Methods For Stochastic Differential Equations Research Articles

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.

Mean square stability of stochastic theta method for stochastic differential equations driven by fractional Brownian motion

In this paper, we study the mean square stability of the solution and its stochastic theta scheme for the following stochastic differential equations driven by fractional Brownian motion with Hurst parameter H∈(12,1): dX(t)=f(t,X(t))dt+g(t,X(t))dBH(t). Firstly, we consider the special case when f(t,X)=−λκtκ−1X and g(t,X)=μX. Secondly, the stability of the solution and its stochastic theta scheme for nonlinear equations is studied. Due to presence of long memory, even the problem of stability in the mean square sense of the solution has not been well studied, let alone the stability of numerical schemes. A complete new set of techniques to deal with this difficulty are developed. Numerical examples are carried out to illustrate our theoretical results.

Strong and weak convergence rates of logarithmic transformed truncated EM methods for SDEs with positive solutions

To inherit numerically the positivity of stochastic differential equations (SDEs) with non-globally Lipschitz coefficients, we devise a novel explicit method, called logarithmic transformed truncated Euler–Maruyama method. There is however a price to be paid for the preserving positivity, namely that the logarithmic transformation would cause the coefficients of the transformed SDEs growing super-linearly or even exponentially, which makes the strong and weak convergence analysis more complicated. Based on the exponential integrability, truncation techniques and some other arguments, we show that the strong convergence rate of the underlying numerical method is 1/2, and the weak convergence rate can be arbitrarily close to 1. To the best of our knowledge, this is the first result establishing the weak convergence rate of numerical methods for the general SDEs with positive solutions. Numerical experiments are finally reported to confirm our theoretical results.

Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry.

Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.

An efficient and spectral accurate numerical method for computing SDE driven by multivariate Gaussian variables

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

Numerical Modeling of Spectral Hardening at a Finite-width Shock

Spectral hardening has been identified in solar flare hard X-ray observations for several decades and remains a puzzle. We examine spectral hardening under the diffusive shock acceleration mechanism using numerical simulations. The hardening is related to the finite width of the shock and is controlled by the shock Péclet number. We implement two different types of Monte Carlo simulations. The first is based on the backward stochastic differential equation method, where the Parker transport equation is solved by casting it to a set of stochastic different equations, and by following the trajectories of individual quasiparticles. In the second approach, we follow real particles and particles are assumed to move freely between scatterings from magnetic turbulence in the plasma. The scattering is modeled as either large-angle hard-sphere elastic collision, or small-angle pitch-angle scattering. We show that the results from these two approaches agree well with each other and agree with analytical results. We also use a Pan-spectrum form to fit the resulting spectra.

Deep Weak Approximation of SDEs: A Spatial Approximation Scheme for Solving Kolmogorov Equations

In this paper, we propose a new computation scheme for numerical solutions of Kolmogorov equations based on a high-order weak approximation method of stochastic differential equations and deep learning. The scheme provides a spatial approximation for solving Kolmogorov equations without the curse of dimensionality. We show numerical examples based on the proposed scheme for high-dimensional Kolmogorov equations.

Least-squares estimators based on the Adams method for stochastic differential equations with small Lévy noise

We consider stochastic differential equations (SDEs) driven by small L\'evy noise with some unknown parameters, and propose a new type of least squares estimators based on discrete samples from the SDEs. To approximate the increments of a process from the SDEs, we shall use not the usual Euler method, but the Adams method, that is, a well-known numerical approximation of the solution to the ordinary differential equation appearing in the limit of the SDE. We show the consistency of the proposed estimators as well as the asymptotic distribution in a suitable observation scheme. We also show that our estimators can be better than the usual LSE based on the Euler method in the finite sample performance.

Mean-square convergence and stability of two-step Milstein methods for stochastic differential equations with Poisson jumps

Mean-square convergence and stability of two-step Milstein methods for stochastic differential equations with Poisson jumps

An adaptive strong order 1 method for SDEs with discontinuous drift coefficient

In recent years, a number of results has been proven in the literature for strong approximation of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. In many of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and the diffusion coefficient is Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. For scalar SDEs of that type the best Lp-error rate known so far for approximation of the solution at the final time point is 3/4 in terms of the number of evaluations of the driving Brownian motion and it is achieved by the transformed equidistant quasi-Milstein scheme, see [24]. Recently in [27] it has been shown that for such SDEs the Lp-error rate 3/4 can not be improved in general by no numerical method based on evaluations of the driving Brownian motion at fixed time points. In the present article we construct for the first time in the literature a method based on sequential evaluations of the driving Brownian motion, which achieves an Lp-error rate of at least 1 in terms of the average number of evaluations of the driving Brownian motion for such SDEs.

A note on the asymptotic stability of the semi-discrete method for stochastic differential equations

Abstract We study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of ℒ 2 {\mathcal{L}^{2}} -convergence of the truncated SD method and showed that it can be arbitrarily close to 1 2 {\frac{1}{2}} ; see [I. S. Stamatiou and N. Halidias, Convergence rates of the semi-discrete method for stochastic differential equations, Theory Stoch. Process. 24 2019, 2, 89–100]. We show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE. Motivated by a numerical example, we also propose a different SD scheme, using the Lamperti transformation to the original SDE. Numerical simulations support our theoretical findings.

Statistical Error for Cosmic Rays Modulation Evaluated by SDE Backward in Time Method for 1D Model

The propagation of cosmic rays through the heliosphere has been solved for more than half a century by stochastic methods based on Ito’s lemma. This work presents the estimation of statistical error of solution of Fokker–Planck equation by the 1D backward in time stochastic differential equations method. The error dependence on simulation statistics and energy is presented for different combinations of input parameters. The 1% precision criterion in mean value units of intensity standard deviation is defined as a function of solar wind velocity and diffusion coefficient value.

Stability of Cyber-Physical Systems of Numerical Methods for Stochastic Differential Equations: Integrating the Cyber and the Physical of Stochastic Systems

This paper presents the cyber-physical system (CPS) of a numerical method (the widely-used Euler- Maruyama method) and establishes a foundational theory of the CPSs of numerical methods for stochastic differential equations (SDEs), which transforms the way we understand the relationship between the numerical method and the underlying dynamical system. Unlike in the literature where they are treated as separate systems linked by inequalities, the CPS is a seamless integration of the SDE and the numerical method and we construct a new and general class of stochastic impulsive differential equations (SiDEs) that can serve as a canonic form of the CPSs of numerical methods. By the CPS approach, we show the equivalence and intrinsic relationship between the stability of the SDE and the stability of the numerical method using the Lyapunov stability theory we develop for our class of SiDEs. Applying our established theory, we present the CPS Lyapunov inequality that is the necessary and sufficient condition for meansquare stability of the CPS of the Euler-Maruyama method for linear SDEs. The proposed CPS and theory initiate the study of systems numerics and provoke many open and interesting problems for future work.