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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]

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 "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]

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 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 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]

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]

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]

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]

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]

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]

When the evolution family is hyperbolic and satisfies the Acquistapace-Terreni conditions, the existence and uniqueness of an almost automorphic mild solution and a weighted pseudo almost automorphic mild solution in distribution of mean-filed nonautonomous stochastic evolution equations driven by fractional Brownian motion is proved. Examples illustrating the main results are included. [ABSTRACT FROM AUTHOR]

We study the unique solvability of the Cauchy problem for a mixed-type stochastic differential equation driven by the standard Brownian motion and fractional Brownian motions with Hurst exponents . We prove a theorem on the existence and uniqueness of strong solutions of these stochastic differential equations. [ABSTRACT FROM AUTHOR]

The contributions of this paper are twofold: we define and investigate the properties of a short rate model driven by a general Gaussian Volterra process and, after defining precisely a notion of convexity adjustment, derive explicit formulae for it. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven by a fractional Brownian motion with a Hurst parameter H ∈ (0 , 1) . We establish the Malliavin differentiability of the fHt model and derive an expression for the expected payoff function, revealing potential discontinuities. Simulation experiments are conducted to illustrate the dynamics of the stock price process and option prices. [ABSTRACT FROM AUTHOR]

In this paper, a class of second-order neutral impulsive stochastic evolution equations with infinite delay driven by fractional Brownian motion and Poisson jumps is considered. We utilise the Carathéodory approximation technique, the Burkhölder–Davies–Gundy inequality and the Gronwall inequality to present the existence and uniqueness of mild solutions for the addressed system under Carathéodory conditions. Furthermore, we derive an averaging principle for the proposed system which is approximated by the simplified equation in the sense of mean square. Finally, an application is used to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Green measure for a class of non-Gaussian processes in R d . These measures are associated with the family of generalized grey Brownian motions B β , α , 0 < β ≤ 1 , 0 < α ≤ 2 . This family includes both fractional Brownian motion, Brownian motion, and other non-Gaussian processes. We show that the perpetual integral exists with probability 1 for d α > 2 and 1 < α ≤ 2 . The Green measure then generalizes those measures of all these classes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and finite-time blow-up for the solution of a reaction–diffusion system of semilinear stochastic partial differential equations (SPDEs) subjected to a two-dimensional fractional Brownian motion given by d u 1 (t , x) = [ Δ α u 1 (t , x) + γ 1 u 1 (t , x) + u 2 1 + β 1 (t , x) ] d t + k 1 1 u 1 (t , x) d B 1 H (t) + k 1 2 u 1 (t , x) d B 2 H (t) , d u 2 (t , x) = [ Δ α u 2 (t , x) + γ 2 u 2 (t , x) + u 1 1 + β 2 (t , x) ] d t + k 2 1 u 2 (t , x) d B 1 H (t) + k 2 2 u 2 (t , x) d B 2 H (t) , for x ∈ ℝ d ,   t ≥ 0 , along with u i (0 , x) = f i (x) , x ∈ ℝ d , where Δ α is the fractional power − (− Δ) α 2 of the Laplacian, 0 < α ≤ 2 and β i > 0 ,   k i j ≥ 0 ,   i , j = 1 , 2 and γ i ,   i = 1 , 2 are real constants. We provide sufficient conditions for the existence of a global weak solution. Under the assumption that β 1 ≥ β 2 > 0 with Hurst index 1 / 2 ≤ H < 1 , we obtain the blow-up times for an associated system of random partial differential equations in terms of an integral representation of exponential functions of Brownian motions. Moreover, we provide lower and upper bounds for the finite-time blow-up of the above system of SPDEs and obtain the upper bounds for the probability of non-explosive solution to our considered system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonparametric local linear estimator for the drift function of stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H ∈ (1 / 4 , 1). The drift function is one-sided dissipative Lipschitz that ensures the ergodic property of the SDE. We derive the strong consistency of the proposed estimator with proper bandwidth selectors associated with the determined Hurst parameter H. The main tools are the ergodic theorem, Malliavin calculus, and a maximum inequality for It o ̂ –Skorohod integrals. [ABSTRACT FROM AUTHOR]

We show that the tail probability of the rough line integral ∫ 0 1 ϕ (X t) d Y t , where (X, Y) is a 2D fractional Brownian motion with Hurst parameter H ∈ (1 / 4 , 1 / 2) and ϕ is a C b ∞ -function satisfying a mild non-degeneracy condition on its derivative, cannot decay faster than a γ -Weibull tail with any exponent γ > 2 H + 1 . In particular, this produces a simple class of examples of differential equations driven by fBM, whose solutions fail to have Gaussian tail even though the underlying vector fields are assumed to be of class C b ∞ . This also demonstrates that the well-known upper tail estimate proved by Cass–Litterer–Lyons in 2013 is essentially sharp. [ABSTRACT FROM AUTHOR]

Roller bearing degradation features fractal characteristics such as self-similarity and long-range dependence (LRD). However, the existing remaining useful life (RUL) prediction models are memoryless or short-range dependent. To this end, we propose a RUL prediction model based on fractional Brownian motion (FBM). Bearing faults can happen in different places, and thus their degradation features are difficult to extract accurately. Through variational mode decomposition (VMD), the original degradation feature is decomposed into several components of different frequencies. The monotonicity, robustness and trends of the different components are calculated. The frequency component with the best metric values is selected as the training data. In this way, the performance of the prediction model is hugely improved. The unknown parameters in the degradation model are estimated by the maximum likelihood algorithm. The Monte Carlo method is applied to predict the RUL. A case study of a bearing is presented and the prediction performance is evaluated using multiple indicators. [ABSTRACT FROM AUTHOR]

We investigate a class of statistical arbitrage strategies under the assumption that stock prices are driven by fractional Brownian motions. Specifically, the buy-and-hold with a stop-profit threshold strategies are analysed to demonstrate the existence of statistical arbitrage opportunities. Our analysis establishes the conditions for the considered strategy class to yield statistical arbitrage. The Hurst parameter in the fractional Brownian motion-based asset price model is shown to be a determining factor. The analysis is confirmed by a Monte Carlo simulation study. Furthermore, a modified Thompson sampling method is proposed for optimizing the strategy parameters of the selling-threshold and its growth rate to maximize investment performance. [ABSTRACT FROM AUTHOR]

The existence of solutions of non-instantaneous impulsive Hilfer–Katugampola fractional differential equations of order $ 1/2 \lt \alpha \lt 1 $ 1 / 2 < α < 1 and parameter $ 0\leq \beta \leq 1 $ 0 ≤ β ≤ 1 with fractional Brownian motion (fBm) and Poisson jumps is investigated in this paper. The required results are obtained based on fractional calculus, stochastic analysis, semigroups, and the fixed point theorem. In the end of the paper, an example is provided to illustrate the applicability of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of efficient estimation for the drift parameter θ ∈ R d in the linear model Z t : = θ t + σ 1 B H 1 (t) + σ 2 B H 2 (t) , t ∈ [ 0 , T ]. Where B H 1 and B H 2 are two independent d-dimensional fractional Brownian motions with Hurst indices H1 and H2 such that 1 2 ≤ H 1 < H 2 < 1. The main goal is firstly to define the maximum likelihood estimator (MLE) of the drift θ, and secondly to provide a sufficient condition for the James-Stein type estimators which dominate, under the usual quadratic risk, the usual estimator (MLE). [ABSTRACT FROM AUTHOR]

We analyze the multiscale behaviors of high-frequency intraday prices, with a focus on how asset prices are correlated over different timescales. The multiscale approach proposed in this paper is designed for the analysis of high-frequency intraday prices. It incorporates microstructure noise into the stochastic price process. We consider a noisy fractional Brownian motion model and illustrate its various statistical properties. This leads us to introduce new latent correlation and noise estimators. New numerical algorithms are developed for model estimation using empirical high-frequency data. For a collection of stocks and exchange-traded funds, examples are provided to illustrate the relationship between multiscale correlation and sampling frequency as well as the evolution of multiscale correlation over time. [ABSTRACT FROM AUTHOR]

Over the years, the research of backward stochastic differential equations (BSDEs) has come a long way. As a extension of the BSDEs, the BSDEs with time delay have played a major role in the stochastic optimal control, financial risk, insurance management, pricing, and hedging. In this paper, we study a class of BSDEs with time-delay generators driven by Caputo fractional derivatives. In contrast to conventional BSDEs, in this class of equations, the generator is also affected by the past values of solutions. Under the Lipschitz condition and some new assumptions, we present a theorem on the existence and uniqueness of solutions. [ABSTRACT FROM AUTHOR]

This paper analyses the attributes and the significance of the roughness of oil market volatility. We employ unspanned stochastic volatility models driven by rough Brownian motions that yield semi-analytical prices for future options entailing efficient calibration applications. By performing a Monte Carlo simulation study, we show that the semi-analytical pricing performs well thus establishing its efficiency for calibration applications. Thus we calibrate option prices written on oil futures and provide empirical evidence of the roughness in oil volatility. Introducing just one additional parameter, the Hurst parameter, indicating the roughness of the volatility improves the calibration by almost a factor of 10. The calibrated option-implied Hurst parameter varies over time, but the entire set of parameters becomes more stable than in the non-rough case corresponding to a fixed Hurst parameter 1/2. These results underscore the importance to model the time dependency of the roughness of oil market volatility. [ABSTRACT FROM AUTHOR]

We consider a stochastic control problem for a non-linear forward-backward stochastic differential equation driven by fractional Brownian motion, with Hurst parameter H ∈ (0 , 1) , in the case where the set of the control domain is convex. We provide an estimation of the solution and establish the necessary and sufficient optimality conditions in the form of the stochastic maximum principle. We apply the theory to solve a linear quadratic stochastic control problem. [ABSTRACT FROM AUTHOR]

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 H > 1 2 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 τ for the multiplicative case, but remains an open question beyond τ. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic parabolic problem involving a nonlocal diffusion operator associated with nonlocal Robin-type boundary conditions. The stochastic dynamics under consideration is driven by a mixture of a classical Brownian and a fractional Brownian motion with Hurst index $ H\in(\frac{1}{2}, 1). $ We first establish local in time existence results and then explore conditions under which the resulting SPDE exhibits finite-time quenching. Using results on the probability distribution of perpetual integral functionals of Brownian motion as well as tail estimates for the fractional Brownian motion we provide analytic estimates for certain quantities of interest, such as upper bounds for quenching times and the corresponding quenching probabilities. The existence of global in time solutions is also investigated and as a consequence a lower estimate of the quenching time is also derived. Our analytical results demonstrate the non-trivial impact of the noise on the dynamics of the system. The analytic results are complemented with a detailed numerical study of the model under Dirichlet boundary conditions. A possible application concerning MEMS technology is considered and the implications of the results in this context are commented upon. [ABSTRACT FROM AUTHOR]

In this paper, we consider stochastic evolution equations with finite delay driven by fractional Brownian motion (fBm) with Hurst index $ H\in(0,1) $. First, the global existence and uniqueness of mild solutions are established by using a temporally weighted norm, the semigroup theory, the Banach fixed point theorem, and a new estimate of the stochastic integral with respect to fBm. Then after extending the integral inequality to the general case, we obtain the mean-square exponential stability of mild solutions. In particular, the convolution method and the Young convolution inequality are utilized to deal with the difficulty of the noise term caused by fBm with Hurst index $ H\in(0,1) $. Finally, the stochastic evolution equation with multiple time-varying delays and the reaction diffusion neural networks system with time-varying delay are considered, and the global existence, uniqueness and exponential stability of mild solutions are proved by using the abstract results. [ABSTRACT FROM AUTHOR]

This paper extends a novel piecewise mathematical model of the COVID-19 epidemic using fractional and variable-order differential equations and fractional stochastic derivatives in three intervals of time. The deterministic models are augmented with hybrid fractional order and variable order operators, while the stochastic differential equations incorporate fractional Brownian motion. To probe the behavior of the proposed models, we introduce two numerical techniques: the nonstandard modified Euler Maruyama method for the fractional stochastic model, and the Caputo proportional constant-Grünwald-Letnikov nonstandard finite difference method for the fractional and variable-order deterministic models. Several numerical experiments corroborate the theoretical assertions and demonstrate the efficacy of the proposed approaches. [ABSTRACT FROM AUTHOR]

This paper deals with a class of backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1/2) with time-delayed generators. In this type of equation, a generator at time t can depend on the values of a solution in the past, weighted with a time-delay function, for instance, of the moving average type. We establish an existence and uniqueness result of solutions for a sufficiently small time horizon or for a sufficiently small Lipschitz constant of a generator. The stochastic integral used throughout the paper is the divergence operator-type integral. [ABSTRACT FROM AUTHOR]

In crowded fluids, polymer segments can exhibit anomalous subdiffusion due to the viscoelasticity of the surrounding environment. Previous single-particle tracking experiments revealed that such anomalous diffusion in complex fluids (e.g., in bacterial cytoplasm) can be described by fractional Brownian motion (fBm). To investigate how the viscoelastic media affects the diffusive behaviors of polymer segments without resolving single crowders, we developed a novel fractional Brownian dynamics method to simulate the dynamics of polymers under confinement. In this work, instead of using Gaussian random numbers ("white Gaussian noise") to model the Brownian force as in the standard Brownian dynamics simulations, we introduce fractional Gaussian noise (fGn) in our homemade fractional Brownian dynamics simulation code to investigate the anomalous diffusion of polymer segments by using a simple "bottle-brush"-type polymer model. The experimental results of the velocity autocorrelation function and the exponent that characterizes the subdiffusion of the confined polymer segments can be reproduced by this simple polymer model in combination with fractional Gaussian noise (fGn), which mimics the viscoelastic media. [ABSTRACT FROM AUTHOR]

This article introduces a new class of diffusive processes with rough mutually exciting jumps for modeling financial asset returns. The novel feature is that the memory of positive and negative jump processes is defined by the product of a dampening factor and a kernel involved in the construction of the rough Brownian motion. The jump processes are nearly unstable because their intensity diverges to + ∞ for a brief duration after a shock. We first infer the stability conditions and explore the features of the dampened rough (DR) kernel, which defines a fractional operator, similar to the Riemann-Liouville integral. We next reformulate intensities as infinite-dimensional Markov processes. Approximating these processes by discretization and then considering the limit allows us to retrieve the Laplace transform of asset log-return. We show that this transform depends on the solution of a particular fractional integro-differential equation. We also define a family of changes of measure that preserves the features of the process under a risk-neutral measure. We next develop an econometric estimation procedure based on the peak over threshold (POT) method. To illustrate this work, we fit the mutually exciting rough jump-diffusion to time series of Bitcoin log-returns and compare the goodness of fit to its non-rough equivalent. Finally, we analyze the influence of roughness on option prices. [ABSTRACT FROM AUTHOR]

As defaultable options are subject to default risks in over-the-counter (OTC) markets, vulnerable options are one of the financial securities closely linked with the risk caused by the situation that the option issuer may be unable to execute their contractual obligations. In this study, taking account of the changes in stock prices show different degrees of long-term correlation and autocorrelation at different times in the real-world situations of the financial market, contrary to the assumption of Black-Scholes settings, we propose the closed-form pricing formulas for the vulnerable options based on a fractional Brownian environment. First of all, from the model dynamics for the underlying asset under the fractional Brownian motions, we derive partial differential equations (PDEs) for vulnerable option prices, and subsequently derive the closed-form solutions for the options based on the issuer's credit risk via double Mellin transforms. In addition, we evaluate the accuracy of vulnerable option pricing by comparing the solutions obtained using the aforementioned closed formulas with those obtained via Monte Carlo simulations. Finally, we examine the price sensitivities of vulnerable options in fractional Brownian markets with respect to the model parameters. [ABSTRACT FROM AUTHOR]

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of d x a (s) + g (s , x a (s - ω (s))) = I x a (s) + f (s , x a (s - ϱ (s))) d s + ς (s) d ϖ H (s) , 0 ≤ s ≤ T , x a (s) = ζ (s) , - ρ ≤ s ≤ 0. The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples. [ABSTRACT FROM AUTHOR]

Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) modify the power-law kernel in the moving average representation of fractional Brownian motion by introducing exponential tempering. We construct least-square estimators for the unknown drift parameters within Vasicek models that are driven by these processes. To demonstrate their strong consistency, we establish asymptotic bounds with probability 1 for the rate of growth of trajectories of tempered fractional processes. [ABSTRACT FROM AUTHOR]

In this paper, we study the rate in the Smoluchowski–Kramers approximation for the solution of the following distribution-dependent SDE driven by fractional Brownian motion X t = X 0 + ∫ 0 t b (s , X s , ℒ X s ) d s + ∫ 0 t σ (s , ℒ X s ) B s H , where ℒ X t denotes the law of X t , { B t H , t ∈ [ 0 , T ] } is a d -dimensional fractional Brownian motion with Hurst parameter H ∈ (1 / 2 , 1). Based on the techniques of multiple integrals and Malliavin calculus, we provide an explicit bound on total variation distance for the rate of convergence. [ABSTRACT FROM AUTHOR]

Brownian motion (BM) is the archetypal model of regular diffusion. BM is a Gaussian and Markov process, whose increments are stationary, and whose non-overlapping increments are independent. Elevating from regular diffusion to anomalous diffusion, fractional Brownian motion (FBM) and scaled Brownian motion (SBM) are arguably the two most popular Gaussian anomalous-diffusion models. Each of these two models maintains some BM properties, abandons other, and displays certain anomalous behaviors. This paper explores a Gaussian anomalous-diffusion model— Power Brownian Motion (PBM)—that is attained by a coupled amplitudal and temporal 'tinkering' with BM. The PBM model combines 'the better of FBM and SBM'. Indeed, as FBM, PBM displays the anomalous behaviors of persistence and anti-persistence. And, as SBM, PBM is a Markov process that displays the anomalous behaviors of aging and anti-aging. On their own, neither FBM nor SBM can provide the 'features package' that PBM provides. The PBM 'features package' on the one hand, and its simple construction on the other hand, render PBM a compelling anomalous-diffusion model. [ABSTRACT FROM AUTHOR]

In this paper we study the stochastic control problem of a partially observed (multidimensional) stochastic system driven by both Brownian motions and fractional Brownian motions. In the absence of the powerful tool of Girsanov transformation, we introduce and study new stochastic processes which are used to transform the original problem to a "classical one"". The adjoint backward stochastic differential equations and the necessary condition satisfied by the optimal control (maximum principle) are obtained. [ABSTRACT FROM AUTHOR]