Fractional Brownian Model Research Articles

Investigating long and short memory in cryptocurrency time series by stochastic fractional Brownian models

Over the last years, the world of cryptocurrencies has undergone a tumultuous development, mostly characterized by speculative behaviors, and thus one might argue that it does not satisfy the Efficient Market Hypothesis. Since the efficiency of a financial market can be assessed by checking whether the times series of the assets traded on it are persistent/anti-persistent, we investigate the presence of memory in the price of seven among the most important cryptocurrencies. To this aim, we employ an original approach based on two fractional models, namely the geometric fractional Brownian motion and the geometric mixed fractional Brownian motion, which are tested against the Markovian geometric Brownian motion to detect the presence of memory. The above fractional models are estimated by employing an innovative maximum likelihood procedure that exploits the Toeplitz structure of the covariance matrix of log-returns. The null assumption of absence of memory in the time series is tested based on confidence ellipses for the estimated parameters. We validate the proposed procedure on artificial data and we apply it to the historical series of crypto-prices. Results suggest the presence of memory effects only for some of the considered cryptocurrencies. A possible explanation of such a persistence/anti-persistence phenomenon is provided.

Option pricing under mixed hedging strategy in time-changed mixed fractional Brownian model

In this paper, we obtain a mixed hedging strategy and a pricing formula in a discrete time setting for a European call option in the time-changed mixed fractional Brownian model. In this manner, we generalize the mixed hedging and the pricing formula in Brownian motion to the time-changed mixed fractional Brownian motion. Finally, through some numerical experiments and empirical analysis, we show that our mixed hedging is better than delta hedging on the hedging error ratio in some cases.

A Robust Authentication Algorithm for Medical Images Based on Fractal Brownian Model and Visual Cryptography

In this paper, we aimed to discuss the security authentication requirements of medical images in the medical network, and a security authentication method is designed based on fractal and visual cryptography. Based on the discrete fractal Brownian random field model, the gray-level statistical information and spatial structure information of medical images is fully mined. The gray distribution of medical images is expressed in the form of fractal features. By using the spatial data mining methods, the data of fractal structure space is analyzed, and by using the stability of the energy structure, the authentication features are formed. Using the visual cryptography (VC), the robustness of the authentication method is further enhanced. Through the centralized test of common medical images and the comparison analysis with existing methods, it is further verified that the method is effective against common attacks such as JPEG compression, scaling, rotation operation, clipping, added noise, filtering, and blurring.

Proportional reinsurance for fractional Brownian risk model

This paper investigates the ruin probabilities for a two-dimensional fractional Brownian risk model with a proportional reinsurance scheme. The author focused on the joint and simultaneous ruin probabilities in a finite-time horizon. The risk processes of both insurance and reinsurance companies are composed of a large number of i.i.d. sub-risk processes, representing independent businesses. The asymptotics were derived as the initial capital tends to infinity

Diffusing diffusivity: Fractional Brownian oscillator model for subdiffusion and its solution

Diffusing diffusivity: Fractional Brownian oscillator model for subdiffusion and its solution

Ruin problem of a two-dimensional fractional Brownian motion risk process

ABSTRACTThis paper investigates ruin probability and ruin time of a two-dimensional fractional Brownian motion risk process. The net loss process of an insurance company is modeled by a fractional Brownian motion. The two-dimensional fractional Brownian motion risk process models the surplus processes of an insurance and a reinsurance company, where the net loss is divided between them in some specified proportions. The ruin problem considered is that of the two-dimensional risk process first entering the negative quadrant, that is, the simultaneous ruin problem. We derive both asymptotics of the ruin probability and approximations of the scaled conditional ruin time as the initial capital tends to infinity.

Detrended fluctuation analysis for spatial characterisation of landscapes

The interactions among abiotic, biotic, and anthropic factors and their influence at different scales create a complex dynamic in landscape evolution. Scaling and multifractal analysis have the potential to characterise landscapes in terms of the statistical signature of the selected measure, in this case, altitude. This work evaluates the multifractality of altitude data points along transects that are obtained in several directions using Detrended Fluctuation Analysis (DFA) in a protected area adjacent to Madrid. The study data set consist of a matrix 2048 × 2048 pixels obtained at a 5 m resolution and extracted from a digital terrain model (DTM) using a Geographic Information System (GIS). We found that the distribution of altitude fluctuations at small scales revealed a non-Gaussian character in the statistical moments, indicating that Fractional Brownian modelling is not appropriate. Generalised Hurst dimensions (H(q)) were calculated on several transects crossing the area under study, all of which exhibited multifractality within a certain scale range. The results show a persistent behaviour in all directions because all of the H(q) values exceeded 0.5 and because there were differences in the intensities of the multifractality. The analysis of the directionality by means of a generalised Hurst rose plot showed differences in the scaling characteristics both along and across rivers and reservoirs. This indicates a clear anisotropy that is mainly due to the directions of the two river basins located in the area and the basement movement as a consequence of gradual tectonic displacement, which must be considered in two-dimensional DFAs.

Brownian Integrated Covariance Functions for Gaussian Process Modeling: Sigmoidal Versus Localized Basis Functions

ABSTRACTGaussian process modeling, or kriging, is a popular method for modeling data from deterministic computer simulations, and the most common choices of covariance function are Gaussian, power exponential, and Matérn. A characteristic of these covariance functions is that the basis functions associated with their corresponding response predictors are localized, in the sense that they decay to zero as the input location moves away from the simulated input sites. As a result, the predictors tend to revert to the prior mean, which can result in a bumpy fitted response surface. In contrast, a fractional Brownian field model results in a predictor with basis functions that are nonlocalized and more sigmoidal in shape, although it suffers from drawbacks such as inability to represent smooth response surfaces. We propose a class of Brownian integrated covariance functions obtained by incorporating an integrator (as in the white noise integral representation of a fractional Brownian field) into any stationary covariance function. Brownian integrated covariance models result in predictor basis functions that are nonlocalized and sigmoidal, but they are capable of modeling smooth response surfaces. We discuss fundamental differences between Brownian integrated and other covariance functions, and we illustrate by comparing Brownian integrated power exponential with regular power exponential kriging models in a number of examples. Supplementary materials for this article are available online.

A very efficient approach for pricing barrier options on an underlying described by the mixed fractional Brownian motion

We deal with the problem of pricing barrier options on an underlying described by the mixed fractional Brownian model. To this aim, we consider the initial-boundary value partial differential problem that yields the option price and we derive an integral representation of it in which the integrand functions must be obtained solving Volterra equations of the first kind. In addition, we develop an ad-hoc numerical procedure to solve the integral equations obtained. Numerical simulations reveal that the proposed method is extremely accurate and fast, and performs significantly better than the finite difference method.

The price of granularity and fractional finance

The purpose of this paper is to assess the risk premium a fractional financial lognormal (Black-Scholes or BS) process relative to a non-fractional and complete financial pricing model. While fractional Brownian BS models based on the Duncan and Wicks calculus were shown to define a no arbitrage financial model, this paper claim is that this martingale need not be the pricing martingale. There may be many martingales corresponding to a no-arbitrage financial model. In this vein, the intent of this paper are twofold. On the one hand, provide a definition of the risk premium implied by the discount rate applied to future fractional returns as well as justify this premium compared to a non-fractional financial model. To do so, an insurance rationale for the risk implied by the financial asset volatility is used when defining a no-arbitrage risk neutral probability measure. A “granular risk premium”, is then defined relative to the model granularity. On the other, the paper highlights the effects of a model granularity and its Hurst index on financial risk management. In particular, we argue that that such an index is defined by both the granularity underlying the model as well as the effects of mixed models (combining both diffusion and jump processes) on relative range and volatility variations (and thus the measurement of the Hurst index). To present simply the ideas underlying this paper, we price an elementary fractional risk free bond and its risk premium relative to a known spot interest rate. Similarly, the Black-Scholes no arbitrage model is presented in both its non-fractional conventional form and in its fractional framework. The granularity risk premium is then calculated.

Asymptotic normality of randomized periodogram for estimating quadratic variation in mixed Brownian–fractional Brownian model

We study asymptotic normality of the randomized periodogram estimator of quadratic variation in the mixed Brownian--fractional Brownian model. In the semimartingale case, that is, where the Hurst parameter $H$ of the fractional part satisfies $H\in(3/4,1)$, the central limit theorem holds. In the nonsemimartingale case, that is, where $H\in(1/2,3/4]$, the convergence toward the normal distribution with a nonzero mean still holds if $H=3/4$, whereas for the other values, that is, $H\in(1/2,3/4)$, the central convergence does not take place. We also provide Berry--Esseen estimates for the estimator.

The Price of Granularity and Fractional Finance

The purpose of this paper is to assess the risk premium a fractional financial lognormal (Black-Scholes or BS) process relative to a non-fractional and complete financial pricing model. While fractional Brownian BS models based on the Duncan and Wicks calculus were shown to define a no arbitrage financial model, this paper claim is that this martingale need not be the pricing martingale. There may be many martingales corresponding to a no-arbitrage financial model. In this vein, the intent of this paper are twofold. On the one hand, provide a definition of the risk premium implied by the discount rate applied to future fractional returns as well as justify this premium compared to a non-fractional financial model. To do so, an insurance rationale for the risk implied by the financial asset volatility is used when defining a no-arbitrage risk neutral probability measure. A “granular risk premium”, is then defined relative to the model granularity. On the other, the paper highlights the effects of a model granularity and its Hurst index on financial risk management. In particular, we argue that that such an index is defined by both the granularity underlying the model as well as the effects of mixed models (combining both diffusion and jump processes) on relative range and volatility variations (and thus the measurement of the Hurst index). To present simply the ideas underlying this paper, we price an elementary fractional risk free bond and its risk premium relative to a known spot interest rate. Similarly, the Black-Scholes no arbitrage model is presented in both its non-fractional conventional form and in its fractional framework. The granularity risk premium is then calculated.

Occurrence of solar flares viewed with GPS: Statistics and fractal nature

AbstractIn this paper we describe the statistical properties of the EUV solar flux sudden variation. The solar flux variation is modeled as a time series characterized by the subsolar Vertical Total Electron Content double difference in time, computed with dual‐frequency GNSS (Global Navigation Satellite Systems) measurements in the daylight hemisphere (GNSS solar flare indicator rate parameter). We propose a model that explains its characteristics and the forecasting limitations. The sudden overionization pattern is assumed to be of solar origin, and the data used in this study was collected during the last solar cycle. The two defining characteristics of this time series are an extreme variability (i.e., in a solar cycle one can find events at 400σ from the mean value) and a temporal correlation that is independent of the timescale. We give a characterization of a model that explains the empirical results and properties such as (a) the persistence and presence of bursts of solar flares and (b) their long tail peak values of the solar flux variation. We show that the solar flux variation time series can be characterized by a fractional Brownian model for the long‐term dependence, and a power law distribution for the extreme values that appear in the time series.

Fractal evolution of MHz electromagnetic signals prior to earthquakes: results collected in Greece during 2009

This paper addressed a fractal evolution of 11 one-month lasting MHz electromagnetic disturbances, recorded in Greece prior to nine significant earthquakes of 2009. Time-space wavelet-based power spectral techniques were employed in the analysis. All investigated signals evolved naturally to epochs of fractal organization in space and time. Continuous organization was detected in seven signals. Significant number of successive () power-law -values were observed lying between 1.5 and 3.0 or above. The majority of fractal segments exhibited anti-persistent () or persistent () behaviour. Switching between persistency and anti-persistency was also found. Locality and sensitivity were traced. Findings were considered indicative of self-organized critical states of the last stages of preparation of the investigated earthquakes. Results implied fractional Brownian modelling. Explanations were proposed in view of the asperity model. Persistent–anti-persistent MHz anomalies were due to self-organized micro-cracking of the heterogeneous medium of the earth's crust which may have led the system's evolution towards global failure. The precursory value of the signals was discussed.

The Impact Mechanism of Fractal Noise on PN Code Detection System

Taking the pseudo-random phase modulated CW radar for example, this paper studies the impact mechanism of a class of non-stationary fractal noise on PN code detection system, especially signal mixing and matching filter. The cross correlation function, power spectrum function and average power of pseudo-random signal and fractal noise are deduced, compared with the impact of white noise on the pseudo code detection system. We analyze the impact mechanism of three kinds of sea clutter model, namely fractal Brownian model (FBM), the multifractal (MF) model and the non-stationary random fractal model (e.g., infinitely divisible cascades, IDC), on the pseudo-random code detection system, and demonstrate the reason why the multi-scale filtering method in wavelet domain and the MF methods fail to eliminate the effect of sea clutter. Based on the natural sea clutter data, we simulate and analyze the influence of white noise and fractal noise comparatively on detection system, which indicates that the effect of fractal noise cannot be inhibited effectively by the traditional correlation detection and MF analysis, and finally we put forward possible solutions.

Traces of self-organisation and long-range memory in variations of environmental radon in soil: comparative results from monitoring in Lesvos Island and Ileia (Greece)

This paper addresses issues of self-affinity, long-memory and self-organisation in variations of radon in soil recorded in Lesvos Island, Greece. Several techniques were employed, namely (a) power-law wavelet spectral fractal analysis, (b) estimation of Hurst exponents through (b1) rescaled-range, (b2) roughness-length, (b3) variogram and (a), (c) detrended fluctuation analysis, (d) investigation of fractal dimensions and (e) analysis of five block entropies: (e1) Shannon entropy, (e2) Shannon entropy per letter, (e3) conditional entropy, (e4) Tsallis entropy, and (e5) normalised Tsallis entropy. Long-lasting antipersistency was identified during a period of anomalous radon variations following fractional Brownian modelling. Remaining variations did not exhibit analogous behaviour and followed fractional Gaussian modelling. Antipersistent power-law-beta-exponent-values between 1.5 and 2.0 were detected during anomalies. Persistent values were also found. Hurst exponents were mainly within 0 < H < 0.5. Some persistent exponents (0.5 < H < 1) were also observed. Fractal dimensions were within 1.5 < D < 2. Radon anomalies presented lower fractal dimensions. Shannon entropy ranged between 0.77 ≤ H(n) ≤ 2.38, Shannon entropy per letter, between 0.19 ≤ h(n) ≤ 0.59, conditional entropy, between 0.01 ≤ h(n) ≤ 0.58, Tsallis entropy, between 0.55 ≤ Sq ≤ 1.01 and normalised Tsallis entropy between, 0.98 ≤ \(\hat{S}\) ≤ 5.42 (block-size n = 4). Entropies were lower during anomalies, indicating strong self-organisation. Persistency–antipersistency switching was observed, consistent with long-memory dynamics. Potential geological sources were discussed. The asperity-model was proposed. Findings were compared to results obtained under analogous methodologies in Ileia, Greece.

Pricing currency options in the mixed fractional Brownian motion

This paper deals with the problem of pricing European currency options in the mixed fractional Brownian environment. Both the pricing formula and the mixed fractional partial differential equation for European call currency options are obtained. Some Greeks and the estimator of volatility are also provided. Empirical studies and simulation results confirm the theoretical findings and show that the mixed fractional Brownian pricing model is a reasonable one.

Detecting Aircraft With a Low-Resolution Infrared Sensor

Existing computer simulations of aircraft infrared signature (IRS) do not account for dispersion induced by uncertainty on input data, such as aircraft aspect angles and meteorological conditions. As a result, they are of little use to estimate the detection performance of IR optronic systems; in this case, the scenario encompasses a lot of possible situations that must be indeed addressed, but cannot be singly simulated. In this paper, we focus on low-resolution infrared sensors and we propose a methodological approach for predicting simulated IRS dispersion of poorly known aircraft and performing aircraft detection on the resulting set of low-resolution infrared images. It is based on a sensitivity analysis, which identifies inputs that have negligible influence on the computed IRS and can be set at a constant value, on a quasi-Monte Carlo survey of the code output dispersion, and on a new detection test taking advantage of level sets estimation. This method is illustrated in a typical scenario, i.e., a daylight air-to-ground full-frontal attack by a generic combat aircraft flying at low altitude, over a database of 90,000 simulated aircraft images. Assuming a white noise or a fractional Brownian background model, detection performances are very promising.

Arbitrage-free Models In Markets With Transaction Costs

In the paper [7], Guasoni studies financial markets which are subject to proportional transaction costs. The standard martingale framework of stochastic finance is not applicable in these markets, since the transaction costs force trading strategies to have bounded variation, while continuous- time martingale strategies have infinite transaction cost. The main question that arises out of [7] is whether it is possible to give a convenient condition to guarantee that a trading strategy has no arbitrage. Such a condition was proposed and studied in [6] and [1], the so-called stickiness property, whereby an asset's price is never certain to exit a ball within a predetermined finite time. In this paper, we define the multidimensional extension of the stickiness property, to handle arbitrage-free conditions for markets with multiple assets and proportional transaction costs. We show that this condition is sufficient for a multi-asset model to be free of arbitrage. We also show that d-dimensional fractional Brownian models are jointly sticky, and we establish a time-change result for joint stickiness.

Imaging of Fractal Profiles

In this paper, a model for radar images of fractal (topologically 1-D) profiles is introduced. A twofold approach is followed: on one hand, we analytically solve the problem whenever small-slope profiles are in order; on the other hand, we present a partly analytical and partly numerical setup to cope with the general-slope case. By means of the analytical approach, we evaluate in closed form both the structure function and the power density spectrum of the radar signal. An appropriately smoothed (physical) fractional Brownian model (fBm) process is employed; its introduction is justified by the finite sensor resolution. A fractal scattering model is employed. It is shown that for a fractal profile modeled as an fBm stochastic process, the backscattered signal turns out to be strictly related to the associated fractional Gaussian noise process if a small-slope regime for the observed profile can be assumed. In the analytical–numerical framework, a profile with prescribed fractal parameters is first synthesized; then, fractal scattering methods (applicable to wider slope regimes with respect to the previous case) are employed to compute the signal backscattered toward the sensor. Finally, the power density spectrum of the acquired radar image is estimated. The obtained spectra are favorably compared with the theoretical results, and a parametric study is performed to assess the overall method behavior.