STOCHASTIC VOLTERRA EQUATIONS FOR THE LOCAL TIMES OF SPECTRALLY
POSITIVE STABLE PROCESSES
BYWEI XU
Department of Mathematics, Humboldt-Universität zu Berlin, xuwei@math.hu-berlin.de, xuwei.math@gmail.com
This paper is concerned with the macroevolution mechanism of local times of a spectrally positive
stable process in the spatial direction. The main results state that conditioned on the finiteness of the
first time at which the local time at zero exceeds a given value, the local times at positive half line
are equal in distribution to the unique solution of a stochastic Volterra equation driven by a Poisson
random measure whose intensity coincides with the Lévy measure. This helps us to provide not only
a simple proof for the Hölder regularity, but also a uniform upper bound for all moments of the
Hölder coefficient as well as a maximal inequality for the local times. Moreover, in collaboration
with the stochastic Volterra equation, we extend the method of duality to establish an exponential-
affine representation of the Laplace functional in terms of the unique solution of a nonlinear Volterra
integral equation associated with the Laplace exponent of the stable process.
1. Introduction. Local times of Lévy processes not only have wide applications in various fields; see
[6,43], but they also have been studied in depth with abundant of interesting results obtained, e.g. various
constructions (see [5,17,59]), Hilbert transform (see [8,25]), Hölder regularity (see [4,13,26]) and so on.
We refer to [9,29,62] for survey on local times and their applications. In particular, to understand thoroughly
the dependence of Brownian local times in the space variable, Ray [60] and Knight [40] independently proved
the well-known Ray-Knight theorem that links Brownian local times to Bessel processes. Later, the Ray-Knight
theorem was generalized in [22,64] to strongly symmetric Markov processes with finite 1-potential densities.
For a general spectrally positive Lévy process, Le Gall and Le Jan [49,50] considered the reflected processes
of its time-reversed processes. Associated to the local times at 0, they introduced an exploration process to
describe the genealogy of a continuous-state branching process (CB-process) and generalized an analogue of
the Ray-Knight theorem for a functional of local times of the Lévy process; see also [20,48] for details.
Because of the lack of Markovianity; see [21], local times (not their funtionals) of general spectrally positive
Lévy processes are quite untractable. Their microstructure and evolution mechanism have received considerable
attention in recent years. Specifically, Lambert [44] connected a compound Poisson process with unit negative
drift and killed upon hitting 0to the jumping chronological contour processes of a splitting tree, and then
showed that its local times are equal in distribution to a homogeneous, binary Crump-Mode-Jagers branching
process (CMJ-process). For a general spectrally positive Lévy process, Lambert and Simatos [45] explored the
genealogical structure of their local times preliminarily via an approximating sequence consisting of rescaled
binary CMJ-processes. Later, a detailed genealogical interpretation was given in [47] by considering the cor-
responding totally ordered measured tree that satisfies the splitting property. Meanwhile, Forman et al. [26]
established a locally uniform approximation for the local times of a driftless spectrally positive stable process
by endowing each jump with a random graph. Up to now, the genealogical structure of local times of gener-
al spectrally positive Lévy processes seems to be fairly clear. However, their macroevolution mechanisms, by
contrast, are still incomprehensible.
MSC 2020 subject classifications: Primary 60G52, 60J55, 60H20; secondary 60G22, 60F17, 60G55
Keywords and phrases: Local time,stable process, stochastic Volterra equation, heavy tail, Poisson random measure, marked Hawkes
point measure, Laplace functional, Ray-Knight theorem
This is an updated version of the manuscript “A Ray-Knight Theorem for Spectrally Positive Stable Processes".
1
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The purpose of this work is to establish stochastic equations for the local times of spectrally positive stable
processes and study their macroevolution mechanisms in the spatial direction. In contrast to the genealogical
interpretations given in the aforementioned literature, stochastic equations have many advantages including
•They provide an intuitive description for the evolution of local times in the spatial direction as well as a
detailed interpretation of their perturbations caused by each jump of stable processes.
•They allow us to study the local times by using tools and methods from the modern probability theory, e.g.,
stochastic integral inequalities, stochastic Fubini theorem and extreme value theory.
•They offer a kind of novel non-Markovian models and a convenient way of numerical analysis, which will
benefit greatly the related fields, e.g., processor-sharing queues and stochastic volatility models.
1.1. Overview of main results. Let ξ:= {ξ(t) : t≥0}be a one-dimensional spectrally positive stable
process with index 1 + α∈(1,2) and Laplace exponent
Φ(λ) := bλ +cλα+1 =bλ +Z∞
0e−λy −1 + λyνα(dy), λ ≥0,(1.1)
where b≥0,c > 0and να(dy), known as the Lévy measure, is a σ-finite measure on (0,∞)given by
να(dy) := cα(α+ 1)
Γ(1 −α)·y−α−2·dy.(1.2)
It is recurrent or drifts to −∞ according as b= 0 or >0. Let Wbe the scale function of ξand ¯να(x) :=
να([x, ∞)) the tail function of να. Let Lξ:= {Lξ(x, t) : x∈R, t ≥0}be the local times of ξ, where Lξ(x, t)is
usually interpreted as the amount of time that ξspends at level xup to time t. Denote by τL
ξ(ζ)the first time
that the amount of local time accumulated at level 0exceeds a given value ζ > 0; more accurate definitions can
be found in Section 2.1 and [9,43].
Let Lξ
ζbe the process {Lξ(x, τ L
ξ(ζ)) : x≥0}conditioned on τL
ξ(ζ)<∞. The first main result states that Lξ
ζ
is the unique weak solution1of the following stochastic Volterra equation (SVE)
Lξ
ζ(x) = Z∞
0Zζ
0∇yW(x)N0(dy, dz ) + Zx
0Z∞
0ZLξ
ζ(s)
0∇yW(x−s)e
Nα(ds, dy, dz ), x ≥0,(1.3)
where ∇yW(x) := W(x)−W(x−y),N0(dy, dz )is a Poisson random measure (PRM) on (0,∞)2with inten-
sity ¯να(y)dydz,e
Nα(ds, dy, dz )is a compensated PRM on (0,∞)3with intensity dsνα(dy)dz and independent
of N0(dy, dz ). The first stochastic integral in (1.3) represents the contribution of jumps up-crossing 0to the
local time at level xand the second stochastic integral, known as stochastic Volterra integral (SVI), can be
interpreted as the perturbations caused by jumps with initial positions above 0. Since the convolution kernel
delays the relaxation of its perturbations, the PRM Nα(ds, dy, dz )changes the local times continuously in the
spatial variable. This stands in striking contrast to the jumps in Itô’s stochastic differential equations (Itô’s S-
DEs) driven by PRM. Additionally, because of the joint impact of relative level x−sand jump-size yon the
convolution kernel, the SVE (1.3) cannot be written into the form of SVEs in [1,2,3,57,58].
Based on the SVE (1.3), in the second main result we use stochastic integral inequalities to provide a simple
proof for the Hölder continuity of Lξ
ζand the finiteness of all moments of the Hölder coefficient given in [4,
13,26]. As the novelty, we also establish a uniform upper bound for all moments of the Hölder coefficient and
a maximal inequality for the local times in the spatial variable. With the crucial assistance from the SVE (1.3),
1A continuous process with distribution Pis called a weak solution of (1.3) if there exists a stochastic basis, a PRM N0(dy, dz)
on (0,∞)2with intensity ¯να(y)dydz, a PRM Nα(ds, dy, dz)on (0,∞)3independent of N0(dy , dz)with intensity dsνα(dy)dz and
a continuous process Lξ
ζwith distribution Psuch that (1.3) holds almost surely. We say the weak uniqueness holds if any two weak
solutions are equal in distribution.
3
in the third main result we extend the method of duality developed in [3] to provide an explicit representation
of the Laplace functional E[exp{−λ·Lξ
ζ(x)−g∗Lξ
ζ(x)}]with λ≥0and g∈L∞(R+;R+). It states that the
Laplace exponent can be written as an affine functional of the initial state, in terms of the unique solution of the
nonlinear Volterra integral equation (nonlinear-VIE)
vg
λ(x) = λW 0(x) + g−Vα◦vg
λ∗W0(x), x > 0,(1.4)
where W0is the derivative of Wand Vαis a nonlinear operator acting on a locally integrable function fby
Vα◦f(x) := Z∞
0exp n−Zx
(x−y)+
f(r)dro−1 + Zx
(x−y)+
f(r)drνα(dy), x ≥0.(1.5)
Finally, we provide an alternative fractional integration and differential equation for the process Lξ
ζand its
Laplace exponent. In contrast to the SVE (1.3), the alternative equation takes it a step further and extracts the
impact of drift bon the local times from that of jumps. It also uncovers the remarkable similarity between Lξ
ζand
CB-processes in the evolution mechanism, which, together with the genealogical interpretations in [26,44,47],
tells that the SVE (1.3) defines a novel non-Markovian CB-process.
To illustrate the strength of these results, we use the SVE (1.3) to establish a stochastic equation for the heavy-
traffic limit of recaled queue-length processes of M/G/1 processor-sharing queues with unit service capacity,
heavy-tailed service distribution and stopped upon becoming empty. It can be seen as a continuation of [45],
where the weak convergence of rescaled queue-length processes was proved. In a sense, this helps to partially
answer Problem 2 stated by Zwart in [67] about the heavy-traffic limit of heavy-tailed processor-sharing queues;
readers may refer to the references of Zwart and his coauthors for details. enlightened by the self-exciting
property observed in the SVE (1.3), in the forthcoming preprint [34] we use the evolution mechanism of local
times of stable processes to model the sharp-raise clusters in rough volatilities and introduce a novel fractional
stochastic volatility model with self-excited sharp-raises.
1.2. Methodologies. We start the construction of the SVE (1.3) from the result that the local times of
nearly recurrent compound Poisson processes with unit negative drift, Pareto-distributed jumps are equal in
distribution to a class of nearly critical binary CMJ-processes, which converge weakly to the process Lξ
ζafter
rescaling; see [44,45]. Enlightened by the Hawkes representation of general branching particle systems estab-
lished in [32,65], we reconstruct the binary CMJ-processes as the intensity processes of nearly unstable marked
Hawkes point measures (MHPs) by translating the birth time, life-length and survival state of each individual
into the arrival time, random mark and kernel of an event respectively. Furthermore, we write each intensity
process into a SVE driven by an infinite-dimensional martingale in which the integrand is a functional of the
resolvent function related to the life-length distribution. Consequently, it suffices to prove the weak convergence
of these SVEs after rescaling to the desired SVE (1.3). Unfortunately, the Pareto-distributed life-length gives
raise to long-range dependence in the pre-limit SVEs, which derives a series of challenges and difficulties in
the proof including
•Along with the inseparable impact of time and life-length on the convolution kernel, the infinite-dimensional
driving noises not only lead to the failure of the approximation method and the integral-derivative method
developed in [2,3,38], but also make it hard to seek an approximation for the pre-limit SVEs.
•The resolvent function fluctuates drastically and explodes around 0after rescaling. This leads to the sharp
swings in the cumulative impact of infinite short-lived events on the pre-limit SVEs and also makes the
uniform control on the error processes challengeable.
•The resolvent function inherits long-range dependence from the life-length distribution. It prevents us from
transforming the pre-limit SVEs into the form of Itô’s SDEs and obtaining the weak convergence similarly
as in [37,65] by using the weak convergence results established in [41,42] for Itô’s SDEs.
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To overcome the first two difficulties, we start by analyzing in depth the direct and indirect impact of each event
on the pre-limit SVEs. Our analyses show that the cumulative direct impact of all events can be asymptoti-
cally ignored and a suitably rescaled version of their indirect impact asymptotically behaves as the backward
difference of scale function. This motivates us to approximate the SVIs in the pre-limit SVEs by replacing the
integrands with the backward difference of scale function. For the uniform control on the error processes, we
first split them into several parts according to the source and then prove the finite-dimensional convergence of
each part to 0separately. Based on a deep analysis about the backward difference of scale function, we prove
the C-tightness2of the approximating processes, which, together with the C-tightness result given in [45] for
the local times of nearly recurrent compound Poisson processes, yields the tightness of error processes. To over-
come the third difficulty, we establish a weak convergence result for SVIs with respect to infinite-dimensional
martingales, whose tightness and finite-dimensional convergence are obtained from the foregoing tightness
results and the weak convergence of the related Itô’s stochastic integrals respectively. More precisely, for a
given finite sequence of time points, we first introduce a sequence of Itô’s stochastic integrals with respective
to infinite-dimensional martingale satisfying that their finite-dimensional distributions at the given time points
are equal to those of the corresponding SVIs, and then prove their weak convergence to a limit process whose
finite-dimensional distribution at the given time points is equal to that of the desired limit SVI.
In the proof of existence and uniqueness of solutions of the nonlinear-VIE (1.4), the next two main difficulties
steam from the nonlinear operator Vαand the singularity of the function W0at the origin
•The interplay between the singularity of W0and Vαmakes the existence of local solutions of (1.4) around 0
quite difficult.
•Since Vαis path-dependent and does not satisfy the Lipschitz condition, it is difficult to identify the non-
explosion of local solutions and extend them into global solutions.
To bypass the first difficulty, we first prejudge the behavior of solutions near the origin with the help of an upper
bound estimate of Vαand the expansion given in [14] for solutions of fractional Riccati equations. In a specified
closed set in some Lebesgue space, we then find a local solution of (1.4) successfully by using Banach’s fixed
point theorem. To overcome the second difficulty, associated with a fractional differential equation related to
Vαwe first provide an upper bound estimate for a functional of each local solution, and then, along with the
comparison principle for fractional differential equations, establish a uniform control on the local solutions.
1.3. Related Literature. Let us comment on the relationship between the present work and the existing
literature. Firstly, based on the Markov property, Brownian local times were linked to Bessel processes via their
transition semigroups in [40,60] or their infinitesimal generators in [39,52]. However, the lack of Markovianity
of Lξ
ζmakes it impossible to establish the SVE (1.4) similarly as in the preceding references. Even if it could
be established successfully, the SVE (1.4) is beyond the scope of all existing literature [1,2,3,57,58] and the
existence of its solutions seems to be quite difficult to be proved in the standard way. On the other hand, the
present work establishes the well-posedness of the novel SVE (1.4). Secondly, the main results, as mentioned
above, are obtained by establishing a weak convergence result for the corresponding long-range dependent
MHPs. The first scaling limit theorem for Hawkes processes was established by Jaisson and Rosenbaum [37] in
the study of the asymptotic behavior of Hawkes-based price-volatility models in the context of high-frequency
trading. Their results state that under the short-memory condition, the rescaled intensity processes of nearly
unstable Hawkes processes converge weakly to the well-known CIR-model. The analogous scaling limits were
established for multivariate (marked) Hawkes processes in [23,65] and a jump-diffusion limit was given in [33]
for MHPs with exponential kernel. When the kernel is heavy-tailed, Jaisson and Rosenbaum [38] proved the
weak convergence of the integral of rescaled intensity process to the integral of a fractional diffusion process,
2Readers may refer to Definition 3.25 in [36, p.351] for the definition of C-tightness.
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see also [23,63] for the multivariate case. Because of many difficulties deriving from long-range dependence,
they left the weak convergence of rescaled intensity processes as an open problem. However, we stress that the
weak convergence result in this work is established for the intensity processes of MHPs. As the final remark,
we need to point out that different to the analogous version in [49,50], the Ray-Knight theorem in this work is
established for the local times rather than their functionals.
Organization of this paper. In Section 2, we first introduce general notation and properties of spectrally
positive stable processes, and then formulate the main results. In Section 3, we introduce some elementary
results and a SVE for the local times of a compound Poisson process with negative drift by linking them to a
MHP. Section 4is devoted to proving that Lξ
ζsolves the SVE (1.3). Its Hölder continuity is proved in Section 5.
In Section 6, we prove the exponential-affine transform formula as well as the existence and uniqueness of
solutions of the nonlinear-VIE (1.4). The proof for the alternative representation of Lξ
ζare given in Section 7.
Applications to processor-sharing queues are given in Section 8. Additional proofs and supporting results are
presented in the Appendices.
Notation. For any x∈R, let x+:= x∨0,x−:= x∧0and [x]be the integer part of x. For a Banach space V
with a norm k·kV, let D([0,∞),V)be the space of all cádlág V-valued functions endowed with the Skorokhod
topology and C([0,∞),V)the space of all continuous V-valued functions endowed with the uniform topology.
For any T ⊂ [0,∞)and p∈(0,∞], let Lp(T;V)be the space of V-valued measurable functions fon T
satisfying that kfkp
Lp
T:= RTkf(x)kp
Vdx < ∞. We also write kfkLp
Tfor kfkLp
[0,T ]and kfkLpfor kfkLp
∞. We
make the conventions that for x, y ∈Rwith y≥x,
Zy
x
=−Zx
y
=Z(x,y]
,Zy−
x−
=Z[x,y)
and Z∞
x
=Z(x,∞)
.
Denote by f∗gthe convolution of two functions f, g on R+. Let ∆hand ∇hbe the forward and backward
difference operators with step size h > 0, i.e., ∆hf(x) := f(x+h)−f(x)and ∇hf(x) := f(x)−f(x−
h).Let u.c.
→,a.s.
→,d
→and p
→be the uniform convergence on compacts, almost sure convergence, convergence
in distribution and convergence in probability respectively. We also use a.s.
=,d
=and p
=to denote almost sure
equality, equality in distribution and equality in probability respectively.
For a probability measure µon R, denote by Pµand Eµthe law and expectation of the underlying process
with initial state distributed as µ. When µis a Dirac measure at point x∈R, we write Pxfor Pµand Exfor Eµ.
For simplicity, we also write Pfor P0and Efor E0. For two σ-finite measures µ1, µ2on R, we say µ1≤µ2if
for any non-negative function fon R,
ZR
f(x)µ1(dx)≤ZR
f(x)µ2(dx).
We use Cto denote a positive constant whose value might change from line to line.
Acknowledgements. The author is grateful to Matthias Winkel who noticed the inaccuracy on the Hölder
continuity and recommended several helpful references. The author also like to thank the three professional ref-
erees for their careful and insightful reading of the paper, and for comments, which led to many improvements.
2. Preliminaries and main results.
2.1. Spectrally positive stable processes. Suppose that the spectrally positive stable process ξis defined on
a complete probability space (Ω,F,P)endowed with a filtration {Ft}t≥0satisfying the usual hypotheses. For
every t≥0, let µξ,t(dy)be the occupation measure of ξon the time interval [0, t]given for every non-negative
6
and measurable function fon Rby
Zt
0
fξ(s)ds a.s.
=ZR
f(y)µξ,t(dy).
The measure µξ,t is absolutely continuous with respect to the Lebesgue measure and the density, denoted by
{Lξ(x, t) : x∈R}, is square integrable; see Theorem 1 in [9, p.126]. The quantity Lξ(x, t)is called the local
time of ξat level xand time t. The two-parameter process Lξ:= {Lξ(x, t) : x∈R, t ≥0}is jointly continuous
and satisfies the occupation density formula
Zt
0
f(ξ(r))dr a.s.
=ZR
f(x)Lξ(x, t)dx, t ≥0,(2.1)
see Theorem 15 in [9, p.149]. Moreover, for any (Ft)-stopping time τ, it is easy to identify that
inf x≥0 : Lξ(x, τ ) = 0a.s.
= sup ξ(t) : t∈[0, τ ].(2.2)
The process {Lξ(0, t) : t≥0}is continuous and non-decreasing. This allows us to define the inverse local time
τL
ξ:= {τL
ξ(ζ) : ζ≥0}at level 0by τL
ξ(ζ) = ∞if ζ > Lξ(0,∞)and
τL
ξ(ζ) := inf s≥0 : Lξ(0, s)≥ζ,if ζ∈[0, Lξ(0,∞)].
From Proposition 4 in [9, p.130], the process τL
ξis a subordinator, killed at an independent exponential time if
ξis transient (b > 0), and its Laplace transform is of the form
Eexp −λ·τL
ξ(ζ)= exp −ζ/uλ(0), λ > 0, ζ ≥0,(2.3)
where uλ:= {uλ(y) : y∈R}is the density of the λ-resolvent kernel of ξ. When b= 0, we have Lξ(0,∞)a.s.
=∞
and τL
ξ(ζ)<∞a.s. When b > 0, the potential density u0is well-defined as the limit case λ= 0 for uλ. In this
case, we consider the limit case λ→0+ for (2.3) to get
PLξ(0,∞)≥ζ= 1 −PτL
ξ(ζ) = ∞= exp −ζ/u0(0),
which induces that Lξ(0,∞)is exponentially distributed with mean u0(0) and PτL
ξ(ζ) = ∞>0for any
ζ > 0. For a, θ > 0, let a·ξ(θ·) := {aξ(θt) : t≥0}. The equality (2.1), along with the change of variables,
implies the following two equivalences3
La·ξ(θ·)
a.s.
=(aθ)−1·Lξ(x/a, θt) : x∈R, t ≥0and τL
a·ξ(θ·)(ζ)a.s.
=θ−1·τL
ξ(aθζ), ζ ≥0.(2.4)
Let {W(x) : x∈R}be the scale function of ξ, which is identically zero on (−∞,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.(2.5)
The scale function Wis continuous on Rand differentiable on (0,∞)with derivative denoted as W0; see
Theorem 8 in [9, p.194]. Applying the integration by parts to (2.5), we have
Z∞
0
e−λxW0(x)dx =Z∞
0
λe−λxW(x)dx =1
b+cλα, λ > 0.(2.6)
The Laplace transform of Mittag-Leffler function4yields that W0has the representation
W0(x) = c−1xα−1·Eα,α−b/c ·xα, x > 0.
3Actually, these two equivalences hold for any Lévy process.
4The Mittag-Leffler function Eα,α on R+is defined by Eα,α(x) := P∞
n=0
xn
Γ(α(n+1)) ; see [31] for a precise definition and a survey
of its properties, e.g., Z∞
0
e−λxaxα−1Eα,α (−a·xα)dx =a
a+λα, a, λ ≥0.
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The smoothness of Eα,α induces that Wis infinitely differentiable on (0,∞). When b= 0, we have Eα,α(0) =
1/Γ(α)and
W(x) = xα
c·Γ(α+ 1), W 0(x) = xα−1
c·Γ(α), W 00(x) = (α−1)xα−2
c·Γ(α), x > 0.(2.7)
When b > 0, the function bW 0is a Mittag-Leffler density function and 1−bW (x)→0as x→∞. The properties
of Mittag-Leffler distribution/density function; see [31,54,56], yield that the scale function Wis Hölder
continuous with index α. Moreover,
W(x)∼xα
c·Γ(α+ 1), W 0(x)∼xα−1
c·Γ(α), W 00(x)∼(α−1)xα−2
c·Γ(α),as x→0+,
and
W(x)∼1
b−c·x−α
b2·Γ(1 −α), W 0(x)∼cα ·x−α−1
b2·Γ(1 −α), W 00(x)∼ −cα(α+ 1) ·x−α−2
b2·Γ(1 −α)as x→∞.
A direct consequence of these asymptotic properties and (2.7) is that uniformly in x > 0,
W(x)≤C·xα,|W0(x)|≤C·xα−1and |W00(x)|≤C·xα−2.(2.8)
By the mean-value theorem, it is easy to identify that uniformly in x, y > 0,
|∇yW(x)|=|∆yW(x−y)|≤C·xα∧|(x−y)+|α−1y.(2.9)
In addition to the scale function, we will also need a Sonine pair (K, LK)on (0,∞)defined by
K(x) := xα−1
c·Γ(α)and LK(x) := c·x−α
Γ(1 −α), x > 0,(2.10)
which satisfies the Sonine equation, i.e.,
K∗LK=LK∗K≡1.(2.11)
In the theory of Volterra equations; see [30], the function LKis also said to be the resolvent of the first kind
related to Kand vice versa. When b > 0, a simple calculation shows that the function bW 0is the resolvent of
the second kind corresponding to bK, which is usually introduced by means of the resolvent equation
bW 0=bK −(bK)∗(bW 0).(2.12)
The function bK is usually referred as the resolvent associated to bW 0. Convolving both sides of (2.12) by LK
and then dividing them by b, we have
LK∗W0=W0∗LK= 1 −bW.(2.13)
Actually, this equality also holds when b= 0, since W0=Kin this case; see (2.7) and (2.10).
2.2. Main results. We now formulate the main results for the local times of ξat the stopping time τL
ξ(ζ)for
a given value ζ > 0. For convenience, we write Lξ
ζfor the process {Lξ(x, τ L
ξ(ζ)) : x≥0}under P(·|τL
ξ(ζ)<
∞). Since τL
ξ(ζ)<∞a.s. when b= 0, this conditional probability law turns to be P. When b > 0, the stopping
time τL
ξ(Lξ(0,∞)) is finite almost surely and equal to the last time that ξhits 0. In this case, we are also
interested in the process
Lξ
∞:= Lξ(x, ∞) : x≥0a.s.
=Lξx, τ L
ξ(Lξ(0,∞)):x≥0,(2.14)
under P. Let %be an exponential random variable with mean u0(0), independent of Nα(ds, dy, dz )and
N0(dy, dz ). Our first main theorem establishes SVEs for Lξ
ζand Lξ
∞.
THEOREM 2.1. We have the following:
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(1) For each ζ≥0, the process Lξ
ζis a weak solution of (1.3).
(2) If b > 0, the process Lξ
∞is a weak solution of (1.3) with ζ=%.
(3) The weak uniqueness of non-negative solutions holds for (1.3).
REMARK 2.2. By the change of variables and Proposition A.1 with p= 2, there exists a constant C > 0
such that for any x≥0,
Zx
0
ds Z∞
0∇yW(x−s)2να(dy) = Zx
0
ds Z∞
0∇yW(s)2να(dy)≤C·xα.
Taking expectations on both sides of (1.3) and then using Fubini’s theorem along with (2.13), we have
ELξ
ζ(x)=EhZ∞
0Zζ
0∇yW(x)N0(dy, dz )i
=ζZ∞
0∇yW(x)¯να(y)dy
=ζZ∞
0
¯να(y)dy Zx
(x−y)+
W0(s)ds =ζ·W0∗LK(x) = ζ1−bW (x)≤ζ, x ≥0.(2.15)
The SVI in (1.3) has finite quadratic variation and is well defined as an Itô integral; see [35, p.59-63].
REMARK 2.3. By the exponential formula for PRMs; see [9, p.8], we have for any λ≥0,
Ehexp n−λZ∞
0Zζ
0∇yW(x)N0(dy, dz )oi= exp n−ζZ∞
0
(1 −e−λ∇yW(x))¯να(y)dyo.
From (2.15) and the fact that ∇yW(x)→0uniformly in yas x→0, we have
Ehexp n−λZ∞
0Zζ
0∇yW(x)N0(dy, dz )oi∼exp n−ζ λ Z∞
0∇yW(x)¯να(y)dyo→e−ζλ.
Thus the first term on the right side of (1.3) converges to ζa.s. as x→0+. We make the convention that it is
equal to ζa.s. when x= 0, which is consistent with the fact that Lξ
ζ(0) a.s.
=ζ.
REMARK 2.4. By (2.15), the SVE (1.3) can be written as
Lξ
ζ(x) = ζ1−bW (x)+Z∞
0Zζ
0∇yW(x)e
N0(dy, dz )
+Zx
0Z∞
0ZLξ
ζ(s)
0∇yW(x−s)e
Nα(ds, dy, dz ), x ≥0,(2.16)
where e
N0(dy, dz ) := N0(dy, dz)−¯να(y)dydz. Here the first term on the right side of this equality represents
the average local time at level x. The second term can be interpreted as the perturbations caused by jumps
up-crossing 0; the third term can be translated into the perturbations caused by jumps with initial positions
above 0but below x. More precisely, the convolution kernel ∇yW(x−s)describes the impact of a jump with
initial position sand size yon the local time at level x. Notice that ∇yW(x−s)increases when x∈[s, s +y]
and decreases as x→∞. It would be sensible to consider the jump size of each jump as its life-length/residual-
life during which it perturbs the local times directly. This interpretation is consistent with the genealogical
interpretations in [26,47].
REMARK 2.5. Because of the delayed and smooth relaxation of its perturbations, the PRM Nα(ds, dy, dz )
fails to make solutions of (1.3) jump. This phenomena cannot be observed in Ito’s SDEs driven by PRM, since
the PRM releases its perturbations instantaneously that give raise to jumps in the solutions. Consequently,
the continuity of driving noises is a necessary condition for the continuity of solutions of Ito’s SDEs; see [35,
Chapter III-IV] and [59, Chapter II-V].
9
REMARK 2.6. It is necessary to specify that the SVE (1.3) is beyond the scope of the existing literature, e.g.
[1,2,3,23,38,57,58]. More precisely, all SVEs studied in these literature are driven by finite-dimensional
semimartingale and always can be written as
X(t) = H(t) + Zt
0
K(t, s, Xs)dZ(t), t ≥0,(2.17)
where His a given function, Kis a d×kmatrix-valued convolution kernel on R2
+×Rand Zis a k-dimensional
Itô’s semimartingale whose differential characteristics are functions of X. Differently, the SVI in (1.3) is driven
by an infinite-dimensional martingale; see [42] and Appendix C. Since the impact of time ton the convolution
kernel ∇yW(t)is tightly intertwined with that of mark y, one cannot write (1.3) into the form of (2.17). Con-
sequently, it is difficult to prove the existence of solutions of (1.3) by using the approximation method used in
[1,3] or the martingale problem theory developed in [2].
REMARK 2.7. Ito’s SDEs with non-negative solutions have been widely studied in [7,10,15,16,27] under
two key conditions: (i) when solutions hit 0, the diffusion vanishes and the drift turns to be non-negative; (ii)
solutions cannot jump into the negative half-line. In particular, it is the strong Markovianity that turns the state
0to be a tripper or a reflecting boundary, which results in the existence of non-negative solutions. However, the
convolution kernel in (1.3) results in the lack of (strong) Markovianity of the solutions and makes the standard
stopping time method fail to prove the existence of non-negative solutions. Fortunately, thanks to Theorem 2.1,
the existence of non-negative solutions of (1.3) follows directly from the non-negativity of Lξ
ζ.
REMARK 2.8. The point 0is an absorbing state 5for the process Lξ
ζ(and also Lξ
∞), i.e., once it hits 0, it
will stay at 0forever. Indeed, the equivalence (2.2) shows that conditioned on τL
ξ(ζ)<∞,
τ0:= inf x≥0 : Lξ
ζ(x) = 0<∞, a.s. and Lξ
ζ(τ0+x)a.s.
= 0, x ≥0.
Usually, the lack of Markovianity makes it difficult to obtain this property from the SVE (1.3). Even for the SVE
(2.17), the absorbing states and polarity are also unclear up to now.
The SVE (1.3) makes it possible to study the local times of ξby using tools and methods from stochastic
analysis, e.g., stochastic integral inequalities, stochastic Fubini theorem and martingale problem theory. To
illustrate this, the next main theorem proves the Hölder continuity of Lξ
ζby using the Kolmogorov continuity
theorem and also provides a uniform upper bound for all moments of the Hölder coefficients by using the
Garsia-Rodemich-Rumsey inequality. For κ∈(0,1] and x > 0, the κ-Hölder coefficient of a Hölder continuous
function fon [0, x]is defined by
kfkC0,κ
x:= sup
0≤y<z≤x
|f(y)−f(z)|
|y−z|κ.
THEOREM 2.9 (Hölder continuity). For each ζ≥0, we have the following:
(1) The process Lξ
ζis Hölder-continuous of any order strictly less than α/2.
(2) For each κ∈(0, α/2) and p≥0, there exists a constant C > 0such that for any x≥0,
EkLξ
ζkp
C0,κ
x≤C·(1 + x)p(α−κ).
5Although the two terminologies absorbing state and polarity are initially introduced for Markov processes, it is sensible to use them
to describe the analogous phenomena in other stochastic processes. Precisely, a state in a process is said to be an absorbing state if once
it is entered, it is impossible to leave. A set is said to be a polar set for a process if it cannot be entered in finite time.