# Fault Localization in Cloud using Centrality Measures

Author Affiliation Country Contact Details

###### Abstract

Fault localization is an imperative method in fault tolerance in a distributed
environment that designs a blueprint for continuing the ongoing process even
when one or many modules are non-functional. Visualizing a distributed
environment as a graph, whose nodes represent faults (fault graph), allows us
to introduce probabilistic weights to both edges and nodes that cause the
faults. With multiple modules like databases, run-time cloud, etc. making up a
distributed environment and extensively, a cloud environment, we aim to
address the problem of optimally and accurately performing fault localization
in a distributed environment by modifying the Graph optimization approach to
localization and centrality, specific to fault graphs.

#### Keywords:

Fault Localization, Centrality Measures, Cloud Computing

##  1 INTRODUCTION

Cloud computing is the on-demand availability and delivery of various computer
system resources, cloud storage and computing power, without requiring direct
management by the user. Today, large clouds, predominantly have functions
distributed over various locations from central servers. Cloud computing
delivers three types of computing resources - hardware, software, and software
development framework, also known as a platform. Recently, the extensive use
of cloud services for hosting enterprise applications is leading to service
reliability and availability issues for both service users and providers.
Several types of faults may occur in the cloud environment leading to failures
and performance degradation [1].

Failures, generally lead to breakdown or shut down of a system. In distributed
computing architecture, the notion of failure is characterized by partial
failures. A fault may occur in any constituent node, process, or network
component of the distributed system. This leads to a partial failure and
consequently, the performance of the system degrades until there is a complete
breakdown. Fault tolerance in a system is the capability of a system to
continue performing its expected function regardless of faults. This way, a
fault-tolerant system is expected to be reliable and operate successfully in
the case of a failure or breakdown. In case of a fault occurring from any
particular node, a fault-tolerant system must be able to identify potentially
vulnerable modules in a distributed system to replicate their functionality to
prevent complete failure of the system.

State-of-the-art fault localization approaches rely on long training sessions
based on simulated faults, which is an expensive practice, since it is hard to
inject multiple classes of faults in several machines with different access
controls, and iterate this process while the system evolves [2]. Fault
detection using deep learning is also studied in the case of small amount of
cloud fault feature data [3].

Centrality measures in a graph are a direct indication of influential nodes
and links in the network. In most cases, they show the relative significance
of nodes and edges in the system. It is a direct indication of influential
nodes and links in the network. Researchers have developed many algorithms for
identifying relatively important nodes in a network. The basic and locally
computed centrality measure is degree centrality. Other commonly used metrics
are betweenness centrality, closeness centrality, PageRank, and HITS
eigenvector centrality, which are more relevant in single-layer networks,
limiting their applications in multi-layer networking systems [4]. Among
these, PageRank is a popular and frequently used metric, designed for ranking
of web pages, it is widely applied in other applications also.

In this paper, we propose a fault localization method that uses centrality
measures and works within the constraints of Cloud systems to help us
accurately identify important components in the cloud and localize them in
case of a breakdown or failure, triggered by itself or another component it
depends on.

The contributions of this work are: (i) a novel approach to identify highly
important and vulnerable components in cloud environments to localize the
fault triggering components (ii) an experimental evaluation of the suggested
method for identifying and locating different combinations of faults arising
from different modules.

The paper is organized as follows. Section 2 overviews the existing work on
fault localization techniques. Section 3 discusses the theoretical background
required. Section 4 proposes a centrality measure-based localization method.
Section 5 illustrates the experimental results of the approach. Section 6
summarizes the main contribution of the paper and a few future work possible.

##  2 RELATED WORK

A failure occurs when the user perceives that the software has ceased to
deliver the expected result. The user may need to identify the severity of the
levels of failures such as catastrophic, critical, major, or minor, depending
on their impact on the systems. Omission, Hardware, Software, Network,
Response, Virtual Machine (VM), Application failures are various types of
failures in Cloud systems. An abnormal condition or defect at the equipment,
component or sub-system level which may lead to a failure is termed as a
Fault. Omission faults, Aging-related faults like denial of services, Response
faults, Software Faults, Timing faults, Transient faults are various types of
faults [5]. The fault localization process is to be undertaken by developers
to identify the fault responsible for the failure. To handle the fault that
may occur there are different types of techniques of fault tolerance in the
literature that is classified into two categories:

###  2.1 Proactive Techniques

In Proactive Fault Tolerance, the problems are anticipated before they occur
and are avoided based on the influence on the task. One of the methods in
Proactive Fault Tolerance is preemptive migration, which uses a control system
where the tasks are proctored. When a task is found to be vulnerable, it is
shifted to another virtual machine instance. The disadvantage of this method
is the wait time for calling another virtual machine [6]. Another example of a
proactive fault-tolerance method is Proactive Co-ordinated fault tolerance or
PCFT. The PCFT approach aims at monitoring a deteriorating physical machine.
Given a deteriorating physical machine, the PCFT approach searches for some
optimal target physical machines for the virtual machines hosted on the
deteriorating physical machine [7].

###  2.2 Reactive Techniques

Reactive fault tolerance approaches handle the faults after their occurrence.
The working of reactive approaches is response-based rather than anticipation.
Reactive approaches are usually conservative and need not examine the system
behavior. Steps involved in reactive fault tolerance include (a) Replication:
this is a technique used to replicate the task on more than one resource, (b)
Task re-submission: when the task fails it will run again in either the same
node or on a different resource, this increases the time taken for execution
over the expected time because of the repetition of task (c) Retry: re-
execution of the failed task in the same resource, but it consumes the same
execution time as the Task re-submission, (d) Checkpointing or Restart: It
takes snapshots at different times during task execution. This allows the task
to be re-executed from the last fault-free state [8].

Other methods including the meta-heuristic-based method have also been used
for improving fault tolerance methods in cloud systems to enhance their
performance and to make the systems robust [9]. Failure risk based Cloud
Classification System reliability and efficiency improvement methods have been
developed by combining risk identification with proactive actions [10].

##  3 THEORETICAL BACKGROUND

In this section, we discuss some basic concepts about the types of faults in
cloud architecture and centrality measures in a graph.

###  3.1 Cloud Computing Infrastructure

Cloud computing infrastructure includes the computers, network facilities,
storage devices, and other connected components necessary for offering cloud
computing resources and services to users as mentioned in Table 1. These
hardware components are located within enterprise data centers. These
encompass solid-state drives, multi-core servers, and hard disk drives
offering stable storage and network devices, such as firewalls, switches, and
routers; all on a large scale [11][12].

Various components in a distributed system
---
Servers |  The physical machines that act as host machines for one or more virtual machines.
Virtualization |  Technology that abstracts physical components such as servers, storage, and networking and provides these as logical resources.
Storage |  In the form of Storage Area Networks (SAN), network-attached storage (NAS), disk drives etc. along with facilities such as archiving and backup.
Network |  To provide interconnections between physical servers and storage.
Management |  Various software for configuring, management and monitoring of cloud infrastructure including servers, network, and storage devices.
Security |  Components that provide integrity, availability, and confidentiality of data and security of information, in general.
Table 1: Table 1: Various components in a distributed system

Many factors could trigger a fault including data corruption, hardware or
software or network outage, unreliable clocks, method failure, parity error,
head crash, denial of service, disk full, etc. Faults, in general, are
classified into the below-mentioned categories:

##### Network faults:

Since cloud computing resources are accessed over the internet, one of the
most important causes of failure in cloud computing is network faults. These
faults may occur due to partitions in the network, packet loss or corruption,
congestion, failure of the destination node or link, etc.

##### Physical faults:

Physical faults mainly occur in hardware resources, such as faults in CPUs, in
memory, in storage, failure of power etc.

##### Process faults:

These faults may occur in processes because of resource shortage, bugs in
software, incompetent processing capabilities, etc.

##### Service expiry faults:

If the service time of a resource expires while an application that leased it
is using it, it leads to service faults.

##### Why Perform Fault Localization?

Fault localization is essential to examine the cloud architecture and help
developers improve the reliability and safety of information systems. Some of
the examples of fault symptoms include unreachable hosts or networks and in
most cases, the origin of the path traced by the defect is important to
understand the root cause of a fault.

###  3.2 Centrality Measures

In graph theory and network analysis, indicators of centrality identify the
most important vertices and edges within a graph. They are scalar values given
to each node or edge in the graph to quantify its importance based on some
assumptions. Centrality concepts were originally developed for social network
analysis, and many of the terms used to measure centrality reflect their
sociological origin. There are a vast number of centrality measures used by
analysts. In our approach, we use closeness, eigenvector, and alpha centrality
measures.

##### Closeness Centrality:

In a connected graph, closeness centrality (or closeness) of a node is a
measure of centrality in a network, calculated as the reciprocal of the sum of
the length of the shortest paths between the node and all other nodes in the
graph. Thus, the more central a node is, the closer it is to all other nodes
[13].

##### Eigenvector Centrality:

In graph theory, eigenvector centrality is a measure of the influence of a
node in a network where relative scores are assigned to all nodes in the
network based on the concept that connections to high-scoring nodes contribute
more to the score of the node in question than equal connections to low-
scoring nodes. A high eigenvector score denotes that a node is connected to
many nodes who themselves have high scores [14].

##### Alpha Centrality:

In graph theory and social network analysis, alpha centrality is an alias to
Katz centrality [15]. It is a measure of the centrality of nodes within a
graph. It is an extension of eigenvector centrality where the degree by which
the nodes are impacted by the importance of external sources are taken into
account.

##  4 PROPOSED MODEL

Any graph-theoretic model relies on a precise establishment of relationships
between nodes involved in the graph and require profound understanding of the
underlying system behavior [16]. Fault propagation graphs, usually an input to
clustering techniques, are common ways to trace the components affected by the
failure of a subset of components in a network of nodes [17]. We define a
directed fault graph FG(V, E) whose V represents different faults associated
with components and E represents whether or not will occurrence of a fault
would trigger another fault. The proposed centrality measure-based fault
localization method assigns the importance of modules in a given cloud
architecture with computed rank values of different faults associated with a
component, obtained from the fault graph model generated. Computation is based
on iterations over fault nodes with weights on edges and nodes calculated with
certain probabilistic values.

We initially consider the list of all the cloud components like Runtime, pod,
database, etc., labeled C1, C2, C3, … , Cn that could potentially become
faulty. Further, we derive a list of all faults fki, the suffix indicating the
ith fault arising from component k. With this established, one-to-many mapping
between components and faults associated with the components is made as shown
in Figure 4.1. In any given cloud environment, components in the environment
can fail due to multiple faults. Subsequently, a fault could trigger a subset
of other faults as the cloud could be seen as one connected graph.

Every fault fki, initially, has an independent probability Pfki of arising
regardless of being triggered by any other fault. Identification of a
particular fault in a network is carried out using various failure detection
methods including signature and anomaly-based IDS [18]. These probabilistic
weights are assigned to all faults which are mapped to cloud modules. These
weights are based on the nature of the module and other external factors,
which by default are uniform for all the faults and can be assigned based on
the log of fault data.

We introduce an impact factor value ifv(fki, fmj) that tells how much of an
impact the fault fki has on fault fmj where the relationship from fki to fmj
is one-to-one. These values are specific to architectures and they are
calculated based on normalized conditional probability values of a pair of
faults and the odds of a fault occurring relative to the occurrence of another
fault from the log data of faults. These values represent the initial edge
weights of the fault graph FG. As mentioned earlier, the nodes of the directed
fault graph FG, have values that represent the initial independent
probabilities of occurrence of the corresponding fault. Figure 4.2 shows a
schematic graph consisting of two distributed components C1 and C2, fault f1i
occurring with independent probability 0.179 and f2j with probability 0.232
are caused by C1 and C2 respectively where ifv(f2j, f1i) = 0.34 is the impact
factor value the fault node f2j has on f1i. The influence value is the
normalized conditional probability values of the pair f1i and f2j

Figure 1: Figure 4.1 A graphical representation of different modules in a
distributed computing model - C1, C2, C3, …, Cn where multiple faults - f11,
f12, f21, f22, etc. are caused by the failure or breakdown of modules C1 to Cn
Figure 2: Figure 4.2 A schematic graph consisting of two distributed
components C1 and C2 where f1i and f2j are the faults caused by them
respectively where ifv(f2j, f1i) is the influence fault node f2j has on f1i

Subsequently, the edge weight of any edge from f1 to f2 with an initial edge
weight, impact factor ifv(f1, f2) and the node weight of a node, Pfi,
representing the probability of occurrence of the fault it represents, f2,
impacted by f1 in the fault graph FG are updated to

$E\textsubscript{f1,
f2}=P\textsubscript{f\textsubscript{1}}*ifv\textsubscript{f\textsubscript{1},f\textsubscript{2}}+Z,$
$P\textsubscript{f2}=P\textsubscript{f\textsubscript{1}}*ifv\textsubscript{f\textsubscript{1},f\textsubscript{2}}$

where Z representing normalization constant which is added to all the nodes
that are involved in a cycle to stop the iteration when updating the node
weights is

Different edges in the fault graph have different edge weights which are
differentiated using edge thickness as shown in Figure 4.3. It is
experimentally seen that the efficiency and the accuracy of the node finding
algorithm depend on the accuracy of the probabilistic assumption of each node.

Figure 3: Figure 4.3 A graphical representation of subset of fault node graph
with faults of different components connected with probabilistic edge weight
denoted by networkX thickness

###  4.1 Identification of nodes based on centrality measures

Most site reliability engineering teams use monitoring and alerting systems in
which a node’s signal value is used to determine the subsequent actions to be
taken. The four golden signals in monitoring are traffic, latency, saturation,
and errors. In our model, we record the cause and signal values emitted by the
components and we set threshold values for the signal values emitted by the
cloud components, and whenever the signal values cross the threshold value,
with the type of fault recorded, fault propagation graph FP from the original
directed fault graph FG, is initialized as shown in Figure 4.4.

Figure 4: Figure 4.4 A schematic graph in which black colored node represents
a potential faulty node from which a fault propagation graph is constructed

We run eigenvector-based and closeness-based centrality indices on the fault
propagation graph to find the possibly faulty nodes. With the nodes indicating
the type of fault with the possibility of occurrence known, we can then find
the vulnerable cloud components from the mapping between faults and cloud
modules. For every fault node $f\textsubscript{j}$ $\in$ V, the centrality
measure values are calculated using the three centrality measures,

$EVR(f\textsubscript{n})=\sum_{f\textsubscript{j}\in
V}\frac{EVR(f\textsubscript{j})}{L(f\textsubscript{j})}*ifv(f\textsubscript{j},f\textsubscript{n})$

where EVR(fn) is the eigenvector based rank for a fault node fn, and L(fj) is
the number of nodes that are connected to fj and,

$CR(f\textsubscript{n})=\sum_{f\textsubscript{j}\in
V}\frac{1}{d(f\textsubscript{n},f\textsubscript{j})}*ifv(f\textsubscript{n},f\textsubscript{j})$

where CR(fn) is the closeness centrality based rank for a fault node fn,
calculated for any node fj that fn connects to and d(fn, fj) is the shortest
distance between fn and fj.

Different choices of centrality measure rank fault nodes differently. With the
important nodes, representing the faults of high importance and high
likelihood of occurring ranked and number of nodes to consider for fault
localization known, the faults can be mapped back to the components that raise
them to know which get an insight of whose replicas are needed [19].

Alpha Centrality on weighted fault network graph begins with the
identification of high centrality score nodes using alpha centrality measure
where we consider the weighted adjacency matrix AFP of the fault propagation
graph generated where the edge weight between any fi and fj, WEi,j, denotes
the degree of influence of fi on fj.

With the weighted adjacency matrix for the fault graph FG known, alpha
centrality measure values for the nodes representing faults is calculated.
Alpha centrality measure for the graph FG is given by,

$C\textsubscript{Katz}(f\textsubscript{i})=\alpha\sum_{j=1}^{n}A\textsubscript{FG}({\textsubscript{j,
i}}).C\textsubscript{Katz}(f\textsubscript{j})+\beta$

where CKatz(fi) is the alpha-centrality based rank for a fault node (fi), AFG
represents the adjacency matrix of the fault propagation graph, $\alpha$ is
called attenuation factor that lies within the range [0, 1] and $\beta$ is the
initial weight given to vertex as a way to differentiate vertices using some
quality not modelled by adjacencies, which in our case is the probability of
occurrence. Rewriting the equation in a vector form, we get,

$C\textsubscript{Katz}=\alpha A\textsuperscript{T}C\textsubscript{Katz}+\beta$
$C\textsubscript{Katz}=\beta(I-\alpha
A\textsuperscript{T})\textsuperscript{-1}$

When $\alpha$ = 0, the eigenvector centrality is nullified and all the nodes
have the same centrality value $\beta$. It can also be observed that as
$\alpha$ value increases, the impact of $\beta$ reduces. We select 0 $\leq$
$\alpha$ $<$ $\frac{1}{\lambda}$, where $\lambda$ represents the numerically
largest eigenvalue of AT. In the case of alpha centrality, calculation of node
centrality values in weighted networks lets us consider the edge weights which
is a direct measure of the degree of influence of a fault on another fault.

##  5 EXPERIMENTAL RESULTS

We conducted tests with over a hundred simulated faults originating from
various cloud modules. With the module-fault mapping established, initial
probabilities of fault occurrence for independent faults were taken from the
recent datasets [20]. Further, the impact factor mapping between fault nodes
was assigned based on the frequency of dependency of one fault triggering
module over another. With these values computed and the faulty node from which
the fault propagation graph is constructed known, the relative importance of
fault nodes are calculated based on the three centrality measures. In a real-
time system, this could be input to what requires replication for fault
tolerance [19]. Table 2 represents the list of simulated faults considered
along with the module they originate from. The results of the computation
directly relate to the modules that are more vulnerable than the rest, in an
occasion of a fault occurring.

Fault List
---
Fault | Module
log.fault.vm.steps.smf.svc.maintenance | VM
log.fault.vm.steps.cpu.generic-sparc.strand | VM
log.fault.proxyDeployment.steps.InvalidPattern | Proxy
log.fault.runtime.steps.jwt.KeyParsingFailed | Runtime
. . . |
log.fault.dbauth.steps.QuotaViolation | Database
log.fault.runtime.steps.jwt.TokenNotYetValid | Runtime
Table 2: Table 2: List of faults and the module they originate from Centrality Threshold |  0.6 |  0.7 |  0.8
---|---|---|---
(a) Closeness |  |  |
(b) Eigen |  |  |
(c) Alpha |  |  |
Figure 5.1 Graphical representation of fault node graphs with 100 faults using networkX o X[c]X[c]X[c] Threshold = 0.6 |  Threshold = 0.7 |  Threshold = 0.8
---|---|---
figure figure figure Figure 5.2 Graphical comparison of Closeness, Eigenvector
and Alpha
centrality measures for various threshold values

Vulnerable modules were found from the faults that were most likely to occur
based on the centrality rankings for different threshold values. The test was
run with 85, 90, 95, 100, 105, 110 faults and vulnerable modules were observed
for threshold values of 0.6, 0.7, and 0.8 using the three centrality measures.
Figure 5.1 shows the graphical representation of the fault node graph
constructed using networkX with 100 faults considered for closeness
centrality, eigenvector centrality, and alpha centrality for threshold values
of 0.8, 0.7, and 0.6 respectively. It is observed that alpha centrality finds
the vulnerable modules for all the considered threshold values more accurately
than eigenvector and closeness centrality measures. Figure 5.2 shows the
comparison of observations with the three different centrality measures for
threshold values 0.6, 0.7, and 0.8 respectively, against the ideal values.
This high accuracy observed in alpha centrality can be explained by the
attribution given to the edge weight calculated with the degree of impact the
failure of one node has on the other. It is also observed that the closeness
centrality measure produces the highest number of false-positive faults among
the three considered centrality measures. Since the efficiency with which the
faults propagate is less relevant in finding the vulnerable fault modules, a
high number of false-positive cases are observed in closeness centrality
measure. The accuracy values recorded for the considered faults using alpha,
eigenvector, and closeness centrality measures for threshold values 0.6, 0.7,
and 0.8 are shown in Table 3.

Centrality/Threshold | 0.6 | 0.7 | 0.8
---|---|---|---
Alpha | 96.37 | 93.8 | 93.17
Eigenvector | 88.89 | 65.3 | 60.3
Closeness | 76.69 | 61.51 | 42.32
Table 3: Accuracy for alpha, eigenvector and closeness centrality measures for
threshold values 0.6, 0.7 and 0.8

For the simulated faults considered, it is also observed that the accuracy
dropping with the threshold value increasing as the number of potential faulty
nodes decreases.

_
Research Questions_
In this paper, we address two questions that help in identifying faults in a
distributed environment:

Q1. _Does the choice of centrality index affect the accuracy of identifying
the subset of potential faulty nodes for fault localization?
_
As discussed, we have implemented three centrality-based indices used in fault
localization, namely closeness, eigenvector, and alpha centrality measures. It
is interesting to observe that alpha-centrality performs better for various
threshold values than eigenvector and closeness centrality measures. Using
alpha centrality, the calculation of node centrality values in weighted
networks lets us consider the edge weights, a direct measure of the degree of
influence of a fault on another fault. This results in better accuracy since
the computation of centrality values is fault-specific and not cloud module-
specific. It was alarming to observe that the closeness centrality measure
produces the highest number of false-positive faults among the three
considered centrality measures.

Q2. _In the graph model, how does the representation of faults as nodes
improve the accuracy when compared to the representation of components as
nodes?
_
Even though having faults representing the vertices increases the complexity
of the overall computation significantly due to one-to-many mapping between
components and their faults, having a fault propagation-based graph reduces
the number of false-positive cases as different faults in a component have
different effects on other components. Faults representing nodes make the
centrality calculation more specific to a fault, and hence we get a more
accurate subset of potentially faulty nodes. This approach ensures both higher
accuracy in choosing other modules when one faulty node is observed as well as
prevents the selection of components with lesser probabilities of failing thus
reducing the number of false-positive cases.

##  6 FUTURE WORK

The cloud archetype has become more practical than ever and has been adopted
by researchers, the IT industry, and other organizations. Fault-tolerance
approaches are expected to improve services in the cloud environment. However,
reliability issues of the cloud workflows have not been extensively considered
in the existing approaches and evaluation has been done using some basic
metrics, such as response time, availability, and throughput. Many studies
model the system behaviour that includes execution paths and component
interactions. A more complex, but interesting variant of this research work
includes link failure between cloud components and fault tolerance of the same
as well as employing more precise methods to compute $\alpha$ values in the
centrality measure. Deep Learning approaches incorporated with centrality
measures may be studied to predict fault-tolerance when the volume of the data
increases.

##  7 CONCLUSION

This research work has implemented centrality measure-based fault localization
techniques to locate faults in distributed systems with a probabilistic
method. In our experiments, we conducted tests using the suggested method with
simulated faults in a distributed computing environment. The choice of
centrality index impacts the precision of fault localization. The results of
these experiments with eigenvector, closeness, and alpha centrality measures
were analyzed. Among the three measures, it is observed that the
identification of vulnerable cloud components has the highest accuracy with
alpha-centrality measure and the least with closeness centrality measure and
this sets a reliable way to prioritize vulnerable cloud endpoints and modules
in an event of a fault occurring to make the architecture more robust.

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#  $M$-Estimation based on quasi-processes from discrete samples of Lévy
processes

Yasutaka Shimizu111Department of Applied Mathematics, Waseda University.
<EMAIL_ADDRESS>and Hiroshi Shiraishi222Department of Mathematics,
Keio University<EMAIL_ADDRESS>
(November 4, 2021)

###### Abstract

We consider $M$-estimation problems, where the target value is determined
using a minimizer of an expected functional of a Lévy process. With discrete
observations from the Lévy process, we can produce a “quasi-path” by shuffling
increments of the Lévy process, we call it a quasi-process. Under a suitable
sampling scheme, a quasi-process can converge weakly to the true process
according to the properties of the stationary and independent increments.
Using this resampling technique, we can estimate objective functionals similar
to those estimated using the Monte Carlo simulations, and it is available as a
contrast function. The $M$-estimator based on these quasi-processes can be
consistent and asymptotically normal.

Keywords: $M$-estimation; Lévy processes; resampling; quasi-process; discrete
observations.

MSC2010: 62G20; 62M20.

##  1 Introduction

On a stochastic basis, $(\Omega,\mathcal{F},{\mathbb{P}};\mathbb{F})$ with a
filtration $\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$, we consider a
$\mathbb{F}$-Lévy process $X=(X_{t})_{t\geq 0}$ starting at $X_{0}=u$ of the
form $X_{t}=u+\widetilde{X}_{t}$, where $u\in\mathbb{R}$ and $\widetilde{X}$
are the Lévy process with the characteristic exponent

$\displaystyle\log\mathbb{E}[e^{it\widetilde{X}_{1}}]=i\mu
t-\frac{\sigma^{2}}{2}t^{2}+\int_{\mathbb{R}}\left(e^{itz}-1-itz\mathbf{1}_{\\{|z|\leq
1\\}}\right)\,\nu(\mathrm{d}z).$ (1.1)

Let $\mathbb{D}_{T}:=D([0,T])\ (0<T\leq\infty)$ be a family of càdlàg
functions on $[0,T]$ (or $\mathbb{R}_{+}:=[0,\infty)$ as $T=\infty$), ${\cal
D}_{T}$ be the Borel field on $\mathbb{D}_{T}$ generated by the Skorokhod
topology, which is metrizable with a suitable metric, called the Billingsley
metric, and $\mathbb{D}_{T}$ can be a Polish space; see Billingsley [2],
Section 12 for details. Hereafter, a stochastic process $X$ is a measurable
map $X:(\Omega,\mathcal{F})\to(\mathbb{D}_{T},{\cal D}_{T})$. Throughout this
study, we denote the distribution of $X$ on ${\cal D}_{T}$ by
$P:={\mathbb{P}}\circ X^{-1}$ and write

$Pf:=\int_{\mathbb{D}_{T}}f(x)\,P(\mathrm{d}x)=\mathbb{E}[f(X)]$

for a measurable function, $f:\mathbb{D}_{T}\to\mathbb{R}$.

Let $\Theta$ be a subset of $\mathbb{R}^{d}$, and suppose the following
$\vartheta_{0}$ is well-defined: for a real-valued function,
$h_{\vartheta}(x):=h(x,\vartheta)$, defined on $\mathbb{D}_{T}\times\Theta$,

$\displaystyle\vartheta_{0}=\arg\min_{\vartheta\in\Theta}Ph_{\vartheta}.$
(1.2)

This study aims to statistically infer $\vartheta_{0}$ from the sampled data
of $X$.

This inference problem is often found in some applications in finance and
insurance. For instance, consider a strategy of striking a perpetual American
put option for a stock price $X=(X_{t})_{t\geq 0}$ such that we strike the
right when $X_{t}<\vartheta$ for a certain threshold $\vartheta>0$, and the
strike time is given by a stopping time
$\tau^{\vartheta}:=\inf\\{t>0:X_{t}<\vartheta\\}$. To search for the optimal
$\vartheta$, say $\vartheta_{*}$, it would be natural to consider

$\displaystyle\vartheta_{*}:=\arg\max_{\vartheta\in[0,K]}\mathbb{E}\left[e^{-r\tau^{\vartheta}}(K-X_{\tau^{\vartheta}})_{+}\mathbf{1}_{\\{\tau^{\vartheta}<\infty\\}}\right],$
(1.3)

where $r>0$ is the interest rate, and $K$ is the strike price; see Gerber and
Shiu [6]. Here, it is not easy to estimate the objective expectation from the
observations of the stock price data $X=(X_{t})_{t\geq 0}$ because it depends
on the entire path of $X$, and it is implicit with respect to the general
characteristics of $X$.

There is another example from insurance risk theory. Assuming that process $X$
is a surplus for an insurance contract and considering the present value of a
ruin-related ‘loss’ up to the time of ruin,

$\displaystyle
h_{\vartheta}(X)=\int_{0}^{\tau^{\vartheta}}e^{-rt}U_{\vartheta}(X_{t})\,\mathrm{d}t,\quad\tau^{\vartheta}:=\inf\\{t>0\,|\,X_{t}<f(\vartheta)\\},$

where $f:\Theta\to\mathbb{R}$ and $\vartheta$ are the parameter that controls
the ‘loss’; see Feng [3] and Feng and Shimizu [4]. Hereafter, $\vartheta_{0}$
given by (1.2) is an optimal parameter for minimizing the expected discounted
‘risk’ for surplus $X$. For instance, consider a dividend strategy that pays
ratio $\alpha\in(0,1)$ when the insurance surplus is over threshold
$\vartheta>0$. The total dividends up to the ruin is given by
$U_{\vartheta}(x)=\alpha\mathbf{1}_{[\vartheta,\infty)}(x)$

$\displaystyle
Ph_{\vartheta}=\alpha\int_{0}^{\infty}e^{-rt}{\mathbb{P}}(X_{t}\geq\vartheta,\,t\leq\tau^{\vartheta})\,\mathrm{d}t.$
(1.4)

In this quantity, the probability of paying the dividends is small when
$\vartheta$ is large, although large dividends are paid, and vice versa.
Therefore, the expectation can be optimized to a suitable level.

In these examples, we need to estimate the expected functionals (1.3) and
(1.4) from observations of the Lévy process $X=(X_{t})_{t\geq 0}$. However, it
is often difficult to estimate these functionals because they include a random
time, $\tau^{\vartheta}$, which is often not observable in practice. If we
observed many paths of the true process, we could estimate it using the Monte
Carlo simulations, but this is impossible in practice.

Contrary to generating paths, we will produce “quasi-paths” from discrete
observations of the Lévy process. This idea is similar to bootstrapping. Given
discrete samples of $X=(X_{t})_{t\geq 0}$, say
$(X_{t_{0}},X_{t_{1}},\dots,X_{t_{n}})$ with $0=t_{0}<t_{1}<\dots<t_{n}=T$ and
$h_{n}\equiv t_{k}-t_{k-1}\ (k=1,2,\dots,n)$, we approximate the path by a
step function that jumps at $t_{k}\ (k=1,2,\dots,n)$ with jump size
$\Delta_{k}X=X_{t_{k}}-X_{t_{k-1}}$; see Section 2 for details. By shuffling
the order of the increments, we can create different step functions, which can
be regarded as different paths from the true distribution. Using these paths,
we can estimate the expected functions of $X$ and construct an estimator of
$\vartheta_{0}$ as an $M$-estimator. Additionally, we can provide sufficient
conditions to ensure that the $M$-estimator can be consistent and
asymptotically normal, which are the main results of this study.

The remainder of this paper is organized as follows. In Section 2, we define
the quasi-processes from discrete observation of the Lévy process and the
$M$-esimator for (1.2). In Section 3, we show the weak convergence of quasi-
processes to the true process based on the Skorokhod topology under the high-
frequency sampling in the long term (HFLT), where the sampling interval,
$h_{n}$, goes to zero and the terminal of observation goes to infinity
simultaneously: $T=nh_{n}\to\infty$ as the sample size $n$ increases.
Additionally, we confirmed this phenomenon through simulations. Section 4
discusses the main results, weak consistency, and asymptotic normality of the
$M$-estimators. We provided some sufficient conditions for the results, and we
confirmed that the conditions can be satisfied in some concrete examples in
Section 5. In particular, we consider an example in which $h_{\vartheta}$ in
(1.2) is dividends paid up to ruin, which is a popular problem in insurance
mathematics. We will make a further paper dealing with more examples in detail
using our techniques; Shimizu and Shiraishi [12].

Before proceeding to the next section, we shall make some notations used
throughout the paper.

### Notation

  * •

$A\lesssim B$ means that there exists a universal constant, $c>0$, such that
$A\leq c\cdot B$.

  * •

For a $k$-th order tensor
$x=(x_{i_{1},i_{2},\dots,i_{k}})_{i_{1},\dots,i_{k}=1,\dots,d}\in\mathbb{R}^{d}\otimes\dots\otimes\mathbb{R}^{d}$,
denote by

$|x|=\sqrt{\sum_{i_{1}=1}^{d}\dots\sum_{i_{k}=1}^{d}x^{2}_{i_{1},i_{2},\dots,i_{k}}}.$

  * •

For $T>0$ and $x\in\mathbb{D}_{T}\cup\mathbb{D}_{\infty}$,

$\|x\|_{T}=\sup_{t\in[0,T]}|x(t)|;\quad\|x\|_{\infty}=\sup_{t\in[0,\infty)}|x(t)|,$

  * •

For any $x,y\in\mathbb{D}_{T}$, we denote by $\varrho_{T}$ the Skorokhod
metric:

$\varrho_{T}(x,y)=\inf_{\lambda\in\Lambda_{T}}\max\\{\|x\circ\lambda-y\|_{T},\|\lambda-I\|_{T}\\},$

where $\Lambda_{T}$ is a family of monotonically increasing functions on
$[0,T]$ and $I$ is the identity $I(s)=s$.

  * •

For $p\geq 1$ and a measure $Q$ on ${\cal X}$, we denote $L^{p}(Q)$ using the
family of functions $f:{\cal X}\to\mathbb{R}$ such that

$\|f\|_{L^{p}(Q)}:=\left(\int_{\cal
X}|f(x)|^{p}\,Q(\mathrm{d}x)\right)^{1/p}<\infty.$

Moreover, we write $Qf:=\int_{\cal X}f(x)\,Q(\mathrm{d}x)$ for $f\in
L^{1}(Q)$.

  * •

For function $f:{\cal X}\times{\cal Y}\to\mathbb{R}$, $f\in C^{n,m}({\cal
X}\times{\cal Y})$ means that $f(x,y)$ is of class $C^{n}$ w.r.t. $x\in{\cal
X}$ and $C^{m}$ w.r.t. $y\in{\cal Y}$. In particular, $C^{0}$ represents a set
of continuous functions. Moreover, $f\in C^{n,m}_{b}$ means that $f\in
C^{n,m}$ and $f$ are bounded up to all possible derivatives.

##  2 Contrast functions via “quasi-processes”

We observed the Lévy process, $X$, starting from $u$ equidistantly in time:
the observations consist of $\\{X_{t_{k}}\\}_{k=0,1,\dots,n}$, where

$0=t_{0}<t_{1}<\dots<t_{n}=T;\quad h_{n}\equiv t_{k}-t_{k-1}.$

Let $\mathbb{X}=(\Delta_{1}X,\Delta_{2}X,\dots,\Delta_{n}X)$ be a vector of
increments with $\Delta_{k}X:=X_{t_{k}}-X_{t_{k-1}}$, and let

$\Lambda_{n}:=\left\\{i_{m}=\begin{pmatrix}1&2&\dots&n\\\
i_{m}(1)&i_{m}(2)&\dots&i_{m}(n)\end{pmatrix}\,\bigg{|}\,m=1,2,\dots,n!\right\\}$

be a family of all the permutation of $(1,2,\dots,n)$. Because $\Delta_{k}X$’s
$(1\leq k\leq n)$ are i.i.d. for each $n$, $\mathbb{X}$ is exchangeable; that
is, it follows for any permutation $i\in\Lambda_{n}$ that

$i(\mathbb{X}):=(\Delta_{i(1)}X,\dots,\Delta_{i(n)}X),\quad i\in\Lambda_{n}.$

has the same distribution as $\mathbb{X}$.

######  Definition 2.1.

For given $\mathbb{X}$ and $i\in\Lambda_{n}$, a stochastic process
$\widehat{X}^{i,n}=(\widehat{X}^{i,n}_{t})_{t\geq 0}$ given by

$\displaystyle\widehat{X}^{i,n}_{t}=u+\sum_{k=1}^{n}\Delta_{i(k)}X\cdot\mathbf{1}_{[t_{k},\infty)}(t)$

is said to be a quasi-process of $X$ for a permutation $i\in\Lambda_{n}$.

Notably, the path of the quasi-process
$\widehat{X}^{i,n}=(\widehat{X}^{i,n}_{t})_{t\geq 0}$ belongs to
$\mathbb{D}_{\infty}$, a right continuous step function that has a jump at
$t=t_{k}\ (k=1,2,\dots,n)$ with amplitude $\Delta_{i(k)}X$. For a given
subset, $A\subset\Lambda_{n}$, we denote an empirical measure of the quasi-
process, $\widehat{X}^{i,n}$, associated with $A$

$\displaystyle{\mathbb{P}}_{A}=\frac{1}{\\#A}\sum_{i\in
A}\delta_{\widehat{X}^{i,n}},$

where $\delta_{x}$ is the delta measure concentrated on
$x\in\mathbb{D}_{\infty}$. In particular, as we fix a sequence of permutation
sets $\\{A_{n}\\}_{n\in\mathbb{N}}$, we simply write

$\displaystyle{\mathbb{P}}_{n}:={\mathbb{P}}_{A_{n}}.$ (2.1)

######  Definition 2.2.

Given a vector of increments, $\mathbb{X}$, and permutation sets,
$\\{A_{n}\\}_{n\in\mathbb{N}}$, we denote the minimum contrast estimator of
$\vartheta_{0}$ in (1.2) as

$\displaystyle\widehat{\vartheta}_{n}=\arg\min_{\vartheta\in\Theta}{\mathbb{P}}_{n}h_{\vartheta},$
(2.2)

where $h_{\vartheta}(x)=h(x,\vartheta)$ for each $x\in\mathbb{D}_{\infty}$.

In some special cases where $Ph_{\vartheta}$ is a function of the Lévy measure
$\nu$ of the form

$Ph_{\vartheta}=\int_{\mathbb{R}}H_{\vartheta}(z)\,\nu(\mathrm{d}z)$

for some $H_{\vartheta}:\mathbb{R}\to\mathbb{R}$, $Ph_{\vartheta}$ can be
estimated as

$\widehat{Ph_{\vartheta}}=\frac{1}{nh_{n}}\sum_{k=1}^{n}H_{\vartheta}^{n}(\Delta_{k}X)\stackrel{{\scriptstyle
p}}{{\longrightarrow}}Ph_{\vartheta}$

with some $H^{n}_{\vartheta}\to H_{\vartheta}$ under some regularities; Comte
and Genon-Catalot [5]; Jacod [7]; Shimizu [11]; Kato and Kuris [8]. In this
case, the quasi-process does not make sense because the estimator is
exchangeable with respect to increments $(\Delta_{k}X)_{k=1,2,\dots}$. Our
method is particularly useful when function $h_{\vartheta}$ depends on the
entire path of $X$. For instance, when we consider an example in (1.4), we may
need many data such that $X_{t}\geq\vartheta$ in the past. Using quasi-
processes $\widehat{X}^{i,n}\,(i\in A_{n})$, we can select many “quasi-
samples” such that $\widehat{X}^{i,n}_{t}\geq\vartheta$, which gives us a
better approximation of ${\mathbb{P}}(X_{t}\geq\vartheta)$.

##  3 Weak convergence for quasi-processes

###  3.1 Theoretical results

Here, we consider a high-frequency sampling in the long term:

(HFLT) $h_{n}\to 0$, and $T\to\infty$ as $n\to\infty$.

######  Theorem 3.1.

Under the sampling scheme (HFLT), it holds for any sequence of permutations
$\\{i^{n}\\}\subset\Lambda_{n}$ that

$\widehat{X}^{i^{n},n}\leadsto X\quad\mbox{ in $\mathbb{D}_{\infty}$},\quad
n\to\infty.$

###### Proof.

Notably, process $X$ generally has the following decomposition:

$X_{t}=u+\mu t+\sigma W_{t}+S_{t},$

where $\mu\in\mathbb{R}$, $\sigma\geq 0$, and $W$ are the Wiener process and
$S$ is the pure-jump Lévy process independent of $W$, with the characteristic
exponent

$\displaystyle\log\mathbb{E}[e^{itS_{1}}]=\int_{\mathbb{R}}\left(e^{itz}-1-itz\mathbf{1}_{\\{|z|\leq
1\\}}\right)\,\nu(\mathrm{d}z).$

Hereafter, we fix permutation $i^{n}\in\Lambda_{n}$ arbitrarily, and let
$X^{n}:=\widehat{X}^{i^{n},n}$ for simplicity of notation

$\displaystyle X^{n}_{t}=u+\sum_{k=1}^{n}\left(\mu
h_{n}+\sigma\Delta_{i^{n}(k)}W+\Delta_{i^{n}(k)}S\right)\mathbf{1}_{[t_{k},\infty)}(t),$

where $\Delta_{k}W=W_{t_{k}}-W_{t_{k-1}}$, and so does $\Delta_{k}S$. Because
the limit process $X$ is a Lévy process, which is stochastically continuous,
the claim of the statement is equivalent to the following conditions (a) and
(b) hold true:

  * (a)

Any finite dimensional distribution of $X^{n}$ converges weakly to the
corresponding finite dimensional distribution of $X$: for any $d$ time points
$0<s_{1}<s_{2}<\dots<s_{d}$, it holds that
$(X^{n}_{s_{1}},\dots,X^{n}_{s_{d}})\leadsto(X_{s_{1}},\dots,X_{s_{d}})$ as
$n\to\infty$ because the probability that the Lévy process jumps at a fixed
time points is zero;

  * (b)

The tightness of $X^{n}$ in $\mathbb{D}_{\infty}$:
$\displaystyle\lim_{M\to\infty}\sup_{n\in\mathbb{N}}{\mathbb{P}}(\varrho(X^{n},X)>M)=0$;

see Billingsley [2], Section 13 for details.

For (a), we show the case where $d=2$ for simplicity of discussion, and for
any $s<t$, we show that

$\displaystyle(X^{n}_{s},X^{n}_{t}-X^{n}_{s})\leadsto(X_{s},X_{t}-X_{s}),\quad
n\to\infty,,$ (3.1)

which leads to $(X^{n}_{s},X^{n}_{t})\leadsto(X_{s},X_{t})$ by the continuous
mapping theorem. Here, when $n$ is sufficiently large, we assume that
$s\in[t_{i-1},t_{i})$ and $t\in[t_{j-1},t_{j})$ for $t_{i}\leq t_{j-1}$
without loss of generality. Hereafter, the components of $i(\mathbb{X})$ are
independent and stationary; it suffices for (3.1) to show that

$\displaystyle X^{n}_{s}$ $\displaystyle\leadsto X_{s};\qquad
X^{n}_{t}-X^{n}_{s}\leadsto X_{t}-X_{s},$ (3.2)

separately. Moreover, for $t_{k_{0}}<s<t_{k_{0}+1}<t$

$X^{n}_{t}-X^{n}_{s}=\sum_{k=1}^{n}\Delta_{i^{n}(k_{0}+k)}X\cdot\mathbf{1}_{[t_{k},\infty)}(t)$

and $\Delta_{i^{n}(k)}X\,(k=1,2,\dots,n)$ are identically distributed, it
suffices to show the first convergence in (3.2) for any initial value $u\geq
0$. Hence, we show only the first one.

Fixing any $t\in(0,T]$, we can assume that there exists some $t_{k}$ such that
$t\in(t_{k-1},t_{k}]$ and $t_{k}\downarrow t$, $t_{k-1}\uparrow t$ as
$n\to\infty$. For $t\in[t_{k-1},t_{k})$,

$X^{n}_{t}=^{d}X_{t}+(X_{t_{k-1}}-X_{t}).$

because $X^{n}_{t}=^{d}X_{t_{k-1}}$ based on the property of independent and
stationary increments of $X$. Hence, we obtain $X^{n}_{t}\leadsto X_{t}$ from
the stochastic continuity of the Lévy processes:
$X_{t_{k-1}}-X_{t}\stackrel{{\scriptstyle p}}{{\longrightarrow}}0$ as
$t_{k-1}\to t$ and Slutsky’s lemma; van der Vaart [13], Theorem 2.7, (iv).

For the proof of (b): the tightness of $X^{n}$, we use the following lemma
obtained from Theorem 16.8 by Billingsley [2] and its corollary.

######  Lemma 3.1.

A sequence of $\mathbb{D}_{\infty}$-valued random elements $\\{X^{n}\\}_{n}$
is uniformly tight when the following conditions (i)–(iii) hold true:

  * (i)

For any $t\in[0,\infty)$,

$\lim_{a\to\infty}\limsup_{n\to\infty}{\mathbb{P}}(|X^{n}_{t}|\geq a)=0.$

  * (ii)

For any $t>0$,

$\lim_{a\to\infty}\limsup_{n\to\infty}{\mathbb{P}}\left(\sup_{s\leq
t}|X^{n}_{s}-X^{n}_{s-}|\geq a\right)=0$

  * (iii)

For each ${\epsilon},\eta,t>0$, there exists some $\delta_{0}$ and $n_{0}$
such that, for any $\delta\leq\delta_{0}$, $n\geq n_{0}$ and some discrete
$(\mathcal{F}_{u}^{n})_{u\geq 0}$-stopping time $\tau$ with $\tau\leq t$,

${\mathbb{P}}(|X^{n}_{\tau+\delta}-X^{n}_{\tau}|\geq{\epsilon})\leq\eta.$

First, we show (i). Fixing any $t>0$, $t\in(t_{k-1},t_{k}]$ for some
$k\in\\{1,\dots,n\\}$ and $n$ are sufficiently large because $T\to\infty$ as
$n\to\infty$. Further, we suppose that $t_{k-1}\uparrow t$ and
$t_{k}\downarrow t$ as $n\to\infty$, as in the proof of (a).

Notably, for any $a>|u|+|\mu|t_{k-1}$, it follows that

$\displaystyle{\mathbb{P}}(|X^{n}_{t}|\geq a)$
$\displaystyle\leq{\mathbb{P}}\left(\left|\sigma\sum_{j=1}^{k-1}\Delta_{i^{n}(j)}W\right|\geq\frac{a-|u|-|\mu|t_{k-1}}{2}\right)$
$\displaystyle\qquad+{\mathbb{P}}\left(\left|\sum_{j=1}^{k-1}\Delta_{i^{n}(j)}S\right|\geq\frac{a-|u|-|\mu|t_{k-1}}{2}\right).$

Through infinite divisibility of $W$ and $S$ and their stochastic continuity,
we have

$\displaystyle\sum_{j=1}^{k-1}\Delta_{i^{n}(j)}W$
$\displaystyle\big{(}=^{d}W_{t_{k-1}}\big{)}\leadsto
W_{t};\qquad\sum_{j=1}^{k-1}\Delta_{i^{n}(j)}S\big{(}=^{d}S_{t_{k-1}}\big{)}\leadsto
S_{t}.$

Hence, we observe that

$\displaystyle\limsup_{n\to\infty}{\mathbb{P}}(|X^{n}_{t}|\geq a)$
$\displaystyle\leq{\mathbb{P}}\left(\left|\sigma
W_{t}\right|\geq(a-|u|-|\mu|t_{k-1})/2\right)$
$\displaystyle\qquad+{\mathbb{P}}\left(\left|S_{t}\right|\geq(a-|u|-|\mu|t_{k-1})/2\right)$
$\displaystyle\to 0\quad(a\to\infty),$

by the tightness of the random variables $\sigma W_{t}$ and $S_{t}$.

Second, we show (ii). Fixing $t>0$ in the proof of (i), for any $a>0$ and
${\epsilon}\in(0,a)$, it follows that

$\displaystyle\limsup_{n\to\infty}{\mathbb{P}}\left(\sup_{s\leq
t}|X^{n}_{s}-X^{n}_{s-}|\geq a\right)$
$\displaystyle\leq\limsup_{n\to\infty}{\mathbb{P}}\left(\max_{1\leq j\leq
k-1}|\Delta_{i^{n}(j)}S|\geq a\right)$
$\displaystyle\leq\limsup_{n\to\infty}{\mathbb{P}}\left(\sum_{j=1}^{k-1}|\Delta_{i^{n}(j)}S|\mathbf{1}_{\\{|\Delta_{i^{n}(j)}S|>{\epsilon}\\}}\geq
a\right)$ $\displaystyle\leq{\mathbb{P}}\left(\sum_{u\leq t}|\Delta
S_{u}|\mathbf{1}_{\\{|\Delta S_{u}|>{\epsilon}\\}}\geq a\right).$

Notably, $\sum_{u\leq t}|\Delta S_{u}|\mathbf{1}_{\\{|\Delta
S_{u}|>{\epsilon}\\}}$ is a compound Poisson process with intensity
$\lambda_{\epsilon}:=\int_{|z|>{\epsilon}}\nu(\mathrm{d}z)$, and it is almost
finite for each $t>0$. Therefore, the following holds:

$\lim_{a\to\infty}{\mathbb{P}}\left(\sum_{u\leq t}|\Delta
S_{u}|\mathbf{1}_{\\{|\Delta S_{u}|>{\epsilon}\\}}\geq a\right)=0.$

Finally, we present (iii). Fix any $s\in[0,t]$ and considering the stopping
time as $\tau\equiv s$. Notably, for any sufficiently small $\delta>0$, there
exist some numbers $n_{1}\in\mathbb{N}$ and $p<q$ such that for any $n\geq
n_{1}$,

$t_{p-1}<s\leq t_{p}<t_{q-1}<s+\delta/2\leq t_{q}<s+\delta,$

and, as in the above argument, we assume that $t_{p}\downarrow s$ and
$t_{q}\downarrow s+\delta/2$ as $n\to\infty$ without loss of generality.

We note that for any $n\geq n_{1}$ and the above $\delta$,

$|X^{n}_{s+\delta}-X^{n}_{s}|\leq|\mu|\delta+\left|\sigma\sum_{j=p}^{q-1}\Delta_{i^{n}(j)}W\right|+\left|\sum_{j=p}^{q-1}\Delta_{i^{n}(j)}S\right|.$

Because

$\sum_{j=p}^{q-1}\Delta_{i^{n}(j)}S=^{d}[S_{s+\delta}-S_{s}]-[S_{t_{p}}-S_{s}]-[S_{s+\delta}-S_{t_{q}}],$

and

$\sigma\sum_{j=p}^{q-1}\Delta_{i^{n}(j)}W\sim
N(0,\sigma^{2}(t_{q}-t_{p}));\quad t_{q}-t_{p}<\delta,$

we have

$\displaystyle{\mathbb{P}}(|X^{n}_{s+\delta}-X^{n}_{s}|\geq{\epsilon})$
$\displaystyle\leq\mathbf{1}_{\\{|\mu|\delta\geq{\epsilon}/5\\}}+{\mathbb{P}}(|N(0,\sigma^{2}\delta)|\geq{\epsilon}/5)+{\mathbb{P}}\left(|S_{s+\delta}-S_{s}|\geq{\epsilon}/5\right)$
$\displaystyle\qquad+{\mathbb{P}}\left(|S_{t_{p}}-S_{s}|\geq{\epsilon}/5\right)+{\mathbb{P}}\left(|S_{s+\delta}-S_{t_{q}}|\geq{\epsilon}/5\right).$

For the above ${\epsilon}>0$, there exists a $\delta_{1}>0$ with
$|\mu|\delta_{1}<{\epsilon}/5$ such that

$\mathbf{1}_{\\{|\mu|\delta\geq{\epsilon}/5\\}}=0,\quad{\mathbb{P}}(|N(0,\sigma^{2}\delta)|\geq{\epsilon}/5)<\frac{\eta}{4}$

for any $\delta\leq\delta_{1}$. Moreover, it follows from the stochastic
continuity of the Lévy process $S$ that, for any $\eta>0$, there exist
$\delta_{2}>0$ and $n_{2}\in\mathbb{N}$ such that

$\displaystyle{\mathbb{P}}(|S_{s+\delta}-S_{s}|>{\epsilon}/5)$
$\displaystyle<\frac{\eta}{4};\quad{\mathbb{P}}\left(|S_{s+\delta}-S_{s+\delta/2}|\geq{\epsilon}/10\right)<\frac{\eta}{8}$

for any $\delta\leq\delta_{2}$, and

$\displaystyle{\mathbb{P}}\left(|S_{t_{p}}-S_{s}|\geq{\epsilon}/5\right)$
$\displaystyle<\frac{\eta}{4};\quad{\mathbb{P}}\left(|S_{s+\delta/2}-S_{t_{q}}|\geq{\epsilon}/10\right)<\frac{\eta}{8}$

for any $n\geq n_{2}$; therefore, it also holds that

$\displaystyle{\mathbb{P}}\left(|S_{s+\delta}-S_{t_{q}}|\geq{\epsilon}/5\right)$
$\displaystyle\leq{\mathbb{P}}\left(|S_{s+\delta}-S_{s+\delta/2}|\geq{\epsilon}/10\right)+{\mathbb{P}}\left(|S_{s+\delta/2}-S_{t_{q}}|\geq{\epsilon}/10\right)$
$\displaystyle\leq\frac{\eta}{4}$

for any $\delta\leq\delta_{2}$ and $n\geq n_{2}$. As a consequence, it follows
for any $n\geq n_{0}:=\max\\{n_{1},n_{2}\\}$ and
$\delta\leq\delta_{0}:=\min\\{\delta_{1},\delta_{2}\\}$ that

${\mathbb{P}}(|X^{n}_{s+\delta}-X^{n}_{s}|\geq{\epsilon})<\eta.$

. ∎

###  3.2 Numerical experiments

Let us numerically confirm the weak convergence of the quasi-processes when
$h_{n}\to 0,\ nh_{n}\to\infty$.

Consider a jump-diffusion process of the form

$\displaystyle X_{t}=u+\mu t+\sigma W_{t}-\sum_{i=1}^{N_{t}}\xi_{i},$ (3.3)

where $W$ is a Wiener process, $N$ is a Poisson process with intensity
$\lambda>0$, and $\xi_{i}$’s are i.i.d. exponential variables with mean $m$.
As shown in Figure 1, we simulate 100 paths of the process with parameters
$(\mu,\sigma,\lambda,m,u)=(20,10,5,3,0)$. Assuming that we observe the blue
path, the other scenarios are shown in gray. From this perspective, we can
image the distribution of the proecss.

Some quai-paths based on observations from the blue line on $[0,T]$ are shown
in Figures 2 – 4, where the graphs on $[0,10]$ are extracted from the paths on
$[0,T]$ with $T=10,50$, and $100$. In each figure, the left shows the quasi-
paths for $h_{n}=1.0$, and the right shows the ones for smaller $h_{n}$ (e.g.,
0.1, 0.05, and 0.005). In the simulation, we randomly select
$A_{n}=\\{i_{m_{j}}\,|\,j=1,2,\dots,\alpha_{n}\\}\subset\Lambda_{n}$ randomly
from as a subset of $\\{1,2,\dots,n!\\}$ with $\alpha_{n}=100$, that is, 100
sample paths were generated.

We can observe that the larger the $T$ and the smaller the $h_{n}$, the
distribution of the quasi-paths (gray) seems to be similar to the distribution
of the true paths in Figure 1 (compare Figure 1 and the bottom in Figure 4),
which is a visual confirmation of the weak convergence of the quasi-paths.

Figure 1: Paths of the jump-diffusion process (3.3) with
$(\mu,\sigma,\lambda,m,u)=(20,10,5,3,0)$. We assume that the blue line is an
observed process, and the gray liens are other scenarios.

Figure 2: Gray paths are $100(=\alpha_{n})$ quasi-processes based on the
discrete samples from the blue observed process with $T=10$. (top:
$h_{n}=1.0$; bottom: $h_{n}=0.1$).

Figure 3: Gray paths are $100(=\alpha_{n})$ quasi-processes based on the
discrete samples from the blue observed process with $T=50$. (top $h_{n}=1.0$;
bottom: $h_{n}=0.05$).

Figure 4: Gray paths are $100(=\alpha_{n})$ quasi-processes based on the
discrete samples from the blue observed process with $T=100$. (top:
$h_{n}=1.0$; bottom: $h_{n}=0.005$).

Next, let us we compare the marginal distribution of $X_{1}$ and that of the
quasi-process, $\widehat{X}_{1}^{i,n}$, under the same parameter values as in
the previous experiments. Figures 5 – 7 show histograms of 1000 values of
$X_{1}$ from the true distribution and its estimated density (blue solid
curve) with $T=10,50$ and $100$. At the same time, estimated densities of
$\widehat{X}_{1}^{i,n}$ with $h_{n}=1$ (purple dotted curve), $h_{n}=0.01$
(green dashed curve), and $h_{n}=0.005$ (red solid curve). From those, we can
observe the convergence of the marginal distribution when $h_{n}\to 0$ and
$T\to\infty$.

Figure 5: The blue curve is the estimated density of $X_{1}$. The purple
dotted curve, the green dashed curve and the red solid curve are estimated
densities of quasi-processes when $T=10$ with $h_{n}=1,0.01$ and $0.005$,
respectively. Figure 6: The blue curve is the estimated density of $X_{1}$.
The purple dotted curve, the green dashed curve and the red solid curve are
estimated densities of quasi-processes when $T=50$ with $h_{n}=1,0.01$ and
$0.005$, respectively. Figure 7: The blue curve is the estimated density of
$X_{1}$. The purple dotted curve, the green dashed curve and the red solid
curve are estimated densities of quasi-processes when $T=100$ with
$h_{n}=1,0.01$ and $0.005$, respectively.

##  4 Main results for $M$-estimation

###  4.1 Consistency results

Here, we assume that the empirical measure
${\mathbb{P}}_{n}:={\mathbb{P}}_{A_{n}}$ in (2.1) is given by permutation set
$A_{n}\subset\Lambda_{n}$ such that $\alpha?n:=\\#A_{n}\to\infty$ as
$n\to\infty$. Moreover, when ${\cal H}$ is a family of measurable functions
$f:\mathbb{D}_{\infty}\times\Theta\to\mathbb{R}$, we assume that
$\widehat{\vartheta}_{n}$ is given by (2.2) for a given
$h_{\vartheta}(x):=h(x,\vartheta)\in{\cal H}$.

######  Theorem 4.1.

Suppose that $\Theta$ is compact in $\mathbb{R}^{d}$ and

$\displaystyle{\cal H}\subset C^{0,1}_{b}(\mathbb{D}_{\infty}\times\Theta),$
(4.1)

Moreover, for $h_{\vartheta}(x)\in{\cal H}$, suppose that there exists
$\vartheta_{0}\in\Theta$ such that for any ${\epsilon}>0$,

$\displaystyle\inf_{\vartheta\in\Theta:|\vartheta-\vartheta_{0}|>{\epsilon}}Ph_{\vartheta}>Ph_{\vartheta_{0}}.$
(4.2)

Then $\widehat{\vartheta}_{n}$ is weakly consistent to $\vartheta_{0}$:

$\widehat{\vartheta}_{n}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}\vartheta_{0},\quad n\to\infty.$

###### Proof.

Notably,

$\displaystyle\sup_{\vartheta\in\Theta}|({\mathbb{P}}_{n}-P)h_{\vartheta}|$
$\displaystyle\leq\sup_{\vartheta\in\Theta}|({\mathbb{P}}_{n}h_{\vartheta}-\mathbb{E}[h_{\vartheta}(\widehat{X}^{i^{n},n})]|+\sup_{\vartheta\in\Theta}|\mathbb{E}[h_{\vartheta}(\widehat{X}^{i^{n},n})]-\mathbb{E}[h_{\vartheta}(X)]|$
$\displaystyle=:I_{n}^{(1)}+I_{n}^{(2)},$ (4.3)

where $I_{n}^{(1)}$ and $I_{n}^{(2)}$ are the first and second terms on the
right-hand side of the last inequality, respectively.

To show the convergence of $I_{n}^{(2)}$, we show that

$\displaystyle\lim_{n\to\infty}\mathbb{E}\left[\sup_{\vartheta\in\Theta}|h_{\vartheta}(\widehat{X}^{i^{n},n})-h_{\vartheta}(X)|\right]=0,$

for any sequence of $\\{i^{n}\\}\subset\Lambda_{n}$. Notably, $\Theta$ admits
the Sobolev inequality for embedding $W^{1,p}(\Theta)\hookrightarrow
C(\Theta)$ for $p>d$. For every $f\in W^{1,p}(\Theta)$, it holds that

$\displaystyle\sup_{\vartheta\in\Theta}|f(\vartheta)|^{p}\lesssim\int_{\Theta}\left(|f(\vartheta)|^{p}+|\nabla_{\vartheta}f(\vartheta)|^{p}\right)\,\mathrm{d}\vartheta;$

see Adams and Fournier (2003) for details. We used this inequality to obtain

$\displaystyle\mathbb{E}\left[\sup_{\vartheta\in\Theta}|h_{\vartheta}(\widehat{X}^{i^{n},n})-h_{\vartheta}(X)|^{p}\right]\lesssim\sum_{k=0}^{1}\int_{\Theta}\mathbb{E}\left[|\nabla_{\vartheta}^{k}h_{\vartheta}(\widehat{X}^{i^{n},n})-\nabla_{\vartheta}^{k}h_{\vartheta}(X)|^{p}\right]\,\mathrm{d}\vartheta,$
(4.4)

and the last terms go to zero as $n\to\infty$ by Theorem 3.1 and the bounded
convergence theorem with the Skorokhood representation theorem. Hence, we have

$\displaystyle 0\leq
I_{n}^{(2)}\leq\mathbb{E}\left[\sup_{\vartheta\in\Theta}|h_{\vartheta}(\widehat{X}^{i^{n},n})-h_{\vartheta}(X)|\right]\to
0,\quad n\to\infty.$

To discuss the convergence of $I_{n}^{(1)}$, let
$\\{Z_{\vartheta}^{n}\\}_{n\in\mathbb{N}}$ be a sequence of random functions

$Z_{\vartheta}^{n}:={\mathbb{P}}_{n}h_{\vartheta}-\mathbb{E}[h_{\vartheta}(\widehat{X}^{i^{n},n})],$

which is differentiable in $\vartheta\in\Theta$ because
$\nabla_{\vartheta}h_{\vartheta}$ is bounded. Hereafter, $Z_{\vartheta}^{n}$
is a random element that values in a metric space $C(\Theta)$ equipped with a
sup-norm on $\Theta$. Hence, to show that $I_{n}^{(1)}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0$, it suffices to show the following two conditions:

  * (i)

For each $\vartheta\in\Theta$, $Z_{\vartheta}^{n}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0$ as $n\to\infty$;

  * (ii)

$\displaystyle\sup_{n\in\mathbb{N}}\mathbb{E}\left[\sup_{\vartheta\in\Theta}|\nabla_{\vartheta}Z_{\vartheta}^{n}|\right]<\infty$.

Condition (ii) is significant because $\nabla_{\vartheta}h_{\vartheta}$ is
bounded.

To show (i), we evaluate the variance of $Z_{\vartheta}^{n}$. Notably,
according to the law of total variance,

$\displaystyle Var(Z_{\vartheta}^{n})$
$\displaystyle=\mathbb{E}[Var(Z_{\vartheta}^{n}|X)]+Var(\mathbb{E}[Z_{\vartheta}^{n}|X]).$

Currently, it follows that
$\mathbb{E}\left[\mathbb{E}[Z_{\vartheta}^{n}|X]\right]=0$ because

$\displaystyle\mathbb{E}[Z_{\vartheta}^{n}|X]=\frac{1}{\alpha_{n}}\sum_{i\in
A_{n}}\mathbb{E}[h_{\vartheta}(\widehat{X}^{i,n})|X]-\mathbb{E}[h_{\vartheta}(\widehat{X}^{i^{n},n})];$

therefore,

$\displaystyle Var(\mathbb{E}[Z_{\vartheta}^{n}|X])$
$\displaystyle=\mathbb{E}\left[\left(\frac{1}{\alpha_{n}}\sum_{i\in
A_{n}}\left\\{\mathbb{E}[h_{\vartheta}(\widehat{X}^{i,n})|X]-\mathbb{E}[h_{\vartheta}(\widehat{X}^{i^{n},n})]\right\\}\right)^{2}\right]=O(\alpha_{n}^{-1}),$

because $h_{\vartheta}$ is bounded and $\widehat{X}^{i,n}\ (i=1,\dots,n)$ are
IID conditional on $X$. Similarly, we have that

$\displaystyle\mathbb{E}\left[Var(Z_{\vartheta}^{n}|X)\right]$
$\displaystyle=\frac{1}{\alpha_{n}}\mathbb{E}\left[Var(h_{\vartheta}(\widehat{X}^{i^{n},n})|X)\right]=O(\alpha_{n}^{-1}),$

as $n\to\infty$. Therefore,

$Var(Z_{\vartheta}^{n})=O(\alpha_{n}^{-1})\to 0,\quad n\to\infty.$

As a result, we obtain

$\sup_{\vartheta\in\Theta}|({\mathbb{P}}_{n}-P)h_{\vartheta}|\stackrel{{\scriptstyle
p}}{{\longrightarrow}}0,\quad n\to\infty.$

Combining this and (4.2), we conclude from Theorem 5.7 by van der Vaart [13].
∎

###  4.2 Asymptotic normality

For class ${\cal H}$, we denote the bracketing number of ${\cal H}$ by
$N_{[\,]}({\epsilon},{\cal H},L^{r}(P))$, which is the minimum number of
${\epsilon}$-brackets, including $[l,u]$, that is a family of measurable
functions $f$ satisfying $l\leq f\leq u$ with $\|u-l\|_{L^{r}(P)}<{\epsilon}$,
covering ${\cal H}$.

######  Theorem 4.2.

Considering the same assumptions as in Theorem 4.1, $h_{\vartheta}\in{\cal
H}\cap C^{0,2}_{b}(\mathbb{D}_{T}\times\Theta)$ with matrix
$V_{\vartheta_{0}}:=P\nabla_{\vartheta}^{2}h_{\vartheta_{0}}$ is invertible.
Moreover, suppose that there exists constant $p>d$ and sequence $\\{r_{n}\\}$
with $r_{n}\uparrow\infty$ and $\alpha_{n}/r_{n}^{2}=o(1)$ for
$\alpha_{n}:=\\#A_{n}$ such that, for $k=0,1$ and any $i^{n}\in A_{n}$,

$\displaystyle\left\|\nabla_{\vartheta}^{k}h_{\vartheta}(\widehat{X}^{i^{n},n})-\nabla_{\vartheta}^{k}h_{\vartheta}(X)\right\|_{L^{p}(P)}=o(r_{n}^{-1}),\quad
n\to\infty.$ (4.5)

Then the $M$-estimator $\widehat{\vartheta}_{n}$ is asymptotically normal:

$\displaystyle\sqrt{r_{n}}(\widehat{\vartheta}_{n}-\vartheta_{0})\leadsto
N(0,\Sigma),\quad n\to\infty,$

with
$\Sigma=V_{\vartheta_{0}}^{-1}P[\nabla_{\vartheta}h_{\vartheta_{0}}\nabla_{\vartheta}h_{\vartheta_{0}}^{\top}]V_{\vartheta_{0}}^{-1}$.

######  Remark 4.1.

Notably, $r_{n}$ in Theorem 4.2 is not necessarily unique. Generally, when
some $r^{\prime}_{n}$ satisfies (4.5), $r_{n}=\log
r^{\prime}_{n}\uparrow\infty$ also satisfies (4.5). Because we can freely
control $\alpha_{n}$, $r_{n}$-consistency can be attained. When
$r_{n}=r_{n}^{*}$ satisfies (4.5) and it holds for
$\forall\,\\{r^{\prime}_{n}\\}$ with $r^{\prime}_{n}/r_{n}^{*}\to\infty$,

$\liminf_{n\to\infty}r^{\prime}_{n}\|\nabla_{\vartheta}^{k}h_{\vartheta}(\widehat{X}^{i^{n},n})-\nabla_{\vartheta}^{k}h_{\vartheta}(X)\|_{L^{p}}>0,\quad
k=0,1.$

Then $r_{n}^{*}$ is optimal. In addition, the example in Section 5.1; see
Remark 5.1, indicates that several $r_{n}$ can be found and the optimal rate
of convergence is attained by choosing $\alpha_{n}$ suitably.

######  Remark 4.2.

We can control the rate of convergence for $\widehat{\vartheta}_{n}$ by
choosing the permutation set $A_{n}$ with $\alpha_{n}=\\#A_{n}$ suitably
within the restriction $\alpha_{n}/r_{n}^{2}=o(1)$. The order condition
$\alpha_{n}/r_{n}^{2}=o(1)$ is not necessary but sufficient. Therefore,
Theorem 4.2 does not imply that $\alpha_{n}$ should be slower; therefore, the
order condition $\alpha_{n}/r_{n}^{2}=o(1)$ holds true. The resampling number
$\alpha_{n}$ can be chosen for the order condition to be broken, but a rapid
$\alpha_{n}$ cannot improve the rate of convergence better than
$\sqrt{r_{n}}$.

###### Proof.

Let $(X^{(\beta)})_{\beta=1,2,\dots,B}$ be independent copies of process $X$
defined on the same probability space $(\Omega,\mathcal{F},{\mathbb{P}})$, and
denote its empirical measure as

${\mathbb{P}}^{*}_{B}=\frac{1}{B}\sum_{\beta=1}^{B}\delta_{X^{(\beta)}},$

and let $\alpha_{n}:=\\#A_{n}$ for $A_{n}\subset\Lambda_{n}$ of
${\mathbb{P}}_{n}$ in (2.1). Under condition (4.1), class ${\cal H}$ is
$P$-Glivenko-Cantelli:

$\sup_{h\in{\cal H}}|({\mathbb{P}}^{*}_{\alpha_{n}}-P)h|\to 0\quad a.s.,\quad
n\to\infty.$

Because any function in ${\cal H}$ is Lipschitz with respect to
$\vartheta\in\Theta$, it follows from, for example, argument as in Example
19.7 in van der Vaart [13], p.271, that the bracketing number of ${\cal H}$ is
finite. For every ${\epsilon}>0$,

$\displaystyle N_{[\,]}({\epsilon},{\cal
H},L^{1}(P))\lesssim{\epsilon}^{-d}<\infty,$

which implies that class ${\cal H}$ is $P$-Donsker, and therefore,

$\sup_{h\in{\cal
H}}|({\mathbb{P}}^{*}_{\alpha_{n}}-P)h|=O_{p}(\alpha_{n}^{-1/2}),\quad
n\to\infty.$

It follows from the decomposition (4.3) that

$\sup_{h\in{\cal
H}}|({\mathbb{P}}_{\alpha_{n}}^{*}-{\mathbb{P}}_{n})h|\leq\sup_{h\in{\cal
H}}|({\mathbb{P}}^{*}_{\alpha_{n}}-P)h|+I_{n}^{(1)}+I_{n}^{(2)},$

where $I_{n}^{(1)}$ and $I_{n}^{(2)}$ are given in the proof of Theorem 4.1.
Under assumption (4.5), it follows the same argument as in (4.4) that

$I_{n}^{(2)}=o(r_{n}^{-1}).$

Moreover, from the argument for $Var(Z_{\vartheta}^{n})$ in the proof of
Theorem 4.1, we observe that

$I_{n}^{(1)}=O_{p}(\alpha_{n}^{-1/2}).$

Because we take $\alpha_{n}$ such that $r_{n}^{2}/\alpha_{n}=o(1)$, it holds
that

$\sup_{h\in{\cal
H}}|({\mathbb{P}}_{\alpha_{n}}^{*}-{\mathbb{P}}_{n})h|=o(r_{n}^{-1})+O_{p}(\alpha_{n}^{-1/2})=o(r_{n}^{-1}).$

Considering $|\inf_{n}\alpha_{n}-\inf_{n}b_{n}|\leq\sup_{n}|\alpha_{n}-b_{n}|$
and the definition of $\widehat{\vartheta}_{n}$, we have

$\displaystyle\left|{\mathbb{P}}_{\alpha_{n}}^{*}h_{\widehat{\vartheta}_{n}}-{\mathbb{P}}_{\alpha_{n}}^{*}h_{\vartheta_{0}}\right|$
$\displaystyle\leq\left|({\mathbb{P}}_{\alpha_{n}}-{\mathbb{P}}_{n})h_{\widehat{\vartheta}_{n}}\right|+\left|{\mathbb{P}}_{n}h_{\widehat{\vartheta}_{n}}-\inf_{\vartheta\in\Theta}{\mathbb{P}}_{n}h_{\vartheta}\right|+\left|\inf_{\vartheta\in\Theta}{\mathbb{P}}_{n}h_{\vartheta}-\inf_{\vartheta\in\Theta}{\mathbb{P}}^{*}_{\alpha_{n}}h_{\vartheta}\right|$
$\displaystyle\leq 2\sup_{h\in{\cal
H}}|({\mathbb{P}}_{\alpha_{n}}^{*}-{\mathbb{P}}_{n})h|=o(r_{n}^{-1}),$

and

$\displaystyle{\mathbb{P}}_{\alpha_{n}}^{*}h_{\widehat{\vartheta}_{n}}\leq{\mathbb{P}}_{\alpha_{n}}^{*}h_{\vartheta_{0}}+o_{p}(r_{n}^{-1})$

Here, we assume that $h_{\vartheta}\in{\cal H}$ is Lipschitz continuous with
respect to $\vartheta\in\Theta$. The bracketing number satisfies
$N_{[\,]}({\epsilon},{\cal H},L^{2}(P))<\infty$; see Example 19.7 in van der
Vaart [13]. Then, using the same argument as in the proof of Corollary 5.53 in
van der Vaart [13] with ${\mathbb{P}}_{n}$ replaced by our
${\mathbb{P}}^{*}_{\alpha_{n}}$, we established that
$\sqrt{r_{n}}(\widehat{\vartheta}_{n}-\vartheta_{0})$ is bounded in
probability.

Similarly, we have

$\displaystyle\left|{\mathbb{P}}_{\alpha_{n}}^{*}h_{\widehat{\vartheta}_{n}}-\inf_{\vartheta\in\Theta}{\mathbb{P}}_{\alpha_{n}}^{*}h_{\vartheta}\right|$
$\displaystyle\leq\left|{\mathbb{P}}_{\alpha_{n}}^{*}h_{\widehat{\vartheta}_{n}}-\inf_{\vartheta\in\Theta}{\mathbb{P}}_{n}^{*}h_{\vartheta}\right|+\left|\
\inf_{\vartheta\in\Theta}{\mathbb{P}}_{n}^{*}h_{\vartheta}-\inf_{\vartheta\in\Theta}{\mathbb{P}}_{\alpha_{n}}^{*}h_{\vartheta}\right|$
$\displaystyle\leq\left|{\mathbb{P}}_{\alpha_{n}}^{*}h_{\widehat{\vartheta}_{n}}-{\mathbb{P}}_{n}h_{\widehat{\vartheta}_{n}}\right|+\sup_{h\in{\cal
H}}|({\mathbb{P}}_{\alpha_{n}}^{*}-{\mathbb{P}}_{n})h|$ $\displaystyle\leq
2\sup_{h\in{\cal
H}}|({\mathbb{P}}_{\alpha_{n}}^{*}-{\mathbb{P}}_{n})h|=o_{p}(r_{n}^{-1}),$

which yields

$\displaystyle{\mathbb{P}}_{\alpha_{n}}^{*}h_{\widehat{\vartheta}_{n}}\leq\inf_{\vartheta\in\Theta}{\mathbb{P}}_{\alpha_{n}}^{*}h_{\vartheta}+o_{p}(r_{n}^{-1}).$
(4.6)

Hence, $\widehat{\vartheta}_{n}$ is approximately a minimum contrast estimator
for the contrast function ${\mathbb{P}}^{*}_{\alpha_{n}}h_{\vartheta}$.
Hereafter, we can obtain

$\displaystyle\sqrt{r_{n}}(\widehat{\vartheta}_{n}-\vartheta_{0})$
$\displaystyle=V_{\vartheta_{0}}^{-1}\frac{1}{\sqrt{\alpha_{n}}}\sum_{i\in
A_{n}}\nabla_{\vartheta}h_{\vartheta_{0}}(X^{(i)})+o_{p}(1),\quad n\to\infty.$

using the same argument as in the proof of Theorem 5.23 in van der Vaart [13]
with the condition (4.6): replace ${\mathbb{P}}_{n}$ with
${\mathbb{P}}_{\alpha_{n}}^{*}$;
$n{\mathbb{P}}_{n}(m_{\vartheta_{0}+\widetilde{h}_{n}/\sqrt{n}}-m_{\vartheta_{0}})$
with
$\sqrt{\alpha_{n}r_{n}}{\mathbb{P}}_{\alpha_{n}}^{*}(h_{\vartheta_{0}+\widetilde{h}_{n}/\sqrt{r_{n}}}-h_{\vartheta_{0}})$;
$\mathbb{G}_{n}f$ with
$\mathbb{G}_{\alpha_{n}}^{*}h_{\vartheta}:=\sqrt{\alpha_{n}}({\mathbb{P}}_{\alpha_{n}}^{*}-P)h_{\vartheta}$
in the proof.

In particular, because $X^{(i)}$’s are IIDs, we conclude by the central limit
theorem that

$\frac{1}{\sqrt{\alpha_{n}}}\sum_{i\in
A_{n}}\nabla_{\vartheta}h_{\vartheta_{0}}(X^{(i)})\leadsto
N(0,P[\nabla_{\vartheta}h_{\vartheta_{0}}\nabla_{\vartheta}h_{\vartheta_{0}}^{\top}]),\quad\alpha_{n}\to\infty.$

∎

##  5 Examples

###  5.1 Default-related discounted losses

Here, we consider concrete examples for $h_{\vartheta}$ and investigate
sufficient conditions that ensure the conditions in Theorems 4.1 and 4.2.

Suppose the Lévy process $X=(X_{t})_{t\in[0,T]}$ with (1.1) represents a
‘loss’ process of a company and we define the time of default as

$\tau^{X}:=\inf\\{t\in[0,T]\,:\,X_{t}<\xi\\}\wedge T,$

for constants $\xi\in\mathbb{R}$ and $T\in(0,\infty]$. $\tau^{X}$ is
interpreted as the time of default (resp. ruin) with default levels $\xi$ in
the context of finance (resp. insurance). In this section, we consider a
discounted loss up to default. For $x=(x_{t})_{t\in[0,T]}\in\mathbb{D}_{T}$,

$\displaystyle
h_{\vartheta}(x)=h(x,\vartheta)=\int_{0}^{\tau^{x}}e^{-rt}U_{\vartheta}(t,x_{t})\,\mathrm{d}t,.$
(5.1)

where $r>0$ and $U_{\vartheta}:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$
are a bounded function with parameter
$\vartheta\in\Theta\subset\mathbb{R}^{d}$ to be controlled. Then,
$Ph_{\vartheta}$ is the expected discounted loss up to default

$\displaystyle
Ph_{\vartheta}=\mathbb{E}\left[\int_{0}^{\tau^{X}}e^{-rt}U_{\vartheta}(t,X_{t})\,\mathrm{d}t\right];$
(5.2)

see Feng and Shimizu [4] for details on this quantity.

In the sequel, we consider the Skorokhod space $(\mathbb{D}_{T},\varrho_{T})$,
where $\varrho_{T}$ is the Skorokhod metric given in Section 1.

######  Lemma 5.1.

Let $T\in(0,\infty]$. The functional $\tau^{\cdot}$ defined on
$\mathbb{D}_{T}$ is continuous with respect to $\varrho_{T}$; that is, for any
$x,y\in\mathbb{D}_{T}$ and ${\epsilon}>0$, there exists a $\delta>0$ such that

$\varrho_{T}(x,y)<\delta\quad\Rightarrow\quad|\tau^{x}-\tau^{y}|<{\epsilon}.$

###### Proof.

Notably, the functional

$F(x;T_{1},T_{2})=\inf_{T_{1}\leq s\leq T_{2}}x_{s},\quad 0\leq
T_{1}<T_{2}\leq T,$

for $x\in\mathbb{D}_{T}$ is continuous in the sense of the Skorokhod metric
$\varrho_{T}$, and it follows for each $y\in\mathbb{D}_{T}$ and $\eta>0$,
there exists a constant $\delta:=\delta(y,\eta)>0$ such that when
$\varrho_{T}(x,y)<\delta$,

$\left|F(x;\tau^{y}-{\epsilon},\tau^{y}+{\epsilon})-F(y;\tau^{y}-{\epsilon},\tau^{y}+{\epsilon})\right|<\eta,$

which implies that

$\inf_{\tau^{y}-{\epsilon}\leq
s\leq\tau^{y}+{\epsilon}}x_{s}<\inf_{\tau^{y}-{\epsilon}\leq
s\leq\tau^{y}+{\epsilon}}y_{s}+\eta.$

Because $\inf_{\tau^{y}-{\epsilon}\leq s\leq\tau^{y}+{\epsilon}}y_{s}<d$ by
the definition of $\tau^{y}$, when $\eta>0$ is sufficiently small, we have

$\inf_{\tau^{y}-{\epsilon}\leq s\leq\tau^{y}+{\epsilon}}x_{s}<d,$

which is equivalent to $|\tau^{x}-\tau^{y}|<{\epsilon}$. ∎

######  Lemma 5.2.

Let $N\geq 2$ be an integer and $T\in(0,\infty]$. For bounded function
$U_{\vartheta}:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ with parameter
$\vartheta\in\Theta\subset\mathbb{R}^{d}$, suppose that the derivatives up to
$N$th order with respect to $\vartheta$ are uniformly bounded

$\displaystyle\sup_{(t,z)\in\mathbb{R}_{+}\times\mathbb{R};\vartheta\in\Theta}|\nabla_{\vartheta}^{k}U_{\vartheta}(t,z)|\leq
C,\quad k=1,\dots,N,$ (5.3)

for constant $C>0$. Moreover, suppose the Lipschitz continuity:

$\displaystyle\sup_{t\in\mathbb{R}_{+};\vartheta\in\Theta}|\nabla_{\vartheta}^{k}U_{\vartheta}(t,x)-\nabla_{\vartheta}^{k}U_{\vartheta}(t,y)|\leq
L\|x-y\|_{T},\quad x,y\in\mathbb{D}_{T},\ k=1,\dots,N.$ (5.4)

Then the functional (5.1) satisfies that

$h\in C_{b}^{0,k}(\mathbb{D}_{T}\times\Theta),\quad k=1,\dots,N-1.$

###### Proof.

In this proof, we arbitrarily fix $\vartheta\in\Theta$ and simply write
$h(x):=h(x,\vartheta)$.

For any $x,y\in\mathbb{D}_{T}$ and any increasing surjection
$\lambda:[0,T]\to[0,T]$

$\displaystyle|h(x)-h(y)|$
$\displaystyle=\left|\int_{0}^{\tau^{x}}e^{-rt}U_{\vartheta}(t,x_{t})\,\mathrm{d}t-\int_{0}^{\tau^{y}}e^{-rt}U_{\vartheta}(t,y_{t})\,\mathrm{d}t\right|\left(\mathbf{1}_{\\{\tau^{x}\geq\tau^{y}\\}}+\mathbf{1}_{\\{\tau^{x}<\tau^{y}\\}}\right)$
$\displaystyle=\left|\int_{\tau^{y}}^{\tau^{x}}e^{-rt}U_{\vartheta}(t,x_{t})\,\mathrm{d}t+\int_{0}^{\tau^{y}}e^{-rt}\left[U_{\vartheta}(t,x_{t})-U_{\vartheta}(t,y_{t})\right]\,\mathrm{d}t\right|\mathbf{1}_{\\{\tau^{x}\geq\tau^{y}\\}}$
$\displaystyle\quad+\left|\int_{\tau^{x}}^{\tau^{y}}e^{-rt}U_{\vartheta}(t,y_{t})\,\mathrm{d}t+\int_{0}^{\tau^{x}}e^{-rt}\left[U_{\vartheta}(t,y_{t})-U_{\vartheta}(t,x_{t})\right]\,\mathrm{d}t\right|\mathbf{1}_{\\{\tau^{x}<\tau^{y}\\}}$
$\displaystyle\leq
C|\tau^{x}-\tau^{y}|+\int_{0}^{T}e^{-rt}|U_{\vartheta}(t,x_{t})-U_{\vartheta}(t,y_{t})|\,dt$
$\displaystyle=C|\tau^{x}-\tau^{y}|+\int_{0}^{T}e^{-r\lambda(t)}|U_{\vartheta}(\lambda(t),x\circ\lambda(t))-U_{\vartheta}(\lambda(t),y\circ\lambda(t))|\lambda^{\prime}(t)\,dt$
$\displaystyle\leq
C|\tau^{x}-\tau^{y}|+\int_{0}^{T}e^{-r\lambda(t)}|U_{\vartheta}(\lambda(t),x\circ\lambda(t))-U_{\vartheta}(\lambda(t),y_{t})|\lambda^{\prime}(t)\,dt$
$\displaystyle\quad+\int_{0}^{T}e^{-r\lambda(t)}|U_{\vartheta}(\lambda(t),y\circ\lambda(t))-U_{\vartheta}(\lambda(t),y_{t})|\lambda^{\prime}(t)\,dt$
$\displaystyle\leq
C|\tau^{x}-\tau^{y}|+L\max\\{\|x\circ\lambda-y\|_{T},\|\lambda-I\|_{T}\\}\int_{0}^{T}\lambda^{\prime}(t)e^{-r\lambda(t)}\,dt$
$\displaystyle\quad+L\max\\{\|y\circ\lambda-y\|_{T},\|\lambda-I\|_{T}\\}\int_{0}^{T}\lambda^{\prime}(t)e^{-r\lambda(t)}\,dt$
$\displaystyle\leq
C|\tau^{x}-\tau^{y}|+2L\max\\{\|x\circ\lambda-y\|_{T},\|\lambda-I\|_{T}\\}$
$\displaystyle\quad+2L\max\\{\|y\circ\lambda-y\|_{T},\|\lambda-I\|_{T}\\},$

where $I$ is the identity map of $[0,T]$. In the last inequality, we use that
$U_{\vartheta}$ is Lipschitz continuous with the Lipschitz constant $L>0$.
Consider the infimum with respect to $\lambda\in\Lambda_{T}$ on both sides to
obtain

$|h(x)-h(y)|\lesssim|\tau^{x}-\tau^{y}|+\varrho_{T}(x,y).$

This inequality with Lemma 5.1 implies that $h$ is continuous on
$(\mathbb{D}_{T},\varrho_{T})$.

Because
$\nabla_{\vartheta}^{k}h(x,\vartheta)=\int_{0}^{\tau^{x}}e^{-rt}\nabla_{\vartheta}^{k}U_{\vartheta}(x_{t})\,\mathrm{d}t\
(k\leq N)$ under this assumption, the same argument is valid for
$\nabla_{\vartheta}^{k}h(\cdot,\vartheta)\ (k\leq N-1)$. ∎

######  Lemma 5.3.

Suppose we use the same assumptions as in Lemma 5.2, and that there exist
constants $p>d=\dim(\Theta)$ and $q>0$ as well as sequence $\\{\gamma_{n}\\}$
with $\gamma_{n}\uparrow\infty$ such that

$\displaystyle\sup_{i\in A_{n}}\|\widehat{X}_{t}^{i,n}-X_{t}\|_{L^{p}}\leq
t^{q}\cdot o(\gamma_{n}^{-1}),\quad n\to\infty.$ (5.5)

For each $t>0$, where $o(\gamma_{n}^{-1})$-term is independent of $t$. Then it
follows for each $i^{n}\in A_{n}$ and any $p>d$ that

$\displaystyle\left\|\nabla_{\vartheta}^{k}h_{\vartheta}(\widehat{X}^{i^{n},n})-\nabla_{\vartheta}^{k}h_{\vartheta}(X)\right\|_{L^{p}}=o(\gamma_{n}^{-1}),\quad
k\leq N-1$

###### Proof.

For any integer $k\leq N-1$, $\nabla_{\vartheta}^{k}U_{\vartheta}$ is the
Lipschitz continuous with respect to $\vartheta$ uniformly in $x$ under the
assumptions in Lemma 5.2. Hence, it follows for any $p\geq 1$ that

$\displaystyle\Big{\|}\nabla_{\vartheta}^{k}h_{\vartheta}(\widehat{X}^{i^{n},n})-\nabla_{\vartheta}^{k}h_{\vartheta}(X)\Big{\|}_{L^{p}}$
$\displaystyle\leq\int_{0}^{\infty}e^{-rt}\left\|\nabla_{\vartheta}^{k}U(\widehat{X}_{t}^{i^{n},n})-\nabla_{\vartheta}^{k}U(X_{t})\right\|_{L^{p}}\,\mathrm{d}t$
$\displaystyle\lesssim\int_{0}^{\infty}e^{-rt}\|\widehat{X}_{t}^{i^{n},n}-X_{t}\|_{L^{p}}\,\mathrm{d}t$
$\displaystyle\leq
o(\gamma_{n}^{-1})\int_{0}^{\infty}t^{q}e^{-rt}\,\mathrm{d}t,$

which implies (4.5) in Theorem 4.2. ∎

We can check the condition (5.5) by taking the sampling interval $h_{n}$
suitably, as in the following Lemma:

######  Lemma 5.4.

Suppose that $h_{n}=n^{-\beta}$ for some $\beta\in(0,1)$. Hereafter, the
condition (5.5) holds with $\gamma_{n}=n^{\beta/p}$ for any $p>d$.

###### Proof.

Recall that

$\displaystyle\widehat{X}_{t_{k}}^{i^{n},n}=\mu+\sum_{l=1}^{k-1}\Delta_{i^{n}(l)}X,\quad
X_{t_{k}}=\mu+\sum_{l=1}^{k-1}\Delta_{l}X,\quad t_{k}=kh_{n}$

Hereafter, it follows that

$\displaystyle\hat{X}_{t_{k}}^{i^{n},n}-X_{t_{k}}$
$\displaystyle=\sum_{l=1}^{k-1}\Delta_{i^{n}(l)}X-\sum_{l=1}^{k-1}\Delta_{l}X=\sum_{c=0}^{k-1}\sum_{i^{n}\in
B_{n,k,c}}(\Delta_{i^{n}(l)}-\Delta_{l^{\prime}}X)$
$\displaystyle=^{d}\sum_{c=0}^{k-1}\sum_{i\in
B_{n,k,c}}(k-c)\bm{1}_{\\{i^{n}=i\\}}(\Delta_{1}X-\Delta_{2}X),$

where
$B_{n,k,c}=\\{i\in\Lambda_{n}:\\#\\{i(l):\\{\\{i(1),\ldots,i(k)\\}\cap\\{1,\ldots,k\\}\\}=c\\}$,
which is the number of pairs $(i^{n}(l),l^{\prime})$with $i^{n}(l)=l^{\prime}$
in $l,l^{\prime}\in\\{1,\ldots,k\\}$ is equal to $c$. Here, we claim that the
number of $i\in\Lambda_{n}$ such that
$\\{\\{i(1),\ldots,i(k)\\}\cap\\{1,\ldots,k\\}\\}=c$ is

$\displaystyle{k\choose c}\cdot{{n-k}\choose{k-c}}\cdot k!(n-k)!.$

Generally, it follows from the Vandermonde’s formula $\sum_{j=0}^{k}{n\choose
j}{m\choose k-j}={n+m\choose k}$, which

$\displaystyle\sum_{c=0}^{k}{k\choose c}{k\choose c}k!(n-k)!={n\choose
k}\frac{k!(n-k)!}{n!}n!=\\#\Lambda_{n},$

for each $k$. Therefore, we have

$\displaystyle\sum_{c=0}^{k-1}$ $\displaystyle{k\choose c}{n-k\choose
k-c}k!(n-k)!\times(k-c)$
$\displaystyle=\sum_{c=0}^{k-1}\frac{k!^{2}(n-k)!^{2}}{c!(k-c)!^{2}(n-2k+c)!}\times(k-c)$
$\displaystyle=\sum_{c=0}^{k-1}\frac{(k-1)!^{2}(n-k+1)!^{2}}{c!(k-1-c)!^{2}(n-2k+2+c)!}\times\frac{k^{2}(n-2k+2+c)(n-2k+1+c)}{(k-c)(n-k+1)^{2}}$
$\displaystyle\leq\sum_{c=0}^{k-1}\frac{(k-1)!^{2}(n-k+1)!^{2}}{c!(k-1-c)!^{2}(n-2k+2+c)!}\times\frac{k^{2}(n-k+1)(n-k)}{(n-k+1)^{2}}$
$\displaystyle=\sum_{c=0}^{k-1}{k-1\choose c}{n-k+1\choose
k-1-c}(k-1)!(n-k+1)!\times\frac{k^{2}(n-k)}{(n-k+1)}$
$\displaystyle=n!\times\frac{k^{2}(n-k)}{(n-k+1)}\leq n!\times k^{2},$

and that

$\displaystyle\|\hat{X}_{t_{k}}^{i^{n},n}-X_{t_{k}}\|_{L^{p}}$
$\displaystyle=\left\|\sum_{c=0}^{k-1}\sum_{i\in
B_{n,k,c}}(k-c)\bm{1}_{\\{i^{n}=i\\}}(\Delta_{1}X-\Delta_{2}X)\right\|_{L^{p}}$
$\displaystyle\leq\sum_{c=0}^{k-1}\sum_{i\in
B_{n,k,c}}(k-c)\mathbb{P}\left(i^{n}=i\right)\left\|(\Delta_{1}X-\Delta_{2}X)\right\|_{L^{p}}$
$\displaystyle\leq n!\times
k^{2}\mathbb{P}(i^{n}=i)\left\|(\Delta_{1}X-\Delta_{2}X)\right\|_{L^{p}}$
$\displaystyle=n!\times
k^{2}\times\frac{1}{n!}\left\|(\Delta_{1}X-\Delta_{2}X)\right\|_{L^{p}}$
$\displaystyle\lesssim k^{2}\cdot O(r_{n}^{-1}).$

Taking $h_{n}=n^{-\beta}$ ($0<\beta<1$) and $q>2$, we observe that

$\displaystyle\frac{\|\hat{X}_{t_{k}}^{i^{n},n}-X_{t_{k}}\|_{L^{p}}}{(t_{k})^{q}}\lesssim\frac{k^{2}}{(kh_{n})^{q}}O(r_{n}^{-1})=k^{2-q}\cdot
n^{q\beta}O(r_{n}^{-1})\leq n^{2-q+q\beta}O(r_{n}^{-1})$

Hence, when we take $q>2/(1-\beta)$, it holds that

$\displaystyle\|\hat{X}_{t_{k}}^{i^{n},n}-X_{t_{k}}\|_{L^{p}}=(t_{k})^{q}\cdot
o(r_{n}^{-1}).$

∎

######  Remark 5.1.

Notably, the rate $\gamma_{n}$ in the above proof depends on $p>d$, which
means that, by Theorem 4.2, we can obtain the rate of convergence
$\sqrt{n^{\beta/p}}$ for each $p>d$:

$\sqrt{n^{\beta/p}}(\widehat{\vartheta}_{n}-\vartheta_{0})\leadsto
N(0,\Sigma),\quad n\to\infty,$

by controlling the resampling number $\alpha_{n}$ such that
$\alpha_{n}/n^{\beta/p}=o(1)$, where $\Sigma$ is given in Theorem 4.2.
Hereafter, we can access the optimal rate $\sqrt{n^{\beta/d}}$ by letting
$p\downarrow d$.

###  5.2 Dividends up to ruin

In the dividends problem, we can consider a case where $U_{\vartheta}$ in
(5.2) is of the form

$U_{\vartheta}(x,t)=\alpha\mathbf{1}_{\\{x\geq\vartheta\\}}\mathbf{1}_{\\{t\leq
g(\vartheta)\\}},$

where $\alpha>0$ is a constant and $g$ is an increasing function in
$\vartheta\in\Theta$. Assuming that a surplus of an insurance company
$x=(x_{t})_{t\geq 0}$ is given, the functional

$h_{\vartheta}(x)=\int_{0}^{\tau^{x}}e^{-rt}U_{\vartheta}(t,x_{t})\,\mathrm{d}t$

is interpreted as the aggregate dividends paid up to ruin $x_{t}<d$ or
maturity $g(\vartheta)$ depending on the parameter $\vartheta$, where the
dividend $\alpha$ is paid when the surplus $x_{t}$ is over the threshold
$\vartheta$, and the maturity depends on the threshold. That is, when the
threshold level is high (the dividends are hard to pay), the maturity for
dividends will be longer, but the maturity will be shorter when the threshold
level is low (the dividends are easy to pay). Because the ruin level is $d>0$,
the threshold $\vartheta$ should be set over $d$, and we assume that
$\Theta=[d,M]$ for constant $M>0$.

For technical reasons, we replace the indicator function $\mathbf{1}_{\\{u\geq
z\\}}$ with $\varphi_{\epsilon}(u,z)\in
C^{\infty}(\mathbb{R}\times\mathbb{R})$ with bounded derivatives such that

$\varphi_{\epsilon}(u,z)=\left\\{\begin{array}[]{cc}1&(u\geq z+{\epsilon})\\\
0&(u\leq z-{\epsilon})\end{array}\right.$

for a ‘small’ constant ${\epsilon}>0$. That is, we can replace the following
indicators as:

$\displaystyle\mathbf{1}_{\\{x_{t}\geq\vartheta\\}}\quad$
$\displaystyle\Rightarrow\quad\varphi_{\epsilon}(x_{t},\vartheta)$
$\displaystyle\mathbf{1}_{\\{t\leq g(\vartheta)\\}}\quad$
$\displaystyle\Rightarrow\quad\varphi_{\epsilon}(\vartheta,g^{-1}(t))$

Hereafter, the functional $h$ can be approximated by

$h_{\vartheta}^{\epsilon}(x)=\alpha\int_{0}^{\tau^{x}}e^{-rt}\varphi_{\epsilon}(x_{t},\vartheta)\varphi_{\epsilon}(\vartheta,g^{-1}(t))\,\mathrm{d}t$

This approximation is valid because it follows by the bounded convergence
theorem that

$\lim_{{\epsilon}\to
0}h_{\vartheta}^{\epsilon}(X)=\alpha\int_{0}^{\tau^{x}}e^{-rt}\lim_{{\epsilon}\to
0}\left[\varphi_{\epsilon}(x_{t},\vartheta)\varphi_{\epsilon}(\vartheta,g^{-1}(t))\right]\,\mathrm{d}t=h_{\vartheta}(x),\quad
x\in\mathbb{D}_{T}.$

Moreover, because the derivatives of $\varphi_{\epsilon}$ are uniformly
bounded, it follows that for fixed ${\epsilon}>0$ and $k\in\mathbb{N}$,

$\nabla_{\vartheta}^{k}U_{\vartheta}^{\epsilon}(x,t)=\alpha\nabla_{\vartheta}^{k}\left[\varphi_{\epsilon}(x_{t},\vartheta)\varphi_{\epsilon}(\vartheta,g^{-1}(t))\right]$

satisfies conditions (5.3) and (5.4) in Lemma 5.2. Hereafter, Theorem 4.2 is
applicable for $h^{\epsilon}_{\vartheta}(X)$ with the Lévy process
$X=(X_{t})_{t\geq 0}$. Further examples will be discussed in a different
paper; Shimizu and Shiraishi [12].

#### Acknowledgments

The first author was partially supported by JSPS KAKENHI Grant Numbers
JP21K03358 and JST CREST JPMJCR14D7, Japan. The second author was supported by
JSPS KAKENHI Grant Number JP16K00036.

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