Search

In this paper, we investigate properties of standard and multilevel Monte Carlo methods for weak approximation of solutions of stochastic differential equations (SDEs) driven by infinite-dimensional Wiener process and Poisson random measure with Lipschitz payoff function. The error of the truncated dimension randomized numerical scheme, which depends on two parameters i.e., grid density n ∈ N and truncation dimension parameter M ∈ N , is of the order n − 1 / 2 + δ (M) such that δ (⋅) is positive and decreasing to 0. We derive a complexity model and provide proof for the complexity upper bound of the multilevel Monte Carlo method which depends on two increasing sequences of parameters for both n and M. The complexity is measured in terms of upper bound for mean-squared error and is compared with the complexity of the standard Monte Carlo algorithm. The results from numerical experiments as well as Python and CUDA C implementation details are also reported. [ABSTRACT FROM AUTHOR]

The accurate knowledge of cosmic ion fluxes is essential for fundamental physics, deep space missions, and exploration activities in the solar system. In the HelMod-4 model the Parker transport equation is solved using a Monte Carlo approach to evaluate the solar modulation effect on local interstellar spectra of Galactic Cosmic Rays (GCRs). This work presents the latest updates to the HelMod-4 model parameters, focusing on the descending phase of solar cycle 24. The updates are motivated by the latest high-precision measurements from the AMS-02 detector which revealed for the first time with high accuracy the features of GCR fluxes' evolution during a period of positive interplanetary magnetic field polarity. Furthermore, we present HelMod-4/CUDA , a GPU-accelerated approach for solving the Parker equation in the heliosphere using a stochastic differential equation method. The code is an evolution of the HelMod-4 code, porting the algorithm to GPU architecture using the CUDA programming language. This approach achieves significant speedup compared to a CPU implementation. The HelMod-4/CUDA code has been validated by comparing its results with the most precise and updated experimental GCR spectra observed during high and low solar activity periods, both in the inner and outer heliosphere, at the Earth location, and outside the ecliptic plane. The comparison shows that HelMod-4 and HelMod-4/CUDA can be equivalently used to provide solar-modulated spectra with a similar degree of accuracy in reproducing observed data. [ABSTRACT FROM AUTHOR]

Forward–backward doubly stochastic differential equations (FBDSDEs) serve as a probabilistic interpretation of stochastic partial differential equations (SPDEs) with diverse applications. Coupled FBDSDEs encounter numerous challenges in numerical approximation compared to forward–backward stochastic differential equations (FBSDEs) and decoupled FBDSDEs, including ensuring the measurability of the numerical solutions, accounting for the mutual influences between forward and backward processes, and considering the relationship with respect to SPDEs rather than PDEs. This paper introduces, for the first time, a numerical method for solving multi-dimensional coupled FBDSDEs. By integrating an optimal control-based approach with deep neural networks, it effectively addresses the coupling-related challenges between forward and backward equations. Computational examples of coupled FBDSDEs with explicit solutions demonstrate that the proposed deep learning-based numerical algorithm achieves commendable performance in terms of both accuracy and efficiency. [ABSTRACT FROM AUTHOR]

The present work focuses on the convergence order of one point large deviations rate functions for the backward Euler method of stochastic delay differential equations (SDDEs) with small noise. The drift and diffusion coefficients of SDDEs are allowed to grow super-linearly with respect to both the state variables and the delay variables. It is shown that the backward Euler method satisfies the one point large deviations principle (LDP) with good rate function. Further, the local uniform convergence orders of the one point large deviations rate function are derived by means of the equivalent characterizations for the original and numerical rate functions. These theoretical results are finally supported by a series of numerical experiments. [ABSTRACT FROM AUTHOR]

While scanning or shooting a document, factors like ink density and paper transparency may cause the content from the reverse side to become visible through the paper, resulting in a digital image with a 'seen-through' phenomenon, which will affect practical applications. In addition, document images can be affected by random factors during the imaging process, such as differences in the performance of camera equipment and variations in the physical properties of the document itself. These random factors increase the noise of the document image and may cause the seen-through phenomena to become more complex and diverse. To tackle this issue, we propose the Fuzzy Diffusion Model (FDM), which combines fuzzy logic with diffusion models. It effectively models complex seen-through effects and handles uncertainties in document images. Specifically, we gradually degrade the original image with mean-reverting stochastic differential equation(SDE) to transform it into seen-through mean state with fixed Gaussian noise version. Following this, fuzzy operations are introduced into the noise network. Which helps the model better learn noise and data distributions by reasoning about the affiliation relationship of each pixel point through fuzzy logic. Eventually, in the reverse process, the low-quality image is gradually restored by simulating the corresponding reverse-time SDE. Extensive quantitative and qualitative experiments conducted on various datasets demonstrate that the proposed method significantly removes the seen-through effects and achieves good results under several metrics. The proposed FDM effectively solves the seen-through effects of document images and obtains better visual quality. [ABSTRACT FROM AUTHOR]

This study mainly concerns the controllability of semilinear noninstantaneous impulsive neutral stochastic differential equations via the Atangana-Baleanu (AB) Caputo fractional derivative (FD). The essential findings are created using methods and concepts from semigroup theory, stochastic theory, fractional calculus, K -set contraction, and measure of noncompactness. Finally, an example is provided to demonstrate the applications of the key findings. [ABSTRACT FROM AUTHOR]

In 2014, Neuenkirch and Szpruch established the drift-implicit Euler-Maruyama method for a class of SDEs which take values in a given domain. However, expensive computational cost is required for implementation of an implicit numerical method. A competitive positivity preserving explicit numerical method for SDEs which take values in the positive domain is the logarithmic truncated Euler-Maruyama method. However, assumptions for the logarithmic truncated Euler-Maruyama method used in previous work are restrictive which exclude some important SDE models with specific parameters. The main aim of this paper is to use weaker assumptions to establish strong convergence theory for the logarithmic truncated Euler-Maruyama method. [ABSTRACT FROM AUTHOR]

We propose a new class of weak approximation schemes for stochastic differential equations with coefficients of suplinearly growth. Both the modified weak Euler schemes and the drift-implicit weak Euler scheme are studied. Under certain non-globally Lipschitz conditions, the proposed schemes are proved to have first-order convergence in the weak sense. Numerical experiments are included to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

We investigate existence, uniqueness and approximation of solutions to stochastic delay differential equations (SDDEs) under Carathéodory-type drift coefficients. Moreover, we also assume that both drift f = f (t , x , z) and diffusion g = g (t , x , z) coefficient are Lipschitz continuous with respect to the space variable x , but only Hölder continuous with respect to the delay variable z. We provide a construction of randomized Euler scheme for approximation of solutions of Carathéodory SDDEs, and investigate its upper error bound. Finally, we report results of numerical experiments that confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

For robot manipulators composed of Lagrange subsystems driven by direct current (DC) motors under stochastic environments, multi-stage trajectory tracking is investigated in this paper. The main challenge is how to achieve the end-effector drive of manipulators from a given initial state to a final state. First, the inverse kinematics method and the partition of the task space are adopted to tackle multi-stage trajectory planning. Second, the adaptive backstepping technique is used to design tracking controller for stochastic Lagrangian subsystems. Then, based on the state-dependent switching signal, a multi-stage switched controller is designed for trajectory tracking of robot manipulators. All signals in the close-loop error switched system are bounded in probability, and the tracking error in mean square can be made arbitrarily small enough by parameters-tuning The effectiveness of the proposed control method is illustrated by simulation results. • The multi-stage trajectory planning is proposed by the partition of the task space. • The adaptive backstepping controller is designed to track the pre-planned trajectory. • The multi-stage switched controller is proposed for performing a grab task. [ABSTRACT FROM AUTHOR]

We propose a new method, called a deep-genetic algorithm (deep-GA), to accelerate the performance of the so-called deep-BSDE method, which is a deep learning algorithm to solve high dimensional partial differential equations through their corresponding backward stochastic differential equations (BSDEs). Recognizing the sensitivity of the solver to the initial guess selection, we embed a genetic algorithm (GA) into the solver to optimize the selection. We aim to achieve faster convergence for the nonlinear PDEs on a broader interval than deep-BSDE. Our proposed method is applied to two nonlinear parabolic PDEs, i.e., the Black-Scholes (BS) equation with default risk and the Hamilton-Jacobi-Bellman (HJB) equation. We compare the results of our method with those of the deep-BSDE and show that our method provides comparable accuracy with significantly improved computational efficiency. • We introduce a deep-GA method. • The method provides a better accuracy and efficiency than the deep-BSDE solver. • Efficiency and accuracy is obtained by embedding a genetic algorithm (GA). • We solve the Black-Scholes with default risk and Hamilton-Jacobi-Bellman equations. [ABSTRACT FROM AUTHOR]

• A data-driven model identification method is devised to extract the non-Gaussian governing laws of the multistable VEH. • The Lévy, drift and diffusion terms can be approximately expressed by the sample trajectories of system. • The data-driven identified results agree well with the original system. • The dynamics of VEH can be further explored based on the data-driven stochastic differential equation. In engineering, due to the complex structural characteristics of system and the non-Gaussian properties of random excitation, it is difficult to establish an accurate stochastic dynamic model for the strongly nonlinear multistable vibration energy harvester (VEH), especially for these driven by non-Gaussian Lévy noise. From the view of machine learning, a data-driven model identification method is devised to extract the non-Gaussian governing laws of the multistable VEH with the aid of the observed sample trajectory data. Based on the Nonlocal Kramers-Moyal formulas, the Lévy, drift and diffusion terms can be approximately expressed by the sample trajectories of the system. By implementing the least square method and the stepwise sparse regressor algorithm, the optimal drift and diffusion coefficients can be identified, and then the non-Gaussian stochastic differential equation of VEH is extracted. Two examples are utilized to verify the feasibility and effectiveness of the data-driven modelling method in VEH, which indicates that the identified results agree well with the original system. Finally, the stochastic dynamic behaviors induced by non-Gaussian Lévy noise are explored based on the data-driven penta-stable VEH. The proposed method can provide the theoretical guidance for the modelling and dynamics research of VEH in engineering. [ABSTRACT FROM AUTHOR]

In this paper we focus on a general type of mean-field stochastic control problem with partial observation, in which the coefficients depend in a non-linear way not only on the state process X t and its control u t but also on the conditional law E [ X t | F t Y ] of the state process conditioned with respect to the past of observation process Y. We first deduce the well-posedness of the controlled system by showing weak existence and uniqueness in law. Neither supposing convexity of the control state space nor differentiability of the coefficients with respect to the control variable, we study Peng's stochastic maximum principle for our control problem. The novelty and the difficulty of our work stem from the fact that, given an admissible control u , the solution of the associated control problem is only a weak one. This has as consequence that also the probability measure in the solution P u = L T u Q depends on u and has a density L T u with respect to a reference measure Q. So characterizing an optimal control leads to the differentiation of non-linear functions f (P u ∘ { E P u [ X t | F t Y ] } − 1) with respect to (L T u , X t). This has as consequence for the study of Peng's maximum principle that we get a new type of first and second order variational equations and adjoint backward stochastic differential equations, all with new mean-field terms and with coefficients which are not Lipschitz. For their estimates and for those for the Taylor expansion new techniques have had to be introduced and rather technical results have had to be established. The necessary optimality condition we get extends Peng's one with new, non-trivial terms. [ABSTRACT FROM AUTHOR]

• Three new stochastic integration schemes for solving stochastic differential equations are proposed. • The proposed schemes are derived by applying a weak correction to the existing Milstein schemes. • The weak correction is based on the concept of change of measure and Girsanov theorem. • The proposed approach performs equivalently as higher-order Ito-Taylor schemes. • Various popular stochastic oscillators and a practical problem involving ring-type gyroscope are considered. This work proposes Girsanov corrected explicit, semi-implicit, and implicit Milstein approximations for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the change of measures , aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensures that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation that is accounted in the Milstein trajectories as an additive term. Numerical demonstrations using parametrically and non-parametrically excited stochastic oscillators, a practical problem involving ring type gyroscope, and a 7-degree-of-freedom nonlinear structural system subjected to both stationary and non-stationary excitation demonstrate the improvement in the solution accuracy for the proposed schemes with much coarser time steps when compared with the classical Milstein approximation with finer time steps. [ABSTRACT FROM AUTHOR]

In this paper, we study small-noise asymptotic behaviors for a class of distribution dependent stochastic differential equations driven by fractional Brownian motions with Hurst parameter H ∈ (1 / 2 , 1) and magnitude ϵ H. By building up a variational framework and two weak convergence criteria in the factional Brownian motion setting, we establish the large and moderate deviation principles for these types of equations. Besides, we also obtain the central limit theorem, in which the limit process solves a linear equation involving the Lions derivative of the drift coefficient. [ABSTRACT FROM AUTHOR]

This paper concerns the convergence of empirical measures in high dimensions. We propose a new class of probability metrics and show that under such metrics, the convergence is free of the curse of dimensionality (CoD). Such a feature is critical for high-dimensional analysis and stands in contrast to classical metrics (e.g. , the Wasserstein metric). The proposed metrics fall into the category of integral probability metrics, for which we specify criteria of test function spaces to guarantee the property of being free of CoD. Examples of the selected test function spaces include the reproducing kernel Hilbert spaces, Barron space, and flow-induced function spaces. Three applications of the proposed metrics are presented: 1. The convergence of empirical measure in the case of random variables; 2. The convergence of n -particle system to the solution to McKean–Vlasov stochastic differential equation; 3. The construction of an ɛ -Nash equilibrium for a homogeneous n -player game by its mean-field limit. As a byproduct, we prove that, given a distribution close to the target distribution measured by our metric and a certain representation of the target distribution, we can generate a distribution close to the target one in terms of the Wasserstein metric and relative entropy. Overall, we show that the proposed class of metrics is a powerful tool to analyze the convergence of empirical measures in high dimensions without CoD. [ABSTRACT FROM AUTHOR]

We consider the symmetric tridiagonal matrix-valued process associated with Gaussian beta ensemble (G β E) by putting independent Brownian motions and Bessel processes on the diagonal entries and upper (lower)-diagonal ones, respectively. Then, we derive the stochastic differential equations that the eigenvalue processes satisfy, and we show that eigenvalues of their (indexed) principal minor sub-matrices appear in the stochastic differential equations. By the Cauchy's interlacing argument for eigenvalues, we can characterize the sufficient condition that the eigenvalue processes never collide with each other almost surely, by the dimensions of the Bessel processes. [ABSTRACT FROM AUTHOR]

Incorporating uncertainty quantification into the modeling of deep learning-based model has become a research focus in the deep learning community. Within this group of methods, stochastic differential equation (SDE)-based models have demonstrated advantages in their ability to model uncertainty quantification. However, the use of Euler's method in these models introduces imprecise numerical solutions, which limits the accuracy of SDE systems and weakens the performance of the network. In this study, we build a more precise Milstein-driven SDE network (MDSDE-Net) to improve the network performance. In addition, we analyze the convergence of the Milstein scheme and theoretically guarantee the feasibility of MDSDE-Net. Experimental and theoretical results show that the MDSDE-Net outperforms existing models. • We propose to construct a more precise uncertainty quantification network using a Milstein method of SDE mathematically. • Based on Milstein method, our MDSDE-Net improves interpretability and performance in DNN uncertainty quantification. • Our MDSDE-Net enhances performance, outperforming existing methods on different tasks with a simple term addition. [ABSTRACT FROM AUTHOR]

We prove the existence of maximal (and minimal) solution for one-dimensional generalized doubly reflected backward stochastic differential equation (RBSDE for short) with irregular barriers and stochastic quadratic growth, for which the solution Y has to remain between two rcll barriers L and U on [ 0 , T [ , and its left limit Y − has to stay respectively above and below two predictable barriers l and u on ] 0 , T ]. This is done without assuming any P − integrability conditions and under weaker assumptions on the input data. In particular, we construct a maximal solution for such a RBSDE when the terminal condition ξ is only F T − measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z. Our result is based on a (generalized) penalization method. This method allows us find an equivalent form to our original RBSDE where its solution has to remain between two new rcll reflecting barriers Y ¯ and Y ̲ which are, roughly speaking, the limit of the penalizing equations driven by the dominating conditions assumed on the coefficients. A standard and equivalent form to our initial RBSDE as well as a characterization of the solution Y as a generalized Snell envelope of some given predictable process l are also given. [ABSTRACT FROM AUTHOR]

We propose two Euler–Maruyama (EM) type numerical schemes in order to approximate the invariant measure of a stochastic differential equation (SDE) driven by an α -stable Lévy process (1 < α < 2): an approximation scheme with the α -stable distributed noise and a further scheme with Pareto-distributed noise. Using a discrete version of Duhamel's principle and Bismut's formula in Malliavin calculus, we prove that the error bounds in Wasserstein-1 distance are in the order of η 1 − ϵ and η 2 α − 1 , respectively, where ϵ ∈ (0 , 1) is arbitrary and η is the step size of the approximation schemes. For the Pareto-driven scheme, an explicit calculation for Ornstein–Uhlenbeck α -stable process shows that the rate η 2 α − 1 cannot be improved. [ABSTRACT FROM AUTHOR]

This paper investigates multidimensional extended type-I BSVIEs and infinite families of BSDEs in the case of quadratic generators. We establish existence and uniqueness results in the case of fully quadratic as well as Lipschitz-quadratic generators. As a preliminary step, we establish the well-posedness of a class of infinite families of BSDEs, as introduced in Hernández and Possamaï (2021), which are of interest in their own right. Our approach relies on the strategy developed by Tevzadze (2008) for quadratic BSDEs and the treatment of Lipschitz extended type-I BSVIEs in Hernández and Possamaï (2021). We also present and discuss a type of flow property satisfied by this family of BSVIEs. We motivate the analysis of both of these objects by a series of practical applications. [ABSTRACT FROM AUTHOR]

In this article, we study the Finite-Time Stability (FTS) of Linear Stochastic Fractional Differential Equations of Itô-Doob Type with Delay (LSFDEIDTwD) for a derivative order ρ ∈ (0 , 1). In fact, the (FTS) here consists in studying the stability of the (LSFDEIDTwD) in a finite-time domain [ 0 , T ]. To our knowledge, this work represents the first successful attempt to study the (FTS) for the (LSFDEIDTwD), where the fractional order ρ ∈ (0 , 1). According to the generalized Gronwall Inequality (GWI) and stochastic calculus theory, the (FTS) of (LSFDEIDTwD) is investigated. We illustrate the main results by two numerical examples. [ABSTRACT FROM AUTHOR]

We consider a one-dimensional stochastic differential equation with jumps driven by a Brownian motion and an independent Lévy process with finite Lévy measure, whose drift and diffusion coefficients depend on an unknown parameter. Under smoothness and non-degeneracy assumptions on the drift and diffusion coefficients and integrability assumption of jump size distribution, we prove the Local Asymptotic Mixed Normality property when the solution process is observed discretely at high frequency on a fixed time interval. The proof is essentially based on Malliavin calculus techniques and an analysis of the jump structure of the Lévy process. • LAMN property for jump diffusion processes. • Parameter estimation for high frequency data on a fixed time interval. • Expression of the logarithm derivative of the transition density of jump diffusion processes via Malliavin Calculus. [ABSTRACT FROM AUTHOR]

• Considering the randomness of the equivalent conicity of the wheelset system. • The analysis of wheelset system under non-smooth condition is more appropriate. • Considering the time delay displacement feedback control in the primary suspension. • Stochastic D(P)-bifurcation occur in the probabilistic sense in the wheelset system. • The hunting stability of the wheelset system during operation were analyzed. Considering not only the stochastic track irregularity and the possible effect of stochastic parameter excitation, but also the time delay of spring, that is, its response is in place, but the force generated is not in place, a stochastic delay wheelset system is established. The infinite dimensional system is reduced to the finite dimensional stochastic differential equation by using the center manifold, and further reduced to a one-dimensional diffusion process by using the stochastic averaging method. The stability of the wheelset system is obtained by analyzing the singular boundary theory and calculating the maximum Lyapunov exponent. The conditions and types of stochastic bifurcation in wheelset system are obtained by combining probability density function. The numerical simulation verifies the correctness of the theoretical analysis and shows that the time delay affects the critical hunting instability speed of wheelset. The stochastic term affects the lateral displacement of the stochastic delay wheelset system. [ABSTRACT FROM AUTHOR]

• Higher-order Ito-Taylor stochastic integration scheme for preserving geometry of manifold is proposed. • Formulation for stochastic Hamiltonian systems on S 2 manifold is developed. • New expressions of Kolmogorov operators for geometric SDEs are demonstrated. • Comparison of error with existing lower order geometric integration schemes is shown. • Application of the proposed scheme for physical systems on manifolds is provided. Naturally occurring systems often have inherent uncertainties and often evolve on complicated smooth and differentiable hypersurfaces that are not necessarily Euclidean. The dynamics of such systems can be potentially described using a stochastic Hamiltonian formulation on differentiable manifold. Solution of these stochastic Hamiltonian systems using the current state-of-the-art methods, either fails to preserve the geometry of the manifold, or, ignore the stochastic nature of the differential equations. To address this, we develop a framework which preserves geometry while accounting for stochasticity in input and states. A new development for higher order geometry preserving Ito-Taylor expansion based stochastic integration scheme is demonstrated. Detailed mathematical development of the scheme is laid down and numerically demonstrated through the application of this method on a number of practical physical systems such as spherical pendulum on a cart and two quadrotors transporting a mass point. The accuracy of the method is established by comparing the global error of the solution with the existing geometric Euler-Maruyama integration scheme, geometric Milstein scheme and the non-geometric Ito-Taylor based stochastic integration schemes. [ABSTRACT FROM AUTHOR]

In this work, four problems for stochastic fractional pseudo-parabolic containing bounded and unbounded delays are investigated. The fractional derivative and the stochastic noise we consider here are the Caputo operator and the fractional Brownian motion. For the two problems involving bounded delays, we aim at establishing global existence, uniqueness, and regularity results under integral Lipschitz conditions for the non-linear source terms. Such behaviors of mild solutions are also analyzed in the unbounded delay cases but under globally and locally Lipschitz assumptions. We emphasize that our results are investigated in the novel spaces C ([ − r , T ] ; L p (Ω , W l , q (D))) , C μ ((− ∞ , T ] ; L p (Ω , W l , q (D))) , and the weighted space F μ ɛ ((− ∞ , T ] ; L p (Ω , W l , q (D))) , instead of usual ones C ([ − r , T ] ; L 2 (Ω , H)) , C μ ((− ∞ , T ] ; L 2 (Ω , H)). The main technique allowing us to overcome the rising difficulties lies on some useful Sobolev embeddings between the Hilbert space H = L 2 (D) and W l , q (D) , and some well-known fractional tools. In addition, we also study the Hölder continuity for the mild solutions, which can be considered as one of the main novelties of this paper. Finally, we consider an additional result connecting delay stochastic fractional pseudo-parabolic equations and delay stochastic fractional parabolic equations. We show that the mild solution of the first model converges to the mild solution of the second one, in some sense, as the diffusion parameter β → 0 + . [ABSTRACT FROM AUTHOR]

In the present paper, we consider multidimensional nonlinear backward stochastic differential equations (BSDEs) with a driver depending on the martingale part M of a solution. We assume that the nonlinear term is merely monotone continuous with respect to the state variable. As to the regularity of the driver with respect to the martingale variable, we consider a very general condition which permits path-dependence on "the future" of the process M as well as a dependence of its law (McKean–Vlasov-type equations). For such drivers, we prove the existence and uniqueness of a global solution (i.e. for any maturity T > 0) to a BSDE with data satisfying natural integrability conditions. [ABSTRACT FROM AUTHOR]

Petrochemical equipment is subjected to frequent discontinuous operation and underwent alternate loads, leading to fatigue failure of metal structure. Due to the fact that petrochemical equipment operates in a harsh environment, it's necessary to assess the fatigue state of metal materials to ensure production safety. The residual stress of metal materials is closely related to the fatigue evolution. Physical model is unable to express the possibility of the evolution of residual stress. Data-driven model requires large-scale data to optimize the model and may not conform to the laws of physical evolution. Therefore, this paper proposes a probabilistic model integrating physical model and Gaussian Process Flow (GPF) for fatigue assessment of metal structure of petrochemical equipment under scarce data. Firstly, different evolutionary trajectories with stochastic parameters are obtained by applying stochastic differential equation (SDE). The uncertainty of SDE is assessed by Gaussian process (GP). Then, from the perspective of a particle filter, the physical model is combined to develop a fusion-driven model to infer the importance of the residual analysis. An example of an irreplaceable metal material in petrochemical equipment is studied to demonstrate the effectiveness of this model. The model proposed could provide a promising solution for preventing fatigue failure at petrochemical processing facilities. [ABSTRACT FROM AUTHOR]

Charged microparticles confined in electrodynamic traps evolve into strongly coupled Coulomb systems (SCCS) which are the subject of current investigation. Recent results with respect to particle dynamics in linear and nonlinear Paul traps are reviewed, including the case of a confined microparticle in presence of an acoustic wave. An analytical model is used to discuss dynamical stability for a system of two coupled ions confined in a Paul trap. The model is then extended to discuss quantum stability for many-body systems of trapped ions. Dynamical stability for many-body systems of identical ions confined in 3D quadrupole ion traps (QIT) is studied locally, in the neighbourhood of minimum configurations that characterize ordered structures. The analytical model is particularized to the case of a combined trap. It is demonstrated that Paul (ion) traps are versatile instruments to investigate one-component strongly coupled Coulomb systems (microplasmas). Exciting physical phenomena associated to Coulomb systems are reported such as autowave generation, phase transitions, defect formation, system self-locking at the edges of a linear Paul trap, self-organization in layers, or pattern formation and scaling. The dynamics of ordered structures consisting of highly nonideal similarly charged solid particles with coupling parameter of the order Γ = 1 0 8 is explored. The approach used enables one to explore the interaction of microparticle structures in presence and in absence of the neutralizing plasma background, as well as to investigate various types of phenomena and physical forces experienced by these patterns. Brownian dynamics (BD) is used to characterize charged particle evolution in time and thus identify regions of stable trapping. Analytical models are used to explain the experimental results. Numerical modelling considers stochastic forces of random collisions with neutral particles, viscosity of the gas medium, regular forces produced by the a.c. trapping voltage, and gravitational force. Microparticle dynamics is characterized by a stochastic Langevin differential equation. Laser plasma acceleration of charged particles is also discussed, with an emphasis on Paul traps employed to investigate collective effects in space-charge-dominated (relativistic) beams, and for target micropositioning. This review paper is both an add-on as well as an update on late progress in SCCS confined in electrodynamic traps. [ABSTRACT FROM AUTHOR]

The paper gives a novel path integral formula inspired by large deviation theory and Malliavin calculus. The proposed finite-dimensional approximation of integrals on path space will be a new higher-order weak approximation of multidimensional stochastic differential equations where the dominant part of the local expansion is governed by Varadhan's geodesic distance and the correction terms are given as Malliavin weights. An optimal truncation of asymptotic expansion is used to reduce computational effort. Kusuoka's estimate is applied to justify the finite-dimensional approximation of path integrals. An efficient simulation method is provided with the algorithm. Numerical results are shown to verify the effectiveness. [ABSTRACT FROM AUTHOR]

We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the match between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. To this end, we use certain local discrepancy functions that do not involve sampling random variables. Precise directions for designing suitable discrepancy functions and for selecting starting step-sizes are given. Second, we introduce a variable step-size Euler scheme, together with a variable step-size second order weak scheme via extrapolation. Finally, numerical simulations are presented to show the potential of the introduced variable step-size strategy and the adaptive scheme to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations. [ABSTRACT FROM AUTHOR]

• A collection of Lévy motion induced stochastic differential equations equipped with neural networks are proposed. • The convergence of the proposed model is proved. • The model can achieve the goal of multi-step prediction under the attention mechanism. • This accurate model can help investors to formulate appropriate timing or stock selection strategies. In this article, we employ a collection of stochastic differential equations with drift and diffusion coefficients approximated by neural networks to predict the trend of chaotic time series which has big jump properties. Our contributions are, first, we propose a model called Lévy induced stochastic differential equation network, which explores compounded stochastic differential equations with α -stable Lévy motion to model complex time series data and solve the problem through neural network approximation. Second, we theoretically prove that the numerical solution through our algorithm converges in probability to the solution of corresponding stochastic differential equation, without curse of dimensionality. Finally, we illustrate our method by applying it to real financial time series data and find the accuracy increases through the use of non-Gaussian Lévy processes. We also present detailed comparisons in terms of data patterns, various models, different shapes of Lévy motion and the prediction lengths. [ABSTRACT FROM AUTHOR]

In this paper a relative analysis of moments reversion of the class of theta methods is provided for an stochastic differential equation with Poisson-driven jumps. We first determine under which conditions the first and second moments revert to steady state values. Second, we consider two different classes of implicit theta methods; theta-Euler method, and compensated theta-Euler method, and derive closed-form expressions for the conditional and asymptotic means and variances of considered methods. We provide a full analysis about the possibility to find methods able to replicate such long-terms quantities. Finally, to verify our theoretical results numerical experiments are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to the existence, uniqueness and comparison theorem on unbounded solutions of a scalar backward stochastic differential equation (BSDE) whose generator grows (with respect to both unknown variables y and z) in a super-linear way like | y | | ln | y | | δ + | z | | ln | z | | λ for some δ ∈ [ 0 , 1 ] and λ ≥ 0. Let k be the maximum of δ , λ + 1 / 2 and 2 λ. For the following four different ranges of the growth power parameter k : k = 1 / 2 , k ∈ (1 / 2 , 1) , k = 1 and k > 1 , we give reasonably weakest possible different integrability conditions on the terminal value for ensuring existence and uniqueness of the unbounded solution to the BSDE. In the first two cases, they are stronger than the L ln L -integrability and weaker than any L p -integrability with p > 1 ; in the third case, the integrability condition is just some L p -integrability for p > 1 ; and in the last case, the integrability condition is stronger than any L p -integrability with p > 1 and weaker than any exp (L ɛ) -integrability with ɛ ∈ (0 , 1). We also establish three comparison theorems, which yield immediately the uniqueness, when either one of generators of both BSDEs is convex (or concave) in both unknown variables (y , z) , or satisfies a one-sided Osgood condition in the first unknown variable y and a uniform continuity condition in the second unknown variable z. Finally, we give an application of our results in mathematical finance. [ABSTRACT FROM AUTHOR]

We study multiplicative SDEs perturbed by an independent additive fractional Brownian motion. Provided the Hurst parameter is chosen in a specified regime, we establish existence of probabilistically weak solutions to the SDE if the measurable diffusion coefficient merely satisfies an integrability condition. In particular, this allows to consider certain singular diffusion coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and smoothness of the densities of stochastic functional differential equations with jumps whose proof are based on recently well-developed lent particle method introduced by Bouleau and Denis. [ABSTRACT FROM AUTHOR]