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

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

Two-Step Scheme for Backward Stochastic Differential Equations

In this paper, a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations (BSDEs). A necessary and sufficient condition is given to judge the $\mathbb{L}_2$-stability of our numerical schemes. This stochastic linear two-step method possesses a family of $3$-order convergence schemes in the sense of strong stability. The coefficients in the numerical methods are inferred based on the constraints of strong stability and $n$-order accuracy ($n\in\mathbb{N}^+$). Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.     

Composition of stochastic B-series with applications to implicit Taylor methods

In this article, we construct a representation formula for stochastic B-series evaluated in a B-series. This formula is used to give for the first time the order conditions of implicit Taylor methods in terms of rooted trees. Finally, as an example we apply these order conditions to derive in a simple manner a family of strong order 1.5 Taylor methods applicable to Itô SDEs.     

The Numerical Approximation of Stochastic Partial Differential Equations

The numerical solution of stochastic partial differential equations (SPDEs) is at a stage of development roughly similar to that of stochastic ordinary differential equations (SODEs) in the 1970s, when stochastic Taylor schemes based on an iterated application of the Itô formula were introduced and used to derive higher order numerical schemes. An Itô formula in the generality needed for Taylor expansions of the solution of a SPDE is however not available. Nevertheless, it was shown recently how stochastic Taylor expansions for the solution of a SPDE can be derived from the mild form representation of the SPDE, which avoid the need of an Itô formula. A brief review of the literature is given here and the new stochastic Taylor expansions are discussed along with numerical schemes that are based on them. Both strong and pathwise convergence are considered.     

Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes

Abstract

Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Finally, the stochastic Taylor formulas based on coloured trees turn out to be a generalization of the deterministic Taylor formulas using plain trees as recommended by Butcher for the solutions of ordinary differential equations.     

Numerical schemes for random ODEs with affine noise

Numerical schemes for random ordinary differential equations, abbreviated RODEs, with an affine structure can be derived in a similar way as for affine control systems using Taylor expansions that resemble stochastic Taylor expansions for Stratonovich stochastic differential equations. The driving noise processes can be quite general, such as Wiener processes or fractional Brownian motions with continuous sample paths or compound Poisson processes with piecewise constant sample paths, and even more general noises. Such affine-Taylor schemes of arbitrarily high order are constructed here. It is shown how their structure simplifies when the noise terms are additive or commutative. A derivative free counterpart is given and multi-step schemes are derived too. Numerical comparisons are provided for various explicit one-step and multi-step schemes in the context of a toggle switch model from systems biology.     

Higher-order semi-implicit Taylor schemes for Itô stochastic differential equations

The paper considers the derivation of families of semi-implicit schemes of weak order N=3.0 (general case) and N=4.0 (additive noise case) for the numerical solution of Itô stochastic differential equations. The degree of implicitness of the schemes depends on the selection of N parameters which vary between 0 and 1 and the families contain as particular cases the 3.0 and 4.0 weak order explicit Taylor schemes. Since the implementation of the multiple integrals that appear in these theoretical schemes is difficult, for the applications they are replaced by simpler random variables, obtaining simplified schemes. In this way, for the multidimensional case with one-dimensional noise, we present an infinite family of semi-implicit simplified schemes of weak order 3.0 and for the multidimensional case with additive one-dimensional noise, we give an infinite family of semi-implicit simplified schemes of weak order 4.0. The mean-square stability of the 3.0 family is analyzed, concluding that, as in the deterministic case, the stability behavior improves when the degree of implicitness grows. Numerical experiments confirming the theoretical results are shown.     

Euler estimates for rough differential equations

We consider controlled ordinary differential equations and give new estimates for higher order Euler schemes. Our proofs are inspired by recent work of A.M. Davie who considers first and second order schemes. In order to implement the general case we make systematic use of geodesic approximations in the free nilpotent group. Such Euler estimates have powerful applications. By a simple limit argument they apply to rough path differential equations (RDEs) in the sense of T. Lyons and hence also to stochastic differential equations driven by Brownian motion or other random rough paths with sufficient integrability. In the context of the latter, we obtain strong remainder estimates in stochastic Taylor expansions a la Azencott, Ben Arous, Castell and Platen. Although our findings appear novel even in the case of driving Brownian motion our main insight is the genuine rough path nature of (quantitative) remainder estimates in stochastic Taylor expansions. There are several other applications of which we discuss in detail Lq-convergence in Lyons' Universal Limit Theorem and moment control of RDE solutions.     

Implicit Stochastic Runge–Kutta Methods for Stochastic Differential Equations

In this paper we construct implicit stochastic Runge–Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

Highly Accurate Numerical Schemes for Stochastic Optimal Control via FBSDEs

This work is concerned with numerical schemes for stochastic optimal control problems (SOCPs) by means of forward backward stochastic differential equations (FBSDEs). We first convert the stochastic optimal control problem into an equivalent stochastic optimality system of FBSDEs. Then we design an efficient second order FBSDE solver and an quasi-Newton type optimization solver for the resulting system. It is noticed that our approach admits the second order rate of convergence even when the state equation is approximated by the Euler scheme. Several numerical examples are presented to illustrate the effectiveness and the accuracy of the proposed numerical schemes.     

The geometric structure of the expected/observed likelihood expansions

Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor expansions, rearranging terms according to their asymptotic order. The resulting expansions are called hereexpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, expected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation producesinvariant expected/observed expansions.This research was partially supported by the Italian National Research Council grant n.93.00824.CT10.     

Single machine stochastic JIT scheduling problem subject to machine breakdowns

In this paper we research the single machine stochastic JIT scheduling problem subject to the machine breakdowns for preemptive-resume and preemptive-repeat.The objective function of the problem is the sum of squared deviations of the job-expected completion times from the due date.For preemptive-resume,we show that the optimal sequence of the SSDE problem is V-shaped with respect to expected processing times.And a dynamic programming algorithm with the pseudopolynomial time complexity is given.We discuss the difference between the SSDE problem and the ESSD problem and show that the optimal solution of the SSDE problem is a good approximate optimal solution of the ESSD problem,and the optimal solution of the SSDE problem is an optimal solution of the ESSD problem under some conditions.For preemptive-repeat,the stochastic JIT scheduling problem has not been solved since the variances of the completion times cannot be computed.We replace the ESSD problem by the SSDE problem.We show that the optimal sequence of the SSDE problem is V-shaped with respect to the expected occupying times.And a dynamic programming algorithm with the pseudopolynomial time complexity is given.A new thought is advanced for the research of the preemptive-repeat stochastic JIT scheduling problem.     

非Lipschitz条件下由Lévy过程驱动的倒向随机微分方程解的存在唯一及其稳定性

本文研究了由满足某种矩条件下Lévy过程相应的Teugel鞅及与之独立的布朗运动驱动的倒向随机微分方程,给出了飘逸系数满足非Lipschitz条件下解的存在唯一及稳定性结论.解的存在性是通过Picard迭代法给出的.解的L2收敛性是在飘逸系数弱于L2收敛意义下所得到的.     

随机延迟微分方程的全隐式Euler方法

研究随机延迟微分方程数值解具有重要的意义,目前已有显式和半隐式两种数值方法,还没有全隐式的数值方法.本文构造了一种全隐式Euler方法,在该方法中用一些截断的随机变量代替维纳过程增量△W<,n>,接着证明了全隐式方法是1/2阶收敛的并通过数值实验验证了该方法的收敛性.最后,用数值实验表明在某些情况下全隐式方法的稳定性比半隐式方法好一些.     

随机变延迟微分方程平衡方法的均方收敛性与稳定性

针对一类变延迟微分方程,应用全隐式方法一平衡方法,研究了其收敛性和稳定性.结果表明平衡方法以1/2 γ,γ∈(0,1]阶收敛到精确解;并且强平衡方法和弱平衡方法都能保持解析解的均方稳定性;进一步数值实验验证了算法理论分析的正确性,并且表明全隐式的平衡方法比显式方法—Euler方法具有更好的稳定性.     

Finite Element Methods for Parabolic Stochastic PDE’s

We study the rate of convergence of some explicit and implicit numerical schemes for the solution of a parabolic stochastic partial differential equation driven by white noise. These include the forward and backward Euler and the Crank–Nicholson schemes. We use the finite element method. We find, as expected, that the rates of convergence are substantially similar to those found for finite difference schemes, at least when the size of the time step k is on the order of the square of the size of the space step h: all the schemes considered converge at a rate on the order of h^1/2+k^1/4, which is known to be optimal. We also consider cases where k is much greater than h², and find that only the backward Euler method always attains the optimal rate; other schemes, even though they are stable, can fail to convergence to the true solution if the time step is too long relative to the space step. The Crank–Nicholson scheme behaves particularly badly in this case, even though it is a higher-order method. Mathematics Subject Classifications (2000) 60H15, 60H35, 65N30, 35R60.     

An Ito–Taylor weak 3.0 method for stochastic dynamics of nonlinear systems

A new framework for development of order 3.0 weak Taylor scheme towards stochastic modeling and dynamics of coupled nonlinear systems is presented. The proposed method is derived by including third order multiple stochastic integral terms of Ito–Taylor expansion and developing them for a wide class of stochastic nonlinear systems. For computing the system responses of linear and a wide class of nonlinear structural systems, the use of lower order integration schemes is sufficient. But for highly non-linear stochastically driven systems like base isolated hysteretic systems and degrading stochastic systems the evaluation of higher order terms is necessary. Additionally, the use of higher order integration schemes for stochastic dynamics of higher dimensional nonlinear systems remains a challenge due to the arising mathematical complexities with the increase in the number of DOFs (degrees-of-freedom) which really necessitates the development of the proposed algorithm. The proposed algorithm is verified using a representative class of coupled nonlinear system in presence and absence of nonlinear degradation and hysteretic oscillators. The efficiency of the proposed numerical scheme over classical integration schemes is demonstrated through a practical engineering problem. Finally, an automated extension of the proposed algorithm is presented by generalizing it for a system of N-DOFs.     

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretize the SPDE in space by the finite element method and propose a novel scheme called stochastic Rosenbrock-type scheme for temporal discretization. Our scheme is based on the local linearization of the semi-discrete problem obtained after space discretization and is more appropriate for such equations. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise and obtain optimal rates of convergence. Numerical experiments to sustain our theoretical results are provided.     

Economical Runge-Kutta methods with strong global order one for stochastic differential equations

Economical Runge-Kutta schemes for the numerical solution of Stratonovich stochastic differential equations are proposed. The methods have strong global order 1. Numerical stability is studied and some examples are presented to support the theoretical results.     

Stable strong order 1.0 schemes for solving stochastic ordinary differential equations

The Balanced method was introduced as a class of quasi-implicit methods, based upon the Euler-Maruyama scheme, for solving stiff stochastic differential equations. We extend the Balanced method to introduce a class of stable strong order 1.0 numerical schemes for solving stochastic ordinary differential equations. We derive convergence results for this class of numerical schemes. We illustrate the asymptotic stability of this class of schemes is illustrated and is compared with contemporary schemes of strong order 1.0. We present some evidence on parametric selection with respect to minimising the error convergence terms. Furthermore we provide a convergence result for general Balanced style schemes of higher orders.