Search Results (814)

We investigate dynamic weighted fractional information-theoretic measures for linear stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/2,1)$. Motivated by recent constructions of fractional Deng entropy and building upon explicit Gaussian solutions and closed-form fractional moments derived in previous work, we establish fully analytical expressions for the Shannon entropy, Rényi entropy, Tsallis entropy, extropy, and a continuous weighted fractional entropy $\mathbb{E}\left[X_t^p(-log{p}_{X_t}\left(X_t\right))\right]$ for $p\ge 0$, expressed directly in terms of known fractional moments without density estimation. All derived measures share a universal asymptotic scaling law growing as $Hlogt$, establishing a precise quantitative link between long-memory effects and information dynamics. The weighted fractional entropy further reveals remarkable structural properties as a function of the weighting order p, exposing a dual role of long memory on the system’s informational content. As a concrete application, we characterize anomalous diffusion in aging soft materials through an explicit critical time linking maximal uncertainty to the memory exponent H and the macroscopic aging rate. All results are validated through extensive Monte-Carlo simulations, demonstrating excellent agreement with the closed-form expressions across a wide range of Hurst exponents H and weighting orders p. Full article

The main contribution of this paper is to prove the existence and uniqueness of almost automorphic solutions in distribution for a class of McKean–Vlasov SDEs driven by fractional Brownian motion, under appropriate conditions on the coefficients. The practical relevance of this result is illustrated by analyzing a stochastic heat equation on a bounded domain. Full article

Hybrid carbon nanotube–fullerene architectures provide a controllable setting in which to study irreversibility and information flow in strongly structured quantum environments. We analyze entropy generation in a platform where paramagnetic endohedral fullerenes (PEFs), such as N@C₆₀ and P@C₆₀, are encapsulated inside a suspended carbon nanotube (CNT) resonator, such that selected multi-level PEF spin states define an effective qubit coupled to quantized CNT flexural modes. Motivated by prior work on fullerene-filled CNTs, on spin–phonon manipulation in suspended nanotubes, and on exact phase-space propagators for damped driven oscillators, we formulate a hybrid open-system description that combines a driven quantum Brownian description of the CNT resonator with an effective Jaynes–Cummings type spin–vibrational interaction. The resonator dynamics are represented in phase space through the Wigner function, whose time evolution can be written analytically in terms of the initial Wigner distribution and a Gaussian propagator. This representation makes it possible to separate drive-induced phase space displacement, diffusion, and damping, and to connect these features directly to entropy flow. The coupled spin–mechanical dynamics are then embedded in a Lindblad quantum master equation that includes mechanical damping, spin relaxation, pure dephasing, and thermally activated excitation channels. Within this framework we derive the entropy balance equation—identifying entropy flux and non-negative entropy production—and examine how hybridization between the molecular spin and the nanotube vibration redistributes irreversibility between coherent exchange and dissipative channels. We show that spin–phonon coupling enhanced by a magnetic field gradient, resonant driving, and moderate thermal occupation can produce identifiable crossovers between entropy–production regimes dominated by the oscillator and those dominated by the spin. The resulting framework provides a quantitative basis for using CNT–PEF hybrids as nanoscale platforms for studying nonequilibrium quantum thermodynamics, decoherence, and information loss in structured vibrational environments. Full article

This paper studies Dirichlet problems for one-dimensional Lévy-type nonlocal elliptic equations on the half-line. The equation $L^{\mu }\nu (x)=f(x),x>0,\nu (x)=0,x\le 0$ is transformed into a weighted nonlocal equation associated with a multiplicative jump process. Under basic structural assumptions on the Lévy measure, the transformed generator is realized through a martingale problem, and the associated exponential killing representation gives a probabilistic mild solution with an immediate $L^{\infty }$-estimate. For the one-dimensional fractional Laplacian, the transformed process is exactly multiplicative. This yields a new approach in which solution estimates are derived from the stochastic equation of the transformed process; smooth-data resolvent solutions are estimated in weighted $L_p$-spaces and extended to general data by approximation. For more general Lévy measures, a smooth weighted energy estimate is proved. The key analytic input is a weighted adjoint integral inequality for the transformed generator, verified for subordinate Brownian motions associated with Bernstein functions and for non-unimodal logarithmically perturbed stable-type operators. Full article

This study investigates the growth and evolution of SiO₂-based inclusions in Si-killed steel under stagnant conditions and varying oxygen levels. Deoxidation experiments were conducted in a high-temperature furnace using commercial FeSi, with systematic variations in holding time and total oxygen content. Automated SEM–EDS analysis was employed to quantify inclusion size, number density, and chemical composition. Under stagnant conditions, SiO₂ inclusions were observed to grow and coarsen in the absence of melt agitation, following a t^1/3 scaling law. In high-oxygen melts, rapid inclusion growth was dominated by Stokes collision mechanisms, resulting in the formation of inclusions in the size range of 1–5 μm, which were subsequently removed by flotation. In contrast, low-oxygen melts exhibited slower growth kinetics governed primarily by Brownian motion and Ostwald ripening, producing smaller inclusions with characteristic sizes of 1–2 μm. These results demonstrate that the initial oxygen content plays a decisive role in controlling the dominant growth mechanisms and the extent of inclusion coarsening in non-agitated steel. Full article

This study formulates a corporate abatement decision problem under carbon price uncertainty as a continuous-time stochastic control model. To this end, the carbon price is modeled as a geometric Brownian motion, while abatement capacity is accumulated through costly effort and depreciates over time. Specifically, the firm chooses its abatement effort to maximize expected discounted profits while accounting for allowance purchasing costs, compliance-related penalties, abatement costs, and potential revenues from surplus allowances. The paper contributes by integrating stochastic carbon prices, endogenous abatement-capacity accumulation, allowance-shortage/allowance-surplus asymmetry, and surplus allowance monetization into a unified corporate abatement framework. Applying the dynamic programming principle, the associated Hamilton–Jacobi–Bellman equation is derived, and the bounded optimal abatement effort is characterized in feedback form. Since the resulting nonlinear HJB equation generally does not admit a closed-form solution, a finite-difference scheme with damped policy iteration is used for numerical analysis. The results show that optimal abatement effort is strongly state-dependent. Higher carbon prices strengthen abatement incentives in the allowance-shortage region, whereas effort declines sharply after reaching allowance neutrality if surplus allowances cannot be monetized. Moreover, partial monetization of surplus allowances significantly increases abatement effort in the surplus region and can shift firms’ behavior from passive compliance to active low-carbon investment. Overall, these findings suggest that surplus allowance monetization plays an important role in sustaining firms’ abatement incentives under carbon price uncertainty. Full article

Accurate Remaining Useful Life (RUL) prediction is pivotal for the intelligent operation and maintenance of high-precision equipment. However, existing deep learning-based prognostic methods predominantly focus on point estimations and often overlook the non-Markovian characteristics and stochastic uncertainties inherent in complex mechanical degradation. To bridge this gap, this study proposes a novel uncertainty-aware hybrid prognostic framework by synergizing TCN–Transformer architectures with fractional Brownian motion (FBM). Specifically, a TCN–Transformer hybrid network is developed to adaptively learn a multi-scale drift function, effectively capturing both localized causal features and global long-range temporal dependencies. Concurrently, the FBM component is employed to model the diffusion process, explicitly accounting for the long-range dependence and inherent stochasticity of degradation. By leveraging the first hitting time (FHT) principle, an approximate analytical expression for the RUL probability density function (PDF) is derived based on an established approximation treatment for FBM-driven degradation processes, enabling robust uncertainty quantification. Experimental results on both the XJTU-SY bearing dataset and the servo tool holder power head system dataset demonstrate that the proposed method achieves superior predictive accuracy and reliable uncertainty quantification, thereby providing effective support for condition-based maintenance and intelligent decision-making. Full article

This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby capturing long-range dependence as well as random discontinuities. By combining techniques from fractional calculus, stochastic analysis, and operator theory, we establish sufficient conditions for the existence of mild solutions. The analysis is carried out through the construction of suitable solution operator families and the application of Sadovskii’s fixed point theorem in an appropriate phase space framework. In addition, we investigate the controllability properties of the system and derive criteria ensuring approximate controllability of the underlying fractional neutral dynamics. The proposed approach relies on the structural properties of the higher-order Hilfer fractional derivative, estimates for stochastic integrals with respect to fractional Brownian motion, and compactness arguments adapted to non-instantaneous impulsive effects. The inclusion of Poisson jumps and neutral terms introduces significant analytical difficulties, which are overcome using refined resolvent operator techniques and fractional power estimates. An illustrative example is presented to demonstrate the applicability of the theoretical results. The results obtained generalize and unify several recent developments in the theory of fractional stochastic systems and provide a flexible framework for analyzing controlled dynamical models with memory, randomness, and impulsive behavior. Full article

Significance: This study addresses critical industrial and biomedical applications including glass blowing (thermal management of shrinking sheets), polymer sheet extrusion (controlled cooling), magnetic drug delivery (nanoparticle targeting), and nuclear reactor cooling (enhanced heat transfer). Aim: We present a novel nonlinear analysis of magnetohydrodynamic (MHD) boundary layer flow of a Jeffery Al₂O₃ nanofluid over a shrinking permeable plate with convective boundary conditions, uniquely integrating mixed convection, Ohmic dissipation, heat generation, Brownian motion, and thermophoresis within a non-Newtonian nanofluid framework. Methodology: The governing partial differential equations are transformed using similarity transformations and solved via the Adomian decomposition method (ADM). Comprehensive validation against RK4, RK45, and bvp4c demonstrates excellent agreement with maximum relative errors below $5\times {10}^{-4}$. Key Contribution: (i) Normal velocity decreases by 15–25% as the Biot number increases from $Bi=0.4$ to $0.6$; (ii) tangential velocity decreases by 20–30% as the magnetic parameter increases from $M=5$ to 15; (iii) temperature increases by 30–40% as the Eckert number increases from $Ec=0.5$ to $2.5$; (iv) ADM converges within 12–15 terms with $L_2$ errors $<{10}^{-5}$; (v) skin friction coefficient increases from $C_f=3.02713$ to $3.90082$ as $Q_0$ increases from 1 to 4; (vi) Nusselt number values: $Nu/\sqrt{Re}=0.4621$ at $Pr=0.7$, $0.8954$ at $Pr=2$, $3.2890$ at $Pr=20$. These quantitative findings provide design guidelines for engineers in thermal management and biomedical applications. Full article

The Black–Scholes model laid the mathematical foundation for modern option pricing; however, its assumptions—stationary, independent, and Gaussian returns—are frequently violated in real markets, where long-memory volatility and sudden price jumps are well-documented. Two issues remain open: (1) Few option pricing models comprehensively incorporate long-memory and jump features. (2) The equivalence of the hedging, risk-neutral, and actuarial pricing methods, well-established under the standard Black–Scholes framework, has not been examined under jump–diffusion models. To address these gaps, we developed a sub-mixed fractional Brownian motion with Jumps (smfBm-J) model that jointly captures long memory, nonstationary increments, and jumps and derives a closed-form European call option pricing formula under the smfBm-J framework, highlighting the impact of model choice on valuation in incomplete markets. Full article

The objective of this paper is to offer a mathematical formulation of economic value added (EVA) that incorporates distance-to-default (DD) and thus a default-free capital structure. The latter is extended via the weighted average cost of capital (WACC) to introduce a default-free EVA. The data include the nonfinancial firms listed in the DJIA30 and NASDAQ100 covering the period 1992Q2–2023Q3. The results of standard specification tests and the GMM estimator show that (a) DD causes an increase in WACC and thus, EVA decreases; (b) the interest coverage ratio can be used effectively to compensate for default risk, thus adjusting the default-free EVA positively; (c) both EVA and default-free EVA can effectively be managed via common determinants, namely, net working capital ratio, total liabilities to EBITDA, sales growth rate, debt–equity ratio, and earnings per share; (d) the positive impact of the inflation rate on both EVA and default-free EVA justifies the use of default-free EVA as a metric for equity risk premium; and (e) the robustness of the results via stochastic geometric Brownian motion shows that the determinants of default-free EVA are stable. This paper contributes to related studies by incorporating credit risk via the DD into default-free EVA. Full article

We study finite-horizon first-passage time and boundary non-crossing probabilities for Gaussian martingales, viewed as continuous local martingales obtained by running Brownian motion on a deterministic variance clock associated with deterministic volatility. Our aim is to quantify how the associated survival probability changes when the variance clock is perturbed. Using a deterministic time change representation, we reduce the problem to a Brownian boundary-crossing problem with a transformed horizon and a transformed boundary. This allows us to combine time change arguments with recent differentiability results for boundary-crossing probabilities. Under suitable regularity assumptions, we derive a first-order sensitivity formula with respect to the variance clock. The derivative splits naturally into two components: one produced by the deformation of the transformed boundary and one produced by the variation of the terminal transformed horizon. Several explicit examples are provided, including affine barriers and nonlinear deterministic clocks. These examples show in particular that, for nonconstant boundaries, redistributing variance over calendar time can change the finite-horizon survival probability even when the terminal variance is kept fixed. Full article

The original concept of multiple-point correlation is extended to reveal the fine structure of correlation in mono-variate time series. Specifically, we display the patterns of the three-point correlation in daily volatility series for five stock markets distributed across the world and compare them with those for increment series generated by the fractional Brownian motion model(fBm). For the fBm increment series, with the increase of persistence (Hurst exponent) the macroscopic patterns occur and contribute more and more to the correlation pattern, and the original three-point correlation has a fractal structure whose fractal dimension increases with the Hurst exponent. For the empirical series, whose Hurst exponents turn out to be $0.79$ (SSE), $0.78$ (HSI), $0.73$ (N225), $0.81$ (NSDQ), and $0.72$ (FTSE), a simple comparison with the fBm increment series generated with identical Hurst exponents shows that the fractal structures in the original correlations can be reproduced by the fBm model. However, the fBm model can not reproduce the behaviors of the widely distributed contributions for the first principal components and the characteristics of the topological structures for the original and reconstructed three-point correlations in empirical records. Hence, the three-point correlation can tell us the fine-scale difference between the fBm model and the real dynamical processes for stock markets. Full article

A method is developed for integrating stochastic differential equations (SDEs) with Poisson (PWN) and Gaussian (GWN) white noises interpreted as the formal derivatives of the compound Poisson and Brownian motion processes. In contrast to the current integration schemes, which solve discrete time versions of the posed SDEs, the proposed method solves the posed SDEs for finite dimensional (FD) models of the compound Poisson and Brownian motion processes, i.e., finite sums of deterministic functions of time weighted by random coefficients. Paths of the resulting solutions, referred to as FD solutions, can be generated by standard ordinary differential equation (ODE) solvers since the paths of the FD input models are smooth. We also establish conditions under which the distributions of extremes and other continuous functionals of the solutions of the posed SDEs can be approximated by those of their FD solutions. This is essential in applications since the distributions of functionals of FD solutions can be estimated while those of actual solutions are rarely available analytically and cannot be obtained numerically. Full article

This study presents findings on the existence, uniqueness, averaging principle, and numerical solutions for fractional stochastic systems influenced by both Brownian motion and Poisson jumps within the pth-moment framework. While most existing results for fractional stochastic differential equations are derived using the mean-square approach, this research offers results in the more general pth-moment framework, enhancing applicability. The results are derived using the $\gamma$-Hilfer fractional derivative, a generalized operator defined in relation to another function. This operator enables the memory effect to vary according to a nonlinear time scale. The main motivation behind this work is that there is no research work on $\gamma$-Hilfer fractional stochastic systems with standard Brownian motion and Poisson jumps regarding existence, uniqueness, and averaging principles in the pth-moment. Full article