Methods For Stochastic Differential Equations Research Articles

Statistical analysis of diffusion systems with invariants

Abstract The aim of the paper is the construction and numerical solution of stochastic differential equations whose trajectories are located on a given smooth manifold with probability 1. Second order cylindrical surfaces, i.e., elliptic, hyperbolic, and parabolic cylinders serve as examples of such manifolds for the tree-dimensional space (the phase space is two-dimensional). Classes of stochastic differential equations are constructed for these surfaces and linear equations with multiplicative noise are marked in these classes. The results of modelling were used to estimate the deviations of numerical solutions from the manifold. A comparative analysis of considered examples was carried out for accuracy of eight numerical solution methods for stochastic differential equations.

Modeling a Single SEP Event from Multiple Vantage Points Using the iPATH Model

Abstract Using the recently extended 2D improved Particle Acceleration and Transport in the Heliosphere (iPATH) model, we model an example gradual solar energetic particle event as observed at multiple locations. Protons and ions that are energized via the diffusive shock acceleration mechanism are followed at a 2D coronal mass ejection-driven shock where the shock geometry varies across the shock front. The subsequent transport of energetic particles, including cross-field diffusion, is modeled by a Monte Carlo code that is based on a stochastic differential equation method. Time intensity profiles and particle spectra at multiple locations and different radial distances, separated in longitudes, are presented. The results shown here are relevant to the upcoming Parker Solar Probe mission.

Convergence rate and stability of the truncated Euler–Maruyama method for stochastic differential equations

Abstract Recently, Mao (2015) developed a new explicit method, called the truncated Euler–Maruyama (EM) method, for the nonlinear SDE and established the strong convergence theory under the local Lipschitz condition plus the Khasminskii-type condition. In his another follow-up paper (Mao, 2016), he discussed the rates of L q -convergence of the truncated EM method for q ≥ 2 and showed that the order of L q -convergence can be arbitrarily close to q ∕ 2 under some additional conditions. However, there are some restrictions on the truncation functions and these restrictions sometimes might force the step size to be so small that the truncated EM method would be inapplicable. The key aim of this paper is to establish the convergence rate without these restrictions. The other aim is to study the stability of the truncated EM method. The advantages of our new results will be highlighted by the comparisons with the results in Mao (2015, 2016) as well as others on the tamed EM and implicit methods.

Mean-Square Stability of Split-Step Theta Milstein Methods for Stochastic Differential Equations

The fundamental analysis of numerical methods for stochastic differential equations (SDEs) has been improved by constructing new split-step numerical methods. In this paper, we are interested in studying the mean-square (MS) stability of the new general drifting split-step theta Milstein (DSSθM) methods for SDEs. First, we consider scalar linear SDEs. The stability function of the DSSθM methods is investigated. Furthermore, the stability regions of the DSSθM methods are compared with those of test equation, and it is proved that the methods with θ≥3/2 are stochastically A-stable. Second, the nonlinear stability of DSSθM methods is studied. Under a coupled condition on the drifting and diffusion coefficients, it is proved that the methods with θ>1/2 can preserve the MS stability of the SDEs with no restriction on the step-size. Finally, numerical examples are given to examine the accuracy of the proposed methods under the stability conditions in approximation of SDEs.

The Numerical Invariant Measure of Stochastic Differential Equations With Markovian Switching

The existence and uniqueness of the numerical invariant measure of the backward Euler--Maruyama method for stochastic differential equations with Markovian switching is yielded, and it is revealed that the numerical invariant measure converges to the underlying invariant measure in the Wasserstein metric. The global Lipschitz condition on the drift coefficients required by [J. Bao, J. Shao, and C. Yuan, Potential Anal., 44 (2016), pp. 707--727] and [X. Mao, C. Yuan, and G. Yin, J. Comput. Appl. Math., 174 (2005), pp. 1--27] is released. Under a polynomial growth condition imposed on drift coefficients we show that the convergence rate of the numerical invariant measure is polynomial. Several examples and numerical experiments are given to verify our theory.

Numerical methods for Stochastic differential equations: two examples

The goal of this paper is to present a series of recent contributions arising in numerical probability. First we present a contribution to a recently introduced problem: stochastic differential equations with constraints in law, investigated through various theoretical and numerical viewpoints. Such a problem may appear as an extension of the famous Skorokhod problem. Then a generic method to approximate in a weak way the invariant distribution of an ergodic Feller process by a Langevin Monte Carlo simulation. It is an extension of a method originally developed for diffusions and based on the weighted empirical measure of an Euler scheme with decreasing step. Finally, we mention without details a recent development of a multilevel Langevin Monte Carlo simulation method for this type of problem.

A Fully Parallelizable Space-Time Multilevel Monte Carlo Method for Stochastic Differential Equations with Additive Noise

In this work a combination of parallelizable space-time multigrid methods for deterministic parabolic partial differential equations with multilevel Monte Carlo methods for stochastic differential equations with additive noise is developed. Instead of applying the backward Euler--Maruyama scheme sequentially for every time step, the basic idea for the considered space-time method is to solve a large linear system at once, for which a parallelizable multigrid algorithm is constructed that inherits the space-time hierarchy of the multilevel Monte Carlo method. Overall, this results in a fully parallelizable algorithm with respect to space, time, and probability. As model problems for the numerical testing of the proposed method serve in finite dimensions the Ornstein--Uhlenbeck process and in infinite dimensions the stochastic heat equation in 1, 2, and 3 space dimensions.

Optimal Explicit Stabilized Integrator of Weak Order 1 for Stiff and Ergodic Stochastic Differential Equations

A new explicit stabilized scheme of weak order one for stiff and ergodic stochastic differential equations (SDEs) is introduced. In the absence of noise, the new method coincides with the classical deterministic stabilized scheme (or Chebyshev method) for diffusion dominated advection-diffusion problems and it inherits its optimal stability domain size that grows quadratically with the number of internal stages of the method. For mean-square stable stiff stochastic problems, the scheme has an optimal extended mean-square stability domain that grows at the same quadratic rate as the deterministic stability domain size in contrast to known existing methods for stiff SDEs [A. Abdulle and T. Li. Commun. Math. Sci., 6(4), 2008, A. Abdulle, G. Vilmart, and K. C. Zygalakis, SIAM J. Sci. Comput., 35(4), 2013]. Combined with postprocessing techniques, the new methods achieve a convergence rate of order two for sampling the invariant measure of a class of ergodic SDEs, achieving a stabilized version of the non-Markovian scheme introduced in [B. Leimkuhler, C. Matthews, and M. V. Tretyakov, Proc. R. Soc. A, 470, 2014].

Strong convergence rates of modified truncated EM method for stochastic differential equations

Motivated by truncated Euler–Maruyama (EM) method introduced by Mao (2015), a new explicit numerical method named modified truncated Euler–Maruyama method is developed in this paper. Strong convergence rates of the given numerical scheme to the exact solutions to stochastic differential equations are investigated under given conditions in this paper. Compared with truncated EM method, the given numerical simulation strongly converges to the exact solution at fixed time T and over a time interval [0,T] under weaker sufficient conditions. Meanwhile, the convergence rates are also obtained for both cases. Two examples are provided to support our conclusions.

Implicit Milstein method for stochastic differential equations via the Wong-Zakai approximation

The aim of this paper is to derive a numerical scheme for solving stochastic differential equations (SDEs) via Wong-Zakai approximation. One of the most important methods for solving SDEs is Milstein method, but this method is not so popular because the cost of simulating the double stochastic integrals is high. For overcoming this complexity, we present an implicit Milstein scheme based on Wong-Zakai approximation by approximating the Brownian motion with its truncated Haar expansion. The main advantages of this method lie in the fact that it preserves the convergence order and also stability region of the Milstein method while its simulation is much easier than Milstein scheme. We show the convergence rate of the method by some numerical examples.

Convergence of the Tamed Euler Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Non-global Lipschitz Continuous Coefficients

ABSTRACTIn this paper, we deal with the strong convergence of numerical methods for stochastic differential equations with piecewise continuous arguments (SEPCAs) with at most polynomially growing drift coefficients and global Lipschitz continuous diffusion coefficients. An explicit and time-saving tamed Euler method is used to solve this type of SEPCAs. We show that the tamed Euler method is bounded in pth moment. And then the convergence of the tamed Euler method is proved. Moreover, the convergence order is one-half. Several numerical simulations are shown to verify the convergence of this method.

Modeling Particle Acceleration and Transport at a 2‐D CME‐Driven Shock

AbstractWe extend our earlier Particle Acceleration and Transport in the Heliosphere (PATH) model to study particle acceleration and transport at a coronal mass ejection (CME)‐driven shock. We model the propagation of a CME‐driven shock in the ecliptic plane using the ZEUS‐3D code from 20 solar radii to 2 AU. As in the previous PATH model, the initiation of the CME‐driven shock is simplified and modeled as a disturbance at the inner boundary. Different from the earlier PATH model, the disturbance is now longitudinally dependent. Particles are accelerated at the 2‐D shock via the diffusive shock acceleration mechanism. The acceleration depends on both the parallel and perpendicular diffusion coefficients κ|| and κ⊥ and is therefore shock‐obliquity dependent. Following the procedure used in Li, Shalchi, et al. (), we obtain the particle injection energy, the maximum energy, and the accelerated particle spectra at the shock front. Once accelerated, particles diffuse and convect in the shock complex. The diffusion and convection of these particles are treated using a refined 2‐D shell model in an approach similar to Zank et al. (). When particles escape from the shock, they propagate along and across the interplanetary magnetic field. The propagation is modeled using a focused transport equation with the addition of perpendicular diffusion. We solve the transport equation using a backward stochastic differential equation method where adiabatic cooling, focusing, pitch angle scattering, and cross‐field diffusion effects are all included. Time intensity profiles and instantaneous particle spectra as well as particle pitch angle distributions are shown for two example CME shocks.

Numerical simulation of surface roughness impact on dust optical properties

A stochastic differential equation method is introduced for geometric modeling of dust surface roughness along with discrete differential geometry technique. Optical scattering properties are computed for randomly oriented spheroidal particles with uniformly random surface roughness. Invariant imbedding T-Matrix and geometric optics method are applied to compute light scattering properties of dust particles covering from Rayleigh to geometric optics region. We simulated optical scattering properties of feldspar with these new model particles, which shows better performance than smooth spheroids. In addition, we also introduce entropy and relative entropy as similarity measures of particle scattering properties, especially phase functions.

Corporate Bond Defaults: Models and Simulations

This thesis lies at the intersection of mathematics, finance and numerical methods. The financial question of how to model credit risk, specifically corporate bond defaults, and how to model correlations between defaults, motivates our study of the mathematics behind a specific kind of credit risk models called hazard rate models. We consider corporate bond defaults as random events and apply a probabilistic framework to pricing defaultable corporate bonds and to analyzing the case of multi-entity defaults and default correlations. Using the R programming language, we simulate different algorithms and discretization schemes of stochastic processes, checking our theoretical results with simulation results and numerically solving stochastic differential equations where we do not know the closed-form solutions. This motivates us to study and test performance of different simulation methods of stochastic differential equations. In the end, we give a short summary of the two other types of credit risk models as comparisons to think critically of hazard rate models so as to better answer the financial question of how to model credit risk.

Stochastic Taylor expansion of derivative-free method for stochastic differential equations

This paper demonstrates a derivation of stochastic Taylor methods for stochastic differential equations (SDEs). The stochastic Taylor series is extended and truncated at certain terms to achieve the order of convergence of stochatsic Taylor methods for SDEs. The systematic derivation of the expansion of stochastic Taylor series formula is presented. Numerical methods of Euler, Milstein scheme and stochastic Taylor methods of order 2.0 are proposed.

Some methods of parameter estimation for stochastic differential equations

We propose three methods of parameter estimation based on discrete observation data for stochastic differential equations (SDEs). The first method is designed for linear stochastic differential equations (SDEs). For these equations we deduce distribution of certain operation of the exact solution and assume that the relevant operation of the observed data obey this distribution, from which we estimate the unknown parameters in the drift and diffusion coefficients. In the second method, we suppose that certain operation of the observation data and that of the numerical solution arising from the Euler-Maruyama scheme for the SDEs of Ito sense obey the same distribution, from which the unknown parameters can be estimated. We use the third method for SDEs of Stratonovich sense. For these equations we derive the distribution of relevant operation of the numerical solution produced by the midpoint scheme and let the same operation of the data obey this distribution to get estimation of the unknown parameters. Numerical results show validity of the proposed methods, and illustrate that the estimation error produced by the Euler-Maruyama scheme is about of order O(h0.5) while that by the midpoint scheme is about of order O(h), with h being the time step size of the numerical methods. Furthermore, the numerical results show that our methods are more accurate than the existing EM-MLE estimator.

Stability analysis of explicit and implicit stochastic Runge-Kutta methods for stochastic differential equations

This paper concerns to the stability analysis of explicit and implicit stochastic Runge-Kutta methods in approximating the solution of stochastic models. The stability analysis of the schemes in mean-square norm is investigated. Linear stochastic differential equations are used as test equations to demonstrate the efficiency of the proposed schemes.

A strong order $1/2$ method for multidimensional SDEs with discontinuous drift

In this paper we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order 1/2. Our result is the first one that shows strong convergence for such a general class of SDEs. The proof is based on a transformation technique that removes the discontinuity from the drift such that the coefficients of the transformed SDE are Lipschitz continuous. Thus the Euler-Maruyama method can be applied to this transformed SDE. The approximation can be transformed back, giving an approximation to the solution of the original SDE. As an illustration, we apply our result to an SDE the drift of which has a discontinuity along the unit circle.

Study on efficiency of the Adomian decomposition method for stochastic differential equations

Many time-varying phenomena of various fields in science and engineering can be modeled as a stochastic differential equations, so investigation of conditions for existence of solution and obtain the analytical and numerical solutions of them are important. In this paper, the Adomian decomposition method for solution of the stochastic differential equations are improved. Uniqueness and convergence of their adapted solutions are reviewed. The efficiency of the method is demonstrated through the two numerical experiments.

A Hitch-hiker’s Guide to Stochastic Differential Equations

In this review, an overview of the recent history of stochastic differential equations (SDEs) in application to particle transport problems in space physics and astrophysics is given. The aim is to present a helpful working guide to the literature and at the same time introduce key principles of the SDE approach via “toy models”. Using these examples, we hope to provide an easy way for newcomers to the field to use such methods in their own research. Aspects covered are the solar modulation of cosmic rays, diffusive shock acceleration, galactic cosmic ray propagation and solar energetic particle transport. We believe that the SDE method, due to its simplicity and computational efficiency on modern computer architectures, will be of significant relevance in energetic particle studies in the years to come.