Abstract We show that, under certain conditions, a strongly continuous semigroup admits an almost surely frequently hypercyclic random vector defined as a stochastic integral in Fréchet spaces with respect to the Brownian motion. Two criteria are given. We will apply the second criterion to three examples: translation semigroups on spaces of integrable functions, the exponential of weighted shifts, and the translation operators on the space of entire functions. This last example, with a stochastic approach, seems to be new in the literature. Some other examples are given.

This paper establishes a general parametric integral identity involving (n+1)-times differentiable stochastic processes, formulated entirely in terms of stochastic k-Caputo fractional derivatives. This identity serves as a unifying tool for deriving a broad class of parameter-dependent inequalities for differentiable s-convex stochastic processes. Remarkably, by assigning specific values to the underlying parameter, we have ensured our results specialize to well-known numerical integration inequalities, including those of midpoint, trapezium, Simpson, and Bullen types, in the stochastic fractional context. The findings not only enrich the theory of stochastic fractional calculus but also provide a flexible analytical apparatus for uncertainty quantification in fractional dynamical systems.

Underwater wireless sensor networks (UWSNs) are critical for monitoring environmentally sensitive areas, and operation of such networks is however, very much energy constrained. Conventional deterministic methods do not accurately capture the random and time-varying properties of the underwater acoustic environment. This paper presents a novel routing paradigm, temperature-aware SNC for underwater wireless sensor networks (T-SNC UWSN), which is the combination of stochastic network calculus (SNC) with temperature-based analysis and piezoelectric energy harvesting (PEH) and mobile approaches to improve network flexibility and robustness. Variation in temperature affects the efficiency of energy harvesting and has an immediate impact on power availability at the sensor nodes as well, indicating the occurrence of such underwater catastrophes as seismic events or tsunamis. By incorporating the temperature fluctuation into the SNC model, our model is capable of precisely revealing thermal influence on harvested energy and network stability, which enables efficient adaptive routing and enhanced disaster detection. The performance is analysed through simulations on packet delivery ratio (PDR), end-to-end delay, network throughput and path loss. It is demonstrated that SNC with temperature-aware modelling and mobile technologies manages to improve energy sustainability and disaster preparedness as well as the robustness of the network in unforeseen aquatic environments. This probabilistic model is helpful to practical systems of energy-efficient UWSNs, early warning systems, mobility-assisted monitoring, climate-resilience solutions and so forth.

This paper presents a topology-based approach to the general vector-valued stochastic integral for predictable integrands and semimartingale integrators. The integral is defined as a unique mapping that achieves closure under the semimartingale topology. While the topology and the closedness of the integral operator are well known, the method of defining the integral via this mapping is new and offers a significantly more efficient path to understanding the general stochastic integral compared to existing techniques. Instead of defining a basic integral and then extending it through a sequence of case distinctions, our construction performs a single topological closure: we define the vector stochastic integral as the unique continuous extension of the simple-predictable integral under the Émery topology, within the predictable σ-algebra. This single step yields the general predictable, vector-valued integral without invoking semimartingale decompositions, Doob–Meyer, or detours through H2/quasimartingale frameworks and without re-engineering from the componentwise to the vector case.

In this study, we introduce and rigorously develop the concept of [Formula: see text]-superquadratic stochastic processes, a novel generalization and refinement of classical superquadratic stochastic processes. We systematically investigate their core structural properties and employ these to establish advanced forms of Jensen’s inequality and Hermite–Hadamard ([Formula: see text])-type inequalities within the framework of mean-square stochastic calculus. These inequalities are further extended to their fractional analogs via the stochastic Riemann–Liouville ([Formula: see text]) fractional integrals, providing a deeper analytical toolkit for fractional stochastic analysis. The theoretical results are substantiated through comprehensive graphical visualizations and detailed tabular representations, which are constructed from diverse illustrative examples. Additionally, we demonstrate the applicability of the proposed framework in information theory by formulating new classes of stochastic divergence measures. For reproducibility and computational transparency, we provide direct access to the commands used for generating all graphs and tables, along with the recorded execution times for each computation.

Abstract We provide an analytical framework for analyzing the quality of stochastic Verlet-type integrators for simulating the Langevin equation. Focusing only on basic objective measures, we consider the ability of an integrator to correctly simulate two characteristic configurational quantities of transport, a) diffusion on a flat surface and b) drift on a tilted planar surface, as well as c) statistical sampling of a harmonic potential. For any stochastic Verlet-type integrator expressed in its configurational form, we develop closed form expressions to directly assess these three most basic quantities as a function of the applied time step. The applicability of the analysis is exemplified through twelve representative integrators developed over the past five decades, and algorithm performance is conveniently visualized through the three characteristic measures for each integrator. The GJ set of integrators stands out as the only option for correctly simulating diffusion, drift, and Boltzmann distribution in linear systems, and we therefore suggest that this general method is the one best suited for high quality thermodynamic simulations of nonlinear and complex systems, including for relatively high time steps compared to simulations with other integrators.

We present analytical solutions for particle transport and deposition over a horizontally oscillating plate. The dynamics and deposition of particles with negligible Brownian forces are resolved, and the distribution of particle deposition and the effects of Stokes number, Froude number, and the particle's initial height on the deposition length are analyzed. The analysis reveals nonlinear oscillatory behavior of the particle deposition length in response to variations of the flow and particle properties. Then, following the method previously developed by Wang and Dagan [“Brownian particle diffusion in generalized polynomial shear flows,” Phys. Rev. E 110, 024117 (2024)], the dynamics and anomalous diffusion of Brownian particles are studied by solving the Langevin equation using stochastic calculus. The anomalous diffusion predicted by the analytical formulation is then validated by high-fidelity numerical simulations. We demonstrate that particle diffusion in the streamwise direction is significantly altered due to the coupling between the flow velocity gradient and Brownian motion in the transverse direction. The dynamical response of Brownian particles to both the horizontal periodic flow and the vertical body force is examined, revealing the relative significance of the coupling between Brownian motion, external forcing, and the carrier flow in realizing particle diffusion.

Stochastic integration on stochastic sets of interval type and applications to mathematical finance

Abstract In this work we develop and apply a path integral formulation for the microscopic degrees of freedom obeying stochastic differential equations to an active Brownian particle (ABP) trapped in a harmonic potential. The formalism allows to derive exact analytic expressions for the time-dependent moments, like the mean position and the mean square displacement, including full dependence on initial conditions. In addition, the probability distribution of the particle’s position can be evaluated systematically as a series expansion in the propulsion speed. Compared to previous methods relying on eigenfunction expansions of the equivalent Fokker–Planck equation, our method is easier to generalize to more complex situations: it does not rely on eigenfunctions but on a reference state that can be solved analytically, which in our case is the passive Brownian particle in a harmonic potential. We exemplify this versatility by also briefly treating an ABP with an active torque (Brownian circle swimmer, BCS) in a harmonic potential.

ABSTRACT Stochastic evolution underpins several approaches to the dynamics of open quantum systems, such as random modulation of Hamiltonian parameters, the stochastic Schrödinger equation (SSE), and the stochastic Liouville equation (SLE). These approaches replace the explicit system–environment coupling with an effective system‐only dynamics, where dissipative behavior emerges from ensemble averaging. Stochastic Hamiltonians, in particular, have long served as phenomenological tools in physical chemistry to include environmental effects without recourse to an explicit microscopic derivation. In this work, we aim at a self‐contained and accessible presentation of these approaches to further elaborate on their common roots in essential concepts of stochastic calculus and to delineate the conditions under which they are equivalent. We also discuss how different formulations naturally lead to different numerical time‐integration schemes, better suited for either classical simulation platforms, based on finite‐difference approximations, or quantum algorithms, that employ random unitary maps. Our analysis aims at providing a unified perspective and actionable recipes for classical and quantum implementations of stochastic evolution in the simulation of open quantum systems.

The Dupire formula plays a significant role in pricing financial derivatives. This paper is devoted to deriving a generalized version of the Dupire formula for asset exchange options and its mathematically rigorous proofs. Financial derivatives of this type are not special cases of plain vanilla options. Moreover, the application of Lévy-type stochastic integrals in place of Itô models used in Nowak and Gatarek [Application of Itô processes and Schwartz distributions to local volatility for Margrabe options. Stochastics. 2022;94(6):807–832] allows taking into account jumps in price of the considered underlying assets. Therefore, the generalization is useful in practice and essential from a theoretical point of view. Our approach combines methods of mathematical finance, stochastic analysis and the theory of Schwartz distributions to prove the aforementioned generalized formula in the space of distributions. To illustrate the main theoretical result, we present an example of its application.

ABSTRACT Acoustic propagation delays, bandwidth limits, and node mobility are all big problems for underwater wireless sensor networks (UWSNs). We present a unified analytical framework that combines network function virtualization (NFV) and stochastic network calculus (SNC) to meet the need for reliable communication with low latency. Using VNFs made for underwater environments, our method simulates per‐flow delay limits, queue prioritization, and dynamic service chaining. We test the framework by running many simulations in different UWSN scenarios using OMNeT++. The results show that the end‐to‐end delay, packet delivery ratio, and energy efficiency have all improved significantly. This proves that NFV‐SNC integration can be used in real life for underwater applications like marine farming. To make sure the proposed method works, a comparison of the analytical model and simulation data is done. The goal of this study is to improve network architecture so that it can impose delay limits and improve overall performance. The problem is to achieve reliable and quick communication in UWSNs. People think that dynamic optimization using NFV will greatly improve the reliability and performance of UWSN communication systems.

In this study, we introduce and rigorously formalize the notion of (P, m)-superquadratic stochastic processes, representing a novel and far-reaching generalization of classical convex stochastic processes. By exploring their intrinsic structural characteristics, we establish advanced Jensen and Hermite–Hadamard (H.H)-type inequalities within the mean-square stochastic calculus framework. Furthermore, we extend these inequalities to their fractional counterparts via stochastic Riemann–Liouville (RL) fractional integrals, thereby enriching the analytical machinery available for fractional stochastic analysis. The theoretical findings are comprehensively validated through graphical visualizations and detailed tabular illustrations, constructed from diverse numerical examples to highlight the behavior and accuracy of the proposed results. Beyond their theoretical depth, the developed framework is applied to information theory, where we introduce new classes of stochastic divergence measures. The proposed results significantly refine the approximation of stochastic and fractional stochastic differential equations governed by convex stochastic processes, thereby enhancing the precision, stability, and applicability of existing stochastic models. To ensure reproducibility and computational transparency, all graph-generation commands, numerical procedures, and execution times are provided, offering a complete and verifiable reference for future research in stochastic and fractional inequality theory.

We prove the existence of statistically stationary solutions to the Schrödinger map equation on a bounded open interval, with zero Neumann boundary conditions. The model is also known as the Landau-Lifschitz equation. We deal directly with the equation in its real-valued formulation, u t = u 0 + ∫ 0 t u r × ∂ x 2 u r d r , | u t | R 3 = 1 , for t≥0, without using any transform. To approximate the Schrödinger map equation, we employ the stochastic Landau-Lifschitz-Gilbert equation. By a limiting procedure à la Kuksin, we establish the existence of a random initial datum, whose distribution μ is preserved under the dynamics of the deterministic equation. Among other properties, the corresponding statistically stationary solution u is proved to exhibit non-trivial dynamics in space and time. Moreover, it is genuinely random, i.e., the measure μ is not a Dirac delta in a single static solution to the Schrödinger map equation. With an analogous argument, we prove the existence of stationary solutions to the stochastic Schrödinger map equation u t = u 0 + ∫ 0 t u r × ∂ x 2 u r d r + ∫ 0 t u r × ∘ d W r , | u t | R 3 = 1 , for t≥0, where W is an ℝ 3-valued Brownian motion in time and the stochastic integral is interpreted in the Stratonovich sense. We discuss the relationship between the statistically stationary solutions to the Schrödinger map equation, the binormal curvature flow and the cubic non-linear Schrödinger equation. Additionally, we prove the existence of statistically stationary solutions to the binormal curvature flow, given for t≥0 by v t = v 0 + ∫ 0 t ∂ x v r × ∂ x 2 v r d r , | ∂ x v t | R 3 = 1 ,

Abstract By decoupling forward and backward stochastic trajectories, we construct a family of martingales and work theorems for both overdamped and underdamped Langevin dynamics. Our results are made possible by an alternative derivation of work theorems that uses tools from stochastic calculus instead of path-integration. We further strengthen the equality in work theorems by evaluating expectations conditioned on an arbitrary initial state value. These generalizations extend the applicability of work theorems and offer new interpretations of entropy production in stochastic systems. Lastly, we discuss the violation of work theorems in far-from-equilibrium systems.

In 5G cellular networks, end-to-end data transmission delay is a key metric for evaluating network performance. High-frequency signal fading and complex transmission links often lead to increased delays. Pinching-antenna optimizes signal propagation through directional transmission, enhancing signal quality and reducing delay. Therefore, this paper analyzes the end-to-end transmission delay performance of 5G cellular networks assisted by pinching-antenna. Specifically, the data transmission process is modeled as a two-hop link, where data is first transmitted from the base station to the relay station (RS) via a 5G high-frequency transmission link, and then from the RS to the user equipment via a dielectric waveguide-based pinching-antenna link. We derive the statistical characteristics of the service processes for both the 5G high-frequency transmission link and the dielectric waveguide link. Considering traffic arrivals and service capabilities, we then precisely define the network’s end-to-end delay using stochastic network calculus. Through numerical experiments, we initially evaluate the impact of various network parameters on the performance upper bound and provide system performance. The experimental results show that the pinching-antenna-assisted 5G cellular network significantly reduces end-to-end delay compared with the traditional decode and forward relay, further confirming the substantial advantage of pinching-antenna in optimizing delay performance.

Over the last 30 years, there has been spectacular progress in deriving the well-known hydrodynamic limits from stochastic interacting particle systems, as well as characterizing the fluctuations of locally conserved quantities around this limit. Many interesting results on the aforementioned topic have been derived from stochastic integrability, an approach relying on very specific combinatorial and algebraic properties of the underlying dynamics which allow deriving several scaling limits. However, a microscopic change on the dynamics can dramatically impact the macroscopic level, in the sense that scaling limits are no longer tractable by this methodology. Moreover, microscopic perturbations can lead to evolution equations with a variety of behaviours and at the critical parameter of the underlying dynamics, several universal anomalous laws can emerge, both in hydrodynamics and in fluctuations. More generally, understanding critical points where physical systems undergo phase transitions, and establishing that the phenomenology is described by universal mathematical objects that do not depend on the specific properties of the underlying microscopic dynamics, is a cornerstone of modern probability and mathematical physics, both from a pure and an applied point of view.

Offset analgesia (OA), an endogenous pain inhibition after an abrupt decrease in noxious stimulation, provides a paradigm to study dynamic interaction between ascending and descending pain pathways. Previous studies assumed that this interaction follows deterministic dynamics. In contrast, a recent perspective views pain perception as a Bayesian process: a statistically optimal updating of pain predictions based on noisy sensory input. We examined whether OA is driven by a deterministic interaction between ascending and descending pathways, or by a Bayesian process in which the brain updates pain perception by combining expectations with incoming signals. We modified the conventional OA paradigm by adding high-frequency noise after an abrupt decrease in noxious stimulation and measured pain intensity responses in healthy participants. Pain reports were analyzed using 2 computational models: a deterministic dynamic equation model and a recursive Bayesian integration model. Hypothesis testing was conducted using model selection. Offset analgesia was observed after reduction of noxious stimuli, but pain was disinhibited by high-frequency disturbances. The deterministic model predicted unbounded oscillations depending on disturbance sequence, whereas the Bayesian model predicted gradual OA attenuation by filtering out noise. Model selection favored the Bayesian model. The brain dissociates noise from primary signals, achieving stable pain perception even in the presence of noisy inputs. Thus, OA reflects a stochastic integration between prediction and observation, with noise magnitude modulating pain intensity. Clinically, these results suggest that enhancing endogenous pain inhibition for chronic pain may be achieved through interventions targeting noise recognition mechanisms.

Purpose The main objective of this study is to find out the numerical solution of the 2-D stochastic Ito-Volterra integral equation (SIVIE), characterized by kernels with singularities. Since solving these equations analytically is very complicated. Design/methodology/approach The study presents a novel operational matrix method for solving 2-D stochastic integral equations with weakly singular kernels. Operational matrices for product, weakly singular, and stochastic integrals are constructed using orthonormal Bernoulli polynomials. The method employs collocation at Newton–Cotes nodes, transforming the problem into a solvable system of algebraic equations. After determining these coefficients, we get the approximate solution. Error bounds and convergence analyses are also performed to ensure accuracy and stability. Findings The proposed method demonstrates high computational efficiency and precision in solving stochastic integral equations. Numerical experiments on two benchmark problems confirm its capability to handle singularities and randomness simultaneously. Comparative analysis with the bicubic B-spline method shows that the proposed approach achieves superior accuracy and reliability. Practical implications These equations are widely used in viscoelastic material modeling, financial mathematics (e.g. pricing path-dependent options), image processing (e.g. denoising and texture synthesis), and biological systems (e.g. drug diffusion or population dynamics). In engineering, they help model stress propagation and structural health monitoring. Social implications Two-dimensional stochastic Itô-Volterra integral equations with weakly singular kernels model complex systems influenced by memory effects and randomness, such as in climate dynamics, financial markets, and biomedical processes. Their solutions help predict and analyze behaviors under uncertainty, contributing to better risk assessment and decision-making. Originality/value The originality of this study lies in the development of a 2-D novel operational matrix method based on orthonormal Bernoulli polynomials to solve 2-D stochastic integral equations with weakly singular kernels. The approach uniquely constructs operational matrices for product, weakly singular, and stochastic integrals, enabling efficient numerical implementation through collocation at Newton–Cotes nodes. This integrated framework provides a new and unified strategy for addressing the combined challenges of singularity and randomness in stochastic systems.

This study introduces Itô-RMSProp, a novel extension of the RMSProp optimizer inspired by Itô stochastic calculus, which integrates adaptive Gaussian noise into the update rule to enhance exploration and mitigate overfitting during training. We embed this optimizer within Gated Recurrent Unit (GRU) networks for stock price forecasting, leveraging the GRU’s strength in modeling long-range temporal dependencies under nonstationary and noisy conditions. Extensive experiments on real-world financial datasets, including a detailed sensitivity analysis over a wide range of noise scaling parameters (ε), reveal that Itô-RMSProp-GRU consistently achieves superior convergence stability and predictive accuracy compared to classical RMSProp. Notably, the optimizer demonstrates remarkable robustness across all tested configurations, maintaining stable performance even under volatile market dynamics. These findings suggest that the synergy between stochastic differential equation frameworks and gated architectures provides a powerful paradigm for financial time series modeling. The paper also presents theoretical justifications and implementation details to facilitate reproducibility and future extensions.