Properties Of Fractional Brownian Motion Research Articles

Evaluating plasma fluctuation by collisional-radiative model using Malliavin derivative

Abstract Non-equilibrium plasma has garnered significant attention due to its involvement in short-term phenomena. One example is plasma fluctuation, encompassing variations in electron temperature or density. This phenomenon is explored not only in nuclear fusion but also in semiconductor etching and thin film deposition. This study introduces an evaluation method employing Malliavin derivatives to predict plasma fluctuations based on a collisional-radiative model. This model delineates the chemical kinetics of excited states within the plasma. Given that electron impact excitation and atomic collisional processes are stochastic, they exhibit characteristics similar to a Wiener process. Furthermore, the properties of fractional Brownian motion are applied to the Malliavin derivative. The revised Wiener process is utilized to analyze the changes in excited-level populations within short durations. This approach assesses electron or atomic density fluctuations by considering the contributions of electron collisions and those of ground-state atoms.

An approximate approach to fuzzy stochastic differential equations under sub-fractional Brownian motion

In this paper, we introduce fuzzy stochastic differential equations (FSDEs) driven by sub-fractional Brownian motion (SFBM) which are applied to describe phenomena subjected to randomness and fuzziness simultaneously. The SFBM is an extension of the Brownian motion that retains many properties of fractional Brownian motion (FBM), but not the stationary increments. This property makes SFBM a possible candidate for models that include long-range dependence, self-similarity, and non-stationary increments which is suitable for the construction of stochastic models in finance and non-stationary queueing systems. We apply an approximation method to stochastic integrals, and a decomposition of the SFBM to find the existence and uniqueness of the solutions.

Inhibitory effect induced by fractional Gaussian noise in neuronal system

We discover a phenomenon of inhibition effect induced by fractional Gaussian noise in a neuronal system. Firstly, essential properties of fractional Brownian motion (fBm) and generation of fractional Gaussian noise (fGn) are presented, and representative sample paths of fBm and corresponding spectral density of fGn are discussed at different Hurst indexes. Next, we consider the effect of fGn on neuronal firing, and observe that neuronal firing decreases first and then increases with increasing noise intensity and Hurst index of fGn by studying the time series evolution. To further quantify the inhibitory effect of fGn, by introducing the average discharge rate, we investigate the effects of noise and external current on neuronal firing, and find the occurrence of inhibitory effect about noise intensity and Hurst index of fGn at a certain level of current. Moreover, the inhibition effect is not easy to occur when the noise intensity and Hurst index are too large or too small. In view of opposite action mechanism compared with stochastic resonance, this suppression phenomenon is called inverse stochastic resonance (ISR). Finally, the inhibitory effect induced by fGn is further verified based on the inter-spike intervals (ISIs) in the neuronal system. Our work lays a solid foundation for future study of non-Gaussian-type noise on neuronal systems.

Fractional Barndorff-Nielsen and Shephard model: applications in variance and volatility swaps, and hedging

In this paper, we introduce and analyze the fractional Barndorff-Nielsen and Shephard (BN-S) stochastic volatility model. The proposed model is based upon two desirable properties of the long-term variance process suggested by the empirical data: long-term memory and jumps. The proposed model incorporates the long-term memory and positive autocorrelation properties of fractional Brownian motion with $H>1/2$, and the jump properties of the BN-S model. We find arbitrage-free prices for variance and volatility swaps for this new model. Because fractional Brownian motion is still a Gaussian process, we derive some new expressions for the distributions of integrals of continuous Gaussian processes as we work towards an analytic expression for the prices of these swaps. The model is analyzed in connection to the quadratic hedging problem and some related analytical results are developed. The amount of derivatives required to minimize a quadratic hedging error is obtained. Finally, we provide some numerical analysis based on the VIX data. Numerical results show the efficiency of the proposed model compared to the Heston model and the classical BN-S model.

Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls

Fractional Brownian motion and the fractional Langevin equation are models of anomalous diffusion processes characterized by long-range power-law correlations in time. We employ large-scale computer simulations to study these models in two geometries, (i) the spreading of particles on a semi-infinite domain with an absorbing wall at one end and (ii) the stationary state on a finite interval with absorbing boundaries at both ends and a source in the center. We demonstrate that the probability density and other properties of the fractional Langevin equation can be mapped onto the corresponding quantities of fractional Brownian motion driven by the same noise if the anomalous diffusion exponent α is replaced by 2 − α. In contrast, the properties of fractional Brownian motion and the fractional Langevin equation with reflecting boundaries were recently shown to differ from each other qualitatively. Specifically, we find that the probability density close to an absorbing wall behaves as P(x) ∼ x κ with the distance x from the wall in the long-time limit. In the case of fractional Brownian motion, κ varies with the anomalous diffusion exponent α as κ = 2/α − 1, as was conjectured previously. We also compare our simulation results to a perturbative analytical approach to fractional Brownian motion.

Taylor Based Jump Diffusion Model of Fractional Brownian Motion of Stock Price

Assuming that the stock price follows the stochastic differential equation driven by fractional Brownian motion and jump process, under the condition that the interest rate and volatility are constant. It has nothing to do with the relative importance of the information causing the jump. Generally speaking, the relative jump height of stock price depends on the relative importance of the important information. Fractional Brownian motion not only has various properties of fractional Brownian motion, but also has nonstationary increment. On the basis of fractional Brownian motion model, the fractional Brownian motion model with jump, namely fractional jump-diffusion process, is investigated. It is proposed to use Taylor expansion to deal with the control items in the model. For large-scale numerical simulation, it can reduce the amount of calculation required by the algorithm and save computer memory. The method of guessing value function and Taylor series expansion is introduced to solve the value function. The dialectical relationship between mathematics and finance is shown from one side: on the one hand, mathematics is a powerful tool for financial research; on the other hand, financial practice has promoted the development of mathematical theory itself.

How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?

The present paper investigates the effects of tempering the power law kernel of the moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the processes that are considered in order to investigate the role of tempering. Tempering does not change the local properties of fBm including the sample paths and p-variation, but it has a strong impact on the Breuer–Major theorem, asymptotic behavior of the third and fourth cumulants of fBm and the optimal fourth moment theorem.

APPLICATION OF FRACTAL DIMENSION OF FRACTIONAL BROWNIAN MOTION TO SUPPLY CHAIN FINANCING AND OPERATIONAL COMPREHENSIVE DECISION-MAKING

For a long time, the mismatch between material flow and capital flow in the supply chain operation management practice is very prominent, which has led to the inefficiency of supply chain operation and hindered the exertion of supply chain’s advantages. In the field of supply chain management theory research, the information completeness in capital market has long been a hypothetical premise, which has led to the separation of corporate financing and operational decision-making research. In addition to production, inventory, procurement, pricing and other strategies in supply chain operations, payment options and credit incentives are also important decisions for both parties, especially for products with long production and sales cycles; different payment methods directly affect corporate capital flows and the enterprise’s long-term development. On the basis of summarizing and analyzing previous works, this paper analyzed the research status and significance of supply chain financing and operational comprehensive decision-making, expounded the development background, current situation, and future challenges of the fractal dimension of fractional Brownian motion; elaborated the principles and methods of the scaling properties of fractional Brownian motion and the phase space reconstruction of time series, established a financial management analysis model based on the fractal dimension of fractional Brownian motion, performed the analysis of the agglomeration degree, time series and multi-fractal characteristics of supply chain financing, explored the coupling relationship between the comprehensive operational decision-making and Brownian motion’s scaling properties. The final empirical analysis showed that when own funds are sufficient, the production should be carried out with the goal of maximizing profits, and full consideration should be given customer channel stickiness, relative costs of offline and online channel products, and product profitability; the proposed analysis model can achieve the optimal order quantity in supply chain, and reach risk and benefit sharing among financial institutions, retailers, and suppliers by setting the financing interest rate, wholesale price, repurchase price and other parameters, thereby improving supply chain performance. This study results of this paper provided a reference for further researches on the application of fractal dimension of the fractional Brownian motion to the supply chain financing and operational comprehensive decision-making.

INTERIM ANALYSIS OF CLINICAL TRIALS: SIMULATION STUDIES OF CONDITIONAL POWER UNDER FRACTIONAL BROWNIAN MOTION

Classical Brownian motion (BM) techniques for statistical monitoring of clinical trials have been widely used. The conditional power (CP) and [Formula: see text]-spending function-based boundary crossing probabilities are popular procedures for statistical hypothesis testing under the assumption of BM. However, in some clinical trials, the assumptions of BM may not be fully met for the design and data analysis. Therefore, a more general class of stochastic processes, fractional Brownian motion (FBM), was proposed in the literature to model the test statistics derived from interim analysis of clinical trials. To investigate the properties of FBM, in this paper, we simulated a wide range of FBM data, e.g. [Formula: see text] (BM) versus [Formula: see text] (FBM), with treatment effects versus without treatment effects. Then the performance of CP-based interim analysis was compared by assuming that the data follow BM or FBM. Our simulation study suggested that CP under the FBM assumptions was generally higher than that under the BM assumptions when [Formula: see text] and also matched well with the empirical results.

CONTINUOUS DEPENDENCE ON THE INITIAL DATA OF THE SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH FRACTIONAL BROWNIAN MOTIONS

In the present acticle we consider finite-dimensional stochastic differential equations with fractional Brownian motions having different Hurst indices larger than 1/3 and a drift. These heterogeneous components of the equations are combined into a single process. The solutions of the equations are understood in the integral sense, and the integrals in turn are Gubinelli’s rough path integrals [1] realizing the well-known approach of the rough paths theory [2]. The existence and uniqueness conditions of the solutions of these stochastic differential equations are specified. Such conditions are sufficient to obtain the results related the continuous dependence on the initial data. In this article, we have first proved a continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations under consideration for almost all their trajectories. The result obtained does not depend on the probabilistic properties of fractional Brownian motions, and therefore it can be easily generalized to the case of arbitrary Holder-continuous processes with an exponent greater than 1/3. In this case, the constant arising in the estimates appears to be exponentially dependent on the norms of fractional Brownian motions. Taking into account the last fact and the proved result, an expected logarithmic continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations con - si dered is subsequently derived. This is the major result of this article.

Random attractors for stochastic discrete Klein-Gordon-Schrödinger equations driven by fractional Brownian motions

In this paper, we study the asymptotic behavior of the stochastic discrete Klein-Gordon-Schrodinger equations driven by fractional Brownian motions. With the properties of fractional Brownian motions, we prove the existence of a singleton sets random attractor.

Responses of Linear and Nonlinear Oscillators to Fractional Gaussian Noise With Hurst Index Between 1/2 and 1

The responses of linear and nonlinear oscillators to fractional Gaussian noise (fGn) are studied. First, some preliminary concepts and properties of fractional Brownian motion (fBm) and fGn with Hurst index 1/2<H<1 are introduced. Then, the exact sample solution, correlation function, spectral density, and mean-square value of the response of linear oscillator to fGn are obtained. Based on the sample solution, it is proved that the long-range correlation index of displacement response of linear oscillator is the same as that of excitation fGn, i.e., 2-2H, while the velocity response has no such long-range correlation. An interesting discovery is that the ratio of kinetic energy to total energy decreases as increasing Hurst index H. Finally, for the responses of one and two degrees-of-freedom (DOF) nonlinear oscillators to fGn, the equivalent linearization method is applied to obtain the sample functions, correlation functions and mean-square values of the responses. Plenty of digital simulation results are obtained to support these solutions. It is shown that the approximate solution is effective for weakly nonlinear oscillators and it is feasible to apply the equivalent linearization to study multi-DOF weakly nonlinear oscillators.

Persistence of fractional Brownian motion with moving boundaries and applications

We consider various problems related to the persistence probability of fractional Brownian motion (FBM), which is the probability that the FBM X stays below a certain level until time T. Recently, Oshanin et al (2012, arXiv:1209.3313v2) have studied a physical model, where persistence properties of FBM are shown to be related to scaling properties of a quantity JN, called the steady-state current. It turns out that for this analysis, it is important to determine persistence probabilities of FBM with a moving boundary. We show that one can add a boundary of logarithmic order to an FBM without changing the polynomial rate of decay of the corresponding persistence probability, which proves a result needed in Oshanin et al (2013 Phys. Rev. Lett. at press (arXiv:1209.3313v2)). Moreover, we complement their findings by considering the continuous-time version of JT. Finally, we use the results for moving boundaries in order to improve estimates by Molchan (1999 Commun. Math. Phys. 205 97–111) concerning the persistence properties of other quantities of interest, such as the time when an FBM reaches its maximum on the time interval (0, 1) or the last zero in the interval (0, 1).

A note on the use of fractional Brownian motion for financial modeling

In the second part of the past decade, the usage of fractional Brownian motion for financial models was stuck. The favorable time-series properties of fractional Brownian motion exhibiting long-range dependence came along with an apparently insuperable shortcoming: the existence of arbitrage. Within the last two years, several new models using fractional Brownian motion have been published. However, still the problem remains unsolved whether such models are reasonable choices from an economic perspective.In this article, we take on a straightforward mathematical argument in order to clarify when and why fractional Brownian motion is suited for economic modeling: We provide a fractional analog to the work of Sethi and Lehoczky (1981) thereby confirming that fractional Brownian motion and continuous tradability are incompatible. In the light of a market microstructure perspective to fractional Brownian motion, it becomes clear that the correct usage of fractional Brownian motion inherently implies dynamic market incompleteness.Building a bridge to application, we show that one peculiar, but nevertheless popular result in the literature of fractional option pricing can be well explained by the fact that authors disobeyed this need for compatibility.

Fractional Brownian Motion and the Inherent Exclusion of Arbitrage

Over the last years, the usage of fractional Brownian motion for nancial models was stuck. The favorable time-series properties of fractional Brownian motion exhibiting long-range dependence came along with an apparently insuperable shortcoming: the existence of arbitrage. In this article, we solve this dilemma: We provide a fractional analogue to the work of Sethi and Lehoczky (1981) thereby

Some topics in fractional Brownian motion

Fractional Brownian motion denotes a family of Gaussian processes whose applicability has been demonstrated empirically for a wide variety of physical phenomena. For more than five decades, these processes have described risky outcomes or physical uncertainties and in particular for more than four decades these processes have been used to model fluctuations in economic data. These processes have a selfsimilarity or fractal property in probability law. They can provide a model for long range dependence, rare events and bursty behavior. Kolmogorov [15] initially defined the family of fractional Brownian motions motivated by his study of turbulence [16,17], though Schonberg [25] predated Kolmogorov’s construction by showing that the covariance functions for these processes are positive definite and thereby Gaussian processes can be constructed. Hurst [14] empirically identified an index parameter of these processes in his modeling of rainfall in the Nile River region. The rainfall in this region had a bursty behavior with often long periods of either excessive rainfall or meager rainfall. Mandelbrot [19] motivated by his study of fractals studied fractional Brownian motions to model economic data as well as turbulence [20]. He coined the name fractional Brownian motion and noted that Hurst empirically computed an index for these processes which is now called the Hurst index. Mandelbrot and van Ness [21] investigated some basic properties of fractional Brownian motion. The empirical evidence for models with a fractional Brownian motion has continued in hydrology, finance and turbulence. More recently the empirical evidence for the use of fractional Brownian motions (FBMs) as random models broadened to internet traffic [4,18], medicine [23] and cognition [3,12]. 2. Fractional Brownian motion

Stochastic 2-microlocal analysis

A lot is known about the Hölder regularity of stochastic processes, in particular in the case of Gaussian processes. Recently, a finer analysis of the local regularity of functions, termed 2-microlocal analysis, has been introduced in a deterministic frame: through the computation of the so-called 2-microlocal frontier, it allows us in particular to predict the evolution of regularity under the action of (pseudo-)differential operators. In this work, we develop a 2-microlocal analysis for the study of certain stochastic processes. We show that moments of the increments allow us, under fairly general conditions, to obtain almost sure lower bounds for the 2-microlocal frontier. In the case of Gaussian processes, more precise results may be obtained: the incremental covariance yields the almost sure value of the 2-microlocal frontier. As an application, we obtain new and refined regularity properties of fractional Brownian motion, multifractional Brownian motion, stochastic generalized Weierstrass functions, Wiener and stable integrals.

Burgers equation with self-similar gaussian initial data: Tail probabilities

The statistical properties of solutions of the one-dimensional Burgers equation in the limit of vanishing viscosity are considered when the initial velocity potential is fractional Brownian motion (FBM). We establish the asymptotic power-law order for log-probability of large values, both velocity and shock (amplitude of velocity discontinuity). This confirms the conjecture of U. Frisch and his collaborators. Rigorous results for this problem were previously derived for the case of Brownian motion using Markov techniques. Our approach is based on the intrinsic properties of FBM and the theory of extreme values for Gaussian processes.

A fractal‐based stochastic interpolation scheme in subsurface hydrology

The need for a realistic and rational method for interpolating sparse data sets is widespread. Real porosity and hydraulic conductivity data do not vary smoothly over space, so an interpolation scheme that preserves irregularity is desirable. Such a scheme based on the properties of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) is presented. Following the methodology of Hewett (1986), the authors test for the presence of fGn in a set of 459 hydraulic conductivity (K) measurements. The use of rescaled‐range analysis strongly indicated the presence of fGn when applied to the natural logs of the K data, and the resulting Hurst coefficient (H) was determined to be 0.82. This H value was then used along with the methodology of successive random additions to generate a fBm K interpolation (realization) in the vertical cross section between two wells. The results appeared realistic, and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media. It is now known that fGn‐ and fBm‐related processes are among the most common spatial and temporal variations found in nature, although a common physical origin, if any, remains obscure.

On the spectrum of fractional Brownian motions

Fractional Brownian motions (FBMs) provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order. However, due to the nonstationary nature of these processes, the precise meaning of such spectra remains generally unclear. Two complementary approaches are proposed which are intended to clarify this point. The first one, based on a time-frequency analysis, takes into account the nonstationary nature of FBM and puts emphasis on time-averaged measurements; the second one, based on a time-scale analysis, is matched to self-similarity properties of FBM and reveals an underlying stationary structure relative to each time-scaling.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>