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ABSTRACT At present, inverse random problems are receiving increasing attention, especially the case of inverse random sources. Here, we are interested in an inverse random source problem with biharmonic operator. The same as the existing studies of inverse random source problems, some theoretical results of the corresponding direct and inverse problems have been investigated in this paper. The difference is that the source term is driven by time‐fractional Brownian motion, and the spatial function in the source term needs to be determined. It has been found that, for different problems, the properties of the integrand in the corresponding stochastic integrals are distinct, which has a significant impact on the entire theoretical analysis. For the problem considered here, the lack of monotonicity in the integrand of the stochastic integrals makes the analysis more challenging. Additionally, biharmonic operators will cause some difficulties in numerical calculations. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating Freidlin–Wentzell’s large deviation principle for one (spatial) dimensional nonlinear stochastic wave equation ∂ 2 u ε (t , x) ∂ t 2 = ∂ 2 u ε (t , x) ∂ x 2 + ε σ (t , x , u ε (t , x)) W ˙ (t , x) , where W ˙ is white in time and fractional in space with Hurst parameter H ∈ (1 4 , 1 2) . The variational framework and the modified weak convergence criterion proposed by Matoussi et al. (Appl Math Optim 83(2):849–879, 2021) are adopted here. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic processes in the square mean sense, and establish some basic results on the function space of such processes like completeness, convolution and composition theorems. Under the situation that the functions forcing are Stepanov-like (weighted) pseudo S-asymptotically Bloch type periodic and verify some suitable assumptions, we establish the existence and uniqueness of square-mean (weighted) pseudo S-asymptotically Bloch type periodic mild solutions of some fractional stochastic integrodifferential equations (driven by fractional Brownian motion). Finally, the most important findings are substantiated with the assistance of an illustration. [ABSTRACT FROM AUTHOR]

We investigate the fractional stochastic heat equation, driven by a random noise which admits a covariance measure structure with respect to the time variable and has a spatial covariance given by the Riesz kernel. This class of process includes White-colored noise, fractional colored noise and other related processes. We give a sufficient condition for the existence of the mild solution and we establish some properties of its. Then, we study the self similarity and the path regularity of this solution with respect to time variable on the particular case when the noise behaves as a fractional Brownian motion in time. [ABSTRACT FROM AUTHOR]

This paper deals with a class of Generalized delay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 ). In this type of equation, a generator at time t can depend not only on the present but also the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients. This paper is an extension of the first paper from [S. Aidara and I. Sane, Deplay BSDEs driven by fractional Brownian motion, Random Oper. Stoch. Equ. 30 2022, 1, 21–31]. The stochastic integral used throughout this paper is the divergence-type integral. [ABSTRACT FROM AUTHOR]

The proposed model employs curvelet-based techniques to generate anisotropic Fractional Brownian Fields, simulating systems with orientation-dependent self-similar properties. Curvelets are a mathematical tool that allows for an efficient representation of data with edges and other anisotropic singularities, being essential for capturing the directional complexity in the self-similar properties of the modeled systems. The synthesis procedure involves generating coefficients in curvelet space with a zero-mean Gaussian distribution. This approach is tailored to depict the stochastic behavior of natural systems, particularly in scenarios where angular distributions of correlations are critical. The main contribution of this paper is presenting a novel method for generating 2-D anisotropic Fractional Brownian Fields (AFBFs) using the Curvelet Transform, demonstrating the Curvelet Transform’s efficiency in modeling anisotropic properties. Potential applications include modeling heterogeneous geological structures, anisotropic materials, and complex disordered media. [ABSTRACT FROM AUTHOR]

The "gas fee" paid for inclusion in the blockchain is analyzed in two parts. First, we consider how "effort" in terms of resources required to process and store a transaction turns into a "gas limit," which, through a fee comprised of the "base" and "priority fee" in the current version of Ethereum, is converted into the cost paid by the user. We adhere closely to the Ethereum protocol to simplify the analysis and to constrain the design choices when considering "multidimensional gas." Second, we assume that the "gas" price is given deus ex machina by a fractional Ornstein–Uhlenbeck process and evaluate various derivatives. These contracts can, for example, mitigate gas cost volatility. The ability to price and trade "forwards" in addition to the existing "spot" inclusion into the blockchain could enable users to hedge against future cost fluctuations. Overall, this article offers a comprehensive analysis of gas fee dynamics on the Ethereum blockchain, integrating supply-side constraints with demand-side modelling to enhance the predictability and stability of transaction costs. [ABSTRACT FROM AUTHOR]

At long times, a fractional Brownian particle in a confining external potential reaches a non-equilibrium (non-Boltzmann) steady state. Here we consider scale-invariant power-law potentials V (x) ∼ | x | m , where m > 0, and employ the optimal fluctuation method (OFM) to determine the large- | x | tails of the steady-state probability distribution P (x) of the particle position. The calculations involve finding the optimal (that is, the most likely) path of the particle, which determines these tails, via a minimization of the exact action functional for this system, which has recently become available. Exploiting dynamical scale invariance of the model in conjunction with the OFM ansatz, we establish the large- | x | tails of ln ⁡ P (x) up to a dimensionless factor α (H , m) , where 0 < H < 1 is the Hurst exponent. We determine α (H , m) analytically (i) in the limits of H → 0 and H → 1, and (ii) for m = 2 and arbitrary H, corresponding to the fractional Ornstein-Uhlenbeck (fOU) process. Our results for the fOU process are in agreement with the previously known exact P (x) and autocovariance. The form of the tails of P (x) yields exact conditions, in terms of H and m, for the particle confinement in the potential. For H ≠ 1 / 2 , the tails encode the non-equilibrium character of the steady state distribution, and we observe violation of time reversibility of the system except for m = 2. To compute the optimal paths and the factor α (H , m) for arbitrary permissible H and m, one needs to solve an (in general nonlinear) integro-differential equation. To this end we develop a specialized numerical iteration algorithm which accounts analytically for an intrinsic cusp singularity of the optimal paths for H < 1 / 2 . [ABSTRACT FROM AUTHOR]

This paper proposes a deep neural network (DNN) model to estimate the Hurst exponent, a crucial parameter in modelling stock market price movements driven by fractional geometric Brownian motion. We randomly selected 446 indices from the S&P 500 and extracted their price movements over the last 2010 trading days. Using the rescaled range (R/S) analysis and the detrended fluctuation analysis (DFA), we computed the Hurst exponent and related parameters, which serve as the target parameters in the DNN architecture. The DNN model demonstrated remarkable learning capabilities, making accurate predictions even with small sample sizes. This addresses a limitation of R/S analysis, known for biased estimates in such instances. The significance of this model lies in its ability, once trained, to rapidly estimate the Hurst exponent, providing results in a small fraction of a second. [ABSTRACT FROM AUTHOR]

In this article, we consider the problem of parameter estimation in a power-type diffusion driven by fractional Brownian motion with Hurst parameter in (1 / 2 , 1). To estimate the parameters of the process, we use an approximate bayesian computation method. Also, a particular case is addressed by means of variations and wavelet-type methods. Several theoretical properties of the process are studied and numerical examples are provided in order to show the small sample behavior of the proposed methods. [ABSTRACT FROM AUTHOR]

This study introduces the fractional stochastic diffusion particle tracking model (FSD-PTM), a novel Lagrangian particle tracking model (PTM) designed to simulate stochastic suspended sediment transport by incorporating fractional Brownian motion (FBM). This FBM-based approach introduces memory into particle movements, particularly when particles are influenced by coherent turbulent structures near boundaries, such as ejection and sweep events. These coherent structures exhibit consistent motion over time, leading to correlated increments in particle movements. In scenarios unaffected by coherent structures, the FBM reverts to the Wiener process, preserving the independent increments. The FSD-PTM is applied to a steady, incompressible, fully developed two-dimensional open channel flow, and Monte Carlo simulations yield ensemble statistics results, including mean, variance, skewness of particle positions, particle velocity, and concentrations. These simulation results validate the FSD-PTM against experimental data, demonstrating its superior performance when considering correlated increments. Without such consideration, concentration profiles generated by previous PTMs fail to reach long-term equilibrium. A key innovation of the FSD-PTM is its integration of correlated motion directly within the governing stochastic differential equation (SDE), eliminating the need for additional terms. This model's flexibility also allows for the incorporation of various particle dynamic models, enhancing its utility for sediment transport analysis. The FSD-PTM represents a substantial advancement in modeling stochastic suspended sediment transport, particularly when particles' movements exhibit correlations due to surrounding turbulence. [ABSTRACT FROM AUTHOR]

Human respiratory mucus (HRM) is extremely soft, compelling passive microrheology for linear viscoelastic characterization. We focus this study on the use of passive microrheology to characterize HRM heterogeneity, a phenomenon in normal HRM that becomes extreme during cystic fibrosis (CF) disease. Specifically, a fraction of the mucin polymers comprising HRM phase-separate into insoluble structures, called flakes, dispersed in mucin-depleted solution. We first reconstitute HRM samples to the MUC5B:MUC5AC mucin ratios consistent with normal and CF clinical samples, which we show recapitulate progressive flake formation and heterogeneity. We then employ passive particle tracking with 200 nm and 1 μm diameter beads in each reconstituted sample. To robustly analyze the tracking data, we introduce statistical denoising methods for low signal-to-noise tracking data within flakes, tested and verified using model-generated synthetic data. These statistical methods provide a fractional Brownian motion classifier of all successfully denoised, tracked beads in flakes and the dilute solution. From the ensemble of classifier data, per bead diameter and mucus sample, we then employ clustering methods to learn and infer multiple levels of heterogeneity: (i) tracked bead data within vs. outside flakes and (ii) within-flake data buried within or distinguishable from the experimental noise floor. Simulated data consistent with experimental data (within and outside flakes) are used to explore form(s) of the generalized Stokes–Einstein relation (GSER) that recover the dynamic moduli of homogeneous and heterogeneous truth sets of purely flakelike, dilute solution, and mixture samples. The appropriate form of GSER is applied to experimental data to show (i) flakes are heterogeneous with gel and sol domains; (ii) dilute solutions are heterogeneous with only sol domains; and (iii) flake and dilute solution properties vary with probe diameter. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence of optimal solutions for control problems involving nonlinear stochastic Schrödinger equations perturbed by a multiplicative fractional Brownian motion. Finite-dimensional approximations and a linearization method are employed and lead to ε-optimal solutions for the considered problems. [ABSTRACT FROM AUTHOR]

The valuation of equity-linked annuity (EIA) has been extensively studied, while few papers consider the impact of long-range dependence (LRD) features on financial and mortality dynamics in EIAs' valuation. To characterise the LRD feature, we adopt the fractional Brownian motion (FBM) in modelling the dynamics of asset prices and mortality intensities, in which clustering jumps are captured by the compound Hawkes process. Besides, the interest rate dynamic is driven by the FBM and correlates to the asset price dynamic. Numerical analyses show that ignorance of the LRD feature and the jump component in the modelling framework could lead to a potential deficiency in the reserve and solvency capital of EIA policies. [ABSTRACT FROM AUTHOR]

In this paper, we focus on filtering and parameter estimation in stochastic volatility models when observations arise from high-frequency data. We are particularly interested in rough volatility models where spot volatility is driven by fractional Brownian motion with Hurst index $ H H < 1 2 . Since volatility is not directly observable, we rely on particle filtering techniques for statistical inference regarding the current level of volatility and the parameters governing its dynamics. In order to obtain numerically efficient and recursive algorithms, we use the fact that a fractional Brownian motion can be represented through a superposition of Markovian semimartingales (Ornstein-Uhlenbeck processes). We analyze the performance of our approach on simulated data and we compare it to similar studies in the literature. The paper concludes with an empirical case study, where we apply our methodology to high-frequency data of a liquid stock. [ABSTRACT FROM AUTHOR]

Considering that the degradation of ball screw grease involves fractal characteristics, which exhibit long-term dependency and autocorrelation, a multivariate accelerated degradation modeling and reliability assessment method based on the fractional Brownian motion process model is proposed in this paper. Firstly, a nonlinear accelerated degradation model of grease is established using fractional Brownian motion, considering the heterogeneity of samples as well as the memory effect and long-term dependence in the deterioration process, and realizing parameter estimation. Secondly, a multivariate reliability evaluation model is established by considering multivariate performance indicators in combination with the Frank copula function. Finally, the effectiveness and potential engineering application value of this method are verified through actual degradation data of the grease. [ABSTRACT FROM AUTHOR]

In this article, using conditional power, we extended the widely used classical Brownian motion (Bm) technique for monitoring clinical trial data to a larger class of stochastic processes fractional Brownian motion (fBm) with drift, and compare these results. We applied a Bayesian method to estimate parameters. The simulation study is presented as an example to illustrate scenarios under classical and fractional Brownian motion. We demonstrated the drift fractional Brownian motion in sequential monitoring of clinical trials and provided a formula for calculating conditional power under an alternative hypothesis for interim analysis. We investigated cases commonly used in clinical trials, with the Hurst parameter H = 0.5 and H > 0.5, respectively. Our simulation study suggests that the conditional power under the fBm assumption is generally higher than that under the Bm assumption when H > 0.5 and also matches better with our empirical results. [ABSTRACT FROM AUTHOR]

This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H < 1 / 2 , without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method. [ABSTRACT FROM AUTHOR]

We investigate the problem of joint statistical estimation of several parameters for a stochastic differential equation driven by an additive fractional Brownian motion. Based on discrete-time observations of the model, we construct an estimator of the Hurst parameter, the diffusion parameter and the drift, which lies in a parametrised family of coercive drift coefficients. Our procedure is based on the assumption that the stationary distribution of the SDE and of its increments permits to identify the parameters of the model. Under this assumption, we prove consistency results and derive a rate of convergence for the estimator. Finally, we show that the identifiability assumption is satisfied in the case of a family of fractional Ornstein–Uhlenbeck processes and illustrate our results with some numerical experiments. [ABSTRACT FROM AUTHOR]

This paper deals with asymptotic errors, limit theorems for errors between numerical and exact solutions of stochastic differential equations (SDEs) driven by one-dimensional fractional Brownian motion (fBm). The Euler–Maruyama, higher-order Milstein, and Crank–Nicolson schemes are among the most studied numerical schemes for SDE (fSDE) driven by fBm. Most previous studies of asymptotic errors have derived specific asymptotic errors for these schemes as main theorems or their corollaries. Even in the one-dimensional case, the asymptotic error was not determined for the Milstein or the Crank–Nicolson method when the Hurst exponent is less than or equal to 1/3 with a drift term. We obtain a new evaluation method for convergence and asymptotic errors. This evaluation method improves the conditions under which we can prove convergence of the numerical scheme and obtain the asymptotic error under the same conditions. We completely determine the asymptotic error of the Milstein method for arbitrary orders. In addition, we newly determine the asymptotic error of the Crank–Nicolson method for 1 / 4 < H ≤ 1 / 3 . [ABSTRACT FROM AUTHOR]

This paper considers a class of mixed slow-fast McKean–Vlasov stochastic differential equations that contain the fractional Brownian motion with Hurst parameter H > 1 / 2 and the standard Brownian motion. Firstly, we prove an existence and uniqueness theorem for the mixed coupled system. Secondly, under suitable assumptions on the coefficients, using the approach of Khasminskii's time discretization, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the mean square sense. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion. [ABSTRACT FROM AUTHOR]

This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path (B H , W) with H ∈ (1 / 3 , 1 / 2) as ε → 0. Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the Itô map restricted on the Cameron-Martin space of the mixed fBm. To this end, we embed the Cameron-Martin space into a larger Hilbert space, then the Hessian is computable. Subsequently, the probability representation for the Hessian is shown. Finally, the Laplace approximation is constructed, which asserts the more precise asymptotics in the exponential scale. [ABSTRACT FROM AUTHOR]

We consider a functional stochastic delay semilinear Rayleigh–Stokes equation involving Riemann–Liouville derivative. Our aim is using the resolvent theory, fixed point argument to prove the global solvability and gives some sufficient conditions to ensure the asymptotic stability of mild solutions in the mean square moment. [ABSTRACT FROM AUTHOR]

In this article, we study the quasi-translation-invariant property of a class of non-Gaussian measures. These measures are associated with the family of generalized grey Brownian motions. We identify the Cameron–Martin space and derive the explicit Radon-Nikodym density in terms of the Wiener integral with respect to the fractional Brownian motion. Moreover, we show an integration by parts formula for the derivative operator in the directions of the Cameron–Martin space. As a consequence, we derive the closability of both the derivative and the corresponding gradient operators. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A highly versatile evaluation method is proposed for transient plasmas based on statistical physics. It would be beneficial in various industrial sectors, including semiconductors and automobiles. Our research focused on low-energy plasmas in laboratory settings, and they were assessed via our proposed method, which incorporates relative entropy and fractional Brownian motion, based on a revised collisional–radiative model. By introducing an indicator to evaluate how far a system is from its steady state, both the trend of entropy and the radiative process' contribution to the lifetime of excited states were considered. The high statistical weight of some excited states may act as a bottleneck in the plasma's energy relaxation throughout the system to a steady state. By deepening our understanding of how energy flows through plasmas, we anticipate potential contributions to resolving global environmental issues and fostering technological innovation in plasma-related industrial fields. [ABSTRACT FROM AUTHOR]

We consider an ordinary differential equation driven by rough paths, in the T. Lyons sense (Rev Mat Iberoamer. 1998;14(2)215–310), depending on a small parameter and with a negative long-range random coefficient. We establish sufficient conditions under which the solution of this ordinary differential equation converges to the solution of a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H ∈ (1 4 , 1 2) , depends on the asymptotic behavior of the covariance function of the random coefficient. [ABSTRACT FROM AUTHOR]

The existence condition H < 1 / d for first-order derivative of self-intersection local time for d ≥ 3 dimensional fractional Brownian motion was obtained in Yu (J Theoret Probab 34(4):1749–1774, 2021). In this paper, we establish a limit theorem under the nonexistence critical condition H = 1 / d . [ABSTRACT FROM AUTHOR]

A class of partial sum processes based on a sequence of observations having the structure of finite-order moving averages is studied. The random component of this sequence is formed using a heterogeneous process in discrete time, while the non-random component is formed using a regularly varying function at infinity. The heterogeneous process with discrete time is defined as a power transform of partial sums of a certain stationary sequence. An approximation of the random processes from the above-mentioned class is studied by random processes defined as the convolution of a power transform of the fractional Brownian motion with a power function. Sufficient conditions for -convergence in the Donsker invariance principle are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we focus on fractional stochastic differential equations (FSDEs) with a stochastic forcing term, i.e., to FSDE, we add a stochastic forcing term. Using the implicit scheme of Euler's approximation, the conditions for the existence and uniqueness of the solution of FSDEs with a stochastic forcing term are established. Such equations can be applied to considering FSDEs with a permeable wall. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion. [ABSTRACT FROM AUTHOR]

We consider the (rough) stochastic differential equations driven by a linear multiplicative fractional Brownian motion with Hurst index H ∈ (1 2 , 1) (or a linear multiplicative geometric fractional Brownian rough path with Hurst index H ∈ (1 3 , 1 2 ]), we construct the pullback random attractors for these two type noises via the transformation method. Furthermore, we obtain the invariant probability measures, which are supported on the pullback random attractors. Finally, we obtain the limit behavior of the invariant probability measures. [ABSTRACT FROM AUTHOR]

C a 2 + signals have a crucial role in the immune system for cell functions such as proliferation, differentiation, apoptosis and gene transcription as well as the activation of Nuclear factor of activated T Cell (NFAT) which modulates the immune response of the cells. In this study, a fractional order spatiotemporal calcium dynamics model is formulated using the reaction-diffusion equation incorporating parameters like buffers, source influx, J SScyt and J PMCA . Grünwald estimation is employed to obtain the solution and stability analysis has been performed. The analysis of numerical results provides novel insights about the role of Brownian motion, super diffusion, source influx, buffers, etc., in the regulation of calcium and NFAT concentration levels in T cell and the conditions which might lead to disordered immune responses causing diseases such as HINI, HIV and HBV. [ABSTRACT FROM AUTHOR]

This paper introduces fractional Brownian motion into the study of Maxwell nanofluids over a stretching surface. Nonlinear coupled spatial fractional-order energy and mass equations are established and solved numerically by the finite difference method with Newton's iterative technique. The quantities of physical interest are graphically presented and discussed in detail. It is found that the modified model with fractional Brownian motion is more capable of explaining the thermal conductivity enhancement. The results indicate that a reduction in the fractional parameter leads to thinner thermal and concentration boundary layers, accompanied by higher local Nusselt and Sherwood numbers. Consequently, the introduction of a fractional Brownian model not only enriches our comprehension of the thermal conductivity enhancement phenomenon but also amplifies the efficacy of heat and mass transfer within Maxwell nanofluids. This achievement demonstrates practical application potential in optimizing the efficiency of fluid heating and cooling processes, underscoring its importance in the realm of thermal management and energy conservation. [ABSTRACT FROM AUTHOR]

The Athens Stock Exchange (ASE) is a dynamic financial market with complex interactions and inherent volatility. Traditional models often fall short in capturing the intricate dependencies and long memory effects observed in real-world financial data. In this study, we explore the application of fractional Brownian motion (fBm) to model stock price dynamics within the ASE, specifically utilizing the Athens General Composite (ATG) index. The ATG is considered a key barometer of the overall health of the Greek stock market. Investors and analysts monitor the index to gauge investor sentiment, economic trends, and potential investment opportunities in Greek companies. We find that the Hurst exponent falls outside the range typically associated with fractal Brownian motion. This, combined with the established non-normality of increments, disfavors both geometric Brownian motion and fractal Brownian motion models for the ATG index. [ABSTRACT FROM AUTHOR]

This work focuses on moderate deviations for two-time scale systems with mixed fractional Brownian motion. Our proof uses the weak convergence method which is based on the variational representation formula for mixed fractional Brownian motion. Throughout this paper, the Hurst parameter of fractional Brownian motion is larger than 1/2 and the integral along the fractional Brownian motion is understood as the generalized Riemann-Stieltjes integral. First, we consider single-time scale systems with fractional Brownian motion. The key of our proof is showing the weak convergence of the controlled system. Next, we extend our method to show moderate deviations for two-time scale systems. To this goal, we combine the Khasminskii-type averaging principle and the weak convergence approach. [ABSTRACT FROM AUTHOR]

The upper semicontinuity of random attractors for stochastic/random (partial) differential equations with nonlinear diffusion term is an unsolved problem. In this paper, we first show the existence of random attractor for the random differential equation with nonlinear diffusion term driven by the approximation of the fractional noise, and then prove the upper semicontinuity of the random attractors when the intensity of the approximations tends to zero. The obtained result partly gives an answer to this problem. [ABSTRACT FROM AUTHOR]

The issues of stability and sliding mode control (SMC) for time-varying delay Markov jump systems (MJSs) with structured perturbations constrained by fractional Brownian motion (fBm) are explored. First, constructing a novel Lyapunov–Krasovskii functional (LKF) with exponential terms that contain the double-integral term, the p th moment exponential stability conditions are derived by utilizing the generalized fractional It o ˆ formula and conditional mathematical expectation. Subsequently, by designing the innovative integral sliding mode surface (SMS) associated with time-varying delay and the SMC law, the state trajectories of the dynamic systems can reach the designed SMS within a finite time. Ultimately, the numerical experiment is executed to confirm and ensure the accuracy and reliability of the obtained results. • Markov jump systems with time-varying delay and structured perturbations driven by fractional Brownian motion are studied. • Exponential terms that contain the double-integral term are considered in the Lyapunov–Krasovskii Function. • An innovative integral sliding mode surface associated with time-varying delay is considered. [ABSTRACT FROM AUTHOR]

We establish a practical means for unbiased computation of the marginal probability density function of the dynamics under stochastic resetting. In contrast to conventional dynamics devoid of resetting, the marginal probability density function under resetting may exhibit cusps and, in multi-dimensions, explosions at reset positions, arising from the compelled displacement of mass. Standard numerical techniques, such as kernel density estimation, fall short in accurately reproducing those characteristics due to their inherent smoothing effects. The proposed unbiased estimation formulas are derived using advanced stochastic calculus in their general formulations, yet their implementation in specific problem settings involves only elementary numerical operations, requiring minimal mathematical expertise and marking the very first instance of a numerical method free from bias in this context. We present numerical results throughout to validate the derived estimation formulas and, more broadly, to demonstrate the effectiveness of our approach in accurately capturing the irregularities of the marginal probability density function. [ABSTRACT FROM AUTHOR]

This paper deals on the valuation of European call option price in a stochastic environment by employing three factors which are the stochastic model of the asset value, the stochastic interest rate and the transaction cost. We specify that our underlying asset and the stochastic interest rate, particularly Hull-White model, follows a fractional Brownian Motion governed by Hurst parameter H. We used the hedging and replicating technique to established the zero-coupon bond on the European option. Finally, we give a closed-form formula of the European call option price. [ABSTRACT FROM AUTHOR]

This article investigates the Parisian ruin probability for a class of Gaussian processes with power-asymmetric behavior of the variance near the unique optimal point. We derive the exact asymptotics as the initial capital tends to infinity and extend the previous result[8] to the case when the length of the Parisian interval is on the Pickands scale. As a primary application, we extend the recent result[13] on the many inputs proportional reinsurance fractional Brownian motion risk model to the Parisian ruin. [ABSTRACT FROM AUTHOR]

We propose a nonparametric estimation of random effects from the following fractional diffusions d X j (t) = ψ j X j (t) d t + X j (t) d W H , j (t) , X j (0) = x 0 j , t ≥ 0 , j = 1 , ... , n , where ψ j are random variables and W j , H are fractional Brownian motions with a common known Hurst index H ∈ (0 , 1) . We are concerned with the study of Hermite projection and kernel density estimators for the ψ j 's common density, when the horizon time of observation is fixed or sufficiently large. We corroborate these theoretical results through simulations. An empirical application is made to the real Asian financial data. [ABSTRACT FROM AUTHOR]

This paper deals with a Skorokhod's integral based projection type estimator b ^ m of the drift function b 0 computed from N ∈ N ∗ independent copies X 1 , ⋯ , X N of the solution X of d X t = b 0 (X t) d t + σ d B t , where B is a fractional Brownian motion of Hurst index H ∈ (1 / 2 , 1) . Skorokhod's integral based estimators cannot be calculated directly from X 1 , ⋯ , X N , but in this paper an L 2 -error bound is established on a calculable approximation of b ^ m . [ABSTRACT FROM AUTHOR]

This study aims to investigate a stochastic Volterra integral equation driven by fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). We employ the Wong-Zakai approximation to simplify this intricate problem, transforming the stochastic integral equation into an ordinary integral equation. Moreover, we consider the convergence and the rate of convergence of the Wong-Zakai approximation for this kind of equation. [ABSTRACT FROM AUTHOR]

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) . 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 ). 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. [ABSTRACT FROM AUTHOR]

This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy. [ABSTRACT FROM AUTHOR]