Search Results (48)

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 ($\epsilon$), 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. Full article

This research develops a novel coupled system of nonlinear hybrid stochastic fractional differential equations that integrates neutral effects, stochastic perturbations, and hybrid switching mechanisms. The system is formulated using the Atangana–Baleanu–Caputo fractional operator with a non-singular Mittag–Leffler kernel, which enables accurate representation of memory effects without singularities. Unlike existing approaches, which are limited to either neutral or hybrid stochastic structures, the proposed framework unifies both features within a fractional setting, capturing the joint influence of randomness, history, and abrupt transitions in real-world processes. We establish the existence and uniqueness of mild solutions via the Picard approximation method under generalized Carathéodory-type conditions, allowing for non-Lipschitz nonlinearities. In addition, mean-square Mittag–Leffler stability is analyzed to characterize the boundedness and decay properties of solutions under stochastic fluctuations. Several illustrative examples are provided to validate the theoretical findings and demonstrate their applicability. Full article

The widely used Heun algorithm for the numerical integration of stochastic differential equations (SDEs) is critically re-examined. We discuss and evaluate several alternative implementations, motivated by the fact that the standard Heun scheme is constructed from a low-order integrator. The convergence, stability, and equilibrium properties of these alternatives are assessed through extensive numerical simulations. Our results confirm that the standard Heun scheme remains a benchmark integration algorithm for SDEs due to its robust performance. As a byproduct of this analysis, we also disprove a previous claim in the literature regarding the strong convergence of the Heun scheme. Full article

This study presents a self-contained derivation and solution of the Black and Scholes partial differential equation (PDE), replacing the standard Wiener process with a smoothed Wiener process, which is a differentiable stochastic process constructed via normal kernel smoothing. By presenting a self-contained, Itô-free derivation, this study bridges the gap between heuristic financial reasoning and rigorous mathematics, bringing forth fresh insights into one of the most influential models in quantitative finance. The smoothed Wiener process does not merely simplify the technical machinery but further reaffirms the robustness of the Black and Scholes framework under alternative mathematical formulations. This approach is particularly valuable for instructors, apprentices, and practitioners who may seek a deeper understanding of derivative pricing without relying on the full machinery of stochastic calculus. The derivation underscores the universality of the Black and Scholes PDE, irrespective of the specific stochastic process adopted, under the condition that the essential properties of stochasticity, volatility, and of no arbitrage may be preserved. Full article

A well-known model of infectious diseases, described by a nonlinear system of delay differential equations, is investigated under the influence of stochastic perturbations. Using the general method of Lyapunov functional construction combined with the linear matrix inequality (LMI) approach, we derive sufficient conditions for the stability of the equilibria of the considered system. Numerical simulations illustrating the system’s behavior under stochastic perturbations are provided to support the thoretical findings. The proposed method for stability analysis is broadly applicable to other systems of nonlinear stochastic differential equations across various fields. Full article

In this paper, we investigate the nonlinear dynamic behavior of a cart–double pendulum system with single time delay feedback control. Based on the center manifold theorem and stochastic averaging method, a reduced-order dynamic model of the system is established, with a focus on analyzing the influence of time delay parameters and stochastic excitation on the system’s Hopf bifurcation characteristics. By introducing stochastic differential equation theory, the system is transformed into the form of an Itô equation, revealing bifurcation phenomena in the parameter space. Numerical simulation results demonstrate that decreasing the time delay and increasing the time delay feedback gain can effectively enhance system stability, whereas increasing the time delay and decreasing the time delay feedback gain significantly improves dynamic performance. Additionally, it is observed that Gaussian white noise intensity modulates the bifurcation threshold. This study provides a novel theoretical framework for the stochastic stability analysis of time delay-controlled multibody systems and offers a theoretical basis for subsequent research. Full article

Here, we consider the stochastic graphene sheets model (SGSM) forced by multiplicative noise in the Itô sense. We show that the exact solution of the SGSM may be obtained by solving some deterministic counterparts of the graphene sheets model and combining the result with a solution of stochastic ordinary differential equations. By applying the extended tanh function method, we obtain the soliton solutions for the deterministic counterparts of the graphene sheets model. Because graphene sheets are important in many fields, such as electronics, photonics, and energy storage, the solutions of the stochastic graphene sheets model are beneficial for understanding several fascinating scientific phenomena. Using the MATLAB program, we exhibit several 3D graphs that illustrate the impact of multiplicative noise on the exact solutions of the SGSM. By incorporating stochastic elements into the equations that govern the evolution of graphene sheets, researchers can gain insights into how these fluctuations impact the behavior of the material over time. Full article

We focus on solving stochastic differential equations driven by jump processes (SDEJs) with measurable drifts that may exhibit quadratic growth. Our approach leverages a space transformation and Itô-Krylov’s formula to effectively eliminate the singular component of the drift, allowing us to obtain a transformed SDEJ that satisfies classical solvability conditions. By applying the inverse transformation proven to be a one-to-one mapping, we retrieve the solution to the original equation. This methodology offers several key advantages. First, it extends the well-known result of Le Gall (1984) from Brownian-driven SDEs to the jump process setting, broadening the range of applicable stochastic models. Second, it provides a robust framework for handling singular drifts, enabling the resolution of equations that would otherwise be intractable. Third, the approach accommodates drifts with quadratic growth, making it particularly relevant for financial modeling, insurance risk assessment, and other applications where such growth behavior is common. Finally, the inclusion of multiple examples illustrates the practical effectiveness of our method, demonstrating its flexibility and applicability to real-world problems. Full article

This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often yield intractable expressions, we derive explicit closed-form formulas for these moments. Our methodology leverages the Wick–Itô calculus (fractional Itô formula) and the properties of Hermite polynomials to express moments efficiently. Additionally, we establish a recurrence relation for moment computation and propose an alternative approach based on generalized binomial expansions. To validate our findings, Monte Carlo simulations are performed, demonstrating a high degree of accuracy between theoretical and empirical results. The proposed framework provides novel insights into stochastic processes with long-memory properties, with potential applications in statistical inference, mathematical finance, and physical modeling of anomalous diffusion. Full article

This paper extends classical Markov switching models. We introduce a generalized semi-Markov switching framework in which the system dynamics are governed by an Itô stochastic differential equation. Of note, the optimal control synthesis problem is formulated for stochastic dynamic systems with semi-Markov parameters. Further, a system of ordinary differential equations is derived to characterize the Bellman functional and the corresponding optimal control. We investigate the case of linear dynamics in detail, and propose a closed-form solution for the optimal control law. A numerical example is presented to illustrate the theoretical results. Full article

We establish a relationship between stochastic differential games (SDGs) and a unified forward–backward coupled stochastic partial differential equation (SPDE) with discontinuous Lévy Jumps. The SDGs have q players and are driven by a general-dimensional vector Lévy process. By establishing a vector-form Ito-Ventzell formula and a 4-tuple vector-field solution to the unified SPDE, we obtain a Pareto optimal Nash equilibrium policy process or a saddle point policy process to the SDG in a non-zero-sum or zero-sum sense. The unified SPDE is in both a general-dimensional vector form and forward–backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position parameters over a domain. Since the unified SPDE is of general nonlinearity and a general high order, we extend our recent study from the existing Brownian motion (BM)-driven backward case to a general Lévy-driven forward–backward coupled case. In doing so, we construct a new topological space to support the proof of the existence and uniqueness of an adapted solution of the unified SPDE, which is in a 4-tuple strong sense. The construction of the topological space is through constructing a set of topological spaces associated with a set of exponents $\{{\gamma }_1,{\gamma }_2,\dots \}$ under a set of general localized conditions, which is significantly different from the construction of the single exponent case. Furthermore, due to the coupling from the forward SPDE and the involvement of the discontinuous Lévy jumps, our study is also significantly different from the BM-driven backward case. The coupling between forward and backward SPDEs essentially corresponds to the interaction between noise encoding and noise decoding in the current hot diffusion transformer model for generative AI. Full article

This article examines the stability properties of linear stochastic difference equations with delays. For this purpose, a novel approach is used that combines the theory of inverse-positive matrices and the asymptotic methods developed by N.V. Azbelev and his students for deterministic functional differential equations. Several efficient conditions for p-stability and exponential p-stability $(2\le p<\infty )$ of systems of linear Itô-type difference equations with delays and random coefficients are found. All results are conveniently formulated in terms of the coefficients of the equations. The suggested examples illustrate the feasibility of the approach. Full article

In this article, we construct a new order 2.0 simplified weak Itô–Taylor symmetrical scheme for stochastic delay differential equations. By the new local weak convergence lemma and the connection inequality, we theoretically prove the global weak convergence theorem in two parts on the basis of Malliavin stochastic analysis. Meanwhile, numerical examples are presented to illustrate the error and convergence results. Furthermore, the obtained results display the influence of the delay coefficient on global errors. Full article

The classical Lotka–Volterra predator–prey model is globally stable and uniformly persistent. However, in real-life biosystems, the extinction of species due to stochastic effects is possible and may occur if the magnitudes of the stochastic effects are large enough. In this paper, we consider the classical Lotka–Volterra predator–prey model under stochastic perturbations. For this model, using an analytical technique based on the direct Lyapunov method and a development of the ideas of R.Z. Khasminskii, we find the precise sufficient conditions for the stochastic extinction of one and both species and, thus, the precise necessary conditions for the stochastic system’s persistence. The stochastic extinction occurs via a process known as the stabilization by noise of the Khasminskii type. Therefore, in order to establish the sufficient conditions for extinction, we found the conditions for this stabilization. The analytical results are illustrated by numerical simulations. Full article

We present an integration condition ensuring that a stochastic differential equation $d{X}_t=\mu (t,X_t)dt+\sigma (t,X_t)d{B}_t$, where $\mu$ and $\sigma$ are sufficiently regular, has a solution of the form $X_t=Z(t,B_t)$. By generalizing the integration condition we obtain a class of stochastic differential equations that again have a functional solution, now of the form $X_t=Z(t,Y_t)$, with $Y_t$ an Ito process. These integration conditions, which seem to be new, provide an a priori test for the existence of functional solutions. Then path-independence holds for the trajectories of the process. By Green’s Theorem, it holds also when integrating along any piece-wise differentiable path in the plane. To determine Z at any point $(t,x)$, we may start at the initial condition and follow a path that is first horizontal and then vertical. Then the value of Z can be determined by successively solving two ordinary differential equations. Due to a Lipschitz condition, this value is unique. The differential equations relate to an earlier path-dependent approach by H. Doss, which enables the expression of a stochastic integral in terms of a differential process. Full article