The conditional mean and the conditional variance of $X_t$ are given respectively by $\mathbb{E}\left(X_t\mid {\mathcal{F}}_{t-1}\right)$=$m\left(\rho ;Z_{t-1}\right)$ and $\mathbb{V}\mathrm{ar}\left(X_t\mid {\mathcal{F}}_{t-1}\right)={\sigma }^2\left(\theta ;Z_{t-1}\right)$. From these, one has that for all $z\in {\mathbb{R}}^p$, Therefore, for any bounded measurable functions $g(·)$ and $k(·)$, we have Without loss of generality, in the following we take, for all $z\in {\mathbb{R}}^p,g\left(z\right)=k\left(z\right)=1$. Now, given $X_{-p+1},\dots ,X_{-1},X_0,X_1,\dots ,X_n$ with $n\gg p$, we let ${\mathbb{X}}_n=(X_{-p+1},$$\dots ,$$X_{-1},$$X_0,$$X_1,$$\dots ,$$X_n)$ and consider the sequences of random functions We have the following theorem:

The study of the asymptotic distribution of the test statistic under $H_0$, is based on that of the the process ${\xi }_n(·)$ defined for any $s\in [0,1]$ by where where we recall that $\left[ns\right]$ is the integer part of $ns$. For some $\delta \in (1/n,1/2)$ and for any s in $\left[\delta ,1-\delta \right]$, we define where $q\left(s\right)=\sqrt{s(1-s)}$ and ${\widehat{\sigma }}_w$ is any consistent estimator of For $\delta \in (0,1/2)$, we denote by $D_{\delta }\equiv D\left(\left[\delta ,1-\delta \right]\right)$ the space of all right continuous functions with left limits on $\left[\delta ,1-\delta \right]$ endowed with the Skorohod metric. It is clear that $C_n(·),{\xi }_n(·)\in D_0$ and $T_n(·)\in D_{\delta }$.

It is worth noting that if the change occurs at the very beginning or at the very end of the data, we may not have sufficient observations to obtain consistent LSE estimators of the parameters or these may not be unique. This is why we stress on the truncated version of the test statistic given in [21] that we recall: By Theorem 2, it is easy to see that for any $1⩽\nu <n/2$, which yields the asymptotic null distribution of the test statistic. With this, at level of significance $\alpha \in (0,1)$, $H_0$ is rejected if ${\Lambda }_n>C_{\alpha ,n}$, where $C_{\alpha ,n}$ is the $(1-\alpha )$-quantile of the distribution of the above limit. This quantile can be computed by observing that under $H_0$, for larger values of n one has From the following relation (1.3.26) of [32], for each $h_{\nu }\left(n\right)>0$, and for larger real number x, we have which gives an approximation of the tail distribution of $\underset{h_{\nu }\left(n\right)⩽s⩽1-h_{\nu }\left(n\right)}{\sup }\left|\tilde{B}\left(s\right)\right|/q\left(s\right)$. Thus, using ${\widehat{\sigma }}_w$, an estimation of $C_{\alpha ,n}$ can be obtained from this approximation. Monte Carlo simulations are often carried out to obtain accurate approximations of $C_{\alpha ,n}$. In this purpose, it is necessary to do a good choice of $\nu$. We selected $\nu =0.9\times n^{4/5}$ as our option, which we found to be a suitable choice for all the cases we examined. But, to avoid the difficulties associated with the computation of $C_{\alpha ,n}$, a decision can also be taken by using the p-value method as in [33]. That is using the approximation (16), reject $H_0$ if This idea is used in the simulation section.

For the study of the estimator ${\widehat{t}}^{*}$, we let $\kappa ={\kappa }_n={\vartheta }_2^2-{\vartheta }_1^2$ and assume without loss of generality that ${\kappa }_n>0\left({\vartheta }_2>{\vartheta }_1\right),{\kappa }_n⟶0\mathrm{as}n⟶\infty$ (see, e.g., [34]) and that the unknown change point $t^{*}$ depends on the sample size n. We have the following result:

In this section we study the asymptotic behavior of the location estimator. We make the additional assumptions that ${\kappa }_n>>n^{-\frac{1}{2}}$ and that as $n⟶\infty$, By (10), we have To derive the limiting distribution of ${\widehat{t}}^{*}$, we study the behavior of $n\left({\Delta }_k^2-{\Delta }_{t^{*}}^2\right)$ for those k’s in the neighborhood of $t^{*}$ such that $k=\left[t^{*}+r{\kappa }_n^{-2}\right]$, where r varies in an arbitrary bounded interval $\left[-N,N\right]$. In this purpose, we define where ${\Delta }_n\left(r\right)={\Delta }_{\left[r\right]}$. In addition, we define the two-sided standard Wiener process $\left\{B^{*}\left(r\right),r\in \mathbb{R}\right\}$ as follows: where $B_i\left(r\right),i=1,2$ are two independent standard Wiener processes defined on $\left[0,\infty \right)$ with $B_i\left(0\right)=0,i=1,2$.

First, we identify the limit of the process $P_n\left(r\right)$ on $\left|r\right|⩽N$ for every given $N>0$. We denote by $C\left(\left[-N,N\right]\right)$ the space of all continuous functions on $\left[-N,N\right]$ endow with the uniform metric.

The above results make it possible to achieve a weak convergence result for $n\left({\Delta }_k^2-{\Delta }_{t^{*}}^2\right)$ and then apply the Argmax-Continuous Mapping Theorem (Argmax-CMT). We have:

This result yields the asymptotic distribution of the change location estimator. [35,36] and [37] investigated the density function of the random variable $\mathcal{S}$ (see the Lemma 1.6.3 of [32] for more details). They also showed that $\mathcal{S}$ has a symmetric (with respect to 0) probability density function $\gamma (·)$ defined for any $x\in \mathbb{R}$ by where $\Phi (·)$ is the cumulative distribution function of the standard normal variable. From this result, a confidence interval for the change-point location can be obtained, if one has consistent estimates of ${\kappa }_n^2$ and ${\sigma }_w^2$. With ${\widehat{t}}^{*}$, consistent estimates of ${\vartheta }_1^2$ and ${\vartheta }_2^2$ are given respectively by Thus a consistent estimate of ${\kappa }_n^2$ is given by A consistent estimator of ${\sigma }_w^2$ that we denote by ${\widehat{\sigma }}_w^2$, can be easily obtained by taking its empirical counterpart. So, at risk $\alpha \in (0,1)$, letting $q_{1-\frac{\alpha }{2}}$ be the quantile of order $1-\frac{\alpha }{2}$ of the distribution of the random variable $\mathcal{S}$, an asymptotic confidence interval for $t^{*}$ is given by

These data of length 2022 from the American stock market were recorded daily from January 2nd, 1992 to December 31st, 1999. They represent the daily stock prices of the S&P 500 stock market (SPX). They are among the most closely followed stock market indices in the world and are considered as an indicator of the USA economy. They have also been recently examined by [40] and can be found at: (www.investing.com).

On Figure 2, we observe that the trend of the SPX daily stock price series is not constant over time. We also observe that stock prices have fallen sharply, especially in the time interval between the two vertical dashed blue lines (the period of the 1997-1998 Asian financial crisis).

Denote by $D_t$ the value of the stock price for SPX index at day t, and the first difference of the logarithm of stock price, $X_t$ as $X_t$ is the logarithmic return of stock price for SPX index at day t.

The series $\left(X_t\right)$ is approximately piece-wise stationary on two segments and symmetric around zero (see Figure 3). This brought us to consider a CHARN model with $m(\rho ;x)=0$, ${\delta }_0\left(x\right)=\sqrt{{\theta }_0+{\theta }_1{x}^2}$ for ${\theta }_0=1,{\theta }_1=0$, ${\vartheta }_1$ and ${\vartheta }_2$ estimated by CLS described in Section 2.

Using our procedure, we found an important change point in stock price volatility in March 26, 1997, which is consistent with the date found by [40] (see Figure 3).

The vertical dashed blue line of Figure 3, represents the date at which the change occurred. It should be noted that the change in volatility coincides with the Asian crisis in 1997 when Thailand devalued its currency, the baht, against the US dollar. This decision led to a fall in the currencies and financial markets of several countries in its surroundings. The crisis then spread to other emerging countries with important social and political consequences and repercussions on the world economy.

These data of length 585 are based on Brent oil futures. They represent the prices of Brent oil (USD/barrel) on a daily basis between January 4th, 2021 and April 6th, 2023. They are available at: (www.investing.com).

Figure 4, shows that the evolution of the daily series of Brent oil prices is non-stationary. It also shows that stock prices fell sharply, especially in early March 2022 (the date of the conflict between OPEC and Russia, when the latter refused to reduce its oil production in the face of declining global demand caused by the Covid-19 pandemic).

We follow the same procedure as in the previous example and obtain the logarithmic transformation of the daily rate of return series for Brent oil (see Figure 5 below). Proceeding as for the first data, the application of our procedure allows to find a change at February 25, 2022. The break date is marked by a dashed blue vertical line (see Figure 5).

It also performs well on these data. Indeed, oil volatility was very high in March 2022 due to the Covid-19 pandemic and the conflict between OPEC and Russia. The health crisis led to a significant drop in global oil demand, while Russia refused to cut oil production as proposed by OPEC, which caused oil prices to fall. Brent crude oil fell from over 60 dollars a barrel to less than 20 dollars in a month.

Recall that $\left(\Omega ,\mathcal{F},\mathbb{P}\right)$ is a probability space. Let $X:=\left(X_k,k\in \mathbb{Z}\right)$ be a sequence of random variables (not necessarily stationary). For $-\infty ⩽i⩽j⩽\infty$, define the $\sigma -$field ${\mathcal{F}}_i^j:=\sigma \left(X_k,i⩽k⩽j\right)$ and for each $n⩾1$, define the following mixing coefficients: The random sequence X is said to be "strongly mixing" or "$\alpha$-mixing " if $\alpha \left(n\right)⟶0$ as $n⟶\infty$, "$\varphi$-mixing " if $\varphi \left(n\right)⟶0$ as $n⟶\infty$. One shows that $\varphi \left(n\right)\le \alpha \left(n\right)$, so that an $\alpha$-mixing sequence of random variables is also $\varphi$-mixing. The following corollary serves to prove Lemma A1 below.