Hurst Exponent Research Articles

Eigenvalue distributions of high-dimensional matrix processes driven by fractional Brownian motion

In this paper, we study high-dimensional behavior of empirical spectral distributions [Formula: see text] for a class of [Formula: see text] symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic differential equation driven by fractional Brownian motion with Hurst parameter [Formula: see text]. For Wigner-type matrices, we obtain almost sure relative compactness of [Formula: see text] in [Formula: see text] following the approach in [1]; for Wishart-type matrices, we obtain tightness of [Formula: see text] on [Formula: see text] by tightness criterions provided in Appendix B. The limit of [Formula: see text] as [Formula: see text] is also characterized.

Impact of Subprime and Covid-19 crises on WAEMU stock exchange efficiency: a classic Hurst vs dynamic Hurst approach

Financial markets are alternative sources of financing, above all, in the current context of financial globalization and the development of information and telecommunication technologies. In the WAEMU zone in particular, the size of the economies seemed insufficient to supply national markets without high costs. The union’s authorities therefore set up a regional market that seemed to be a beneficial outcome : the WAEMU stock market. This stock exchange has undergone significant modernizations since its inception, with the aim of achieving optimal resource allocation. The informational efficiency of this stock exchange has become a hot topic of interest among researchers and professionals, giving rise to a growing number of publications on the subject. Our article is set in this context and aims to analyze the WAEMU’s weak-form efficiency. Based on the limitations of traditional methods, we use new approaches. Firstly, the classic Hurst is a tool for characterizing market structure and analyzing efficiency. The results highlight the multifractal nature of the WAEMU’s market and the rejection of the null hypothesis of no memory in the return series. Given the resurgence of crises, it would be interesting to analyze stock market behavior during such events. We also study the impact of crises (subprime and covid-19) on efficiency, using the dynamic Hurst exponent to account for efficient market varying with time. The results show that during periods of crisis, the stock market alternates between efficiency and inefficiency. They also reveal that Covid-19’s impact was faster than subprime crisis. They also reveal that covid impact crisis was faster than that of subprimes. However, despite the big spike linked to the covid context, the composite index picked up and continued in the same vein.

A unifying representation of path integrals for fractional Brownian motions

Fractional Brownian motion (fBm) is an experimentally-relevant, non-Markovian Gaussian stochastic process with long-ranged correlations between the increments, parametrised by the so-called Hurst exponent H; depending on its value the process can be sub-diffusive (0<H<1/2) , diffusive (H=1/2) or super-diffusive (1/2<H<1) . There exist three alternative equally often used definitions of fBm—due to Lévy and due to Mandelbrot and van Ness (MvN), which differ by the interval on which the time variable is formally defined. Respectively, the covariance functions of these fBms have different functional forms. Moreover, the MvN fBms have stationary increments, while for the Lévy fBm this is not the case. One may therefore be tempted to conclude that these are, in fact, different processes which only accidentally bear the same name. Recently determined explicit path integral representations also appear to have very different functional forms, which only reinforces the latter conclusion. Here we develop a unifying equivalent path integral representation of all three fBms in terms of Riemann–Liouville fractional integrals, which links the fBms and proves that they indeed belong to the same family. We show that the action in such a representation involves the fractional integral of the same form and order (dependent on whether H<1/2 or H>1/2 ) for all three cases, and differs only by the integration limits.

Formation of partially embedded Au nanostructures: Ion beam irradiation on thin film.

The Au partially embedded nanostructure (PEN) is synthesized by ion irradiation on an Au thin film deposited on a glass substrate using a 50 keV Ar ion. Scanning electron microscopy results show ion beam-induced restructuring from irregularly shaped nanostructures (NSs) to spherical Au NSs, and further ion irradiation leads to the formation of well-separated spherical nanoparticles. Higuchi's algorithm of surface analysis is utilized to find the evolution of surface morphology with ion irradiation in terms of the Hurst exponent and fractal dimension. The Au PEN is evidenced by Rutherford backscattering spectrometry and optical studies. Also, the depth of the mechanism behind synthesized PEN is explained on the basis of theoretical simulations, namely, a unified thermal spike and a Monte Carlo simulation consisting of dynamic compositional changes (TRIDYN). Another set of plasmonic NSs was formed on the surface by thermal annealing of the Au film on the substrate. Glucose sensing has been studied on the two types of plasmonic layers: nanoparticles on the surface and PEN. The results reveal the sensing responses of both types of plasmonic layers. However, PEN retains its plasmonic behavior as the NSs are still present after washing with water, which demonstrates the potential for reusability. RESEARCH HIGHLIGHTS: Synthesis of PENs by ion irradiation Utilization of Higuchi's algorithm to explore the surface morphology. Unified thermal spike and TRIDYN simulations being used to explain the results. Glucose is only used as a test case for reusability of substrate.

Random and time-persistent depositional processes in turbidite successions: an example from the marine deep-water Aoshima Formation (Neogene, Kyushu Island, southwest Japan)

ABSTRACT The deposits of flood- and earthquake-derived subaqueous sediment gravity flows represent a significant fraction of lacustrine and deep-sea sedimentary successions, thus providing a valuable record of such natural disasters. The magnitude of these events and the thickness of the associated deposits are considered to follow a lognormal or power-law frequency distribution, whilst that of time intervals between subsequent events appear to be best approximated by a Poisson model, indicative of a random, time-independent phenomenon. However, the debate on whether the sedimentary record of these natural disasters is governed by randomness alone or whether there is some underlying stratigraphic ordering is still unsettled and requires detailed time-series analysis. This study consists of a time-series analysis of mudstone- and sandstone-dominated turbidite successions offshore a fan-delta system in the Neogene Aoshima Formation that belongs to the sedimentary fill of the forearc basin of southwest Japan. The formation consists of a monotonous alternation of very fine- to medium-grained sandstones capped by hemipelagic mudstones and, more rarely, by turbidite mudstones. The results show that the autocorrelation function of the time series suggests quasi-periodic variability in the upper sandstone-dominated part, whereas the lower mudstone-dominated part shows a white-noise-like pattern. Rescaled range analysis shows that the number of events per unit time in the lower part is characterized by a random time series, such as Brownian noise with a Hurst exponent of 0.5. In contrast, the thickness of event beds of the lower part and the thickness and the number of events of the upper part are persistent time series with a Hurst exponent &amp;gt; 0.5. These results suggest that the number of turbidite depositional events in the mudstone-dominated part indicates random timing, whereas its thickness time series and the sandstone-dominant part are not governed by simple stochastic processes but are affected by sea-level changes, sediment transport dynamics, and other factors such as, for example, seafloor topography.

The Multifractal Phenomenon of Stock Price Caused by “Tesla Rights Defense Event”

In this paper, we explore the periodicity of the multifractal phenomenon of stock price caused by public relations events using the multifractal detrended fluctuation analysis (MF-DFA) method. The Tesla stock closing price in the context of the Tesla rights defense event is used to obtain the generalized Hurst exponent and multifractal spectrum. We find that the stock price exhibits multifractal characteristics. Afterwards, we conduct a multifractal time-varying analysis using high-frequency data for 12 time periods before and after the Tesla rights defense event, which happened on 19 April 2021. We calculate the [Formula: see text] and [Formula: see text] before and after the rights defense event to measure the degree of multifractal of Tesla stock price. In addition, we also compute the difference between the two [Formula: see text] and two [Formula: see text], which are regarded as [Formula: see text] and [Formula: see text]. The results show that the Tesla stock market efficiency degree raised as the period time increases until the eighth period. And after the eighth period, both [Formula: see text] and [Formula: see text] tend to be stable, and the degree of market efficiency caused by the Tesla rights defense event remains almost unchanged. In addition, we use three broad market indices in the context of the Russia–Ukraine conflict to test the universality. The three broad market indices are Dow Jones Industrial Average (DJIA), Shanghai Stock Exchange Composite Index (SSEC) and Nikkei 225 ETF (N225). The results demonstrate that with the extending of time period, [Formula: see text] and [Formula: see text] gradually stabilize, indicating that public memorability will diminish and the market efficiency almost remains unchanged.

Anomalous diffusion of synaptic vesicles and its influences on spontaneous and evoked neurotransmission.

Neurons in the central nervous system communicate with each other by activating billions of tiny synaptic boutons distributed along their fine axons. These presynaptic varicosities are very crowded environments, comprising hundreds of synaptic vesicles. Only a fraction of these vesicles can be recruited in a single release episode, either spontaneous or evoked by action potentials. Since the seminal work by Fatt and Katz, spontaneous release has been modelled as a memoryless process. Nevertheless, at central synapses, experimental evidence indicates more complex features, including non-exponential distributions of release intervals and power-law behaviour in their rate. To describe these features, we developed a probabilistic model of spontaneous release based on Brownian motion of synaptic vesicles in the presynaptic environment. To account for different diffusion regimes, we based our simulations on fractional Brownian motion. We show that this model can predict both deviation from the Poisson hypothesis and power-law features in experimental quantal release series, thus suggesting that the vesicular motion by diffusion could per se explain the emergence of these properties. We demonstrate the efficacy of our modelling approach using electrophysiological recordings at single synaptic boutons and ultrastructural data. When this approach was used to simulate evoked responses, we found that the replenishment of the readily releasable pool driven by Brownian motion of vesicles can reproduce the characteristic binomial release distributions seen experimentally. We believe that our modelling approach supports the idea that vesicle diffusion and readily releasable pool dynamics are crucial factors for the physiological functioning of neuronal communication. KEY POINTS: We developed a new probabilistic model of spontaneous and evoked vesicle fusion based on simple biophysical assumptions, including the motion of vesicles before they dock to the release site. We provide closed-form equations for the interval distribution of spontaneous releases in the special case of Brownian diffusion of vesicles, showing that a power-law heavy tail is generated. Fractional Brownian motion (fBm) was exploited to simulate anomalous vesicle diffusion, including directed and non-directed motion, by varying the Hurst exponent. We show that our model predicts non-linear features observed in experimental spontaneous quantal release series as well as ultrastructural data of synaptic vesicles spatial distribution. Evoked exocytosis based on a diffusion-replenished readily releasable pool might explain the emergence of power-law behaviour in neuronal activity.

Detection of Gate Valve Leaks through the Analysis Fractal Characteristics of Acoustic Signal

This paper considers the possibility of using monofractal and multifractal analysis of acoustic signals to detect water leaks through gate valves. Detrended fluctuation analysis (DFA) and multifractal detrended fluctuation analysis (MF-DFA) were used. Experimental studies were conducted on a ½-inch nominal diameter wedge valve, which was fitted to a ¾-inch nominal diameter steel pipeline. The water leak was simulated by opening the valve. The resulting leakage rates for different valve opening conditions were 5.3, 10.5, 14, 16.8, and 20 L per minute (L/min). The Hurst exponent for acoustic signals in a hermetically sealed valve is at the same level as a deterministic signal, while the width of the multifractal spectrum closely matches that of a monofractal process. When a leak occurs, turbulent flow pulsations appear, and with small leak sizes, the acoustic signals become anticorrelated with a high degree of multifractality. As the leakage increases, the Hurst exponent also increases and the width of the multifractal spectrum decreases. The main contributor to the multifractal structure of leak signals is small, noise-like fluctuations. The analysis of acoustic signals using the DFA and MF-DFA methods enables determining the extent of water leakage through a non-sealed gate valve. The results of the experimental studies are in agreement with the numerical simulations. Using the Ansys Fluent software (v. 19.2), the frequencies of flow vortices at different positions of gate valve were calculated. The k-ω SST turbulence model was employed for calculations. The calculations were conducted in a transient formulation of the problem. It was found that as the leakage decreases, the areas with a higher turbulence eddy frequency increase. An increase in the frequency of turbulent fluctuations leads to enhanced energy dissipation. Some of the energy from ordered processes is converted into the energy of disordered processes.

Observation of avalanche-like transport in Heliotron J and JT-60U plasmas

The avalanche type of transport can induce a long-radial transport and thus can contribute to the global profile formation. In this study, we observed the heat perturbations exhibiting avalanche-like transport in the stellarator/heliotron device, Heliotron J, and the tokamak device, JT-60U. We found that the electron heat propagation in Heliotron J is mainly generated from the heating source region. The relatively high value of the Hurst exponent, which is a signature of avalanches, depends on the total heating power. On the other hand, the electron and ion heat avalanches measured in JT-60U tend to spread from the local peak of the temperature gradient and are not influenced by the heating source profiles. The contrasting features of avalanches in stellarator/heliotrons and tokamaks potentially imply the difference in the temperature profile formation, such as the presence of stiffness.

Reflected stochastic differential equations driven by standard and fractional Brownian motion

The reflection problem on the positive half-line with reflection at zero for a time-dependent stochastic differential equations driven by standard and fractional Brownian motion with Hurst parameter [Formula: see text] is considered. We prove the existence of weak solutions by using Euler scheme. Moreover, we show that pathwise uniqueness holds and a strong solution exists in the case of additive fractional noise and also up to a stopping time [Formula: see text] for the multiplicative case, but remains an open question beyond [Formula: see text].

The investigation of thickness-dependent mono-fractal, optical and optoelectronics properties of sputtered silver thin film for silicon solar cell

Here we report, the room temperature deposition of silver (Ag) thin films on silicon, glass and silicon solar cell substrates by magnetron sputtering. The effect of thickness on the deposited thin films and their morphological, fractal, optical and solar cell parameters have been extensively investigated. AFM results show that the Ag thin films are composed of different-size of nano-grains and the grain size increases with increasing thickness. Mono-fractal parameters, namely fractal dimension (Df) and Hurst exponent (H), were determined from the power spectral density function. X-ray photoelectron spectroscopy studies indicate that the deposited samples exhibit the presence of oxidation state of Ag (0) in thin films, confirming the stable structure. The optical analysis shows that the transmittance of the deposited thin films decreases with increasing thickness, on the other hand, the silver thin film with thickness of 20 nm shows surface Plasmon resonance (SPR) at 436 nm. Current-voltage (I–V) curves were measured in dark and under illumination mode at standard testing conditions, i.e., 300K, 1000 W/m2 and 1.5 AM global using a solar simulator. These measurement studies indicate that thicknesses are effectively impacts on photovoltaic performance. The calculated values of change in solar efficiency (Δη) are 0.26, −0.20, −0.76 and −6.88 % for thin films with nominal thickness 0.5, 1, 2 and 20 nm respectively.

Seasonality effect on trend and long-term persistence in precipitation and temperature time series of a semi-arid, endorheic basin in Central Anatolia, Turkey

ABSTRACT In climate change research, it is vital to have knowledge about future changes in the trends of climatological time series. The purpose of this study is to look at the effect of seasonality on the trend and long-term persistence of precipitation and temperature time series in the Konya Endorheic Basin, a semi-arid basin in central Anatolia, Turkey. Seasonal-trend decomposition by locally estimated scatterplot smoothing was used to obtain a seasonality adjusted time series, and a comparative analysis with the original series was performed. The Mann–Kendall test and simple linear slope were used to determine monotonic trends. To observe long-term persistence, Hurst exponents were computed using the rescaled range analysis approach. The Onyutha trend test was then used to investigate the sub-trends. As a result, there is a significant increase in temperature. Precipitation is increasing in the west and east of the basin while decreasing in the north and south. The magnitude of the monotonic and sub-trend is enhanced by seasonal adjustment. Meanwhile, in the seasonally adjusted dataset, the Hurst exponent grew dramatically, reinforcing the long-term persistence. Seasonality has less of an impact on precipitation, which is more affected by local variability, than temperature.

Nanostructuring, fractal characterization and wettability of ion irradiated Au thin films and their thickness effect

This study proposes a method for modifying the wettability of Au thin films of thicknesses 5 nm and 10 nm on Si substrates using 8 keV Ne ion beam irradiation at different fluences, thereby enabling surface wettability adjustment at the nano-scale and its thickness dependence. This advancement would be highly beneficial for applications such as high-resolution printing and nano-biotechnology. The contact angle of the irradiated region for the 5 nm thin film changed from hydrophobic to hydrophilic with increasing ion dosage, whereas for the 10 nm thin film, it changed from hydrophilic to hydrophobic under the same conditions. Fractal analysis was performed on pristine and ion beam modified surfaces to investigate the underlying nano-structuring process, with reported values for average roughness (Ra), Hurst exponent (H), fractal dimension (Df), contact angle (CA), and wettability at different ion dosages. Atomic Force Microscopy and Rutherford Backscattering Spectroscopy were used to examine surface roughness and depth profile. The process of sputtering becomes crucial when the ion range approaches the thickness of thin films, with the sputter yield showing dependence on the thickness of thin films at different ion fluence levels. Monte Carlo simulations SRIM/TRIM and TRIDYN were used for comparison with the experimental results, with TRIDYN taking into account dynamic changes in the target's morphology and composition.

Strong convergence analysis of spectral fractional diffusion equation driven by Gaussian noise with Hurst parameter less than [formula omitted]

We propose and analyze an efficient time discretization for the spectral fractional stochastic partial differential equation with Hurst parameter less than 12. By using variable substitution, the original equation is transformed into a system of equations, which includes a partial differential equation and a stochastic integral equation. Then the time discretization is composed of two parts. We discretize the partial differential equation in time by using a semi-implicit Euler scheme. Integration by parts is used to obtain the approximation of stochastic integral. We show the optimal error estimates of the time discretization in the strong sense. Finally, we provide some numerical examples to verify these theoretical results.

A maximum-likelihood-based approach to estimate the long memory parameter using fractional spline wavelets

Fractional spline wavelets act as fractional differentiators for essentially low-pass signals, which is a useful property to analyze time series with fractal behavior. Moreover, under suitable scaling of wavelet coefficients, one can whiten a fractional Gaussian noise inputted in the wavelet channels. In this article, we demonstrate the two former statements and, by employing them, propose a method based on maximum likelihood to estimate the long-memory parameter using fractional spline wavelets. The proposed estimator exhibits competitive behavior in terms of mean squared error in the simulation studies we conducted, dominating all other methods as the sample size increases. In empirical applications, the proposed method outperformed other approaches in 10 of 12 evaluations for three different time series. Moreover, the proposed method allows for estimation in a continuous search grid, even when the Hurst coefficient lies outside of stationarity boundaries, which is a clear advantage over many algorithms: in such cases, the proposed method displays a stable behavior and an increasing outperformance margin. We also propose an identification procedure for the persistence or antipersistence of a given signal, based on the energy concentration of the coefficients of wavelet decomposition and its theoretical properties.

Unravelling the Fractal Complexity of Temperature Datasets across Indian Mainland

Studying atmospheric temperature characteristics is crucial under climate change, as it helps us to understand the changing patterns in temperature that have significant implications for the environment, ecosystems, and human well-being. This study presents the comprehensive analysis of the spatiotemporal variability of scaling behavior of daily temperature series across the whole Indian mainland, using a Multifractal Detrended Fluctuation Analysis (MFDFA). The analysis considered 1° × 1° datasets of maximum temperature (Tmax), minimum temperature (Tmin), mean temperature (Tmean), and diurnal temperature range (DTR) (TDTR = Tmax − Tmin) from 1951 to 2016 to compare their scaling behavior for the first time. Our results indicate that the Tmin series exhibits the highest persistence (with the Hurst exponent ranging from 0.849 to unity, and a mean of 0.971), and all four-temperature series display long-term persistence and multifractal characteristics. The variability of the multifractal characteristics is less significant in North–Central India, while it is highest along the western coast of India. Moreover, the assessment of multifractal characteristics of different temperature series during the pre- and post-1976–1977 period of the Pacific climate shift reveals a notable decrease in multifractal strength and persistence in the post-1976–1977 series across all regions. Moreover, for the detection of climate change and its dominant driver, we propose a new rolling window multifractal (RWM) framework by evaluating the temporal evolution of the spectral exponents and the Hurst exponent. This study successfully captured the regime shifts during the periods of 1976–1977 and 1997–1998. Interestingly, the earlier climatic shift primarily mitigated the persistence of the Tmax series, whereas the latter shift significantly influenced the persistence of the Tmean series in the majority of temperature-homogeneous regions in India.

Spatial-Temporal Evolution Characteristics and Driving Force Analysis of NDVI in Hubei Province, China, from 2000 to 2022

Exploring the characteristics of vegetation dynamics and quantitatively analyzing the potential drivers and the strength of their interactions are of great significance to regional ecological environmental protection and sustainable development. Therefore, based on the 2000–2022 MODIS NDVI dataset, supplemented by climatic, topographic, surface cover, and anthropogenic data for the same period, the Sen+Mann–Kendall trend analysis, coefficient of variation, and Hurst exponent were employed to examine the spatial and temporal characteristics and trends of NDVI in Hubei Province, and a partial correlation analysis and geographical detector were used to explore the strength of the influence of driving factors on the spatial differentiation of NDVI in vegetation and the underlying mechanisms of interaction. The results showed that (1) the mean NDVI value of vegetation in Hubei Province was 0.762 over 23 years, with an overall increasing trend and fluctuating upward at a rate of 0.01/10a (p &lt; 0.005); geospatially, there is a pattern of “low east and high west”; the spatial change in NDVI shows a trend of “large-scale improvement in the surrounding hills and mountains and small-scale degradation in the middle plains”; it also presents the spatial fluctuation characteristics of “uniform distribution in general, an obvious difference between urban and rural areas, and a high fluctuation of rivers and reservoirs”, (2) the future trend of NDVI in 70.76% of the region in Hubei Province is likely to maintain the same trend as that of the 2000–2022 period, with 70.78% of the future development being benign and dominated by sustained improvement, and (3) a combination of partial correlation analysis and geographical detector analysis of the drivers of vegetation NDVI change shows that land cover type and soil type are the main drivers; the interactions affecting the distribution and change characteristics of NDVI vegetation all showed two-factor enhancement or nonlinear enhancement relationships. This study contributes to a better understanding of the change mechanisms in vegetation NDVI in Hubei Province, providing support for differentiated ecological protection and project implementation.

Postural control in gymnasts: anisotropic fractal scaling reveals proprioceptive reintegration in vestibular perturbation.

Dexterous postural control subtly complements movement variability with sensory correlations at many scales. The expressive poise of gymnasts exemplifies this lyrical punctuation of release with constraint, from coarse grain to fine scales. Dexterous postural control upon a 2D support surface might collapse the variation of center of pressure (CoP) to a relatively 1D orientation-a direction often oriented towards the focal point of a visual task. Sensory corrections in dexterous postural control might manifest in temporal correlations, specifically as fractional Brownian motions whose differences are more and less correlated with fractional Gaussian noises (fGns) with progressively larger and smaller Hurst exponent H. Traditional empirical work examines this arrangement of lower-dimensional compression of CoP along two orthogonal axes, anteroposterior (AP) and mediolateral (ML). Eyes-open and face-forward orientations cultivate greater variability along AP than ML axes, and the orthogonal distribution of spatial variability has so far gone hand in hand with an orthogonal distribution of H, for example, larger in AP and lower in ML. However, perturbing the orientation of task focus might destabilize the postural synergy away from its 1D distribution and homogenize the temporal correlations across the 2D support surface, resulting in narrower angles between the directions of the largest and smallest H. We used oriented fractal scaling component analysis (OFSCA) to investigate whether sensory corrections in postural control might thus become suborthogonal. OFSCA models raw 2D CoP trajectory by decomposing it in all directions along the 2D support surface and fits the directions with the largest and smallest H. We studied a sample of gymnasts in eyes-open and face-forward quiet posture, and results from OFSCA confirm that such posture exhibits the classic orthogonal distribution of temporal correlations. Head-turning resulted in a simultaneous decrease in this angle Δθ, which promptly reversed once gymnasts reoriented their heads forward. However, when vision was absent, there was only a discernible negative trend in Δθ, indicating a shift in the angle's direction but not a statistically significant one. Thus, the narrowing of Δθ may signify an adaptive strategy in postural control. The swift recovery of Δθ upon returning to a forward-facing posture suggests that the temporary reduction is specific to head-turning and does not impose a lasting burden on postural control. Turning the head reduced the angle between these two orientations, facilitating the release of postural degrees of freedom towards a more uniform spread of the CoP across both dimensions of the support surface. The innovative aspect of this work is that it shows how fractality might serve as a control parameter of adaptive mechanisms of dexterous postural control.

Functional central limit theorems for rough volatility

The non-Markovian nature of rough volatility makes Monte Carlo methods challenging, and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker’s approximation of Brownian motion to the fractional Brownian case with arbitrary Hurst exponent H∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H \\in (0,1)$\\end{document}. Some of the most relevant consequences of this ‘rough Donsker (rDonsker) theorem’ are functional weak convergence results in Skorokhod space for discrete approximations of a large class of rough stochastic volatility models. This justifies the validity of simple and easy-to-implement Monte Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark hybrid scheme and find remarkable agreement (for a large range of values of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$H$\\end{document}). Our rDonsker theorem further provides a weak convergence proof for the hybrid scheme itself and allows constructing binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan options.

Variance and volatility swaps and options under the exponential fractional Ornstein–Uhlenbeck model

Considering the fair strike values of variance and volatility swaps, we use a stochastic volatility model in which the log volatility is given by a fractional Ornstein–Uhlenbeck process with two versions; a stationary version and a version with a deterministic initial value. Under these versions, the fair strike formulas are obtained in exact form for variance swaps and approximated fair strike formulas are derived for volatility swaps based on the fact that an aggregation of log-normal variables is well-approximated by shifted log-normal or log-normal distribution. In addition, we obtain two approximate pricing formulas for European options on the realized variance and volatility. The accuracy and robustness of the approximated fair strike formulas are examined via Monte-Carlo computations. We conduct calibration experiments to show that the Hurst exponent and the mean-reversion property of the fractional Ornstein–Uhlenbeck process are able to produce various shapes resembling the market term-structures of variance swaps when they are put together.