Approximate analytical solutions for a class of nonlinear stochastic differential equations

An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators

We propose a new method viz., using stochastic partial differential equations to study the pathwise uniqueness of stochastic (ordinary) differential equations. We prove the existence and pathwise uniqueness of a class of stochastic differential equations with coefficients in suitable Hermite-Sobolev class using our approach.

The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments

Digital currency markets exhibit extreme volatility, heavy tails, and bursty order-flow that complicate surveillance and heighten manipulation risk. This systematic review synthesizes the state of the art in stochastic differential equation (SDE)–based approaches for detection and monitoring of abusive practices in cryptocurrencies. Following PRISMA 2020 and PICOS, an Elsevier-indexed search (2015–2025) yielded 273 records; after screening and eligibility assessment, 20 primary studies (2018–2025) were included. The literature clusters into six methodological families: (i) stochastic volatility and jump–diffusion models for heavy-tailed returns and volatility smiles; (ii) Hawkes and related point-process models for clustered order arrivals, spoofing, pump-and-dump, and wash trading; (iii) Markov and regime-switching diffusions that delineate latent “fair” versus manipulated regimes; (iv) hybrid SDE–machine learning frameworks for high-frequency prediction; (v) change-point and sequential detection methods grounded in likelihood ratios and optimal stopping; and (vi) meta-studies consolidating performance trends. Across studies, Hawkes-type intensities consistently outperform Poisson and threshold baselines in event detection; regime-switching models align with known market breaks; and hybrid neural–SDE systems achieve the strongest forecasting but at reduced interpretability. We formalize a taxonomy linking model structure to surveillance objective, and we delineate a practical trade-off between transparency (classical SDEs) and predictive accuracy (hybrid models). The review highlights open needs in explainable hybrid designs, reproducible datasets, and real-time deploy ability for exchanges and regulators. By connecting applied probability, financial engineering, and control, the paper clarifies how SDE frameworks can underpin robust, auditable market-integrity tools for digital assets.

In this paper, we are concerned with strong convergence rate of Euler scheme for time-inhomogeneous one-dimensional stochastic differential equations involving the local time (SDELT) of the unknown process at point zero. We use a space transform in order to remove the local time from this class of stochastic differential equations. We provide the approximation of Euler for the stochastic differential equation without local time. After that the approximation can be transformed back, giving an approximation of Euler to the solution of the original SDELT, and we provide the rate of strong convergence.

Motivation, Aims and Examples.- Stochastic Integral in Hilbert Spaces.- Stochastic Differential Equations in Finite Dimensions.- A Class of Stochastic Differential Equations.

Inspired by the paper Greenhalgh et al. (Appl Math Comput 276:218–238, 2016), we investigate a class of two-dimensional stochastic differential equations related to susceptible–infected–susceptible epidemic models with demographic stochasticity. While preserving the key features of the model considered in Greenhalgh et al. (2016), where an ad hoc approach has been utilized to prove existence, uniqueness and non-explosivity of the solution, we consider an encompassing family of models described by a stochastic differential equation with random and Hölder continuous coefficients. We prove the existence of a unique strong solution by means of a Cauchy–Euler–Peano approximation scheme which is shown to converge in the proper topologies to the unique solution.

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

Parameter estimation for a class of stochastic differential equations driven by small stable noises from discrete observations

This paper proposes a novel weak Simpson method for numerical solution for a class of stochastic differential equations. We show that such a method has weak convergence of order one in general and weak convergence of order three under certain additional assumptions, which improves the weak convergence order of two for the weak trapezoid method developed in (1). The main results are illustrated with numerical examples.

The problem of computing the maximum likelihood estimate of the parameters of a specific class of stochastic differential equation (SDE) models with linear drift whose sample paths are observed at discrete time points is considered. This estimate is obtained as in Cleur and Manfredi (1999) by discretizing the explicit expressions for the estimates which maximize the likelihood function in continuous time, by discretizing the likelihood function through a quadrature approximation before maximizing it, and by maximizing the likelihood function of the Euler scheme approximation to the underlying continuous process. Simulation results indicate that, for the constellation of parameter values considered, all three approaches lead to very similar results.

Asymptotical boundedness and stability for stochastic differential equations with delay driven by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:mi>G</mml:mi></mml:math>-Brownian motion

This chapter is devoted to the study of Langevin equations, first order in time differential equations, which depend on a random noise, and which belong to a class of stochastic differential equations that describe diffusion processes, or random motion. From a Langevin equation, a Fokker–Planck (FP) equation for the probability distribution of the solutions, at given time, of the Langevin equation can be derived. It is shown that observables averaged over the noise can also be calculated from path integrals, whose integrands define automatically positive measures. The path integrals involve dynamic actions that have automatically a Becchi–Rouet–Stora–Tyutin (BRST) symmetry and, when the driving force derives from a potential, exhibit the simplest form of supersymmetry. In some cases, like Brownian motion on Riemannian manifolds, difficulties appear in the precise definition of stochastic equations, quite similar to the quantization problem encountered in quantum mechanics (QM). Time discretization provides one possible solution to the problem.

Variable-stepsize Runge–Kutta methods for stochastic Schrödinger equations

In some problems, partial differential equations are reduced to ordinary differential equations. In special cases, when incorporating randomness, equations can be reduced to systems of stochastic differential Equations (SDEs). Stochastic averaging for a class of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise is considered. Stability criteria for systems of stochastic differential equations with fractional Brownian motion and non-Gaussian Lévy noise do not currently exist. Usually, studies on determining the sensitivity of solutions to the accuracy of setting the initial conditions are being conducted to explain the phenomenon of deterministic chaos. These studies show both convergence in mean square and convergence in probability to averaged systems of stochastic differential equations driven by fractional Brownian motion and Lévy process. The solutions to systems can be approximated by solutions to averaged stochastic differential equations by using the stochastic averaging.