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.     

Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn–Tweedie. Application of these Markov chain results leads to straightforward proofs of geometric ergodicity for a variety of SDEs, including problems with degenerate noise and for problems with locally Lipschitz vector fields. Applications where this theory can be usefully applied include damped-driven Hamiltonian problems (the Langevin equation), the Lorenz equation with degenerate noise and gradient systems.The same Markov chain theory is then used to study time-discrete approximations of these SDEs. The two primary ingredients for ergodicity are a minorization condition and a Lyapunov condition. It is shown that the minorization condition is robust under approximation. For globally Lipschitz vector fields this is also true of the Lyapunov condition. However in the locally Lipschitz case the Lyapunov condition fails for explicit methods such as Euler–Maruyama; for pathwise approximations it is, in general, only inherited by specially constructed implicit discretizations. Examples of such discretization based on backward Euler methods are given, and approximation of the Langevin equation studied in some detail.     

Strong Convergence of Wong-Zakai Approximations of Reflected SDEs in a Multidimensional General Domain

In this paper, we obtain the strong convergence of Wong-Zakai approximations of reflected SDEs in a general multidimensional domain giving an affirmative answer to the question posed by Evans and Stroock (Stoch. Process. Appl. 121, 1464–1491, 2011).     

A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients

We provide a rate for the strong convergence of Euler approximations for stochastic differential equations (SDEs) whose diffusion coefficient is not Lipschitz but only (1/2+α)-Hölder continuous for some α≥0.     

Integration by parts formulas concerning maxima of some SDEs with applications to study on density functions

In this article, we prove integration by parts (IBP) formulas concerning maxima of solutions to some stochastic differential equations (SDEs). We will deal with three types of maxima. First, we consider discrete time maximum, and then continuous time maximum in the case of one-dimensional SDEs. Finally, we deal with the maximum of the components of a solution to multi-dimensional SDEs. Applications to study their probability density functions by means of the IBP formulas are also discussed.     

Pathwise accuracy and ergodicity of metropolized integrators for SDEs

Metropolized integrators for ergodic stochastic differential equations (SDEs) are proposed that (1) are ergodic with respect to the (known) equilibrium distribution of the SDEs and (2) approximate pathwise the solutions of the SDEs on finite‐time intervals. Both these properties are demonstrated in the paper, and precise strong error estimates are obtained. It is also shown that the Metropolized integrator retains these properties even in situations where the drift in the SDE is nonglobally Lipschitz, and vanilla explicit integrators for SDEs typically become unstable and fail to be ergodic. © 2009 Wiley Periodicals, Inc.     

Derivation of Several SDE Systems in One- and Two-Locus Population Genetics

A direct approach is described for deriving stochastic differential equations (SDEs) for the dynamics of evolving populations. Itô SDEs are presented and compared for populations of haploid and diploid individuals with one or more alleles at one locus undergoing pure genetic drift. The results agree with previous investigations in mathematical genetics using diffusion approximations. Furthermore, a stochastic differential equation model is derived for diploid populations with two alleles at two loci. The derived SDE systems provide unifying, consistent models.     

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.     

Modified Equations for Stochastic Differential Equations

We describe a backward error analysis for stochastic differential equations with respect to weak convergence. Modified equations are provided for forward and backward Euler approximations to Itô SDEs with additive noise, and extensions to other types of equation and approximation are discussed.     

A class of degenerate stochastic differential equations with non-Lipschitz coefficients

We obtain sufficient condition for SDEs to evolve in the positive orthant. We use arguments based on comparison theorems for SDEs to achieve this. As an application we prove the existence of a unique strong solution for a class of multidimensional degenerate SDEs with non-Lipschitz diffusion coefficients.     

随机微分方程高阶分裂步(θ1,θ2,θ3)方法的强收敛性

本文首先提出一类高阶分裂步（θ₁,θ₂,θ₃）方法求解由非交换噪声驱动的非自治随机微分方程.其次在漂移项系数满足多项式增长和单边Lipschitz条件下,证明了当1/2 ≤ θ₂ ≤ 1时该方法是1阶强收敛的.此类方法包含很多经典的方法：如随机θ-Milstein方法,向后分裂步Milstein方法等.最后数值实验验证了所得结论.     

Strong comparison result for a class of re-ected stochastic differential equations with non-Lipschitzian coeffcients

In this paper, we are concerned with a class of reflected stochastic differential equations (reflected SDEs) with non-Lipschitzian coeffcients. Under the same coeffcients assumptions as Fang and Zhang [Probab. Theory Relat. Fields, 2005, 132(3): 356 390] for a class of SDEs, we establish the pathwise uniqueness for the reflected SDEs. Furthermore, a strong comparison theorem is proved for the reflected SDEs in a onedimensional case.     

Strong approximation of time-changed stochastic differential equations involving drifts with random and non-random integrators

BIT Numerical Mathematics - The rates of strong convergence for various approximation schemes are investigated for a class of stochastic differential equations (SDEs) which involve a random time...     

Strong comparison result for a class of reflected stochastic differential equations with non-Lipschitzian coefficients

In this paper, we are concerned with a class of reflected stochastic differential equations (reflected SDEs) with non-Lipschitzian coefficients. Under the same coefficients assumptions as Fang and Zhang [Probab. Theory Relat. Fields, 2005, 132(3): 356–390] for a class of SDEs, we establish the pathwise uniqueness for the reflected SDEs. Furthermore, a strong comparison theorem is proved for the reflected SDEs in a one-dimensional case.      

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.     

High strong order stochastic Runge-Kutta methods for Stratonovich stochastic differential equations with scalar noise

This paper concerns the stochastic Runge-Kutta (SRK) methods with high strong order for solving the Stratonovich stochastic differential equations (SDEs) with scalar noise. Firstly, the new SRK methods with strong order 1.5 or 2.0 for the Stratonovich SDEs with scalar noise are constructed by applying colored rooted tree analysis and the theorem of order conditions for SRK methods proposed by Rößler (SIAM J. Numer. Anal. 48(3), 922–952, 2010). Secondly, a specific SRK method with strong order 2.0 for the Stratonovich SDEs whose drift term vanishes is proposed. And another specific SRK method with strong order 1.5 for the Stratonovich SDEs whose drift and diffusion terms satisfy the commutativity condition is proposed. The two specific SRK methods need only to use one random variable and do not need to simulate the multiple Stratonovich stochastic integrals. Finally, the numerical results show that performance of our methods is better than those of well-known SRK methods with strong order 1.0 or 1.5.     

Harnack inequalities for SDEs driven by subordinate Brownian motions

By using regularization approximations of the underlying subordinator and a gradient estimate approach, the dimension-independent Harnack inequalities are established for the inhomogeneous semigroup associated with a class of SDEs with Lévy noise containing a subordinate Brownian motion. Our estimates in Harnack type inequalities improve the corresponding ones in the recent paper by Wang and Wang (2014) [10].     

DISCRETIZATION OF JUMP STOCHASTIC DIFFERENTIAL EQUATIONS IN TERMS OF MULTIPLE STOCHASTIC INTEGRALS

1.IntroductionFOrthestrongdiscretizationofSDEs,anynumericalmethodwhichonlydependsonthevaluesofBrownianpathsorPoissonpathsatthepartitionnodescannotachieveanorderhigherthan0.5ingeneral[')'1'].Thereforetheevaluationofmultiplestochasticintegralsontheintervalsbetweennodesisamajorobstaclethatmustbeovercome.Someattemptshavebeenmadepreviouslyindifferentapproachestoapproximatemul-tiplestochasticintegrals.[2]suggestsanapproximationintermsofFourierGaussiancoefficientsoftheBrownianbridgeprocess.Asthel…     

Discrete approximations, relaxation, and optimization of one-sided Lipschitzian differential inclusions in Hilbert spaces

We study discrete approximations of nonconvex differential inclusions in Hilbert spaces and dynamic optimization/optimal control problems involving such differential inclusions and their discrete approximations. The underlying feature of the problems under consideration is a modified one-sided Lipschitz condition imposed on the right-hand side (i.e., on the velocity sets) of the differential inclusion, which is a significant improvement of the conventional Lipschitz continuity. Our main attention is paid to establishing efficient conditions that ensure the strong approximation (in the W^1,p-norm as p1) of feasible trajectories for the one-sided Lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations. To proceed with the latter issue, we derive a new extension of the Bogolyubov-type relaxation/density theorem to the case of differential inclusions satisfying the modified one-sided Lipschitzian condition. All the results obtained are new not only in the infinite-dimensional Hilbert space framework but also in finite-dimensional spaces.     

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.