Large Time Behavior on the Linear Self-Interacting Diffusion Driven by Sub-Fractional Brownian Motion With Hurst Index Large Than 0.5 I: Self-Repelling Case

Let SH be a sub-fractional Brownian motion with index 12<H<1. In this paper, we consider the linear self-interacting diffusion driven by SH, which is the solution to the equationdXtH=dStH−θ(∫0tXtH−XsHds)dt+νdt,X0H=0,where θ &lt; 0 and ν∈R are two parameters. Such process XH is called self-repelling and it is an analogue of the linear self-attracting diffusion [Cranston and Le Jan, Math. Ann. 303 (1995), 87–93]. Our main aim is to study the large time behaviors. We show the solution XH diverges to infinity, as t tends to infinity, and obtain the speed at which the process XH diverges to infinity as t tends to infinity.

Dependence structure in financial time series: Applications and evidence from wavelet analysis

<p>Conventional time series theory and spectral analysis have independently achieved significant popularity in mainstream economics and finance research over long periods. However, the fact remains that each is somewhat lacking if the other is absent. To overcome this problem, a new methodology, wavelet analysis, has been developed to capture all the information localized in time and in frequency, which provides us with an ideal tool to study non-stationary time series. This paper aims to explore the application of a variety of wavelet-based methodologies in conjunction with conventional techniques, such as the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models and long-memory parameter estimates, in analysing the short and long term dependence structure of financial returns and volatility. Specifically, by studying the long-memory property of these time series we hope to identify the source of their possible predictability. Above all else, we document the indispensable role of trading activities associated with low frequencies in determining the long-run dependence of volatility. It follows that GARCH models incorporating long-memory and asymmetric returns-volatility dynamics can provide reasonably accurate volatility forecasts. Additionally, the persistence parameter of returns, represented by the Hurst index, is observed to be correlated to trading profits obtained from typical technical rules designed to detect and capitalize on existing trending behaviour of stock prices. This implies that the Hurst index can be used as a good indicator of the long-memory characteristic of the market, which in turn drives such trending behaviour.</p>

Dependence structure in financial time series: Applications and evidence from wavelet analysis

<p>Conventional time series theory and spectral analysis have independently achieved significant popularity in mainstream economics and finance research over long periods. However, the fact remains that each is somewhat lacking if the other is absent. To overcome this problem, a new methodology, wavelet analysis, has been developed to capture all the information localized in time and in frequency, which provides us with an ideal tool to study non-stationary time series. This paper aims to explore the application of a variety of wavelet-based methodologies in conjunction with conventional techniques, such as the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models and long-memory parameter estimates, in analysing the short and long term dependence structure of financial returns and volatility. Specifically, by studying the long-memory property of these time series we hope to identify the source of their possible predictability. Above all else, we document the indispensable role of trading activities associated with low frequencies in determining the long-run dependence of volatility. It follows that GARCH models incorporating long-memory and asymmetric returns-volatility dynamics can provide reasonably accurate volatility forecasts. Additionally, the persistence parameter of returns, represented by the Hurst index, is observed to be correlated to trading profits obtained from typical technical rules designed to detect and capitalize on existing trending behaviour of stock prices. This implies that the Hurst index can be used as a good indicator of the long-memory characteristic of the market, which in turn drives such trending behaviour.</p>

Research on Systemic Risk of the Turkish Banking Industry Based on a Systemic Risk Measurement Framework of the Fractional Brownian Motion

Since the 2008 financial crisis, it is an important issue to assess the systemic risk of banks, but there is a lack of research on the assessment of the systemic risk of Turkey’s financial system. In addition, geometric Brownian motion is used in most of the assessment frameworks of systemic risk under the normal financial market state, while the Turkish financial market has the situation of spike and thick tail. Therefore, this paper proposes a fractional Brownian motion measurement framework of systemic risk to study the systemic risk of the Turkish financial system. Firstly, this paper uses the data of 11 Turkish listed banks from 2014 to 2019 to conduct a normality test and demonstrate that its market has the characteristics of a fractal market; that is, there is a spike and thick tail distribution phenomenon in the stock price trend. Then, this paper proposes a fractional Brownian motion systemic risk measurement framework (fBSM). Based on the proposed theoretical framework and the actual data of Turkish listed banks from 2014 to 2019, a dynamically evolving Turkish banking network system is constructed to measure the systemic risk in the Turkish banking system. The research results find that the systemic risk is the highest in 2017, which then improved and gradually recovered. In addition, when analyzing the sensitivity of the Hurst index, it shows that with the increase in Hurst index, the Hurst index elasticity of Turkish banks’ asset value increases gradually and the asset value also increases continuously. Hence, the Hurst index has a greater impact on asset value. Therefore, the measurement framework of systemic risk based on the fBSM can better monitor the systemic risk than the traditional geometric Brownian motion in the Turkish banking system.

Method of Distinguishing Styles by Fractal and Statistical Indicators of the Text as a Sequence of the Number of Letters in Its Words

The method of the analysis of different texts styles is developed in the paper. Integer numerical sequences are used as models. The elements of the sequence are the number of letters in the words of the text. The algorithm for calculating the exact value of the fractal dimension is developed. It provides the determination of the exact value of the Hurst index. The value of the power dependence constant is calculated. The obtained indicators in the aspect of fractality completely describe the objects of research.

High order Anderson parabolic model driven by rough noise in space

In this paper, we concern the fourth parabolic model on [Formula: see text] driven by a multiplicative Gaussian noise which behaves like fractional Brownian motion in time and space with Hurst index [Formula: see text] and [Formula: see text], respectively. The existence and uniqueness of mild solution in Skorohod sense are proved, and the weak intermittency is obtained by estimating [Formula: see text]th ([Formula: see text]) moment of the solution. Moreover, the Hölder continuity can be obtained for the time and space variable.

Lp-Theory for the fractional time stochastic heat equation with an infinite-dimensional fractional Brownian motion

In this paper, we study the [Formula: see text]-theory of the fractional time stochastic heat equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] denotes the Caputo derivative of order [Formula: see text], and [Formula: see text] is a sequence of i.i.d. fractional Brownian motions with a same Hurst index [Formula: see text]. The integral with respect to fractional Brownian motion is the Skorohod integral. By using the Malliavin calculus techniques and fractional calculus, we obtain a generalized Littlewood–Paley inequality, and prove the existence and uniqueness of [Formula: see text]-solution to such equation.