Split-step forward methods for stochastic differential equations

In this paper we discuss split-step forward methods for solving Itô stochastic differential equations (SDEs). Eight fully explicit methods, the drifting split-step Euler (DRSSE) method, the diffused split-step Euler (DISSE) method and the three-stage Milstein (TSM 1a-TSM 1f) methods, are constructed based on Euler-Maruyama method and Milstein method, respectively, in this paper. Their order of strong convergence is proved. The analysis of stability shows that the mean-square stability properties of the methods derived in this paper are improved on the original methods. The numerical results show the effectiveness of these methods in the pathwise approximation of Itô SDEs.     

The composite Milstein methods for the numerical solution of Ito stochastic differential equations

In this paper, we present the composite Milstein methods for the strong solution of Ito stochastic differential equations. These methods are a combination of semi-implicit and implicit Milstein methods. We give a criterion for choosing either the implicit or the semi-implicit scheme at each step of our numerical solution. The stability and convergence properties are investigated and discussed for the linear test equation. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. The stability properties of the composite Milstein methods are found to be more superior compared to those of the Milstein, the Euler and even better than the composite Euler method. This superiority in stability makes the methods a better candidate for the solution of stiff SDEs.     

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

In 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.     

The composite Milstein methods for the numerical solution of Stratonovich stochastic differential equations

In this paper, we present two composite Milstein methods for the strong solution of Stratonovich stochastic differential equations driven by d-dimensional Wiener processes. The composite Milstein methods are a combination of semi-implicit and implicit Milstein methods. The criterion for choosing either the implicit or the semi-implicit method at each step of the numerical solution is given. The stability and convergence properties of the proposed methods are analyzed for the linear test equation. It is shown that the proposed methods converge to the exact solution in Stratonovich sense. In addition, the stability properties of our methods are found to be superior to those of the Milstein and the composite Euler methods. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. Hence, the proposed methods are a good candidate for the solution of stiff SDEs.     

求解随机微分方程的二级Milstein方法(英文)

In this paper we discuss two-stage Miistein methods for solving Ito stochastic differential equations （SDEs）. Six fully explicit methods （TSM 1 -- TSM 6） are given in this paper. Their order of strong convergence is proved. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of Ito SDEs.     

CONVERGENCE ANALYSIS OF PARAREAL ALGORITHM BASED ON MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper, we propose a parareal algorithm for stochastic differential equations (SDEs), which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator. The convergence order of the proposed algorithm is analyzed under some regular assumptions. Finally, numerical experiments are dedicated to illustrating the convergence and the convergence order with respect to the iteration number $k$, which show the efficiency of the proposed method.     

The Milstein Scheme for Stochastic Delay Differential Equations Without Using Anticipative Calculus

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This article employs an elementary method to derive the Milstein scheme and its first order strong rate of convergence for stochastic delay differential equations.     

A novel approach to construct numerical methods for stochastic differential equations

In this paper we propose a new numerical method for solving stochastic differential equations (SDEs). As an application of this method we propose an explicit numerical scheme for a super linear SDE for which the usual Euler scheme diverges.     

Newton–Milstein scheme for stochastic differential equations and its fast uniform convergence

We show that a modified Milstein scheme combined with explicit Newton’s method enables us to construct fast converging sequences of approximate solutions of stochastic differential equations. The fast uniform convergence of our Newton–Milstein scheme follows from Amano’s probabilistic second-order error estimate, which had been an open problem since 1991. The Newton–Milstein scheme, which is based on a modified Milstein scheme and the symbolic Newton’s method, will be classified as a numerical and computer algebraic hybrid method and it may give a new possibility to the study of computer algebraic method in stochastic analysis.     

Continuous weak approximation for stochastic differential equations

A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations (SDEs) by general one-step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second order conditions for a class of continuous stochastic Runge–Kutta methods containing the continuous extension of the second order stochastic Runge–Kutta scheme due to Platen are derived. Further, some coefficients for optimal continuous schemes applicable to Itô SDEs with respect to a multi–dimensional Wiener process are presented.     

Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process

The Euler scheme is a well-known method of approximation of solutions of stochastic differential equations (SDEs). A lot of results are now available concerning the precision of this approximation in case of equations driven by a drift and a Brownian motion. More recently, people got interested in the approximation of solutions of SDEs driven by a general Lévy process. One of the problem when we use Lévy processes is that we cannot simulate them in general and so we cannot apply the Euler scheme. We propose here a new method of approximation based on the cutoff of the small jumps of the Lévy process involved. In order to find the speed of convergence of our approximation, we will use results about stability of the solutions of SDEs.     

Probability density function of SDEs with unbounded and path-dependent drift coefficient

In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path-dependent, and diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama–Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super-linear growth condition), Gaussian two-sided bound and Hölder continuity (under sub-linear growth condition) of a probability density function of a solution of SDEs with path-dependent drift coefficient. As an application of explicit representation, we provide the rate of convergence for an Euler–Maruyama (type) approximation, and an unbiased simulation scheme.     

On the strong convergence rate for the Euler–Maruyama scheme of one-dimensional SDEs with irregular diffusion coefficient and local time

The strong rate of convergence for the Euler–Maruyama scheme of stochastic differential equations (SDEs) driven by a Brownian motion with Hölder continuous diffusion coefficient or irregular drift coefficient have been widely studied. In the case of irregular diffusion coefficient, however, there are few studies. In this article, under Le Gall's condition on the diffusion coefficient, which leads to conclude the pathwise uniqueness for SDEs, we provide the same result on the strong rate of convergence as in the case of 1/2-Hölder continuous diffusion coefficient. The idea of the proof is to use a version of Avikainen's inequality. As an application, we introduce a numerical scheme for SDEs with local time.     

A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise

In this paper a family of fully implicit Milstein methods are introduced for solving stiff stochastic differential equations (SDEs). It is proved that the methods are convergent with strong order 1.0 for a class of SDEs. For a linear scalar test equation with multiplicative noise terms, mean-square and almost sure asymptotic stability of the methods are also investigated. We combine analytical and numerical techniques to get insights into the stability properties. The fully implicit methods are shown to be superior to those of the corresponding semi-implicit methods in term of stability property. Finally, numerical results are reported to illustrate the convergence and stability results.     

First order strong approximations of scalar SDEs defined in a domain

We are interested in strong approximations of one-dimensional SDEs which have non-Lipschitz coefficients and which take values in a domain. Under a set of general assumptions we derive an implicit scheme that preserves the domain of the SDEs and is strongly convergent with rate one. Moreover, we show that this general result can be applied to many SDEs we encounter in mathematical finance and bio-mathematics. We will demonstrate flexibility of our approach by analyzing classical examples of SDEs with sublinear coefficients (CIR, CEV models and Wright–Fisher diffusion) and also with superlinear coefficients (3/2-volatility, Aït-Sahalia model). Our goal is to justify an efficient Multilevel Monte Carlo method for a rich family of SDEs, which relies on good strong convergence properties.     

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.     

Numerical Solution to Hybrid Stochastic Differential Systems

Abstract

In this article numerical methods for solving hybrid stochastic differential systems of Itô-type are developed by piecewise application of numerical methods for SDEs. We prove a convergence result if the corresponding method for SDEs is numerically stable with uniform convergence in the mean square sense. The Euler and Runge–Kutta methods for hybrid stochastic differential equations are specifically described and the order of the error is given for the Euler method. A numerical example is given to illustrate the theory.     

A Runge–Kutta type scheme for nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper a new Runge–Kutta type scheme is introduced for nonlinear stochastic partial differential equations (SPDEs) with multiplicative trace class noise. The proposed scheme converges with respect to the computational effort with a higher order than the well-known linear implicit Euler scheme. In comparison to the infinite dimensional analog of Milstein type scheme recently proposed in Jentzen and Röckner (2012), our scheme is easier to implement and needs less computational effort due to avoiding the derivative of the diffusion function. The new scheme can be regarded as an infinite dimensional analog of Runge–Kutta method for finite dimensional stochastic ordinary differential equations (SODEs). Numerical examples are reported to support the theoretical results.     

A non-standard-Euler–Maruyama scheme

We construct a non-standard finite difference numerical scheme to approximate stochastic differential equations (SDEs) using the idea of weighed step introduced by R.E. Mickens. We prove the strong convergence of our scheme under locally Lipschitz conditions of a SDE and linear growth condition. We prove the preservation of domain invariance by our scheme under a minimal condition depending on a discretization parameter and unconditionally for the expectation of the approximate solution. The results are illustrated through the geometric Brownian motion. The new scheme shows a greater behaviour compared with the Euler–Maruyama scheme and balanced implicit methods which are widely used in the literature and applications.