Limit theorems for stochastic integrals with long memory processes

We address the existence, uniqueness, and averaging principle for Caputo–Hadamard fractional dynamic systems with Dirichlet boundary conditions driven by Rosenblatt process and pure Lévy jumps. First, Lemma 4 establishes the equivalent integral equation representation of our system. Using this foundation, existence and uniqueness are proved by Banach's contraction principle under stochastic calculus, Lipschitz and finite energy conditions. Subsequently, under appropriate averaging assumptions, the system is averaged out with time scale ϵ. Mean-square convergence between original solution and its counterpart is verified by employing tools such as Wiener–Itô double integral, Cauchy–Schwarz, Doob's martingale, and Gronwall–Bellman inequalities. Eventually, computational example with numerical simulations is provided to support the theoretical results.

A risk-averse multi-timescale stochastic integration framework for a community multi-energy system under correlated uncertainties

Underwater Wireless Sensor Networks (UWSNs) enable real-time marine ecosystem monitoring but are constrained by limited energy availability, harsh underwater communication conditions, and high deployment costs. Although energy harvesting from water currents, vibrations, and ambient sources offers a sustainable solution, its efficiency is highly affected by dynamic underwater environments, and existing routing protocols such as Depth-Based Routing (DBR) do not explicitly support energy-harvesting awareness or dependable underwater Internet services. This paper proposes an enhanced DBR routing model that integrates energy-harvesting mechanisms with stochastic worst-case performance guarantees using a Stochastic Network Calculus (SNC)–based analytical framework. The proposed model evaluates end-to-end delay, energy utilization, routing stability, and network resilience under uncertain harvesting and communication conditions. Simulation results demonstrate that the enhanced DBR improves energy harvesting efficiency by approximately 30–35%, extends network lifetime by up to 40%, increases packet delivery ratio by 18–22%, and reduces end-to-end delay variability by around 25% compared to conventional DBR. These improvements enable sustained node operation, enhanced reliability, and more secure underwater Internet connectivity, confirming the suitability of the proposed approach for long-term and dependable UWSN deployment.

Abstract We provide criteria for Itô integration to behave continuously with respect to Skorokhod’s $$J_1$$ J 1 and $$M_1$$ M 1 topologies, when the integrands and integrators converge weakly or in probability. The results are novel in the $$M_1$$ M 1 setting and unify existing theories in the $$J_1$$ J 1 case. Beyond sufficient criteria, we present an example of uniformly convergent martingale integrators for which the continuity breaks down. Moreover, we show that, for families of local martingales, $$M_1$$ M 1 tightness in fact implies $$J_1$$ J 1 tightness under a mild localised uniform integrability condition. Finally, we apply our results to study scaling limits of models of anomalous diffusion driven by continuous-time random walks. This yields new results on weak $$M_1$$ M 1 and $$J_1$$ J 1 convergence to stochastic integrals against subordinated stable processes. In the case of superdiffusive scaling, an interesting counterexample is obtained.

. In this article, we will develop linear and nonlinear filtering methods for a large class of nonlinear wave equations that arise in applications such as quantum dynamics and laser generation and propagation in a unified framework. We consider both stochastic calculus and white noise filtering methods and derive measure-valued evolution equations for the nonlinear filter and prove existence and uniqueness theorems for the solutions. We will also study first-order approximations to these measure-valued evolutions by linearizing the wave equations and characterize the filter dynamics in terms of infinite-dimensional operator Riccati equations and establish solvability theorems.

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.

This paper presents a stochastic $\mathcal{SIS}$ (Susceptible-Infected-Susceptible) epidemic model with generalized incidence function and two separate Brownian noise sources. Using analytical methods from stochastic calculus, we derive explicit threshold conditions governing disease extinction and endemic persistence. These theoretical results are validated through systematic numerical simulations, including phase-space trajectory analysis under varying epidemiological scenarios. A central contribution is the development of a novel computational-analytic framework to rigorously verify the model’s long-term statistical properties: (i) stationarity through distributional convergence tests, and (ii) ergodicity via empirical moment estimation. The combined analytical-numerical approach provides actionable insights for epidemic modeling while advancing methodology for stochastic dynamical systems.

Abstract The statistical properties of non-linear observables of the fractional Gaussian field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ϕ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> of negative Hurst exponent H &lt; 0 in dimension d are revisited with a focus on spatial-averaging observables and on the properties of the finite parts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> of the ill-defined composite operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . For the special case n = 2 of quadratic observables, explicit results include the cumulants of arbitrary order, the Lévy–Khintchine formula for the characteristic function and the anomalous large deviations properties. The case of observables of arbitrary order n &gt; 2 is analyzed via the Wiener–Ito chaos-expansion for functionals of the white noise: the multiple stochastic Ito integrals are useful to identify the finite parts <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> of the ill-defined composite operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mover> <mml:mi>x</mml:mi> <mml:mo stretchy="false">→</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> and to compute their correlations involving the Hurst exponents <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> .

Abstract In this paper we provide sufficient conditions for sequences of stochastic processes of the form ∫ [0, t ] f n ( u ) θ n ( u )d u , to weakly converge, in the space of continuous functions over a closed interval, to integrals with respect to the Brownian motion, ∫ [0, t ] f ( u ) W (d u ), where { f n } n ${\left\{{f}_{n}\right\}}_{n}$ is a sequence of functions converging to f which verify some integrability conditions and { θ n } n ${\left\{{\theta }_{n}\right\}}_{n}$ is a sequence of stochastic processes whose integrals ∫ [0, t ] θ n ( u )d u converge in law to the Brownian motion (in the sense of the finite dimensional distribution convergence), in the multiparameter 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.

We prove an inequality for the spectral norm of matrix valued stochastic integrals. This inequality can be seen either as a non-commutative version of the Burkholder–Davis–Gundy inequality or as an extension of the non-commutative Khintchine inequality of Lust-Piquard to stochastic integrals. The proof relies on a version of Freedman’s inequality for matrix valued martingales.

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

As a typical scenario for the 6th Generation mobile communication systems (6G), Hyper Reliable Low Latency Communication (HRLLC) is expected to ensure extremely low delay and high reliability, while supporting wireless transmission of large-scale massive data. However, existing communication networks face the dual challenges of inadequate performance metrics and limited network resources. Therefore, this paper proposes the Flexible Bit and Semantic on-demand Transmission (FBST) framework, including three key technologies: adaptive transmission mode decision, flexible transmission time interval scheduling, adjustable semantic compression ratio. The FBST framework could satisfy the strict QoS requirements of users and provide on-demand services for users. Based on the Stochastic Network Calculus (SNC) modeling method, we conduct precise delay analysis and provided a general expression for the delay violation probability of the α - κ - μ channel, which could be extended to various complex channels. In addition, the Knowledge-base Parameterized Deep Q-Network (KP-DQN) algorithm is proposed to solve the resource allocation issue, which is a mixed action space problem with complex calculations caused by SNC. Finally, the simulation results show that FBST framework could satisfy extremely strict delay and reliability requirements of users, and the KP-DQN algorithm improving operational efficiency by over 76.8%.

Extended Reality (XR) is defined as an example use case for 6 G networks by the International Telecommunication Union (ITU), where the quality of interactive experience is a crucial factor influencing the XR service performance. In this paper, we focuse on the timeliness and reliability of XR interactive systems. We investigate the peak Age of Information (PAoI) violation probability and backlog violation probability within these systems using Stochastic Network Calculus (SNC). Specifically, we establish a Discrete-Time Markov Modulated Process (MMP) for the frame generation process of XR services and conduct an SNC analysis to characterize the traffic and channel models within XR interactive systems. Additionally, we derive upper bounds for the PAoI violation probability and transmission backlog violation probability in scenarios with finite buffers. Furthermore, we formulate a low-complexity algorithm for solving the involved parameters. Subsequently, we analyze the numerical results to verify the accuracy of our derivations and further discuss why the PAoI violation probability is more suitable than the delay violation probability for characterizing XR interactive performance. Finally, we analyze the impact of system parameters on the PAoI bounds. The proposed violation probability upper bound metrics can be utilized to guide deployment decisions for the XR services.