11affiliationtext: Kellogg School, Northwestern University22affiliationtext:
Department of Economics, Virginia Tech

# Real-Time Detection of Local No-Arbitrage Violations111We are grateful to
Chun He for excellent research assistance and to participants at various
conferences and seminars for useful comments and suggestions.

Torben G. Andersen  Viktor Todorov  Bo Zhou

###### Abstract

This paper focuses on the task of detecting local episodes involving violation
of the standard Itô semimartingale assumption for financial asset prices in
real time that might induce arbitrage opportunities. Our proposed detectors,
defined as stopping rules, are applied sequentially to continually incoming
high-frequency data. We show that they are asymptotically exponentially
distributed in the absence of Itô semimartingale violations. On the other
hand, when a violation occurs, we can achieve immediate detection under infill
asymptotics. A Monte Carlo study demonstrates that the asymptotic results
provide a good approximation to the finite-sample behavior of the sequential
detectors. An empirical application to S&P 500 index futures data corroborates
the effectiveness of our detectors in swiftly identifying the emergence of an
extreme return persistence episode in real time.

JEL classification: C12, C53, G10, G17

Keywords: asset price, high-frequency data, Itô semimartingale violation,
real-time detection, stopping rule

##  1 Introduction

The no-arbitrage principle is central to modern asset pricing theory (Ross
(1976)). Delbaen and Schachermayer (1994) demonstrate, within a frictionless
setting, that arbitrage opportunities are precluded if and only if the price
process constitutes a semimartingale. The standard class of no-arbitrage price
processes in financial economics is the Itô semimartingale, which is a
semimartingale with characteristics that are absolutely continuous in time.
However, recent work document episodic violations of the Itô semimartingale
assumption that, absent transaction costs, might induce arbitrage
opportunities. A prominent example is the gradual jump identified by
Barndorff-Nielsen et al. (2009) and further studied by Christensen et al.
(2014). It occurs when an apparent return jump, identified from lower
frequency data, instead reflects a strongly drifting, yet (near) continuous
price path, when observed at higher frequencies. A related phenomenon is the
so-called flash crash, where a sudden collapse in price is reversed rapidly;
see, e.g., the work on the May 2010 events in the S&P 500 e-mini futures
market by Kirilenko et al. (2017) and Menkveld and Yueshen (2019).222In the
presence of trading costs and uncertainty surrounding the data generating
process, such episodes are not necessarily true arbitrage opportunities, see,
e.g., the discussion in Andersen et al. (2021).

The “explosive” price paths characterizing such events are unlikely to be
generated by an Itô semimartingale. To accommodate these occurrences,
alternative models have been developed for violation episodes, including the
drift burst model proposed by Christensen et al. (2022) and the persistent
noise by Andersen et al. (2021), as a stochastic generalization of the former.
These models contain a parameter $\tau$, indicating a random point in time
located within a neighborhood in which an Itô semimartingale violation occurs.
Such episodes typically involve turbulent market conditions with extreme
realized volatility, raising concerns of evaporating liquidity and general
market malfunction. Moreover, our standard measures for monitoring return
volatility are potentially subject to large biases, when the semimartingale
assumption is violated. Consequently, identification of the onset, indicated
by $\tau$, as well as duration of the extreme return drift episode is of great
interest for regulators, industry practitioners and academics. This is the
goal of this paper.

The detection problem can be addressed from two distinct perspectives. The
more common is the offline approach where the researcher observes the full
dataset, and then conducts a “one-shot” procedure to identify whether and when
violations have occurred. The vast literature on ex-post detection of
structural breaks in macroeconomic and financial time series data, including
Andrews (1993), Bai (1996), Bai and Perron (1998), and Elliott and Müller
(2014), falls within this category. More recently, Bücher et al. (2017)
develop inference procedures for a change point in the jump intensity
parameter for a Lévy process with high-frequency data following this approach.

A more challenging and, for practitioners, investors and regulators, arguably
more relevant perspective is the real-time setting, where data arrive
continuously, and one wishes to detect the violation in a timely manner. This
objective has some resemblance to that of the recent now-casting literature in
macroeconomics, where the aim is to update the assessment of the current and
future state of the economy as new data are received in real time. For early
initial work on a formal statistical framework in this setting, see, e.g.,
Evans (2005) and Giannone et al. (2008), while corresponding work utilizing
financial data can be found in, e.g., Andreou et al. (2013) and Bańbura et al.
(2013). Another related macro-econometric literature is initiated by Diba and
Grossman (1988) regarding the detection of macroeconomic bubbles and crises.
This methodology is extended by Phillips et al. (2011) using a recursive
procedure based on right-tailed unit root tests to detect and locate the
origin and terminal dates for bubbles. A series of subsequent studies provide
further theoretical modifications, e.g., Phillips and Yu (2011), Phillips et
al. (2014), and Phillips et al. (2015b), while empirical applications are
provided by Phillips and Shi (2018) and Phillips et al. (2015a).

Our objective is closely aligned with the latter real-time or online detection
procedures. From a technical perspective, our approach is rooted in the
statistic literature on the sequential detection problem for a change point.
This literature typically deals with the mean of i.i.d samples or with the
drift within a continuous-time model featuring both a drift and a scaled
standard Brownian motion component. The literature goes back to, at least,
Abraham Wald’s sequential analysis. His work inspired the widely-known CUSUM
rule by Page (1954) and subsequently the Bayesian rule developed by Shiryaev
(1963) and Roberts (1966).333For more studies and optimality results, see,
e.g., Moustakides et al. (1986), Ritov (1990), Shiryaev (1996), and
Moustakides et al. (2004). See also Srivastava and Wu (1993) for a comparison
and Figueroa-López and Ólafsson (2019) for the case of a Lévy process. On the
financial econometrics side, Andreou and Ghysels (2006) accommodates the CUSUM
test for strongly dependent financial time series data, and Andreou and
Ghysels (2008) applies real-time detection procedures to the structural
parameters of credit risk models by monitoring the associated Radon-Nikodym
derivative process. We follow the former rule, but also deviate substantially,
because that statistic requires knowledge of the alternative measure after the
change, which is not a natural assumption in our setting. Specifically, our
detector is based on the generalized likelihood ratio (GLR) statistic, which
profiles out the unknown alternative parameter. For studies on this GLR-CUSUM
procedure under an i.i.d setting, see Siegmund and Venkatraman (1995), Pollak
and Siegmund (1975), Lai and Siegmund (1979), and Lai (1995), among others.

Our paper differs from the aforementioned work in two key aspects. First, we
rely on infill asymptotics and exploit the feature of accessing an
asymptotically increasing number of observations of the process locally. This
helps us formalize the notion of rapid detection of local Itô semimartingale
violations. These deviations from the semimartingale dynamics are only local
in nature, unlike the earlier sequential detection literature which deals with
detection of a permanent change. Second, we abandon the use of size versus
power to characterize the properties of our detection procedure because we, by
necessity, must apply our tests sequentially. If a test with fixed critical
value is applied sequentially on a constant flow of newly arriving data, the
null will inevitably be rejected repeatedly, and the traditional type I error
literally explodes (with probability approaching one). To address this issue,
we follow the sequential detection literature to design and evaluate the
performance of our procedure using alternative metrics: the average run length
(ARL) and false detection rate (FDR) or, more comprehensively, an asymptotic
probability bound on the false detection (BFD) rate versus a corresponding
bound on the detection delay (BDD) (see Lorden (1971) and Lai (1995)).
Specifically, for the null probability measure, when there is no Itô
semimartingale violation, ARL measures the expected sample size until the
first false detection, while FDR measures the probability of a false detection
within a given period (and we refer to this period as BFD). For the
alternative, BDD provides a probability bound on the number of observations
before we achieve successful detection following a violation.444The sequential
detection literature typically measures the behavior under the alternative via
the Expected Detection Delay, or EDD. Characterizing the behavior of the
latter is more complicated within our high-frequency setting. Thus, we focus
on the probability estimate BDD instead. Consequently, one strives to develop
a detector that, conditional on a reasonably large ARL/small FDR, achieves the
smallest possible BDD. An alternative would be to develop a time-varying
boundary function – in contrast to a constant threshold – to control test size
uniformly, see, e.g., Chu et al. (1996). Unfortunately, in a real-time
sequential setting, such procedures struggle with the detection of structural
changes that arrive late within the period. There is work seeking to alleviate
this drawback, e.g., Leisch et al. (2000), Horváth et al. (2004), Aue and
Horváth (2004), Aue et al. (2006), and Horvath et al. (2007). However, these
procedures still tend to generate a significant detection delay, and we do not
pursue this direction here.

As noted, our objective is to design statistical devices that detect local Itô
semimartingale violations swiftly and reliably after their occurrence, using
continually incoming high-frequency data. Towards this end, we propose GLR-
CUSUM type detectors as stopping times based on an estimated Brownian motion
component of the latent asset price, which we recover from high-frequency
return data along with short-dated options. We first establish the accuracy of
this Brownian motion estimator, exploiting the option-based spot volatility
estimate of Todorov (2019) to standardize the high-frequency returns. This
approach avoids complications stemming from the inconsistency of standard
volatility measurements from returns, caused by local Itô semimartingale
violations, and it exploits the efficiency offered by option data for
recovering spot volatility. Next, following the asymptotic setting in the
sequential detection literature, we show that our detector is approximately
exponentially distributed under the null (Theorem 1) and develop a bound for
the BDD under the alternative (Theorem 2). The former result implies that the
ARL is the mean and FDR a percentile of the distribution. In turn, the latter
result implies that our detectors achieve immediate detection of Itô
semimartingale violations under infill asymptotics. A Monte Carlo experiment
finds that our theoretical results provide good guidance for the finite-sample
properties of the detectors under a realistically calibrated simulation
setting.

Finally, we apply our detection methods empirically to one-minute S&P 500
equity (SPX) index futures data to investigate the pervasiveness of the Itô
semimartingale violation phenomenon and to assess whether our detector
provides timely alerts regarding the failures. We further extend the procedure
to obtain an identification rule for the duration of the violation. It is
defined as the union of the time intervals on which our GLR-CUSUM statistic
exceeds the chosen threshold. With a threshold specification for our detector
leading to about 6% daily FDR, we find these violations to be common in the
SPX data — with a little more than 1,000 such violations across 3,500 days.
That said, more than half of the violations last for less than $10$ minutes,
and only a small proportion exceeds $20$ minutes. Visual inspection suggests
that, in most instances, our procedure detects such incidents within just a
few minutes of their occurrence.

The remainder of the paper is organized as follows. In Section 2 we describe
the setting and formulate the problem. In Section 3 we introduce our real-time
detectors and study their asymptotic properties. Next, we carry out a Monte
Carlo study in Section 4 to illustrate the finite-sample performances of the
detectors, and we then use them in an empirical application in Section 5.
Section 6 concludes.

##  2 Setup

The observed log-price process, $Y_{t}$, is defined on a filtered probability
space $\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P}\right)$. We assume that it may be decomposed as,

$\displaystyle Y_{t}\leavevmode\nobreak\ =\leavevmode\nobreak\
X_{t}\,+\,H_{t}\,,$ (1)

where $X$ represents an underlying efficient and arbitrage-free log-price. In
addition, the observed price is contaminated by the noise component $H$, which
is the source of brief episodes characterized by extreme return persistence,
such as the so-called gradual jumps and flash crashes. Sections 2.1 and 2.2
introduce our fairly standard specification for the observation scheme and
$X$, while Section 2.3 provides a detailed account of the dynamics for the
noise component $H$.

###  2.1 The Observation Scheme

We assume the price is observed over a given time interval $[0,T]$ at
equidistant times, $t_{i}=i\Delta_{n}$, for all $i=0,1,\dots,nT$, where
$\Delta_{n}=1/n$ is the increment length. Let the $i$-th high-frequency log-
return be denoted by,

$\displaystyle\Delta^{n}_{i}Y\,=\,Y_{i\Delta_{n}}\,-\,Y_{(i-1)\Delta_{n}}.$
(2)

Without loss of generality, we normalize the trading day to unity. Hence, $T$
refers to the number of trading days, and $n=1/\Delta_{n}$ (assumed integer)
is the number of observations per (trading) day. The time span of the data,
$T$, is assumed fixed throughout.

Following the standard infill asymptotic framework, we let the sampling
frequency, $\Delta_{n}$, shrink to $0$ (or, equivalently, $n\to\infty$). This
equidistant sampling scheme can readily be relaxed to a non-equidistant one,
requiring $\max_{\\{i\in{1,\dots,nT\\}}}(t_{i}-t_{i-1})\to 0$.

###  2.2 The efficient log-price $X$

The efficient log-price process $X$ is an Itô semimartingale,

$\displaystyle X_{t}\leavevmode\nobreak\ =\leavevmode\nobreak\
X_{0}\,+\,\int_{0}^{t}b_{s}\,ds\,+\,\int_{0}^{t}\sigma_{s}\,dW_{s}\,+\,\int_{0}^{t}\int_{\mathbb{R}}\delta(s,x)\,\mu(ds,dx)\,,$
(3)

where the initial value $X_{0}$ is $\mathcal{F}_{0}$-measurable, the drift
$b_{t}$ takes value in $\mathbb{R}$, $W=(W_{t})_{t\geq 0}$ is a one-
dimensional standard Wiener process,
$\delta:\,\mathbb{R}_{+}\times\mathbb{R}\mapsto\mathbb{R}$ is a predictable
mapping, and $\mu$ is a Poisson random measure on
$\mathbb{R}_{+}\times\mathbb{R}$ with predictable compensator (or intensity
measure) $\nu(dt,dx)=dt\otimes F(dx)$.

We impose the following assumption on the efficient price process.

######  Assumption 1.

There exists arbitrary small $\mathbb{T}>0$ such that for the process $X$ in
equation (3), we have, for $t\in[0,\mathbb{T}]$ and $\forall p\geq 1$,

$\mathbb{E}_{0}|b_{t}|\,+\,\mathbb{E}_{0}\left(\int_{\mathbb{R}}|\delta(t,x)|F(dx)\right)\,+\,\mathbb{E}_{0}|\sigma_{t}|^{p}\leavevmode\nobreak\
<\leavevmode\nobreak\ C_{0}(p),$ (4)

where $C_{0}(p)$ is $\mathcal{F}_{0}$-adapted random variable that depends on
$p$, and further

$\mathbb{E}_{0}|\sigma_{t}-\sigma_{s}|^{2}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ C_{0}\,|t-s|,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \textrm{for
$s,t\in[0,\mathbb{T}]$},$ (5)

where $C_{0}$ is $\mathcal{F}_{0}$-adapted random variable.

A few comments about this assumption are warranted. First, since our testing
procedure concerns the asset price dynamics during an Itô semimartingale
violation initiated close to time zero, the assumption focuses on the behavior
of $X$ in the vicinity of $0$. Second, the first part of Assumption 1 involves
the existence of conditional moments, but since $\mathbb{T}>0$ can be
arbitrary small, these moment conditions are relatively weak. Third, the jump
part of $X$ is modeled as an integral against a Poisson measure and,
therefore, we restrict attention only to finite variation jumps. Finally, the
“smoothness in expectation” condition for $\sigma$ will be satisfied whenever
$\sigma$ is an Itô semimartingale, which is the standard way of modeling
stochastic volatility. We note, however, that this condition will not hold for
the class of rough stochastic volatility models.555For this class of models,
the right-hand side of the second equation in Assumption 1 will be replaced by
$C_{0}\,|t-s|^{2H}$, for $0<H<1/2$ capturing the degree of roughness of the
volatility path.

######  Remark 2.1.

We refer to the model (3) for the efficient asset price $X$, along with
Assumption 1, as the standard Itô semimartingale model since it embeds most
specifications used in current existing work. We note, however, that there is
a gap between a semimartingale, required to preclude the existence of
arbitrage opportunities, and the standard Itô semimartingale considered above.
The gap includes models in which the semimartingale characteristics are not
absolutely continuous in time as well as models with infinite variation jumps.
We do not consider those in our analysis.

######  Remark 2.2.

We can readily accommodate an empirically relevant extension of the standard
Itô semimartingale model to the case, where the efficient price path features
discontinuities triggered by economic announcements at pre-specified points in
time. Since the announcement times are known a priori, one may remove the
associated price increments from the analysis and proceed as if $X$ follows a
standard Itô semimartingale. To minimize notational complexity, we do not
formally introduce this extension.

###  2.3 The persistent noise $H$

Our main objective is to develop inference tools to detect episodic violations
of the Itô semimartingale assumption, which is almost universally imposed in
standard asset pricing theory. From a practical perspective, the occurrences
of gradual jumps or, in more extreme manifestations, flash crashes, are of
particular interest because the associated drift burst in the returns can
induce severe biases in the high-frequency measurement of the (efficient)
return variation. We include such short-lived episodes of extreme return
persistence in our setup through the Persistent Noise (PN) model of Andersen
et al. (2021), which is designed explicitly to accommodate these types of
empirically observed phenomena. The PN model is a stochastically extended
version of the Drift Burst (DB) model by Christensen et al. (2022), and the
real-time detection procedures developed below applies for either
specification. We initially consider a simplified setting with only a single
episode involving semimartingale violations.

The PN model initiates a violation episode at a random point, $\tau_{n}$,
i.e., $H_{t}=0$ for $t<\tau_{n}$ and in our analysis $\tau_{n}\downarrow 0$.
At $\tau_{n}$, the efficient price may display a discrete jump, $\Delta
X_{\tau_{n}}\neq 0$. If an efficient price jump occurs, it likely reflects the
arrival of new information or unusual ongoing trading activity. To the extent
this information is not common knowledge or not readily interpretable, a gap
will emerge between the efficient price, reflecting rational processing of all
relevant information, and the market price which is determined by the
interaction of risk averse and merely partially informed agents.666For an in-
depth discussion of this feature across a large cross-section of stocks, see
Andersen et al. (2022). In other words, the efficient price may deviate from
the market price, implying that the noise component is active,
$H_{\tau_{n}}\neq 0$. In fact, if the information is not immediately observed
or processed by market participants, we have $H_{\tau_{n}}=\Delta
H_{\tau_{n}}=-\Delta X_{\tau_{n}}$, implying $\Delta Y_{\tau_{n}}=\Delta
X_{\tau_{n}}-\Delta X_{\tau_{n}}=0$. That is, the price reaction is delayed.
An intermediate case is where the news generates a partial response. For
example, if $H_{\tau_{n}}=-\Delta X_{\tau_{n}}/2$, the price jump underreacts
to the new information. Such events may trigger a period of intense
information acquisition and price discovery, inducing rapid price adjustment.
After initiation at $\tau_{n}$, the future path of the noise component is
modeled through a specific functional form,
$g(t),t\in[\tau_{n},\overline{\tau}]$. Finally, we ensure that the PN event
terminates at some random point, $\overline{\tau}>\tau_{n\,}$, so that we have
$H(t)=0$ for $t>\overline{\tau}$.

We are now in position to formally introduce the PN model.

The Persistent Noise (PN) model: For some non-negative sequence
$\tau_{n}\downarrow 0$, we let,

$\displaystyle H_{t}\leavevmode\nobreak\ =\leavevmode\nobreak\ f(\Delta
X_{\tau_{n}},\eta_{\tau_{n}})\,g\left(t\right),\leavevmode\nobreak\
\leavevmode\nobreak\ t\geq\tau_{n},$ (6)

where $\eta_{\tau_{n}}$ is an $\mathcal{F}_{\tau_{n}}$-adapted random
variable, $f$ is a continuous and bounded function, and $g$ is given as,

$\displaystyle
g(s)\,=\,\left\\{1-\left(\frac{s-\tau_{n}}{\overline{\tau}-\tau_{n}}\right)^{\vartheta_{pn}}\right\\}\,\mathbbm{1}_{\\{s\in[\tau_{n},\overline{\tau}]\\}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\vartheta_{pn}\in(0,0.5),$ (7)

for some $\mathcal{F}_{\tau_{n}}$-adapted random $\overline{\tau}>\tau_{n}$.

The strength of the local return drift in the vicinity of the PN initiation
point, $\tau_{n}$, in equation (7) increases as $\vartheta_{pn}$ takes on
lower values, while the duration of the event is controlled by the realization
of $\overline{\tau}$. We also note that $g(s)$ is zero, indicating no Itô
semimartingale violation, until $\tau_{n}$. Finally, $\eta_{\tau_{n}}$ allows
for a random response to the events triggering the permanent noise component,
which may or may not be associated with a jump in the efficient price.

To illustrate the dynamics of the PN model, Figure 1 plots a simulated sample
path for the efficient price $X$ (blue) across the 6.5 hour trading day along
with the associated path for the observed price $Y$ (black). The efficient
price path features a positive jump at $\tau_{n}=0.25$ (at $97$ minute), but
(informational) frictions prevent this from being recognized by market
participants, so there is no instantaneous jump in the observed price. In
terms of the model, we register a corresponding negative latent noise jump at
$\tau_{n}$, namely $f(\Delta X_{\tau_{n}},\eta_{\tau_{n}})=-\Delta
X_{\tau_{n}}$. This is a purely mechanical effect. If there is no observed
price change, but the efficient price jumps, then the deviation between the
two – the noise term $H$ – must offset the efficient price jump.

As noted, Figure 1 is designed to replicate the gradual jump phenomenon. It is
evident that this PN episode also may reflect the recovery from a so-called
flash crash that hits its nadir at $\tau_{n}$. Thus, one may replicate the
typical flash crash pattern by combining this “recovery phase” with an
inverted version of the PN path prior to $\tau_{n}$, see Andersen et al.
(2021) for a more detailed discussion.

Figure 1: Price path samples of the efficient price $X$ (in blue) and observed
price $Y$ (in black) simulated by the Persistent Noise (PN) model.

From a technical perspective, it is convenient to introduce a version of the
DB model by Christensen et al. (2022).777Christensen et al. (2022) allow for
simultaneous volatility and drift bursts. We do not include this feature, as
we find no systematic evidence for significant elevation in option-based
volatility measures at such times.

The Drift Burst (DB) model: For some non-negative sequence $\tau_{n}\downarrow
0$, we have,

$\displaystyle
H_{t}=\int_{0}^{t}c(s-\tau_{n})^{-\vartheta}\mathbbm{1}_{\\{s\in[\tau_{n},\overline{\tau}]\\}}ds,\leavevmode\nobreak\
\leavevmode\nobreak\ \vartheta\in(0.5,1),$ (8)

for a constant $c$ and $\tau_{n}\leq\overline{\tau}$. $\tau_{n}$ and
$\overline{\tau}$ have the same interpretation as in the PN model.

The DB model may be viewed as a differential version of the PN model by
observing that
$\int_{t_{1}}^{t_{2}}(s-\tau)^{-\vartheta}ds=(1-\vartheta)^{-1}(t_{2}-\tau)^{1-\vartheta}-(1-\vartheta)^{-1}(t_{1}-\tau)^{1-\vartheta}$
for $0\leq\tau\leq t_{1}<t_{2}$. Therefore, the asymptotic analysis will be
identical for the PN model and the DB model with $\vartheta=1-\vartheta_{pn}$.
We will exploit this equivalence in our theoretical arguments.

Henceforth, we denote the probability and expectation by
$\mathbb{P}_{\tau_{n}}$ and $\mathbb{E}_{\tau_{n}}$ for the case where the
observed log-price is generated as above, and by $\mathbb{P}_{\infty}$ and
$\mathbb{E}_{\infty}$ if there are no Itô semimartingale violations (that is,
when $\tau_{n}=\infty$).

##  3 Real-Time Detection

In this section, we propose GLR-CUSUM type detectors as stopping rules for the
local Itô semimartingale violations illustrated above. Next, we develop their
theoretical properties. This involves characterizing the accuracy for our
estimate of the Brownian motion driving the price, followed by approximation
results for the distribution of the stopping rules, leading to analytic
formulas for ARL and FDR, and finally a probability result for BDD which, in
turn, implies a probability bound on the speed of detection.

###  3.1 The GLR-CUSUM stopping rule

Our local Itô semimartingale violation detector is built upon a local
estimator of the Brownian motion driving the price, defined as,

$\displaystyle\widehat{W}_{l_{1},\,l_{2}}\leavevmode\nobreak\
=\leavevmode\nobreak\
\sum_{i=l_{1}+1}^{l_{2}}\,\frac{\Delta_{i}^{n}Y}{\leavevmode\nobreak\
\hat{\sigma}_{(i-1)\,\Delta_{n}}\leavevmode\nobreak\
}\,\mathbbm{1}_{\\{|\Delta_{i}^{n}Y|<\zeta\Delta_{n}^{\varpi}\\}}\,,$ (9)

for some $l_{1},l_{2}\in\mathbb{N}$ such that $0\leq l_{1}<l_{2}\leq nT$, and
$\hat{\sigma}_{i\Delta_{n}}$ is an option-based nonparametric spot volatility
estimator defined later. As detection of the semimartingale violation
inevitably is subject to some delay, the option-based $\hat{\sigma}$ enables
us to avoid the bias in volatility estimation induced by the bursting drift
following $\tau_{n}$ (see Andersen et al. (2021)).

Our CUSUM-type detector, employing the generalized likelihood ratio (GLR)
statistic, is now defined by the stopping rule,

$\displaystyle\mathbf{N}^{\rm w}(\xi)\leavevmode\nobreak\
=\leavevmode\nobreak\ \inf\left\\{\,l>0:\max_{l-w_{n}\leq
k<l}\,\frac{|\widehat{W}_{k,l}|}{\leavevmode\nobreak\
\sqrt{(l-k)\,\Delta_{n}}\leavevmode\nobreak\ }\leavevmode\nobreak\
>\leavevmode\nobreak\ \xi\,\right\\}\,,$ (10)

where $w_{n}$ controls the maximum window length for $\widehat{W}_{k,l}$, and
$\xi$ is a chosen threshold.888The terminology of generalized likelihood ratio
statistic here refers to (the square root of $2$ times) the likelihood of a
Brownian motion with drift $\mu$ versus drift of $0$,
$\mu\,W_{k,l}-\mu^{2}(l-k)\Delta_{n}/2$, with the unknown alternative $\mu$
being replaced by its maximum likelihood estimate
$\widehat{\mu}=W_{k,l}/\sqrt{(l-k)\Delta_{n}}$. In words, at time point $l$,
we calculate the absolute value of the Brownian increment estimate
$\widehat{W}_{k,l}$ over the interval $[k,l]$ (normalized by
$1/\sqrt{(l-k)\Delta_{n}\,}$) for each point $k$ before $l$, and then take the
maximum. If the maximum exceeds the threshold $\xi$, the detector reports an
alarm, signaling detection of an Itô semimartingale violation.

###  3.2 Recovering the Driving Brownian Motion for the Price

Our theoretical results concerning detection of local semimartingale
violations hinge on the accuracy of the estimator $\widehat{\sigma}_{t}$. The
following assumption is sufficient.

######  Assumption 2.

For some arbitrary small $\mathbb{T}>0$, we have
$\inf_{t\in[0,\mathbb{T}]}\sigma_{t}>0$ and

$\mathbb{E}_{0\,}|\widehat{\sigma}_{t}-\sigma_{t}|^{2}\leavevmode\nobreak\
\leq\leavevmode\nobreak\ C_{0}\,\delta_{n}^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
t\in[0,\mathbb{T}],$ (11)

for some $\mathcal{F}_{0}$-adapted positive random variable $C_{0}$, and a
deterministic sequence $\delta_{n}\rightarrow 0$.

In addition, we have $\widehat{\sigma}_{t}>C_{0}/l_{n}$, for
$t\in[0,\mathbb{T}]$, some $\mathcal{F}_{0}$-adapted positive random variable
$C_{0}$, and $l_{n}=\log(1/\delta_{n})$.

Several features of the above assumption are noteworthy. First, we impose
conditions on the volatility estimator only in the vicinity of zero. Second,
we require that $\sigma_{t}$ is non-vanishing in a neighborhood of zero. This
is important for the behavior of the volatility estimator, but also for the
overall validity of our detection procedure, as the diffusive component of $X$
drives our asymptotic results. Third, the deterministic sequence $\delta_{n}$
captures the rate of convergence of the option-based volatility estimator.
This rate is determined by the mesh of the strike grid of the available
options as well as their tenor, see Todorov (2019).

The above assumption is natural in the absence of a local semimartingale
violation. Its validity is less clear if there is a violation of the type
discussed in the previous section in the vicinity of $t=0$. However, note that
a drift burst or persistent noise incident will not affect the norm of the
conditional characteristic function of the price increment which forms the
basis for the estimator of Todorov (2019). Hence, the consistency of our spot
volatility estimator should not be affected by the local semimartingale
violation. That said, the quality of the option-based estimator might worsen
during such periods due to poor option data quality. To guard against this
possibility, we impose some filters on the option data in our application.

To facilitate the statement of our next result, we introduce the following
notation,

$\displaystyle\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},\,l_{2}}\leavevmode\nobreak\
=\leavevmode\nobreak\ \sum_{i=l_{1}+1}^{l_{2}}\Delta_{i}^{n}W,$

for some $l_{1},l_{2}\in\mathbb{N}$ such that $0\leq l_{1}<l_{2}\leq nT$. The
issue in this section is how closely $\widehat{W}_{l_{1},\,l_{2}}$
approximates
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},\,l_{2}}$.
The key result is provided by the following proposition.

######  Proposition 1.

Suppose Assumptions 1 and 2 hold and $H\equiv 0$. Let $k_{n}\to\infty$ and
$k_{n}\Delta_{n}\to 0$ as $\Delta_{n}\to 0$. Let $0\leq l_{1}^{n}<l_{2}^{n}$
be such that $l_{1}^{n}\Delta_{n}\rightarrow 0$ as $\Delta_{n}\rightarrow 0$
and, as before, $l_{n}=\log(1/\delta_{n})$. Then, for a positive $\xi>0$, we
have,

$\begin{split}&\mathbb{P}_{0}\left(\max_{|l_{2}^{n}-l_{1}^{n}|\leq
k_{n}}\frac{\leavevmode\nobreak\
|\widehat{W}_{l_{1}^{n},l_{2}^{n}}-\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1}^{n},l_{2}^{n}}|\leavevmode\nobreak\
}{\sqrt{(l_{2}^{n}-l_{1}^{n})\,\Delta_{n}}}\,>\,\xi\right)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leq\leavevmode\nobreak\ C_{0}\,k_{n}\,\left(\frac{\leavevmode\nobreak\
\sqrt{k_{n}}\,l_{n}\,\delta_{n}+\sqrt{k_{n}\,\Delta_{n}}\leavevmode\nobreak\
}{\xi}\,+\,\frac{k_{n}\,\Delta_{n}}{\leavevmode\nobreak\
1\wedge\xi^{2}\leavevmode\nobreak\
}\,+\,\Delta_{n}^{1-\varpi}\right)\,,\end{split}$ (12)

for some $\mathcal{F}_{0}$-adapted positive random variable $C_{0}$ that does
not depend on $k_{n}$, $\Delta_{n}$ or $\xi$.

###  3.3 Average Run Length (ARL)

We first analyze the behavior of the stopping time $\mathbf{N}^{\rm w}(\xi)$
under the null measure, $\mathbb{P}_{\infty}$, where there is no Itô
semimartingale violation, i.e., $\tau_{n}=\infty$. More specifically, our
interest is in evaluating the ARL, formally defined as
$\mathbb{E}_{\infty}[\,\mathbf{N}^{\rm w}(\xi)\,]$. We show in Theorem 1 below
that $\mathbf{N}^{\rm w}(\xi)$, under $\mathbb{P}_{\infty}$, is approximately
exponentially distributed, leading to an approximation result for ARL. Theorem
1 follows by adapting the theoretical framework in Siegmund and Venkatraman
(1995), based on the random fields analysis in Siegmund (1988) to our setting
(see also Yao (1993)). We use the symbol “$x_{n}\sim y_{n}$” to denote
“$x_{n}$ is asymptotic equivalent to $y_{n}$”, that is,
$\lim_{n\to\infty}x_{n}/y_{n}=1$.

######  Theorem 1.

Denote the standard normal CDF and PDF by $\Phi$ and $\phi$. Let $w_{n}$ be
such that $w_{n}\sim a_{n}\,\xi^{2}$, with $a_{n}$ being a deterministic
sequence converging to $a\in(0,\infty]$. Suppose the conditions in Proposition
1 hold. Then, if
$(l_{n}\,\delta_{n}+\sqrt{\Delta_{n\,}})/\,[\xi\,\phi(\xi)]\rightarrow 0$, as
$\xi\to\infty$, $\mathbf{N}^{\rm w}(\xi)$ will be asymptotically exponentially
distributed with expectation,

$\displaystyle\mathbb{E}_{\infty}[\,\mathbf{N}^{\rm
w}(\xi)\,]\leavevmode\nobreak\ \sim\,\frac{1}{\leavevmode\nobreak\
D_{a}\,\xi\,\phi(\xi)\leavevmode\nobreak\ }\,\,.$ (13)

Here $D_{a}$ is a positive constant depending on $a$ such that $D_{a}\to D$,
as $a\to\infty$, where $D=\int_{0}^{\infty}x\,\nu^{2}(x)\,dx$ with
$\nu(x)=2x^{-2}\exp\left[-2\sum_{n=1}^{\infty}n^{-1}\,\Phi(-x\sqrt{n}/2)\right]$
for $x>0$.

The proof of Theorem 1 is provided in Appendix A. Following Siegmund and
Venkatraman (1995), it is useful for numerical evaluation to apply the
approximation $\nu(x)=\exp(-\rho x)+o(x^{2})$ for $x\to 0$, where the value of
the constant $\rho$ is about $0.583$. Based on this, we can numerically
determine $D$ to take the value $0.735$.

We desire for the ARL, under the null, to be as large as possible or,
equivalently, $\xi$ to be as large as possible. The optimal such $\xi$, up to
a log term and for a given $w_{n}$, will be $\xi$ such that
$\phi(\xi)\sim(\delta_{n}\,l_{n}\vee\sqrt{\Delta_{n}})$. It is reasonable to
assume that the size of the error in estimating spot volatility from options
is much smaller than the high-frequency approximation error, that is,
$\delta_{n}\,l_{n}/\sqrt{\Delta_{n}}\rightarrow 0$. Under this condition and
the indicated choice of $\xi$, we obtain from Theorem 1 that,

$\mathbb{E}_{\infty}[\,\mathbf{N}^{\rm w}(\xi)\,]\leavevmode\nobreak\
\sim\,\frac{1}{\leavevmode\nobreak\ \sqrt{\Delta_{n}}\leavevmode\nobreak\
}\leavevmode\nobreak\ \sqrt{\,\log\left(\frac{1}{\Delta_{n}}\right)\,}\,.$
(14)

We note that this ARL value can be obtained with $w_{n}$ (the window size)
taking values within a wide range — from an order of $\xi^{2}$ through using
(almost) all available observations on a given day (retaining
$w_{n}\,\Delta_{n}\rightarrow 0$). For the case of
$w_{n}/\,\xi^{2}\rightarrow\infty$, we may denote our detector
$\mathbf{N}(\xi)$, and replace $D_{a}$ by $D$, as $D_{a}\to D$ with
$a\to\infty$.999For the standard i.i.d. case, more specific guidelines in
selecting $w_{n}$ can be found in Lai (1995).

###  3.4 False Detection Rate (FDR)

Given the nature of our sequential detection procedure, we cannot rely on the
regular notion of test size to control the rejection rate under the null
hypothesis. The preceding subsection instead focuses on the concept of ARL,
establishing this quantity (the ARL) as the mean value of the asymptotic
distribution for stopping time $\mathbf{N}^{\rm w}(\xi)$ under
$\mathbb{P}_{\infty}$. However, this distributional result has wider
implications. For example, it allows us to control the false detection rate
(FDR) as,

$\displaystyle\mathbb{P}_{\infty\,}[\,\mathbf{N}^{\rm
w}(\xi)\,<\,\ell_{n}\,]\leavevmode\nobreak\ \leq\leavevmode\nobreak\
\alpha\,,$ (15)

where $\ell_{n}$ indicates an asymptotically increasing sequence of
observations, while $\alpha$ is the bound for the FDR – the probability of
triggering (at least) one false alarm after observing no more than $\ell_{n}$
price increments. If $\alpha$ is asymptotically shrinking, then we refer to
$\ell_{n}$ as the bound (in probability) on false detection or BFD.

The FDR is often viewed as a more robust measure than ARL. Typically, the
objective is to ensure an expected long duration (ARL) before a false alarm is
triggered. The problem is that a stopping rule may feature a large ARL and
still retain a high probability of false alarms within short periods. Although
this does not apply in our case, we still proceed with a result concerning the
limiting behavior of FDR under the null, since it provides a guideline for
selecting a threshold in empirical applications. Moreover, it renders the
Monte Carlo studies more computationally tractable, as we avoid simulating an
excessively long sample to assess the ARL, especially when we experiment with
large thresholds. Below, we use the symbol “$x_{n}\lesssim y_{n}$” to denote
“$x_{n}$ is asymptotic equivalent to or less than $y_{n}$”, that is,
$\lim_{n\to\infty}x_{n}/y_{n}\leq 1$.

######  Corollary 1.

Suppose the conditions in Theorem 1 hold and let $\ell_{n}$ be such that
$\ell_{n}\,\xi\,\phi(\xi)\to 0$, as $\xi\to\infty$. Then, we have,

$\displaystyle\mathbb{P}_{\infty}[\,\mathbf{N}^{\rm
w}(\xi)<\ell_{n\,}]\leavevmode\nobreak\ \lesssim\leavevmode\nobreak\
\ell_{n}\,D_{a}\,\xi\,\phi(\xi)\,.$ (16)

This corollary follows directly from Theorem 1. Specifically, the approximate
exponential distribution of $\mathbf{N}^{\rm w}(\xi)$ indicates
$\mathbb{P}_{\infty}[\mathbf{N}^{\rm w}(\xi)<\ell_{n}]\sim
1-\exp(-\ell_{n}\,D_{a}\,\xi\,\phi(\xi))$, and the result then follows from
the inequality $1-\exp(-x)\leq x$ for $x$ close to $0$. A sequence $\ell_{n}$
satisfying the conditions of Corollary 1 constitutes a BFD.

###  3.5 Bound on Detection Delay (BDD)

When an Itô semimartingale violation occurs at some time $\tau_{n}\downarrow
0$, we wish to detect it as quickly as possible, using discrete high-frequency
observations of $Y$ starting at time 0. We refer to a deterministic sequence
$T_{n}\rightarrow\infty$ such that $\mathbb{P}\left(\,\mathbf{N}^{\rm
w}(\xi)\,>\,\tau_{n}/\Delta_{n}+T_{n}\,\right)\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ 0$, when $H$ is nontrivial, as a bound (in
probability) on detection delay or BDD. We want $T_{n}$ as small as possible.
Theorem 2 below provides a lower bound on BDD.

######  Theorem 2.

Suppose $b_{s}=b_{0}$, $\sigma_{s}=\sigma_{0}$ and $\Delta X_{s}=0$, almost
surely, for $s$ in a neighborhood of zero and that Assumption 2 holds. Let $H$
be given by the drift burst model in equation (8) for some $c\neq 0$. Let
$w_{n}$ be such that $w_{n}\sim a_{n}\,\xi^{2}$, with $a_{n}$ being a
deterministic sequence converging to $a\in(2/c^{2},\infty]$. Finally, let
$\varpi\in(0,1/2)$ and $\vartheta\in(1/2,1)$ as well as
$\xi\Delta_{n}^{\iota}\rightarrow 0$ for any $\iota>0$. We then have,

$\mathbb{P}\left(\,\mathbf{N}^{\rm
w}(\xi)\,>\,\tau_{n}/\Delta_{n}+T_{n}\,\right)\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ 0\,,$ (17)

for any sequence $T_{n}\rightarrow\infty$ such that
$T_{n}\,\Delta_{n}\rightarrow 0$, $w_{n}/T_{n}\rightarrow 0$,
$T_{n}/\Delta_{n}^{(1/2-\vartheta)/\vartheta}\rightarrow\infty$ and
$\sqrt{w_{n}}\leavevmode\nobreak\
l_{n}\,\delta_{n}\,\Delta_{n}^{\varpi-1/2}/\,(\xi\,T_{n})\rightarrow 0$.

We invoke a number of simplifying assumptions in the derivation of Theorem 2.
Primarily, we assume that $X$ is continuous and features constant drift and
constant diffusive volatility in a (shrinking) neighborhood of zero. This
style of assumption is common in the high-frequency financial econometrics
literature and — as is typically the case — it should be possible to relax at
the cost of more complicated derivations.

Theorem 2 is proved by constructing an alternative measure, say
$\widetilde{\mathbb{P}}$, under which the semimartingale violations are
milder. Specifically, the expected value of the increments under
$\widetilde{\mathbb{P}}$ are smaller in asymptotic order of magnitude than
under $\mathbb{P}$. Consequently, it takes longer to detect the change, and
thus the BDD under $\widetilde{\mathbb{P}}$ is an upper bound for that under
$\mathbb{P}$.

The object characterized by the BDD condition (17) is a diverging sequence of
observations, $(T_{n})$, sufficiently large to ensure almost sure detection,
in our infill asymptotic setting, of an Itô semimartingale violation. It is
natural to require that our detector identifies such violations before
triggering a false alarm with probability approaching one or, equivalently,
BDD $<$ BFD. From Theorem 2, we need
$T_{n}/\Delta_{n}^{(1/2-\vartheta)/\vartheta}\rightarrow\infty$, while BFD,
from Corollary 1 and the discussion thereafter, is of asymptotic order
$1/\,(\xi\,\phi(\xi))$. Thus, we must have
$\xi\,\phi(\xi)\,\Delta_{n}^{(1/2-\vartheta)/\vartheta}\to\,0$ to ensure BDD
$<$ BFD. This condition may be satisfied for any $\vartheta<1$, provided $\xi$
is chosen as large as possible, while still guaranteeing that Theorem 1
applies (recall the discussion after Theorem 1).

The BDD embeds two sources of delay (as shown in our proof): (i) the
truncation of price increments to guard against jumps; (ii) the requisite
cumulation of non-truncated returns in our detection statistic.
Asymptotically, the first delay term dominates the second. Nonetheless, in
finite samples, the BDD will be determined by both effects.

######  Remark 3.1.

We can also introduce a double-window-limited GLR rule, defined as,

$\displaystyle\mathbf{N}^{\rm
ww}(\xi):=\inf\left\\{l>0:\max_{l-w_{n}-r_{n}\leq
k<l-r_{n}}\frac{|\widehat{W}_{k,l}|}{\sqrt{(l-k)\Delta_{n}}}>\xi\right\\},$
(18)

where $r_{n}$ imposes a restriction on the minimum span for the statistic
$\widehat{W}_{k,l}$. This helps reduce the false detection error under the
null induced by extreme values for the statistic $\widehat{W}_{k,l}$ due to
only a few ($\,<r_{n}$) increments. Consequently, this will generate a longer
ARL/smaller FDR under the null, and a slightly larger BDD under the
alternative. We do not analyze $\mathbf{N}^{\rm ww}(\xi)$ theoretically, but
evaluate its performance numerically via simulation. The Monte Carlo study in
Section 4 shows that setting $r_{n}$ to, say, $5$, we obtain effective
protection against outliers without affecting the detection delay by much.

We note that the literature on testing for bubbles in macroeconomic settings
often imposes a similar minimum duration restriction on the bubble period
(see, e.g., Phillips et al. (2011), Phillips and Yu (2011), and Phillips et
al. (2015b)).

##  4 Simulation Study

In this section we carry out a simulation study to assess the finite-sample
performance of our procedure for swift detection of local Itô semimartingale
violations.

###  4.1 Simulation setting

To generate sample paths for the efficient log-price $X$, we simulate the
following Heston type stochastic volatility model with jumps,

$\displaystyle dX_{t}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\ b_{t}\,dt\,+\,\sigma_{t}\,dW_{1,t}\,+\,dJ_{t}\,,$ (19)
$\displaystyle d\sigma_{t}^{2}$ $\displaystyle\leavevmode\nobreak\
=\leavevmode\nobreak\
\kappa\,(\gamma-\sigma_{t}^{2\,})\,dt\,+\,\varsigma\,\sigma_{t}\,W_{2,t}$ (20)

where $W_{1}$ and $W_{2}$ are standard Brownian motions with correlation
$\mathbb{E}(dW_{1,t},dW_{2,t})=\rho\,dt$, and $J_{t}$ denotes the jump term
for which we employ a Poisson process with intensity $p_{X}$ and jump size
distribution $\mathcal{N}(0,\lambda_{X}^{2})$ .

In terms of parameter specification, we set the initial efficient log-price
$X_{0}$ to $\log(1200)$, the drift term $b_{t}$ to zero, and the unit of time
to one trading day. The volatility process $\sigma_{t}^{2}$ is initiated at
its unconditional mean of $\gamma$ on day one while, for other days, it is
initiated at the ending value of the previous day. The annualized parameter
vector for the Heston model is set to
$(\kappa,\gamma,\varsigma,\rho)=(5,0.0225,0.4,-\sqrt{0.5})$. For the jump
components, we let $p_{X}=3/5$, corresponding to 3 jump per week on average,
and $\lambda_{X}=0.5\%$.

For the Itô semimartingale violation term $H$, we employ the DB and PN models
(8) and (6)–(7), respectively, to generate gradual-jump type patterns.
Specifically, for the DB model, we set $c=3$, $\tau_{n}=0.25$ and
$\overline{\tau}=0.4$, and explore $\vartheta\in\\{0.55,0.65,0.75\\}$. For the
PN model, we add a jump in $X$ at $\tau_{n}=0.25$ each day. We let $f(\Delta
X_{\tau},\eta_{\tau})=\eta_{\tau}\,\Delta X_{\tau}$ with $\eta_{\tau}=-1$,
$\overline{\tau}=0.4$, and explore $\vartheta_{pn}\in\\{0.45,0.35,0.25\\}$,
corresponding to $\Delta X_{\tau}=\\{1.4\%,2.0\%,3.0\%\\}$. Finally,
$\widehat{\sigma}_{t}=\sigma_{t}\times(1+0.02\times Z)$, where $Z$ is a
standard normal random variable, serves as proxy for the (noisy) option-based
volatility estimate.

###  4.2 Simulation results

We first present results for the null measure $\mathbb{P}_{\infty}$, i.e., no
Itô semimartingale violations.

Table 1: Average Run Length (ARL) | $\mathbf{N}(\xi)$ | $\mathbf{N}^{\rm w}(\xi)$ | $\mathbf{N}^{\rm ww}(\xi)$
---|---|---|---
Threshold $\xi$ | Theory | Simulation | Theory | Simulation | Simulation
3.4 |  358 |  343 |  393 |  367 |  684
3.5 |  487 |  475 |  536 |  509 |  939
3.6 |  669 |  673 |  740 |  718 | 1298
3.7 |  931 |  956 | 1033 | 1014 | 1792
3.8 | 1310 | 1312 | 1457 | 1432 | 2456
3.9 | 1864 | 1881 | 2082 | 2058 | 3356
4.0 | 2682 | 2609 | 3007 | 2897 | 4406
The ARL is based on simulated $1$-minute data for 1,000 replications. We
choose the window size $w_{n}=30$ minutes for $\mathbf{N}^{\rm w}(\xi)$ and
$(w_{n},r_{n})=(30,5)$ minutes for $\mathbf{N}^{\rm ww}(\xi)$. We set the jump
truncation threshold $\zeta=4\,\widehat{\sigma}_{t-1}^{\rm\,med}$ for all
cases.

In Table 1, we report the ARL values for our proposed detectors
$\mathbf{N}(\xi)$ (with $w_{n}=\infty$), $\mathbf{N}^{\rm w}(\xi)$ ($w_{n}=30$
minutes), and $\mathbf{N}^{\rm ww}(\xi)$ ($(w_{n},r_{n})=(30,5)$ minutes)
based on our simulation setting. We also provide the theoretical ARL values
for the former two detectors based on Theorem 1. Comparing the first two and
the next two columns we find that, for a wide range of threshold values
ranging from $3.4$ to $4.0$, the Monte Carlo values for $\mathbf{N}(\xi)$ and
$\mathbf{N}^{\rm w}(\xi)$ are close to their corresponding theoretical values,
indicating that the asymptotic approximation (13) captures the finite-sample
performance well. When comparing the ARLs of $\mathbf{N}(\xi)$ to those of
$\mathbf{N}^{\rm w}(\xi)$, we find that the latter are slightly larger than
the former. This indicates that the window limit $w_{n}$ does not induce any
major deterioration under the null, even with the relatively small value
(here, $30$ minutes). On the contrary, in the last column, we find the ARL
values for $\mathbf{N}^{\rm ww}(\xi)$ to be significantly larger than the
previous ones within the same settings. Evidently, the window limit $r_{n}$
has a significant impact under the null measure by ignoring false detections
induced by just a couple of increments. Thus, there is room to experiment
along this dimension in order to improve the FDR, although it is critical to
also monitor the associated increase in the detection delay under the
alternative.

Table 2: False Detection Rate (FDR) | $\mathbf{N}(\xi)$ | $\mathbf{N}^{\rm w}(\xi)$ | $\mathbf{N}^{\rm ww}(\xi)$
---|---|---|---
Threshold $\mathbf{\xi}$ | Theory | Simulation | Theory | Simulation | Simulation
3.5 | 0.5227 | 0.5111 | 0.4889 | 0.4880 | 0.3046
3.6 | 0.4160 | 0.4079 | 0.3854 | 0.3868 | 0.2383
3.7 | 0.3207 | 0.3088 | 0.2943 | 0.2903 | 0.1777
3.8 | 0.2403 | 0.2311 | 0.2189 | 0.2143 | 0.1319
3.9 | 0.1756 | 0.1667 | 0.1588 | 0.1508 | 0.0923
4.0 | 0.1256 | 0.1180 | 0.1128 | 0.1066 | 0.0666
4.1 | 0.0881 | 0.0847 | 0.0787 | 0.0775 | 0.0499
4.2 | 0.0608 | 0.0593 | 0.0540 | 0.0541 | 0.0364
4.3 | 0.0413 | 0.0395 | 0.0365 | 0.0355 | 0.0249
4.4 | 0.0276 | 0.0265 | 0.0243 | 0.0233 | 0.0170
4.5 | 0.0183 | 0.0180 | 0.0160 | 0.0157 | 0.0118
FDR results based on simulated $1$-minute data with $\ell_{n}=390$ (a day) for
10,000 replications. We choose the window size $w_{n}=30$ minutes for
$\mathbf{N}^{\rm w}(\xi)$ and $(w_{n},r_{n})=(30,5)$ minutes for
$\mathbf{N}^{\rm ww}(\xi)$. We set the jump truncation threshold
$\zeta=4\,\widehat{\sigma}_{t-1}^{\rm\,med}$.

The above findings are corroborated by the FDR results in Table 2, where we
use a wider range of the threshold values, taking advantage of the fact that
FDR can be assessed through simulations over deliberately chosen shorter time
intervals. In particular, for each replication, we simulate one-minute prices
for a day, and obtain the FDR as the fraction of replications with (at least)
one false detection across our 5,000 replica. In general, the Monte Carlo FDRs
are close to their theoretical values for $\mathbf{N}(\xi)$ and
$\mathbf{N}^{\rm w}(\xi)$ by Corollary 1, implying accurate analytic
approximations. The window limit $w_{n}$ does not impact the FDRs by much,
while a moderate choice for $r_{n}$ may significantly reduce the false
detection rate under the null. Finally, we note that this table can be used as
a guide for choosing $\xi$, as well as $w_{n}$ and/or $r_{n}$, in empirical
applications, perhaps following further simulations for alternative asset
price dynamics.

Figure 2: Histograms for $\mathbf{N}(\xi)$ and $\mathbf{N}^{\rm w}(\xi)$ based
on simulated $1$-minute data and 1,000 replications. The window size is
$w_{n}=30$ minutes for $\mathbf{N}^{\rm w}(\xi)$. The jump truncation
parameter is $\zeta=4\,\widehat{\sigma}_{t-1}^{\rm\,med}$ and the threshold is
$\xi=3.4$ for both cases.

We round off our simulation study for the null measure $\mathbb{P}_{\infty}$
with Figure 2. It displays histograms for the duration until (false) alarm
associated with our detectors $\mathbf{N}(\xi)$ ($w_{n}=\infty$) and
$\mathbf{N}^{\rm w}(\xi)$ ($w_{n}=30$ minutes) based on 1,000 replications.
The plots visually corroborate the approximate exponential distribution
results in Theorem 1.

We now turn to $\mathbb{P}_{\tau_{n}}$ — the probability measure with local
Itô semimartingale violation at time $\tau_{n}$ under the setting of Section
4.1. Table 3 reports the (true) detection rates (denoted $\mathbf{r}$), i.e.,
what fraction of the 1,000 replications contain (at least) one detection, and
the estimated expected detection delay or EDD defined as,

$\displaystyle\mathbb{E}_{\tau_{n}\,}[\,\mathbf{N}^{\rm
w}(\xi)-\tau_{n}\,|\,\mathbf{N}^{\rm w}(\xi)>\tau_{n}\,]\,.$ (21)

The EDD is regarded as the analogue of the ARL for the alternative measure
$\mathbb{P}_{\tau_{n}}$. In fact, ARL versus EDD is the canonical criteria in
the sequential detection literature (see, e.g., Lorden (1971)), as size versus
power for hypothesis testing. Unfortunately, an explicit approximation for the
EDD is not available in our case, and we can only derive BDD, which is a bound
in probability on $\mathbf{N}^{\rm w}(\xi)$. Reporting EDD in the simulation,
however, makes it easier to assess the size of $\mathbf{N}^{\rm w}(\xi)$ under
the alternative.

Table 3: Detection Rate ($\mathbf{r}$) and Expected Detection Delay (EDD) $\mathbf{r}$/EDD | $\mathbf{N}^{\rm w}(\xi)$ | $\mathbf{N}^{\rm ww}(\xi)$
---|---|---
| 10s | 30s | 60s | 10s | 30s | 60s
DB gradual jumps |  |  |  |
0.55 | 0.853/30.37 | 0.772/13.14 | 0.744/ 8.12 | 0.870/28.48 | 0.703/13.90 | 0.514/11.75
0.65 | 0.948/13.28 | 0.948/ 8.77 | 0.932/ 6.03 | 0.948/14.06 | 0.887/ 9.36 | 0.756/ 8.69
0.75 | 1.000/ 7.41 | 0.999/ 5.83 | 0.989/ 5.14 | 0.996/ 9.00 | 0.972/ 8.09 | 0.977/ 6.72
PN gradual jumps |  |  |  |
0.45 | 0.943/14.24 | 0.819/12.24 | 0.645/ 9.16 | 0.938/14.86 | 0.800/13.50 | 0.621/10.65
0.35 | 0.969/10.86 | 0.913/ 9.28 | 0.778/ 7.61 | 0.966/11.46 | 0.909/10.42 | 0.773/ 9.03
0.25 | 0.997/ 7.47 | 0.962/ 7.52 | 0.977/ 5.49 | 0.997/ 8.75 | 0.958/ 8.98 | 0.977/ 6.80
Detection rate ($\mathbf{r}$) and expected detection delay (EDD) results based
on simulated $10$-, $30$\- and $60$-seconds data for $1000$ days. We set
$w_{n}=30$ minutes for $\mathbf{N}^{\rm w}(\xi)$ and $(w_{n},r_{n})=(30,5)$
minutes for $\mathbf{N}^{\rm ww}(\xi)$, with jump truncation threshold
$\zeta=4\,\widehat{\sigma}_{t-1}^{\,\rm med}$.

We consider both DB and PN gradual jump patterns as specified above. In both
cases, we explore our detectors $\mathbf{N}^{\rm w}(\xi)$ with $w_{n}=30$
minutes and $\mathbf{N}^{\rm ww}(\xi)$ with $(w_{n},r_{n})=(30,5)$ minutes
under three sampling frequencies: $10$-seconds, $30$-seconds and $60$-seconds.

Both detectors provide satisfactory performance, delivering high detection
rates and short detection delays. Specifically, comparing vertically within
each panel, we see that $\mathbf{r}$ increases and EDD shrinks, as the
violations grow more severe. Likewise, comparing horizontally within each
panel, we find the detectors improving with higher sampling frequency —
generating larger $\mathbf{r}$’s and shorter detection delays (in seconds,
obtained by multiplying the entries accordingly with $10$, $30$ and $60$).
Experiments featuring even higher sampling frequencies, in turn, confirm our
(asymptotic) immediate detection result. However, implementation at very high
frequencies within an actual market setting is impacted by microstructure
noise, so it is arguably less practically applicable unless one employs a
scheme to explicitly account for ultra high-frequency frictions.

##  5 Empirical Application

###  5.1 Data

We apply our detectors to one-minute S&P 500 equity (SPX) index futures data
covering January 2007 - December 2020. We only use data for the regular
trading hours and eliminate days with reduced trading hours, producing a
sample of $3,524$ days. Following our theoretical framework, we use the
nonparametric option-based volatility estimator, SV, proposed by Todorov
(2019). We rescale SV using the previous day’s truncated volatility (TV) so
that they are at the same level.

###  5.2 Detection of semimartingale violations for S&P 500

We apply $\mathbf{N}^{\rm ww}(\xi)$ with $(w_{n},r_{n})=(30,5)$ minutes and
$\xi=4$ for our one-minute data each day, implying we initiate the procedure
$30$ minutes after the market open. Given this design, we lose power in terms
of detecting violation periods shorter than $5$ minutes. From Table 2, we see
that this detector has a probability of about $6\%$ to trigger a false alarm
at the daily level.

We further equip our detection with an ad hoc identification rule to pin down
Itô semimartingale violation regions. Specifically, we define it as the union
of all (probably overlapping) intervals, $[\,l_{a},l_{b}\,]$, such that

$\displaystyle[\,l_{a},l_{b}\,]=\mathop{\rm arg\leavevmode\nobreak\
max}\limits_{0\leq
l_{a}-w_{n}-r_{n}<l_{b}-r_{n}}\frac{|\widehat{W}_{l_{a},l_{b}}|}{\sqrt{(l_{b}-l_{a})\Delta_{n}}}>\xi.$
(22)

That is, in words, we define the region as the union of the span of GLR-CUSUM
statistics that exceed the chosen threshold $\xi$.

Figures 3 and 4 provide illustrations for two trading days, August 7, 2007 and
August 30, 2019, with detected Itô semimartingale violations. The price paths
are red, $\mathbf{N}^{\rm ww}(\xi)$ rejections are indicated by dark grey
vertical lines, and the violation regions by light grey areas. In the bottom
panel, we also provide $30$-minutes rolling window TV in blue and the option-
based SV (rescaled by the same day’s TV average) in orange.

On August 7, 2007, the price displays a gradual upward trend, until an abrupt
20 point crash over 2:15–2:30 pm, followed by a rapid recovery, taking the
price beyond the original level. Our detector swiftly signals a violation as
the flash-crash pattern emerges, triggering an alarm within a few minutes. As
shown in the bottom plot, this dramatic price pattern induces an explosion in
the local realized volatility measure, which is not consistent with our
option-based SV measure, confirming the potential for dramatic biases in
return-based volatility measurement. Similar findings hold for our second
example on August 30, 2019, where a gradual jump initiates before 11:00 am.
Again, our detector captures it expeditiously.

Figure 3: Upper plot: Price (red), Detection points (black vertical lines) and
PN-regions (in grey) for August 7, 2007. Bottom plot: same day’s rolling
window truncated realized volatility (TV, in blue) and spot volatility (SV, in
orange) standardized by the day’s average TV. Detection based on
$\mathbf{N}^{\rm ww}(\xi)$ with $(w_{n},r_{n})=(30,5)$ minutes, $\xi=4$, and
jump truncation threshold $\zeta=4\,\widehat{\sigma}_{t-1}^{\,med}$.

Table 4 reports the daily count of Itô semimartingale violation regions
detected by $\mathbf{N}^{\rm ww}(\xi)$ with window length
$(w_{n},r_{n})=(30,5)$ minutes under various values for the threshold $\xi$.
We also split the trading day (of $390$ minutes) into three periods — the
“Morning” for the first $130$ minutes, the “Noon” for the middle $130$
minutes, and the “Afternoon” for the rest — and report the associated count of
violation regions initiated within that period. Finally, Figure 5 plots the
corresponding histograms for the duration of these violation regions for the
scenario with $\xi=4$.

Figure 4: Upper plot: Price process (in red), Detection points (black vertical
lines) and PN-regions (in grey) for August 30, 2019. Bottom plot: same day’s
rolling window truncated volatility (TV, in blue) and spot volatility (SV, in
orange) standardized by the day’s average TV. Detection based on
$\mathbf{N}^{\rm ww}(\xi)$ with $(w_{n},r_{n})=(30,5)$ minutes, $\xi=4$, and
jump truncation threshold $\zeta=4\,\widehat{\sigma}_{t-1}^{\,med}$.

We find that “Morning” generates most violations, with the “Noon” containing
only slightly fewer, while “Afternoon” produces the least, even if the
difference is not striking. In sum, the violations are frequent and they are
not particularly prone to occur in specific regions of the trading day.
Finally, from the histograms in Figure 5, we note that the duration of a
typical violation is short, with the clear majority lasting less than 20
minutes. Moreover, the violation regions starting in the morning tend to be
somewhat shorter lived than those initiated during noon or afternoon.

Table 4: Count of Regions with Itô Semimartingale Violations Threshold $\xi$ | Total | Morning | Noon | Afternoon
---|---|---|---|---
4 | 2186 | 824 | 738 | 624
4.1 | 1901 | 714 | 641 | 546
4.2 | 1638 | 620 | 542 | 476
4.3 | 1398 | 520 | 461 | 417
4.4 | 1172 | 438 | 378 | 356
4.5 | 994 | 364 | 318 | 312

The table reports the number of Itô semimartingale violations and their
distribution within the trading day. We split the day ($390$ min) into blocks:
Morning (9:30 am–11:40 am), Noon (11:40 am–1:50 pm), and Afternoon (1:50 pm–4
pm), and count how many PN-regions are initiated within each block. For
$\mathbf{N}^{\rm ww}(\xi)$: window lengths $(w_{n},r_{n})=(30,5)$ minutes and
jump truncation threshold $\zeta=4\,\hat{\sigma}_{t-1}^{med}$.

Figure 5: Histograms of PN-region durations. Specifications: $\mathbf{N}^{\rm
ww}(\xi)$ with $(w_{n},r_{n})=(30,5)$ minutes and $\xi=4$; Jump truncation
threshold $\zeta=4\,\widehat{\sigma}_{t-1}^{\,med}$.

A manifestation of the Itô semimartingale violation during the detected PN
episodes is the divergence between return- and option-based spot volatility
estimators. The bottom panels of Figures 3 and 4 show that the divergences are
large in these two specific cases. Table 5 reports the sample mean of the
return- and option-based volatility estimates $TV$ and $SV$ over the hour
before and after the initiation of a PN episode. The results reveal that the
two volatility proxies are close before the PN episode but feature a
substantial gap over the following hour, with the average $TV$ about $35\%$
higher than the average $SV$. This gap only grows larger, if we exclude days
with FOMC announcements from the calculation, so the discrepancy is not driven
by this type of macroeconomic announcements. These findings demonstrate how
local deviations from the semimartingale assumption can distort the
measurement of volatility.

Table 5: Volatility Estimates during Regions with Itô Semimartingale Violations | 1 hour before | 1 hour after
---|---|---
| TV | SV | TV | SV
ALL | 10.93$\times 10^{-5}$ | 10.58$\times 10^{-5}$ | 12.61$\times 10^{-5}$ | 10.53$\times 10^{-5}$
No FOMC | 11.72$\times 10^{-5}$ | 11.56$\times 10^{-5}$ | 13.72$\times 10^{-5}$ | 11.51$\times 10^{-5}$
This table reports the average volatility measures $TV$ and $SV$, one hour
before and one hour after the starting point of PN episodes.

Finally, Table 6 reports the number of PN episodes for each year across the
sample given different threshold levels for detection. The PN events appear
near uniformly distributed over the sample, with no particular clustering
apparent during periods characterized by generally volatile or tranquil market
conditions. Overall, we conclude that PN episodes are largely idiosyncratic
events with no particular tendency to occur during specific times within the
trading day or during specific market conditions.

Table 6: PN-episodes Yearly Counts Year | $\xi=4$ | $\xi=4.1$ | $\xi=4.2$ | $\xi=4.3$ | $\xi=4.4$ | $\xi=4.5$
---|---|---|---|---|---|---
2007 | 171 | 155 | 132 | 105 | 83 | 71
2008 | 132 | 117 | 98 | 88 | 77 | 63
2009 | 89 | 80 | 65 | 56 | 42 | 30
2010 | 129 | 108 | 94 | 80 | 65 | 53
2011 | 133 | 117 | 100 | 88 | 78 | 57
2012 | 133 | 109 | 92 | 83 | 69 | 56
2013 | 194 | 168 | 142 | 118 | 103 | 92
2014 | 241 | 206 | 165 | 143 | 112 | 96
2015 | 139 | 121 | 116 | 97 | 87 | 76
2016 | 153 | 137 | 118 | 100 | 87 | 75
2017 | 170 | 151 | 132 | 113 | 99 | 84
2018 | 150 | 131 | 117 | 98 | 80 | 71
2019 | 172 | 153 | 133 | 111 | 91 | 83
2020 | 180 | 148 | 134 | 118 | 99 | 87
This table reports annual persistent-noise episode counts from 2007 to 2020
under different threshold values.

##  6 Conclusion

In this paper, we focus on real-time detection of local Itô semimartingale
violations that have drawn increasing attention in the recent literature. We
propose CUSUM-type detectors as stopping rules exploiting high-frequency data.
We show that they possess desirable theoretical properties under infill
asymptotics. Specifically, for a suitably chosen average run length (the
average sample length until a false alarm), our detectors enable “quick”
detection of a violation, once it occurs. Our formal interpretation of rapid
detection is that the bound on the detection delay (BDD) shrinks to zero under
infill asymptotics, as the sampling interval $\Delta_{n}$ goes to zero. These
properties are corroborated through simulations calibrated to reflect key
features of market data. Finally, we apply our detector to S&P 500 equity
(SPX) index futures data. We obtain timely detection of a nontrivial number of
short-lived episodes involving likely semimartingale violations. Such
turbulent market events are critical for real-time decision making. For
example, they may indicate temporary market malfunction motivating exchange or
regulatory action, they may induce the termination of trading strategies and
algorithms among active market participants triggering a period of fleeting
liquidity, and they signal problems in extracting reliable high-frequency
based market volatility and risk measures.

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##  Appendix A Proofs

###### Proof of Proposition 1.

As it will become clear from the proof below, it suffices to conduct the
analysis for the case when $0\leq l_{1}^{n}\leq k_{n}$ and $0\leq
l_{2}^{n}\leq k_{n}$. To simplify the notation, we will use $l_{1}$ and
$l_{2}$ for $l_{1}^{n}$ and $l_{2}^{n}$, respectively, in the proof. We can
write

$\begin{split}&\mathbb{P}_{0}\left(\max_{0<l_{1}<l_{2}\leq
k_{n}}\frac{|\widehat{W}_{l_{1},l_{2}}-\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}>\xi\right)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\leq\sum_{i=1}^{k_{n}}\mathbb{P}_{0}\left(\frac{1}{\sqrt{\Delta_{n}}}\left|\frac{\Delta_{i}^{n}X}{\hat{\sigma}_{(i-1)\Delta_{n}}}\mathbbm{1}_{\\{|\Delta_{i}^{n}X|<\zeta\Delta_{n}^{\varpi}\\}}-\Delta_{i}^{n}W\right|>\frac{\xi}{\sqrt{k_{n}}}\right),\end{split}$
(23)

and further

$\begin{split}&\mathbb{P}_{0}\left(\frac{1}{\sqrt{\Delta_{n}}}\left|\frac{\Delta_{i}^{n}X}{\hat{\sigma}_{(i-1)\Delta_{n}}}\mathbbm{1}_{\\{|\Delta_{i}^{n}X|<\zeta\Delta_{n}^{\varpi}\\}}-\Delta_{i}^{n}W\right|>\frac{\xi}{\sqrt{k_{n}}}\right)\\\
&\leavevmode\nobreak\
\leq\mathbb{P}_{0}\left(\frac{|\Delta_{i}^{n}X|}{\sqrt{\Delta_{n}}}\frac{|\hat{\sigma}_{(i-1)\Delta_{n}}-\sigma_{(i-1)\Delta_{n}}|}{\sigma_{(i-1)\Delta_{n}}}>\frac{C_{0}}{l_{n}}\frac{\xi}{4\sqrt{k_{n}}}\right)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\mathbb{P}_{0}\bigg{(}\frac{|\Delta_{i}^{n}J|+|\int_{(i-1)\Delta_{n}}^{i\Delta_{n}}b_{s}ds|}{\sqrt{\Delta_{n}}\sigma_{(i-1)\Delta_{n}}}>\frac{\xi}{4\sqrt{k_{n}}}\bigg{)}+\mathbb{P}_{0}\left(|\Delta_{i}^{n}X|\geq\zeta\Delta_{n}^{\varpi}\right)\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+\mathbb{P}_{0}\bigg{(}\frac{|\int_{(i-1)\Delta_{n}}^{i\Delta_{n}}(\sigma_{s}-\sigma_{(i-1)\Delta_{n}})dW_{s}|}{\sqrt{\Delta_{n}}\sigma_{(i-1)\Delta_{n}}}>\frac{\xi}{4\sqrt{k_{n}}}\bigg{)}.\end{split}$
(24)

Using Assumption 1 for the smoothness in expectation of $\sigma$, the
integrability conditions of Assumption 1, Assumption 2 and applying
Cauchy–Schwarz inequality, we have

$\mathbb{P}_{0}\left(\frac{|\Delta_{i}^{n}X|}{\sqrt{\Delta_{n}}}\frac{|\hat{\sigma}_{(i-1)\Delta_{n}}-\sigma_{(i-1)\Delta_{n}}|}{\sigma_{(i-1)\Delta_{n}}}>\frac{C_{0}}{l_{n}}\frac{\xi}{4\sqrt{k_{n}}}\right)\leq
C_{0}\left(k_{n}\Delta_{n}+\frac{\sqrt{k_{n}}l_{n}\delta_{n}}{\xi}\right).$
(25)

Using the integrability conditions for the process $b$ and the jump size
function as well as the smoothness in expectation condition for $\sigma$ in
Assumption 1, we have

$\mathbb{P}_{0}\left(\frac{|\Delta_{i}^{n}J|+|\int_{(i-1)\Delta_{n}}^{i\Delta_{n}}b_{s}ds|}{\sqrt{\Delta_{n}}\sigma_{(i-1)\Delta_{n}}}>\frac{\xi}{4\sqrt{k_{n}}}\right)\leq
C_{0}\left(\frac{\sqrt{k_{n}\Delta_{n}}}{\xi}+k_{n}\Delta_{n}\right).$ (26)

Next, using the smoothness in expectation condition for $\sigma$ in Assumption
1, we have

$\mathbb{P}_{0}\left(\frac{|\int_{(i-1)\Delta_{n}}^{i\Delta_{n}}(\sigma_{s}-\sigma_{(i-1)\Delta_{n}})dW_{s}|}{\sqrt{\Delta_{n}}\sigma_{(i-1)\Delta_{n}}}>\frac{\xi}{4\sqrt{k_{n}}}\right)\leq
C_{0}\frac{k_{n}\Delta_{n}}{1\wedge\xi^{2}}.$ (27)

Finally, given the integrability conditions in Assumption 1, we have

$\mathbb{P}_{0}\left(|\Delta_{i}^{n}X|\geq\zeta\Delta_{n}^{\varpi}\right)\leq
C_{0}\Delta_{n}^{1-\varpi}.$ (28)

Combining the above bounds, we get the result to be proved. ∎

###### Proof of Theorem 1.

Recalling that
$\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}$
is defined as a discrete random walk with $N(0,\Delta_{n})$ increments, by
(Siegmund and Venkatraman, 1995, Proposition 1), we have

$\displaystyle\mathbb{P}\left\\{\max_{0<l_{1}<l_{2}<m}\frac{|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}>\xi\right\\}\sim
md(\bar{c})\xi\phi(\xi)$

as $\xi\to\infty$, for $m=\bar{c}\xi^{2}$ with some $\bar{c}>0$ fixed, where

$\displaystyle
d(\bar{c}):=\int_{\bar{c}^{-1/2}}^{\infty}x\nu^{2}(x)dx-\bar{c}^{-1}\int_{\bar{c}^{-1/2}}^{\infty}x^{-1}\nu^{2}(x)dx.$

The above result based on the analysis of the random fields (see Siegmund
(1988)) shows that the probability of the maximum of random fields
$|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},l_{2}}|/\sqrt{l_{2}-l_{1}}$
for $0<l_{1}<l_{2}\leq m$ exceeding $\xi$, when $m$ is of order $\xi^{2}$,
goes to $0$ exponentially fast.

From the statement of Theorem 1, the expected sample size should be of order
$1/(\xi\phi(\xi))\sim\exp(\xi^{2})/\xi$, and it is much bigger than the order
of $m$ (which is $\xi^{2}$). We follow the trick in Siegmund and Venkatraman
(1995) to complete the proof of the theorem. Let the window size $w_{n}$ be
chosen such that $w_{n}\geq m$ and $w_{n}/m$ is a positive integer. More
specifically, we set $m=w_{n}$ if $a$ in the statement of the theorem is
finite and if $a=\infty$ we set $m=w_{n}/\lfloor
w_{n}/(\overline{c}\xi^{2})\rfloor$. We note that with the latter choice,
$m/\xi^{2}\rightarrow\overline{c}$. Picking $M$ with
$M=r\mathbb{E}_{\infty}[\mathbf{N}^{\rm w}(\xi)]\sim r/(D_{a}\xi\phi(\xi))$
for some fixed $r$ bounded away from 0, we have

for some deterministic sequence of positive $\iota_{n}$ satisfying
$\iota_{n}\rightarrow 0$ and $\xi\iota_{n}\rightarrow\infty$.

Consider the first term and let $\epsilon\in(0,1)$, we have

where the last inequality holds for all large $\xi$ due to the result in (A)
with $\bar{c}$ chosen such that $m\lesssim w_{n}$ and thus $d(\bar{c})\lesssim
D_{a}<D$, where $D_{a}:=d(a)$ (see Lai and Shan (1999) and Chan and Lai
(2003)). Note that
$D=\int_{0}^{\infty}x\nu^{2}(x)dx=\lim_{\overline{c}\to\infty}d(\bar{c})=\lim_{a\to\infty}D_{a}$.
Then, because $1-x\leq\exp(-x)$ for $x>0$, we have

$\displaystyle\leavevmode\nobreak\ \mathbb{P}\left\\{\max_{0\leq
j<\frac{M}{m}}\max_{jm<l_{1}<l_{2}<(j+1)m}\frac{|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}<\xi(1+\iota_{n})$
$\displaystyle\leq\exp\left(-MD_{a}\xi(1+\iota_{n})\phi(\xi(1+\iota_{n}))(1-\epsilon)\right)$
$\displaystyle\sim\exp\left(-MD_{a}\xi\phi(\xi)(1-\epsilon)\right)$
$\displaystyle=\exp\left(-r(1-\epsilon)\right).$

Turning to the second term, we show it is asymptotically negligible relative
to the first term. Recalling that $m=\overline{c}\xi^{2}$, we apply
Proposition 1 (note that $M\Delta_{n}\rightarrow 0$ by the assumption of the
theorem) and have

$\displaystyle\begin{split}&\leavevmode\nobreak\ \mathbb{P}\left\\{\max_{0\leq
j<\frac{M}{m}}\max_{jm<l_{1}<l_{2}<(j+1)m}\frac{|\widehat{W}_{l_{1},l_{2}}-\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}\geq\xi\iota_{n}\right\\}\\\
&\leq\sum_{j=0}^{M/m}\mathbb{P}\left\\{\max_{jm<l_{1}<l_{2}<(j+1)m}\frac{|\widehat{W}_{l_{1},l_{2}}-\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}\geq\xi\iota_{n}\right\\}\\\
&\leavevmode\nobreak\ \leq
C_{0}M\left(\frac{\sqrt{m}(l_{n}\delta_{n}+\sqrt{\Delta_{n}})}{\xi\iota_{n}}+\Delta_{n}^{1-\varpi}\right)\leq
C_{0}M\left(\frac{(l_{n}\delta_{n}+\sqrt{\Delta_{n}})}{\iota_{n}}+\Delta_{n}^{1-\varpi}\right).\end{split}$

If we pick $\iota_{n}$ such that $\iota_{n}\xi\rightarrow\infty$ and in
addition
$\left[(l_{n}\delta_{n}+\sqrt{\Delta_{n}})/\iota_{n}+\Delta_{n}^{1-\varpi}\right]/[\xi\phi(\xi)]\rightarrow
0$ (which is possible due to the restriction in the theorem) and noting that
$\Delta_{n}^{1-\varpi}/\sqrt{\Delta_{n}}\to 0$ (because $\varpi<1/2$), then

$\displaystyle\begin{split}&\mathbb{P}\left\\{\max_{0\leq
j<\frac{M}{m}}\max_{jm<l_{1}<l_{2}<(j+1)m}\frac{|\widehat{W}_{l_{1},l_{2}}-\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}\geq\xi\iota_{n}\right\\}\\\
&\bigg{/}\mathbb{P}\left\\{\max_{0\leq
j<\frac{M}{m}}\max_{jm<l_{1}<l_{2}<(j+1)m}\frac{|\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{W}]{\kern-0.3pt\bigwedge\kern-0.3pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.4ex}}\stackon[1pt]{W}{\scalebox{-1.0}{\tmpbox}}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}<\xi(1+\iota_{n})\right\\}\rightarrow
0.\end{split}$

Altogether, therefore, we have, for $\epsilon\in(0,1)$, that

$\displaystyle\mathbb{P}\left\\{\mathbf{N}^{\rm
w}>M\right\\}\leq\exp\left(-MD_{a}\xi\phi(\xi)(1-\epsilon)\right)=\exp\left(-r(1-\epsilon)\right).$
(29)

We next proceed with bounding $\mathbb{P}\left\\{\mathbf{N}^{\rm w}\leq
M\right\\}$. We have

$\displaystyle\mathbb{P}\left\\{\mathbf{N}^{\rm w}\leq M\right\\}\leq$
$\displaystyle\leavevmode\nobreak\ \mathbb{P}\left\\{\max_{0\leq
j<\frac{M}{m}}\max_{jm<l_{1}<l_{2}<(j+1)m}\frac{|\widehat{W}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}>\xi\right\\}$
(30) $\displaystyle\leavevmode\nobreak\ +\mathbb{P}\left\\{\max_{0\leq
j<\frac{M}{m}}\max_{jm<l_{1}<(j+1)m,(j+1)m<l_{2}<(j+1)m+w_{n}}\frac{|\widehat{W}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}>\xi\right\\},$

where in the second term we need to consider only the maximum over
$(j+1)m<l_{2}<(j+1)m+w_{n}$ for each $j$ due to the limited window size. Let
$\epsilon>0$. For the first term, using similar trick to the one for bounding
$\mathbb{P}\left\\{\mathbf{N}^{\rm w}>M\right\\}$ above, we can show

$\displaystyle\mathbb{P}\left\\{\max_{0\leq
j<\frac{M}{m}}\max_{jm<l_{1}<l_{2}<(j+1)m}\frac{|\widehat{W}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}>\xi\right\\}$
$\displaystyle=1-\mathbb{P}\left\\{\max_{0\leq
j<\frac{M}{m}}\max_{jm<l_{1}<l_{2}<(j+1)m}\frac{|\widehat{W}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}\leq\xi\right\\}$
$\displaystyle\leq 1-\exp\left(-MD_{a}\xi\phi(\xi)(1+\epsilon)\right)$
$\displaystyle=1-\exp\left(-r(1+\epsilon)\right).$

If we can further show that the second term in (30) is asymptotically
negligible, relative to $\exp\left(-r(1+\epsilon)\right)$, we will have

$\displaystyle\mathbb{P}\left\\{\mathbf{N}^{\rm
w}>M\right\\}\geq\exp\left(-r(1+\epsilon)\right),$ (31)

which, together with (29), implies

$\displaystyle\mathbb{P}\left\\{\mathbf{N}^{\rm w}\leq M\right\\}\sim
1-\exp\left(-r\right),$

that $\mathbf{N}^{\rm w}(\xi)$ is asymptotically (as $\xi\to\infty$)
exponentially distributed with mean $1/(D_{a}\xi\phi(\xi))$.

For the second term in (30), first we note that

for the same sequence $\iota_{n}$ as above. Again, by Proposition 1 and the
condition that
$[(\delta_{n}l_{n}+\sqrt{\Delta_{n}})/\iota_{n}+\Delta_{n}^{1-\varpi}]/[\xi\phi(\xi)]\rightarrow
0$, the second probability on the right-hand-side of the above inequality is
asymptotically negligible relative to the first one.

For the first of the two probabilities, we can show that it is less than
$\epsilon$, provided $\bar{c}$ is large enough, for all large $\xi$ (i) by
applying again the random field result in Siegmund (1988) if $w_{n}$ is at the
order of $\bar{c}^{1/2}m=\bar{c}^{3/2}\xi^{2}$ or (ii) by Siegmund and
Venkatraman (1995, Lemma 2–8) if $w_{n}$ is at a larger order (say, e.g., of
$1/\xi\phi(\xi)$) or even through infinity. The latter case (ii) thus applies
also to the non-window-limited stopping rule $\mathbf{N}(\xi)$ (where
$w_{n}=\infty$), as well as all above arguments. For this case we replace
$D_{a}$ by $D$ since $D_{a}\to D$ as $a_{n}\to\infty$. ∎

###### Proof of Theorem 2.

Without loss of generality, we can set $\sigma_{0}=1$ and we do so in the rest
of the proof in order to simplify exposition. For some deterministic sequence
$\iota_{n}\downarrow 0$ and such that $\xi\iota_{n}\rightarrow\infty$ and
$\sqrt{w_{n}}l_{n}\delta_{n}\Delta_{n}^{\varpi-1/2}/(\xi\iota_{n}T_{n})\rightarrow
0$ (such a sequence is possible to pick due to the restriction on $w_{n}$ of
the theorem), we have

$\displaystyle\mathbb{P}\left(\mathbf{N}^{\rm w}(\xi)>\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{T}_{n}\right)\leq\mathbb{P}\left(\overline{\mathbf{N}}^{\rm
w}(\xi(1+\iota_{n}))>\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{T}_{n}$ $\displaystyle\leavevmode\nobreak\ +\mathbb{P}\left(\max_{0\leq
j<\frac{\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{T}_{n}}{w_{n}}}\max_{jw_{n}<l_{1}<l_{2}<(j+1)w_{n}}\frac{|\widehat{W}_{l_{1},l_{2}}-\overline{W}_{l_{1},l_{2}}|}{\sqrt{(l_{2}-l_{1})\Delta_{n}}}\geq\xi\iota_{n}\right),$
(32)

where $\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{T}_{n}=\tau_{n}/\Delta_{n}+T_{n}$,
$\overline{W}_{l_{1},l_{2}}=\frac{1}{\sigma_{0}}\sum_{i=l_{1}+1}^{l_{2}}\Delta_{i}^{n}Y\mathbbm{1}_{\\{|\Delta_{i}^{n}Y|<\zeta\Delta_{n}^{\varpi}\\}}$,
and $\overline{\mathbf{N}}^{\rm w}(\xi)$ is the stopping time associated with
$\overline{W}_{l_{1},l_{2}}$. For the second of the two probabilities on the
right-hand side of the above inequality, we can argue as in the proof of
Proposition 1 and given the condition of the theorem conclude that it is
asymptotically negligible. Hence, it suffices to consider the behavior of
$\overline{\mathbf{N}}^{\rm w}(\xi)$ in what follows.

Next, we denote $i_{n}=\inf\\{i=1,...,n:i\Delta_{n}\geq\tau_{n}\\}$ and
introduce

$K_{n}=\inf\left\\{i=0,1,...:\left|c\int_{(i+i_{n}-1)\Delta_{n}}^{(i+i_{n})\Delta_{n}}(s-\tau_{n})^{-\vartheta}ds\right|<\zeta\sqrt{\Delta_{n}}\right\\}.$
(33)

For $n$ large, we have
$K_{n}\asymp\Delta_{n}^{\frac{1/2-\vartheta}{\vartheta}}$. We denote the
stopping time associated with a discretized process constructed from the
increments
$\frac{1}{\sigma_{0}}\Delta_{i}^{n}Y\mathbbm{1}_{\\{|\Delta_{i}^{n}Y|<\zeta\Delta_{n}^{\varpi}\\}}$,
for $i\geq i_{n}+K_{n}$, with $\overline{\mathbf{N}}^{\rm w}_{0}(\xi)$. Since
$w_{n}/K_{n}\rightarrow 0$, we have $\overline{\mathbf{N}}^{\rm
w}(\xi)>\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{T}_{n}$ $\Longrightarrow$ $\overline{\mathbf{N}}^{\rm w}_{0}(\xi)>T_{n}$
and hence it suffices to bound the probability
$\mathbb{P}\left(\overline{\mathbf{N}}^{\rm w}_{0}(\xi)>T_{n}\right)$.

For the increment of $H$, we have

$\displaystyle\Delta_{i}^{n}H$
$\displaystyle=H_{i\Delta_{n}}-H_{(i-1)\Delta_{n}}$
$\displaystyle=\frac{c}{1-\vartheta}\left[(i\Delta_{n}-\tau_{n})^{1-\vartheta}-((i-1)\Delta_{n}-\tau_{n})^{1-\vartheta}\right]$
$\displaystyle\sim
c\left(i-\frac{\tau_{n}}{\Delta_{n}}\right)^{-\vartheta}\Delta_{n}^{1-\vartheta},{\rm\leavevmode\nobreak\
\leavevmode\nobreak\ for\leavevmode\nobreak\ large\leavevmode\nobreak\
}i>\frac{\tau_{n}}{\Delta_{n}},$

and we note that the sequence of $\Delta_{i}^{n}H$ is decreasing in absolute
value (with each element of the same sign), for $i>i_{n}$. Also, for
$i>i_{n}+K_{n}$, we have that
$|\Delta_{i}^{n}H-\Delta_{i-1}^{n}H|\leq\zeta\sqrt{\Delta_{n}}$ for $n$ large
enough. Furthermore, given the behavior of maxima of normal i.i.d. variables
with zero mean, see for instance Example 3.3.29 in Embrechts et al. (2013), we
have that

$\mathbb{P}\left(\max_{i=1,...,\lfloor\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{T}_{n}\rfloor}|\Delta_{i}^{n}W|>\log(1/\Delta_{n})\sqrt{\Delta_{n}}\right)\rightarrow
0.$ (34)

Hence, it suffices to consider only the set on which
$\max_{i=1,...,\lfloor\macc@depth\char
1\relax\frozen@everymath{\macc@group}\macc@set@skewchar\macc@nested@a
111{T}_{n}\rfloor}|\Delta_{i}^{n}W|\leq\log(1/\Delta_{n})\sqrt{\Delta_{n}}$.
On this set, since
$|\Delta_{i}^{n}H-\Delta_{i-1}^{n}H|\leq\zeta\sqrt{\Delta_{n}}$ for
$i>i_{n}+K_{n}$ and $n$ sufficiently large, we have that the truncation is not
triggered for $i>i_{n}+K_{n}$. As a result, we only need to bound
$\mathbb{P}\left(\check{\mathbf{N}}^{\rm w}_{0}(\xi)>T_{n}\right)$ with
$\check{\mathbf{N}}_{0}^{\rm w}(\xi)$ being the stopping time associated with
$\check{W}_{l_{1},l_{2}}=\sum_{i=i_{n}+K_{n}+l_{1}+1}^{i_{n}+K_{n}+l_{2}}(b_{0}\Delta_{n}+\Delta_{i}^{n}W+\Delta_{i}^{n}H)$.

We can then pick $M_{n}=K_{n}+\log(\xi)$. In this case,
$\min_{i>\tau_{n}/\Delta_{n}}|\Delta_{i}^{n}H|>|\mu_{M_{n}}|$, where we set
$\mu_{M_{n}}=cM_{n}^{-\vartheta}\Delta_{n}^{1-\vartheta}$. Note that, given
the definition of $K_{n}$,
$\sup_{i=i_{n}+K_{n},...,i_{n}+M_{n}}|\Delta_{i}^{n}H-\mu_{M_{n}}|\leq
C\sqrt{\Delta_{n}}$, for some random variable $C$. Hence,

$\displaystyle\mathbb{P}\left(\check{\mathbf{N}}^{\rm
w}_{0}(\xi)>T_{n}\right)\leq\mathbb{P}\left(\dot{\mathbf{N}}^{\rm
w}_{0}(\xi)>T_{n}\right)$ $\displaystyle\leavevmode\nobreak\
+\mathbb{P}\left(\sup_{i=i_{n}+K_{n},...,i_{n}+M_{n}}|\Delta_{i}^{n}H+b_{0}\Delta_{n}-\mu_{M_{n}}|\geq\xi\iota_{n}\right),$
(35)

for $\dot{\mathbf{N}}^{\rm w}_{0}(\xi)$ being the stopping time associated
with
$\dot{W}_{l_{1},l_{2}}=\sum_{i=i_{n}+K_{n}+l_{1}+1}^{i_{n}+K_{n}+l_{2}}(\Delta_{i}^{n}W+\mu_{i}^{n})$
with $\mu_{i}^{n}=\mu_{M_{n}}$ for $i=i_{n}+K_{n},...,i_{n}+M_{n}$ and
$\mu_{i}^{n}=\Delta_{i}^{n}H$ otherwise.

Finally, we denote

$\displaystyle\widetilde{\mathbf{N}}(\xi):=\inf\left\\{l>0:\max_{0<k<l}\frac{|\widetilde{W}_{k,l}|}{\sqrt{(l-k)\Delta_{n}}}>\xi\right\\},$
(36)

where $\widetilde{W}_{l_{1},l_{2}}$ is the random walk with
$N(\mu_{M_{n}}\Delta_{n}^{1/2},\Delta_{n})$ increments on the index interval
$[l_{1},l_{2}]$, for $l_{1},l_{2}\in\mathbb{N}_{+}$. Note that
$\dot{\mathbf{N}}^{\rm w}_{0}(\xi)=\widetilde{\mathbf{N}}^{\rm w}(\xi)$ if
$\dot{\mathbf{N}}^{\rm w}_{0}(\xi)<M_{n}$. Hence, we will be done if we can
show that $\mathbb{P}\left(\widetilde{\mathbf{N}}^{\rm
w}(\xi)<M_{n}\right)\rightarrow 1$.

This is what we do next. The rest of our proof is based on the nonlinear
renewal theory of Lai and Siegmund (1979) and the argument for CUSUM-type
statistic by Dragalin (1994).101010One could also consider a very similar
nonlinear renewal theory in Hagwood and Woodroofe (1982) (and verify the
conditions therein). To keep notation simple, we use $\mu$ for $\mu_{M_{n}}$
below.

We decompose the nonlinear (square of the) statistic term into a summation of
a linear term $\mu S_{l}$ and an additional term $V_{l}^{\prime}$, as

$\displaystyle\max_{0<k<l}\frac{(\widetilde{W}_{k,l})^{2}}{2(l-k)\Delta_{n}}$
$\displaystyle=\max_{0<k<l}\left\\{\mu(S_{l}-S_{k})+\frac{(\widetilde{W}_{k,l}/\sqrt{\Delta_{n}}-(l-k)\mu)^{2}}{2(l-k)}\right\\}$
$\displaystyle=\mu S_{l}+V_{l}^{\prime}$

where

$\displaystyle S_{l}$
$\displaystyle:=\widetilde{W}_{0,l}/\sqrt{\Delta_{n}}-\frac{1}{2}l\mu$
$\displaystyle V_{l}^{\prime}$ $\displaystyle:=-\min_{0<k<l}\mu
S_{k}+\max_{0<k<l}\left\\{\frac{(\widetilde{W}_{k,l}/\sqrt{\Delta_{n}}-(l-k)\mu)^{2}}{2(l-k)}\right\\}.$

And the stopping rule can be written as

$\displaystyle\widetilde{\mathbf{N}}(\xi)=\inf\left\\{l>0:\mu
S_{l}+V_{l}^{\prime}>\xi^{2}/2\right\\}$ (37)

For term $V_{l}^{\prime}$, we first introduce $V_{l}^{\prime\prime}$, defined
as

$\displaystyle V_{l}^{\prime\prime}$ $\displaystyle:=-\min_{0<k<l}\mu
S_{k}+V_{l},$ (38) $\displaystyle V_{l}$
$\displaystyle:=(\widetilde{W}_{0,l}/\sqrt{\Delta_{n}}-l\mu)^{2}/(2l).$ (39)

The sequence $\\{V_{l}^{\prime\prime}\\}_{l\geq 1}$ is slowly changing
because, as $l\to\infty$, $\min_{0<k<l}S_{k}$ converges weakly to
$\min_{k>0}S_{k}$, and the second term $V_{l}$ weakly converges to
$\chi_{1}^{2}/2$ random variable ($\chi^{2}$-distributed r.v. with $1$ degree
of freedom). Next, as argued in (Dragalin, 1994, Lemma 1),
$V_{l}^{\prime}=V_{l}^{\prime\prime}$ on a set $\mathcal{A}_{\xi}$ whose
$\mathbb{P}_{\mu}$-probability convergences to $1$ exponentially fast as
$\xi\to\infty$ with the rate $1-O(e^{-\lambda\xi})$ for some $\lambda>0$.
Hence,

$\displaystyle\mathbb{E}[V_{\widetilde{\mathbf{N}}}^{\prime};\mathcal{A}_{\xi}]\to\mathbb{E}[V_{\widetilde{\mathbf{N}}}^{\prime\prime}]=-\mu\mathbb{E}[\min_{k>0}S_{k}]+\frac{1}{2}.$
(40)

Noting that $\mu S_{l}$ corresponds to a random walk with
$N(\mu^{2}/2,\mu^{2})$ increments, Lai and Siegmund (1979, Theorem 3) implies
that

$\displaystyle\frac{1}{2}\mu^{2}\mathbb{E}_{0}[\widetilde{\mathbf{N}}(\xi)]=\frac{1}{2}\xi^{2}-0-\left[-\mu\mathbb{E}[\min_{k>0}S_{k}]+\frac{1}{2}\right]+\frac{\mu^{2}\mathbb{E}[S_{\tau_{+}}^{2}]}{2\mu\mathbb{E}[S_{\tau_{+}}]}+o(1),$
(41)

that is,

$\displaystyle\mathbb{E}_{0}[\widetilde{\mathbf{N}}(\xi)]=\frac{\xi^{2}-1}{\mu^{2}}+\frac{1}{\mu}\left[2\mathbb{E}[\min_{k\geq
0}S_{k}]+\frac{1}{\mu}\frac{\mathbb{E}[S_{\tau_{+}}^{2}]}{\mathbb{E}[S_{\tau_{+}}]}\right]+o(1),$
(42)

for which an approximation provided by Siegmund and Venkatraman (1995) is

$\displaystyle\mathbb{E}_{0}[\widetilde{\mathbf{N}}(\xi)]\approx\frac{\xi^{2}-3}{\mu^{2}}+\frac{4\rho}{\mu},$
(43)

where $\rho\approx 0.583$.

Now we verify the conditions of Lai and Siegmund (1979, Theorem 3) which apply
to our case

  * (A)

For some $\delta>0$,
$\mathbb{P}\\{\widetilde{\mathbf{N}}\leq\delta\xi^{2}\\}=o(\xi^{-2})$ as
$\xi\to\infty$,

  * (B)

Assume that, $\mathbb{E}\,|x_{1}|^{p}<\infty$ for some $p>2$,

  * (C)

$V_{l}$ converges in distribution to a random variable $V$,

  * (D)

$x_{1}$ is non-lattice and there exists $1/2<\alpha\leq 1$ such that
$\mathbb{E}\,|x_{1}|^{2/\alpha}<\infty$,

where $x_{i}:=\mu(\Delta_{i}^{n}\widetilde{W}/\sqrt{\Delta_{n}}-\mu/2)$. In
particular, Condition (A) holds as long as the distribution of
$\Delta_{i}^{n}\widetilde{W}$ belongs to exponential density, see discussion
in (Lai and Siegmund, 1979, Section 4).111111Condition (A) is also used in
Hagwood and Woodroofe (1982), which is their condition (11). The paper shows
that it is satisfied when our Condition (B) holds. Condition (B) can imply
(14), (16) and (17) in Lai and Siegmund (1979) for their Theorem 3, see their
Proposition 1, and it is trivially holds in our case since $x_{i}$’s are
normally distributed. Condition (C) is immediate by the weak convergence of
$V_{l}$. Condition (D) is also met trivially in our case because
$\widetilde{W}$ is a random walk with normal increments.

The result above holds also for the window-limited version of
$\widetilde{\mathbf{N}}(\xi)$ which replaces $0<k<l$ by $l-w_{n}<k<l$ for an
increasing window $w_{n}$ being chosen such that $w_{n}\sim a_{n}\xi^{2}$ with
$a_{n}$ being some deterministic sequence converging to
$a\in(2/c^{2},\infty]$, see Xie and Siegmund (2013, Theorem 2) or Lai and Shan
(1999, Theorem 1).

We are now ready to show $\mathbb{P}\big{(}\widetilde{\mathbf{N}}^{\rm
w}(\xi)<M_{n}\big{)}\rightarrow 1$. Recall that $\mu=\mu_{M_{n}}\sim
cM_{n}^{-\vartheta}\Delta_{n}^{1/2-\vartheta}$, by plugging-in the
$\mu_{M_{n}}$ value we have

$\displaystyle\mathbb{E}_{0}[\widetilde{\mathbf{N}}(\xi)]$
$\displaystyle\approx\frac{\xi^{2}-3}{c^{2}}(M_{n}\Delta_{n})^{2\vartheta}\Delta_{n}^{-1}+\frac{4\rho}{c}(M_{n}\Delta_{n})^{\vartheta}\Delta_{n}^{-1/2}.$
(44)

Now, since $M_{n}\asymp\Delta_{n}^{(1/2-\vartheta)/\vartheta}$, we have that
$\mathbb{E}_{0}[\widetilde{\mathbf{N}}(\xi)]\leq 2\xi^{2}/c^{2}$ for $n$
sufficiently high. From here, since $\xi\Delta_{n}^{\iota}\rightarrow 0$, we
have that $\mathbb{P}\big{(}\widetilde{\mathbf{N}}^{\rm
w}(\xi)<M_{n}\big{)}\rightarrow 1$.

∎