Methods For Stochastic Differential Equations Research Articles

Convergence of Modified Truncated Euler-Maruyama Method for Stochastic Differential Equations with Hölder Diffusion Coefficients

Convergence of Modified Truncated Euler-Maruyama Method for Stochastic Differential Equations with Hölder Diffusion Coefficients

Tamed Stochastic Runge-Kutta-Chebyshev Methods for Stochastic Differential Equations with Non-Globally Lipschitz Coefficients

Tamed Stochastic Runge-Kutta-Chebyshev Methods for Stochastic Differential Equations with Non-Globally Lipschitz Coefficients

Fast and accurate evaluation of deep-space galactic cosmic ray fluxes with HelMod-4/CUDA

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.

The Truncated EM Method for Stochastic Differential Equations Driven by Fractional Brownian Motion

We mainly focus on the numerical method of fractional Brownian motion in this paper. On the basis of the numerical method of general SDEs, an approximation scheme is obtained for the stochastic differential equations about fractional noise. And we get it by using the Lipschitz condition and combining with the truncation function f∆ and g∆. Furthermore, we also prove the moment boundedness and convergence of the solution by some lemma. At last, we apply this method to the Gilpin-Ayala model. The orbital image of the solution and the form of numerical solution are given. The error of solution also has been simulated by MATLAB.

Convergence and stability of modified partially truncated Euler-Maruyama method for stochastic differential equations with piecewise continuous arguments

ABSTRACT This paper constructs a modified partially truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift and diffusion coefficients grow superlinearly. We divide the coefficients of SDEPCAs into global Lipschitz continuous and superlinearly growing parts. Our method only truncates the superlinear terms of the coefficients to overcome the potential explosions caused by the nonlinearities of the coefficients. The strong convergence theory of this method is established and the 1/2 convergence rate is presented. Furthermore, an explicit scheme is developed to preserve the mean square exponential stability of the underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results.

SDE method for cosmic rays modulation in the heliosphere statistical error and solution uniqueness

The cosmic ray modulation in the heliosphere models comes to a new phase of their evolution, where the still higher number of models will be public with published source code. At the dawn of this period, we want to address two topics, the statistical error of their results and the uniqueness of their solutions.We present a method for the evaluation of statistical error for the numerical stochastic differential equation method, which is probably the most used method to numerically solve Parker's equation. We defined a limit of statistical error, for which we present a method to estimate the number of particles needed to be simulated to reach this limit. The estimation of statistical error from a scan of parametric space of two currently available models with public code, for the SOLARPROP model and Geliosphere 2D model, is presented.We present a test of the uniqueness of the solution of Parker's equation for 1D and 2D models of heliospheric modulation. Namely for the 1D B-p model and Geliosphere 2D model. The dependence of solution uniqueness on the selected model's input parameters is presented and discussed. Program summaryProgram Title: CudaHelioCommanderLightCPC Library link to program files:https://doi.org/10.17632/cptpkwphn4.1Developer's repository link:https://github.com/gladiatus55/CudaHelioCommanderLightLicensing provisions: GPLv3Programming language: C#, .NET 6.0Nature of problem: Despite the decades-long history of cosmic ray modulation in the heliosphere models, the topics of the statistical error of their results, and the uniqueness of their solutions were in published articles addressed only in a very limited way. A precise description of the method for statistical error evaluation, with examples applied to available models with published source code, is needed. The uniqueness of the solution of Parker's equation was not addressed in published literature.Solution method: The estimation of statistical error was evaluated for two currently available models with public code, the SOLARPROP model and Geliosphere 2D model, by a scan of their parametric space. The dependence of solution uniqueness on the selected model's input parameters was also done by scanning their parametric space.

The modified truncated Euler–Maruyama method for stochastic differential equations with concave diffusion coefficients

Influenced by Gyöngy and Rásonyi (2011), many scholars established the strong convergence of several numerical methods for scalar stochastic differential equations (SDEs) with superlinearly growing drift and Hölder continuous diffusion coefficients. However, their methods depend on the Yamada-Watanabe method and therefore fail to work for multi-dimensional SDEs. In this paper, we study the strong Lp−convergence, for all p⩾2, of the modified truncated Euler–Maruyama method for multi-dimensional SDEs with superlinearly growing drift and concave diffusion coefficients satisfying the Osgood condition. We also discuss an example with computer simulations to illustrate our theoretical results.

Truncated Euler–Maruyama method for stochastic differential equations driven by fractional Brownian motion with super-linear drift coefficient

In this paper, we propose a truncated Euler–Maruyama scheme for stochastic differential equations driven by fractional Brownian motion with super-linear drift coefficient. Meanwhile, the convergence rate of the numerical method is established. Numerical example is demonstrated to verify the theoretical results.

Convergence and stability of an explicit method for nonlinear stochastic differential equations with piecewise continuous arguments

This paper constructs a new explicit method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift coefficients grow superlinearly and the diffusion coefficients have at most linear growth. We show that this method converges strongly with the convergence rate 1/2 to the exact solutions of SDEPCAs over the finite time interval and demonstrate it can inherit the mean square exponential stability of the underlying SDEPCAs. Several numerical experiments are carried out to support our findings.

Geliosphere - parallel CPU and GPU based models of cosmic ray modulation in the heliosphere

We present Geliosphere, a numerical model of the modulation of cosmic rays in the heliosphere. The current version of Geliosphere implements the stochastic differential equation method to solve Parker's equation. Sets of 1D and 2D models of Geliosphere solve Parker's equation in a backward-in-time manner for 2D models and in a backward-in-time and forward-in-time manner for 1D models. Geliosphere offers GPU-accelerated implementations of all models. Geliosphere also offers multithread CPU implementations of all available models for users who do not have access to GPUs. To verify the Geliosphere 2D model results at different heliospheric latitudes and radii, we show that the Geliosphere 2D model, with implemented heliospheric polar field modification, fits well with measurements provided by the Kiel Electron Telescope (KET), which is part of the Ulysses COsmic ray and Solar Particle INvestigation (COSPIN) experiment. Program summaryProgram title: GeliosphereCPC Library link to program files:https://doi.org/10.17632/9sb2ywg65w.1Developer's repository link:https://github.com/msolanik/GeliosphereLicensing provisions: BSD-3-ClauseProgramming language: C++Nature of problem: Calculation of the modulation of cosmic rays in the heliosphere including drift effects. Implement a heliospheric magnetic field model with polar correction used for simulation to describe nonecliptic measurements. Implementation of modulation models for CPU and GPU environments.Solution method: Stochastic differential equations are used to solve Parker's equation in forward-in-time and backward-in-time manners. The forward-in-time approach is used for the one-dimensional heliosphere model and the backward-in-time approach for one and two-dimensional heliosphere models. The two-dimensional model results are verified with measurements of the Ulysses spacecraft in non-ecliptic regions of the inner heliosphere.

Convergence and exponential stability of modified truncated Milstein method for stochastic differential equations

In this paper, we develop a new explicit scheme called modified truncated Milstein method which is motivated by truncated Milstein method proposed by Guo et al. (2018) and modified truncated Euler–Maruyama method introduced by Lan and Xia (2018). We obtain the strong convergence of the scheme under local boundedness and Khasminskii-type conditions, which are relatively weaker than the existing results, and we prove that the convergence rate could be arbitrarily close to 1 under given conditions. Moreover, pth moment exponential stability of the scheme is also obtained Three numerical experiments are offered to support our conclusions.

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Generalized One-Sided Lipschitz Condition

Strong convergence of an adaptive time-stepping Milstein method for SDEs with monotone coefficients

We introduce an explicit adaptive Milstein method for stochastic differential equations with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly L_2 convergent of order one. This order is inherited by an explicit adaptive Euler–Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.

Spectral Collocation Method for Stochastic Differential Equations Driven by Fractional Brownian Motion

In this paper, a spectral collocation method is developed to numerically approximate a class of stochastic differential equations driven by the fractional Brownian motion. The convergence of the proposed method is proved. Numerical simulations are conducted to illustrate the performance of the proposed method in different cases.

Weak convergence of balanced stochastic Runge–Kutta methods for stochastic differential equations

In this paper, weak convergence of balanced stochastic one-step methods and especially balanced stochastic Runge–Kutta (SRK) methods for Itô multidimensional stochastic differential equations is analyzed. Generalizing a corresponding result obtained by H. Schurz for the standard Euler method, it is shown that under certain conditions, balanced one-step methods preserve the weak convergence properties of their underlying methods. As an application, this allows to prove the weak convergence order of the balanced SRK methods presented in earlier work by A. Rathinasamy, P. Nair and D. Ahmadian.

Conservative Continuous-Stage Stochastic Runge–Kutta Methods for Stochastic Differential Equations

In this paper, we develop a new class of conservative continuous-stage stochastic Runge–Kutta methods for solving stochastic differential equations with a conserved quantity. The order conditions of the continuous-stage stochastic Runge–Kutta methods are given based on the theory of stochastic B-series and multicolored rooted tree. Sufficient conditions for the continuous-stage stochastic Runge–Kutta methods preserving the conserved quantity of stochastic differential equations are derived in terms of the coefficients. Conservative continuous-stage stochastic Runge–Kutta methods of mean square convergence order 1 for general stochastic differential equations, as well as conservative continuous-stage stochastic Runge–Kutta methods of high order for single integrand stochastic differential equations, are constructed. Numerical experiments are performed to verify the conservative property and the accuracy of the proposed methods in the longtime simulation.

Invariant measure of the backward Euler method for stochastic differential equations driven by α$$ \alpha $$‐stable process

The backward Euler method is employed to approximate the invariant measure of a class of stochastic differential equations (SDEs) driven by ‐stable processes. The existence and uniqueness of the numerical invariant measure are proved. Then the numerical invariant measure is shown to converge to the underlying invariant measure. Numerical examples are provided to demonstrate the theoretical results.

Stability in the small moment sense of the backward Euler–Maruyama method for stochastic differential equations with super-linear coefficients

For stochastic differential equations (SDEs) with super-linear drift and diffusion coefficients, the backward Euler–Maruyama (BEM) method is considered to reproduce the stability of the underlying SDEs. The pth moment exponential stability for some small p∈(0,1) and the almost sure exponential stability of the BEM method are proved. The results in this paper partially extend those in Higham, Mao and Yuan (2007). Numerical simulations are provided to illustrate the theoretical results.

Modeling the 2020 November 29 solar energetic particle event using EUHFORIA and iPATH models

Aims.We present the implementation of a coupling between EUropean Heliospheric FORcasting Information Asset (EUHFORIA) and improved Particle Acceleration and Transport in the Heliosphere (iPATH) models. In this work, we simulate the widespread solar energetic particle (SEP) event of 2020 November 29 and compare the simulated time-intensity profiles with measurements at Parker Solar Probe (PSP), the Solar Terrestrial Relations Observatory (STEREO)-A, SOlar and Heliospheric Observatory (SOHO), and Solar Orbiter. We focus on the influence of the history of shock acceleration on the varying SEP time-intensity profiles and investigate the underlying causes in the origin of this widespread SEP event.Methods.We simulated a magnetized coronal mass ejection (CME) propagating in the data-driven solar wind with the EUHFORIA code. The CME was initiated by using the linear force-free spheromak module of EUHFORIA. The shock parameters and a 3D shell structure were computed from EUHFORIA as inputs for the iPATH model. Within the iPATH model, the steady-state solution of particle distribution assuming diffuse shock acceleration is obtained at the shock front. The subsequent SEP transport is described by the focused transport equation using the backward stochastic differential equation method with perpendicular diffusion included.Results.We examined the temporal evolution of shock parameters and particle fluxes during this event and we find that adopting a realistic solar wind background can significantly impact the expansion of the shock and, consequently, the shock parameters. Time-intensity profiles with an energetic storm particle event at PSP are well reproduced from the simulations. In addition, the simulated and observed time-intensity profiles of protons show a similar two-phase enhancement at STA. These results illustrate that modeling a shock using a realistic solar wind is crucial in determining the characteristics of SEP events. The decay phase of the modeled time-intensity profiles at Earth is in good agreement with the observations, indicating the importance of perpendicular diffusion in widespread SEP events. Taking into account the possible large curved magnetic field line connecting to Solar Orbiter, the modeled time-intensity profiles show a good agreement with the observation. We suggest that the broadly distorted magnetic field lines, which are due to a stream interaction region, may be a key factor in helping to improve our understanding of the observed SEPs at Solar Orbiter for this event.

The Development of Tourism Towns with Characteristic Ancient Buildings Based on Partial Differential Model of Competitive Resource Optimization

In this paper, a deep learning-based method for solving high-dimensional nonlinear partial differential equations is proposed, that is, the deep backward stochastic differential equation method. The solution function of the high-dimensional partial differential equation is represented by the corresponding solution function of the backward stochastic differential equation. The substantive carrier of ancient town tourism is the ancient town itself. The essence of resources and the ancient town are highly unified, resource occupiers (suppliers) and tourism participants are highly unified, and tourists need to be highly coupled with the essence of tourism products. The art of ancient architecture is not only an important material basis for the sustainable development of the local tourism industry but also an important experience reference for the traditional architectural design of the creation of artistic architecture in the new era. To create a tourist destination of ancient architecture in a characteristic town, it will contribute to the sustainable development of the local economy and society. Taking the policy support related to tourism of ancient buildings as the starting point, and the internal cultural heritage as the basis for development, we explore the characteristic activities and products, integrate natural tourism resources and modern tourism resources in the whole region, to help ancient buildings become an important driving force to promote the development of the tourism industry.