα−1
cΓ(α), W00(x)∼(α−1)xα−2 cΓ(α) (2.3)
and asx→ ∞,
W(x)∼1
b − cx−α
Γ(1−α), W0(x)∼cα·x−α−1 Γ(1−α) . (2.4)
In conclusion, forb≥0there exists a constantC >0such that for anyx >0, W(x)≤Cxα and W0(x)≤Cxα−1. (2.5)
Recall the functionKdefined by (1.7). Fort >0, define LK(t) := tα−1
cΓ(α).
It is easy to show that the two functionsLKandKsatisfy theSonine equation, i.e.K∗LK≡1, while (LK, K) is known asSonine pair. In addition, when b >0the functionbW0 is theresolventofbLK, which is usually introduced by means of the resolvent equation
bW0=bLK−(bLK)∗(bW0).
(2.6)
Convolving both sides of this resolvent equation byKand then dividing them byb, we have K∗W0= 1−bW.
(2.7)
2.2. Compound Poisson Processes. Let Λ be a Pareto II distribution on R+ with location 0 and shapeα+ 1, i.e.
Λ(dx) = (α+ 1)(1 +x)−α−2dx.
Forn≥1, let {ξ(n)(t) :t≥0} be a compound Poisson process with drift−1, arrival rateγn>0and jump size distributionΛ. It is a spectrally positive Lévy process with Laplace exponent
Φ(n)(λ) :=λ+ Z ∞
0
(e−λx−1)γnΛ(dx), λ≥0.
We are interested in the caseE[ξ(n)(1)] =γn/α−1≤0in which the functionΦ(n) increases strictly to infinity. For any probability measure ν on R, we denote by Pν and Eν the law and expectation of a Lévy process started from ν, respectively. Whenν=δx is a Dirac measure at point x, we write Px=PδxandEx=Eδx. For simplicity, we also writeP=P0andE=E0. For anyA∈F, we denote byP{·|A}the conditional law given the eventA.
Let{Lξ(n)(y, t) :y∈R, t≥0}be the local time process ofξ(n)with Lξ(n)(y, t) := #{s∈(0, t] :ξ(n)(s) =y}.
We have Lξ(n)(y,∞) =∞ a.s. if and only ifE[ξ(n)(1)] = 0; equivalently, if and only if γn=α. The inverse local time{τξL(n)(k) :k= 0,1· · · }at level0is given byτξL(n)(k) =∞ifk > Lξ(n)(y,∞)and
τξL(n)(k) := min{t >0 :Lξ(n)(0, t) =k}, k= 0,1,· · ·, Lξ(n)(y,∞).
We now consider the structure of local time process {Lξ(n)(t, τξL(n)(k)) :t≥0}. Denote by τξ+(n) the first passage time ofξ(n)in[0,∞), i.e.,τξ+(n):= inf{t >0 :ξ(n)(t)≥0}. Note thatξ(n)will move from negative half line into positive half line by jumping, i.e., ifξ(n)(τξ+(n)−)<0we haveξ(n)(τξ+(n))>0a.s.
By Theorem 17(ii) in [4, p.204], we haveξ(n)(τξ+(n))underP{·|τξ+(n)<∞}is distributed as Λ∗(dx) :=αΛ(x)dx¯ =α(1 +x)−1−αdx,
(2.8)
which is known as thesize-biased distributionofΛ. HereΛ(x) := Λ[x,¯ ∞)is the tail distribution ofΛ.
Notice that the sample paths ofξ(n)before the first passage timeτξ+(n) do not make any contribution to the local time process{Lξ(n)(t, τξL(n)(k)) :t≥0}, by the strong Markov property ofξ(n)we can get the following proposition immediately.
PROPOSITION2.1. For anyn, k≥1, the local time process{Lξ(n)(t, τξL(n)(k)) :t≥0}has the same law under eitherP{·|τξL(n)(k)<∞}orPΛ∗{·|τξL(n)(k)<∞}. Moreover, it is equal in distribution to the sum ofki.i.d. copies of{Lξ(n)(t, τξL(n)(1)) :t≥0}underPΛ∗
We now consider under the following condition, the asymptotic behavior of rescaled process {ξ0(n)(t) :t≥0}withξ(n)0 (t) :=n−1ξ(n)(n1+αt)and its local time process{Lξ(n)
0 (y, t) :y, t≥0}.
CONDITION2.2. Assume thatnα(1−γn/α)→β≥0asn→ ∞.
It is easy to see that the Lévy processξ0(n)has Laplace exponentn1+αΦ(n)(λ/n) forλ≥0. Under Condition2.2, a routine computation shows thatn1+αΦ(n)(λ/n)→Φ0(λ) :=βλ+ Γ(1−α)λαasn→
∞. By Corollary 4.3 in [22, p.440], we haveξ0(n)→ξ0weakly inD([0,∞);R), whereξ0is a spectrally positiveα-stable process with Laplace exponentΦ0and Lévy measureν0,α(dy) =α(α+ 1)y−α−2dy.
For the process ξ0, we can define the local time process{Lξ0(y, t) :y∈R, t≥0}, inverse local time
{τξL
0(ζ) :ζ ≥0}, scale function W0 and its derivative W00 in the same way as for ξ. Applying the occupation density formula, we immediately have{Lξ(n)
0 (y, t) :y∈R, t≥0}is equal almost surely to {n−αLξ(n)(ny, n1+αt) :y∈R, t≥0}, i.e. for every nonnegative measurable functionf onR,
Z
R
f(y)Lξ(n)
0 (y, t)dy= Z t
0
f(n−1ξ(n)(n1+αs))ds= Z
R
f(y)·n−αLξ(n)(ny, n1+αt)dy.
(2.9)
Forζ >0, we also haveτL
ξ0(n)
(ζ) =n−1−ατξL(n)(ζnα)a.s. and hence Lξ(n)
0 (t, τξL(n) 0
(ζ)) :t≥0 =
n−αLξ(n)(nt, τξL(n)(ζnα)) :t≥0 , a.s.
(2.10)
The following convergence result for local time processes {Lξ(n) 0 (·, τL
ξ(n)0 (ζ))}n≥1 comes from Theo- rem 2.4 in [29].
LEMMA2.3. Forζ >0, the local time process{Lξ(n) 0 (t, τL
ξ(n)0 (ζ)) :t≥0}underP{·|τL
ξ(n)0 (ζ)<∞}
converges weakly to{Lξ0(t, τξL
0(ζ)) :t≥0}underP{·|τξL
0(ζ)<∞}inD([0,∞);R+)asn→ ∞.
3. Hawkes representation. In this section we provide a new representation for the local time processes of compound Poisson processes in terms of marked Hawkes point processes and then rewrite them as solutions to a class of stochastic Volterra integral equations. This new representation will play the key role in characterizing the local time processes of spectrally positive stable processes.
3.1. Marked Hawkes point processes. Let(Ω,G,P)be a complete probability space endowed with filtration{Gt:t≥0} that satisfies the usual hypotheses andUbe a Lusin topological space endowed with the Borel σ-algebraU. Let{σk:k= 1,2· · · }be a sequence of increasing,(Gt)-adaptable ran- dom times and{ηk:k= 1,2,· · · }be a sequence of i.i.d.U-valued random variables with distribution νH(du). We assume thatηkis independent of{σj:j= 1,· · ·, k}for anyk≥0. In terms of these two sequences we define the(Gt)-random point measure
NH(ds, du) :=
∞
X
k=1
1{σk∈ds,ηk∈du}
(3.1)
on (0,∞)×U. We say NH(ds, du) is amarked Hawkes point measureon U if the embedded point process {NH(t) :t≥0} defined by NH(t) :=NH((0, t],U) has an (Gt)-intensity {γ·Z(t) :t≥0}
withγ >0and
Z(t) =µ(t) +
NH(t)
X
k=1
φ(t−σk, ηk), t≥0,
for somekernelφ:R+×U→[0,∞)and someG0-measurable, nonnegative functional-valued random variable{µ(t) :t≥0}. We usually interpretφ(·, u)andµas the impacts of an event with mark uand all events prior to time0on the arrival of future events respectively.
By the independence betweenηkand{σi:i= 1,· · ·, k}for anyk≥1, we see that the random point measureNH(ds, du)defined by (3.1) has the intensity γZ(s−)dsνH(du). Following the argument in [21, p.93], on an extension of the original probability space we can define a time-homogeneous Poisson random measureN(ds, du, dz)on(0,∞)×U×R+with intensityγdsνH(du)dz such that
NH(ds, du) =
Z Z(s−) 0
N(ds, du, dz) and hence the intensityZ(t)at timetcan be rewritten into
Z(t) =µ(t) + Z t
0
Z
U
Z Z(s−) 0
φ(t−s, u)N(ds, du, dz).
Before proceeding to establish another stochastic Volterra integral representation for the intensity process, we need to introduce several quantities associated to the kernel. Denote by{φH(t) :t≥0}the mean impacts of an event on the arrival of future events with
φH(t) :=γ· Z
U
φ(t, u)νH(du).
We always assumeφH is locally integrable. Let{RH(t) :t≥0}be theresolventofφH defined as the unique solution to
RH(t) =φH(t) +φH ∗RH(t).
It is usual to interpretRH as the mean impacts of an event and its triggered events on the arrivals of future events. In addition, we introduce a two-parameter function onR+×U
R(t, u) =φ(t, u) +RH∗φ(t, u)
to describe the mean impacts of an event with mark u on the arrivals of future events. An argument similar to the one used in Section 2 in [20] induces the following proposition immediately.
PROPOSITION3.1. The intensity processZ satisfies the stochastic Volterra integral equation Z(t) =µ(t) +
Z t 0
RH(t−s)µ(s)ds+ Z t
0
Z
U
Z Z(s−) 0
R(t−s, u) ˜N(ds, du, dz), whereN˜(ds, du, dz) :=N(ds, du, dz)−γdsνH(du)dz.
3.2. Hawkes representation. We now provide the Hawkes representation for the local time pro- cesses of compound Poisson processes defined in Section 2.2based on their connection with a class of homogeneous, binary CMJ-processes; see the next lemma. This connection was first established by Lambert in [28]; see also Theorem 3.2 in [30], based on the observation that the jumping contour pro- cess of a homogeneous, binary CMJ-tree starting from one ancestor is a compound Poisson process with drift−1; conversely, the local time process of a compound Poisson process with drift−1stopped at hitting0is equal in distribution to a homogeneous, binary CMJ-process starting from one ancestor.
LEMMA3.2. Forn, k≥1, the local time process{Lξ(n)(t, τξL(n)(k)) :t≥0}underP{·|τξL(n)(k)<
∞}is equal in distribution to a homogeneous, binary CMJ-process defined by the following properties:
(P1) There arekancestors with residual life distributed asΛ∗; (P2) Offsprings have a common life-length distributionΛ;
(P3) Each individual gives birth to its children following a Poisson process with rateγn.
To establish the Hawkes representation for local time process, we first need to clarify the genealogy of the associated CMJ-process defined properties (P1)-(P3). In the n-th population, denote by`0,ithe residual life of the i-th ancestor at time 0. We sort all offsprings by their birth times. For the i-th offspring, we endow it with the pair(ςi, `i)to represent its birth time and life-length respectively. The population size at timet, denoted asZ(n)(t), can be written as
Z(n)(t) =
k
X
j=1
1{`0,j>t}+X
ςi≤t
1{`i>t−ςi}.
We define a random point measureNH(n)(dt, dy)associated to the sequence{(ςi, `i) :i= 1,2,· · · }by NH(n)(dt, dy) =
∞
X
i=1
1{ςi∈dt,`i∈dy}.
By the branching property, at timeta new offspring would be born at the rateγnZ(n)(t−). Thus the ran- dom point measure NH(n)(ds, dy) has an (Gt)-intensityγnZ(n)(s−)dsΛ(dy). Following the argument in the last section, we can rewrite the population processZ(n)into
Z(n)(t) =
k
X
j=1
1{`0,j>t}+ Z t
0
Z ∞
0
Z Z(n)(s−) 0
1{y>t−s}N0(n)(ds, dy, dz), (3.2)
where N0(n)(ds, dy, dz) is a Poisson random measure on (0,∞)×R2+ with intensity γndsΛ(dy)dz.
Thus NH(n)(ds, dy) is a marked Hawkes point measure onR+. Applying Proposition 3.1 to (3.2), we can immediately get the following Hawkes representation for the population process Z(n) and hence for the local time processLξ(n)(·, τξL(n)(k)).
LEMMA3.3. Forn, k≥1, the local time process{Lξ(n)(t, τξL(n)(k)) :t≥0}underP{·|τξL(n)(k)<
∞}is equal in distribution to the unique solution of the stochastic Volterra integral equation Z(n)(t) =
k
X
j=1
1{`0,j>t}+ Z nt
0
R(n)H (t−s)
k
X
j=1
1{`0,j>s}ds +
Z t 0
Z ∞
0
Z Z(n)(s−) 0
R(n)(t−s, y) ˜N0(n)(ds, dy, dz), (3.3)
whereN˜0(n)(ds, dy, dz) :=N0(n)(ds, dy, dz)−γndsΛ(dy)dz,R(n)H andR(n)are defined by R(n)H (t) =γnΛ(t) +¯ γnΛ¯∗R(n)H (t),
(3.4)
R(n)(t, y) =1{y>t}+ Z t
0
R(n)H (t−s)·1{y>s}ds.
(3.5)
4. Characterizations of local time processes. In this section we first provide a characterization for the local time processLξ0(·, τξL
0(ζ))as the cluster point of the sequence{Lξ(n) 0 (·, τL
ξ0(n)(ζ))}n≥1 and then generalize it toLξ(·, τξL(ζ)). In the sequel of this section, we always assume Condition2.2holds.
Forn≥1, letZ(n)be the unique solution to (3.3) withk=bζnαc. Apparently the equivalence (2.10) along with Lemma3.2implies that the process{Lξ(n)
0 (t, τL
ξ(n)0
(ζ)) :t≥0}underP{·|τL
ξ(n)0
(ζ)<∞}is equal in distribution to the process{Xζ(n)(t) :t≥0}withXζ(n)(t) :=n−αZ(n)(nt). From Lemma2.3, we haveXζ(n)converges weakly to a limit process, denoted asX0,ζ, inD([0,∞);R+)asn→ ∞and the processX0,ζis equal in distribution to{Lξ0(t, τξL
0(ζ)) :t≥0}underP{·|τξL
0(ζ)<∞}. In addition, by Lemma3.3it is not difficult to verify that
Xζ(n)(t) =n−α
bζnαc
X
i=1
1{`0,i>nt}+ Z nt
0
R(n)H (nt−s)n−α
bζnαc
X
i=1
1{`0,i>s}ds +
Z t 0
Z ∞
0
Z Xζ(n)(s−) 0
n−αR(n)(n(t−s), ny) ˜N0(n)(nds, ndy, nαdz).
(4.1)
We now describe the intuitions about characterizing the cluster points of {Xζ(n)}n≥1, which is achieved by analyzing the asymptotics of each terms in (4.1). The detailed proofs will be provided later. By the Glivenko-Cantelli theorem, we may approximate the first two terms on the right side of (4.1) respectively with
Xζ(n)(0)Λ∗(nt) and Xζ(n)(0)·R(n)H ∗Λ∗(nt),
where Λ∗(x) := Λ∗(x,∞). The approximation errors are denoted asε(n)0 (t) and ε(n)1 (t) respectively.
Integrating both sides of (3.4) on(0, nt]and then changing the order of integration, we have α
γnkR(n)H kL1
nt=αkΛk¯ L1
nt+α· kΛ¯∗R(n)H kL1
nt= 1−Λ∗(nt) + 1−Λ∗
∗RH(n)(nt).
Here (2.8) is also used in the second equality above. Moving1andR(n)H (nt)to the left side of the first equality, it turns into
Λ∗(nt) + Λ∗∗RH(n)(nt) = 1− 1−γn
α α
γn
kR(n)H kL1
nt
= 1−nα
1−γn α
α γn
Z t 0
n1−αR(n)H (ns)ds
and hence the sum of the first two terms on the right side of (4.1) can be well approximated by Xˆζ(n)(t) :=Xζ(n)(0)−Xζ(n)(0)·nα
1−γn
α α
γn Z t
0
n1−αR(n)H (ns)ds.
(4.2)
It is observed from this result and (4.1) that the asymptotic behavior of{n1−αR(n)H (nt) :t≥0}plays a crucial role in the weak convergence of{Xζ(n)}n≥1.
LEMMA4.1. For anyT >0, asn→ ∞we have uniformly int∈[0, T], Z t
0
n1−αRH(n)(ns)ds→W0(t) and Xˆζ(n)(t)→ζ 1−βW0(t) .
PROOF. Apparently the second uniform convergence follows immediately by applying the first one and Condition2.2to (4.2). We now prove the first uniform convergence. Forλ≥0, denote byLΛ¯(λ) andLR(n)
H
(λ)the Laplace transform ofΛ¯ andR(n)H respectively. Taking the Laplace transform of both sides of (3.4), we have LR(n)
H (λ) =γnLΛ¯(λ)
1 +LR(n) H (λ)
and hence LR(n)
H (λ) =γnLΛ¯(λ)·(1− γnLΛ¯(λ))−1. Thus
Z ∞
0
e−λtn1−αR(n)H (nt)dt=n−αLR(n) H
(λ/n) = γnLΛ¯(λ/n) nα(1−γnLΛ¯(λ/n))
= γnLΛ¯(λ/n)
nα(1−γαn) + γαn·nα(1− LΛ∗(λ/n)), whereLΛ∗ denotes the Laplace transform of the measureΛ∗. A simple calculation along with Condi- tion2.2shows thatγnLΛ¯(λ/n)→1 asn→ ∞. Using the fact that Λ∗(t) = (1 +t)−α together with Karamata-Tauberian theorem; see, e.g., Theorem 8.1.6 in [6, p.333], we have nα 1− LΛ∗(λ/n)
→ Γ(1−α)λαasn→ ∞and hence
Z ∞
0
e−λtn1−αR(n)H (nt)dt→ 1
β+ Γ(1−α)λα.
By (2.1) the function whose Laplace transform is equal to the last quantity isW00 and hence the desired
result follows.
We now start to analyze the asymptotics of the stochastic Volterra integral on the right side of (4.1).
By (3.5) we first notice that fort, y≥0,
n−αR(n)(nt, ny) =n−α1{y>t}+ Z t
0
n1−αR(n)H (n(t−s))1{y>s}ds.
Apparently the first term on the right side of this equality vanishes uniformly asn→ ∞. Additionally, Lemma4.1suggests us to approximate the second term with
Z t
(t−y)+
n1−αR(n)H (ns)ds∼ ∇yW0(t).
Thus the stochastic Volterra integral in (4.1) can be well approximated by M0(n)(t) :=
Z t 0
Z ∞
0
Z Xζ(n)(s−) 0
∇yW0(t−s) ˜N0(n)(nds, ndy, nαdz)
with the error denoted asε(n)2 (t). In conclusion, based on all estimates and notation above we have Xζ(n)(t) =
2
X
k=0
ε(n)k (t) + ˆXζ(n)(t) +M0(n)(t).
(4.3)
Lemma 4.1shows thatXˆζ(n) converges locally uniformly toζ(1−βW0) asn→ ∞. Also, it will be shown that the random measureN˜0(n)(nds, ndy, nαdz) can be approximated by a compensated Pois- son random measure N˜0(ds, dy, dz) on (0,∞)×R2+ with intensity dsν0,α(dy)dz, which yields that M0(n)(t)is asymptotically equivalent to
M0(t) :=
Z t 0
Z ∞
0
Z X0,ζ(s−) 0
∇yW0(t−s) ˜N0(ds, dy, dz).
Finally, passing to the limit in (4.3), we can obtain that
X0,ζ(t)∼ζ(1−βW0(t)) +M0(t).
In order to turn this approximation into equality, it is necessary to verify that (ε(n)0 , ε(n)1 , ε(n)2 , M0(n)) converges weakly to(0,0,0, M0)inD([0,∞),R4)asn→ ∞.
4.1. The resolvent. As a preparation, we provide some upper bound estimates for the resolvent R(n)H andR(n). They will play an important role in the following proofs and analysis.
PROPOSITION4.2. Forn≥1, the resolventR(n)H is bounded and completely monotone. Moreover, there exists a constantC >0such that for anyt≥0andn≥1,
R(n)H (t)≤C 1 +tα−1
. (4.4)
PROOF. By Theorem 5.3.1 in [17, p.148] we have Λ¯ is completely monotone and so isR(n)H . We now prove the uniform upper bound. LetRΛ be the resolvent ofαΛ, i.e.,¯ RΛ=αΛ +¯ αΛ¯∗RΛ. Since γn≤α, we haveRH(n)(t)≤RΛ(t)for anyn≥1andt≥0. Obviously,RΛis bounded and completely monotone with RΛ(0) =α. Thus it suffices to prove that RΛ(t)≤C 1 +tα−1
for some constant C >0. LetR¯Λ(t) := 1−RΛ(t)/αfor anyt≥0, which is continuous and increasing. Using integration by parts, we have
Z ∞
0
e−λtdR¯Λ(t) =λ Z ∞
0
e−λt[1−Rλ(t)/α]dt= 1−λ
αLRΛ(λ).
(4.5)
where LRΛ denotes the Laplace transform of RΛ. Taking Laplace transform of both sides of RΛ= αΛ +¯ αΛ¯∗RΛ, we haveLRΛ(λ) =αLΛ¯(λ)
1 +LRΛ(λ)
for anyλ >0and hence LRΛ(λ) = αLΛ¯(λ)
1−αLΛ¯(λ).
It is obvious that the numerator goes to 1 as λ→ 0+. On the other hand, from the fact that R∞
t αΛ(s)ds¯ = (1 +t)−αand Theorem 8.1.6 in [6, p.333] we also have asλ→0+, 1−αLΛ¯(λ)∼Γ(1−α)λα and hence LRΛ(λ)∼λ−α/Γ(1−α).
Taking this back into (4.5), we immediately have Z ∞
0
e−λtdR¯Λ(t)∼1− λ1−α αΓ(1−α). Using Theorem 8.1.6 in [6, p.333] again, we also have ast→ ∞,
1−R¯Λ(t)∼ tα−1
αΓ(α)Γ(1−α) and RΛ(t)∼ tα−1 Γ(α)Γ(1−α).
PROPOSITION4.3. Forp >1 +α, there exists a constantC >0such that for anyt≥0andn≥1, Z t
0
ds Z ∞
0
n−αR(n)(ns, ny)
pnα+1Λ(ndy)≤C(1 +t)α(p−1). (4.6)
PROOF. From (3.5) and the Cauchy-Schwarz inequality we have the term on the left side of (4.6) can be bounded by the sum of the following four terms:
J1(n)(t) :=C Z t
0
ds Z ∞
0
n−α1{y>s}
pnα+1Λ(ndy), J2(n)(t) :=C
Z t 0
ds
Z s 0
n1−αRH(n)(nr)dr
p
nα+1Λ(ns),¯ J3(n)(t) :=C
Z t 0
ds Z s/2
0
Z s s−y
n1−αR(n)H (nr)dr
p
nα+1Λ(ndy), J4(n)(t) :=C
Z t 0
ds Z s
s/2
Z s s−y
n1−αR(n)H (nr)dr
p
nα+1Λ(ndy).
Here the constantCdepends only onp. We first have J1(n)(t) =Cn(1−p)α+1
Z t 0
Λ(ns)ds¯ ≤Cn(1−p)α Z ∞
0
Λ(s)ds¯ ≤Cn(1−p)α≤C.
From (4.4) we haveRs
0 R(n)H (r)dr≤Csαand hence J2(n)(t)≤Cn(1−p)α
Z nt 0
s(p−1)α−1ds≤C·t(p−1)α.
Moreover, (4.4) also indicates thatn1−αRH(n)(nr)≤Crα−1for anyr >0, which immediately induces thatRs
s−yn1−αR(n)H (nr)dr≤C|s−y|α−1·yand J3(n)(t)≤C
Z t 0
ds Z s/2
0
|s−y|p(α−1)·yp−α−2dy≤C Z t
0
sp(α−1)ds Z s/2
0
yp−α−2dy≤Ct(p−1)α. ForJ(4)(t), by (4.4) we haveRs
s−yn1−αR(n)H (nr)dr≤CRs
s−yrα−1dr≤Csαand hence J4(n)(t)≤C
Z t 0
spαds Z s
s/2
y−α−2dy≤C Z t
0
s(p−1)α−1ds≤Ct(p−1)α.
Putting all estimates together, we can immediately get the desired result.
