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1992 | Buch

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Numerical Solution of Stochastic Differential Equations

verfasst von: Peter E. Kloeden, Eckhard Platen

Verlag: Springer Berlin Heidelberg

Buchreihe : Stochastic Modelling and Applied Probability

Enthalten in: Professional Book Archive

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Über dieses Buch

The aim of this book is to provide an accessible introduction to stochastic differ ential equations and their applications together with a systematic presentation of methods available for their numerical solution. During the past decade there has been an accelerating interest in the de velopment of numerical methods for stochastic differential equations (SDEs). This activity has been as strong in the engineering and physical sciences as it has in mathematics, resulting inevitably in some duplication of effort due to an unfamiliarity with the developments in other disciplines. Much of the reported work has been motivated by the need to solve particular types of problems, for which, even more so than in the deterministic context, specific methods are required. The treatment has often been heuristic and ad hoc in character. Nevertheless, there are underlying principles present in many of the papers, an understanding of which will enable one to develop or apply appropriate numerical schemes for particular problems or classes of problems.

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Inhaltsverzeichnis

Frontmatter

Preliminaries

Chapter 1. Probability and Statistics

Abstract

The basic concepts and results of probability and stochastic processes needed later in the book are reviewed here. The emphasis is descriptive and PC-Exercises (PC= Personal Computer), based on pseudo-random number generators introduced in Section 3, are used extensively to help the reader to develop an intuitive understanding of the material. Statistical tests are discussed briefly in the final section.

Peter E. Kloeden, Eckhard Platen

Chapter 2. Probability Theory and Stochastic Processes

Abstract

Like Chapter 1, the present chapter also reviews the basic concepts and results on probability and stochastic processes for later use in the book, but now the emphasis is more mathematical. Integration and measure theory are sketched and an axiomatic approach to probability is presented. Apart form briefly perusing the chapter, the general reader could omit this chapter on the first reading.

Peter E. Kloeden, Eckhard Platen

Stochastic Differential Equations

Chapter 3. Ito Stochastic Calculus

Abstract

This chapter provides an introduction to stochastic calculus, in particular to stochastic integration. A fundamental result, the Ito formula, is also derived. This is a stochastic counterpart of the chain rule of deterministic calculus and will be used repeatedly throughout the book. Section 1 summarizes the key concepts and results and should be read by nonspecialists. Mathematical proofs are presented in the subsequent sections.

Peter E. Kloeden, Eckhard Platen

Chapter 4. Stochastic Differential Equations

Abstract

The theory of stochastic differential equations is introduced in this chapter. The emphasis is on Ito stochastic differential equations, for which an existence and uniqueness theorem is proved and the properties of their solutions investigated. Techniques for solving linear and certain classes of nonlinear stochastic differential equations are presented, along with an extensive list of explicitly solvable equations. Finally, Stratonovich stochastic differential equations and their relationship to Ito equations are examined.

Peter E. Kloeden, Eckhard Platen

Chapter 5. Stochastic Taylor Expansions

Abstract

In this chapter stochastic Taylor expansions are derived and investigated. They generalize the deterministic Taylor formula as well as the Ito formula and allow various kinds of higher order approximations of functionals of diffusion processes to be made. These expansions are the key to the stochastic numerical analysis which we shall develop in the second half of this book. Apart from Section 1, which provides an introductory overview, this chapter could be omitted at the first reading of the book.

Peter E. Kloeden, Eckhard Platen

Applications of Stochastic Differential Equations

Chapter 6. Modelling with Stochastic Differential Equations

Abstract

Important issues which arise when stochastic differential equations are used in applications are discussed in this chapter, in particular the appropriateness of the Ito or Stratonovich version of an equation. Stochastic stability, parametric estimation, stochastic control and filtering are also considered.

Peter E. Kloeden, Eckhard Platen

Chapter 7. Applications of Stochastic Differential Equations

Abstract

This chapter consists of a selection of examples from the literature of applications of stochastic differential equations. These are taken from a wide variety of disciplines with the aim of stimulating the readers’ interest to apply stochastic differential equations in their own particular fields of interest and of providing an indication of how others have used models described by stochastic differential equations. Here we simply describe the equations and refer readers to the original papers for the justification and analysis of the models.

Peter E. Kloeden, Eckhard Platen

Time Discrete Approximations

Chapter 8. Time Discrete Approximation of Deterministic Differential Equations

Abstract

In this chapter we summarize the basic concepts and assertions of the numerical analysis of initial value problems for deterministic ordinary differential equations. The material is presented so as to facilitate generalizations to the stochastic setting and to highlight the differences between the deterministic and stochastic cases.

Peter E. Kloeden, Eckhard Platen

Chapter 9. Introduction to Stochastic Time Discrete Approximation

Abstract

To introduce the reader to the main issues concerning the time discrete approximation of Ito processes, we shall examine the stochastic Euler scheme in some detail in this chapter. We shall consider some examples of typical problems that can be handled by the simulation of approximating time discrete trajectories. In addition, general definitions for time discretizations and time discrete approximations will be given, and the strong and weak convergence criteria for time discrete approximations introduced. These concepts will all be developed more extensively in the subsequent chapters.

Peter E. Kloeden, Eckhard Platen

Strong Approximations

Chapter 10. Strong Taylor Approximations

Abstract

In this chapter we shall use stochastic Taylor expansions to derive time discrete approximations with respect to the strong convergence criterion, which we shall call strong Taylor approximations. We shall mainly consider the corresponding strong Taylor schemes, and shall see that the desired order of strong convergence determines the truncation to be used. To establish the appropriate orders of various schemes we shall make frequent use of a technical lemma estimating multiple Ito integrals. This is Lemma 10.8.1, which is stated and proved in Section 8 at the end of the chapter, although it will be used earlier.

Peter E. Kloeden, Eckhard Platen

Chapter 11. Explicit Strong Approximations

Abstract

In this chapter we shall propose and examine strong schemes which avoid the use of derivatives in much the same way that Runge-Kutta schemes do in the deterministic setting. We shall also call these Runge-Kutta schemes, but it must be emphasized that they are not simply heuristic generalizations of deterministic Runge-Kutta schemes to stochastic differential equations. The notation and abbreviations of the last chapter will continue to be used, often without direct reference.

Peter E. Kloeden, Eckhard Platen

Chapter 12. Implicit Strong Approximations

Abstract

In this chapter we shall consider implicit strong schemes which are necessary for the simulation of the solutions of stiff stochastic differential equations. The regions of absolute stability of several of these implicit strong schemes and other explicit strong schemes will also be investigated.

Peter E. Kloeden, Eckhard Platen

Chapter 13. Selected Applications of Strong Approximations

Abstract

Several applications of the strong schemes that were derived in the preceding chapters will be indicated in this chapter. These are the direct simulation of trajectories of stochastic dynamical systems, including stochastic flows, the testing of parametric estimators and Markov chain filters. In addition, some results on asymptotically efficient schemes will be presented.

Peter E. Kloeden, Eckhard Platen

Weak Approximations

Chapter 14. Weak Taylor Approximations

Abstract

In this chapter we shall use truncated stochastic Taylor expansions as we did in Chapter 10, but now to derive time discrete approximations appropriate for the weak convergence criterion. We shall call the approximations so obtained weak Taylor approximations and shall investigate the corresponding weak Taylor schemes. As with strong approximations, the desired order of weak convergence also determines the truncation that must be used. However, this will be different from the truncation required for strong convergence of the same order, in general involving fewer terms. In the final section we shall state and prove a convergence theorem for general weak Taylor approximations. Throughout the chapter we shall use the notation and abbreviations introduced in Chapter 10.

Peter E. Kloeden, Eckhard Platen

Chapter 15. Explicit and Implicit Weak Approximations

Abstract

We saw in the previous chapter that higher order weak Taylor schemes require the determination and evaluation of derivatives of various orders of the drift and diffusion coefficients. As with strong schemes, we can also derive Runge-Kutta like weak approximations which avoid the use of such derivatives. Here too, these will not be simply heuristic generalizations of deterministic Runge-Kutta schemes. We shall also introduce extrapolation methods, implicit schemes and predictor-corrector methods in this chapter.

Peter E. Kloeden, Eckhard Platen

Chapter 16. Variance Reduction Methods

Abstract

In this chapter we shall describe several methods which allow a reduction in the variance of functionals of weak approximations of Ito diffusions. One method changes the underlying probability measure by means of a Girsanov transformation, another uses general principles of Monte-Carlo integration. Unbiased estimators are also constructed.

Peter E. Kloeden, Eckhard Platen

Chapter 17. Selected Applications of Weak Approximations

Abstract

In this final chapter we indicate several examples of applications of weak approximations. We begin with the evaluation of Wiener function space integrals, which generalize stochastic quadrature formulae, and then use weak schemes to approximate invariant measures. Finally, we compute the top Lyapunov exponents for linear stochastic differential equations. We believe that the techniques outlined here bear much potential for the development of effective numerical methods for higher dimensional partial differential equations, in particular nonlinear ones.

Peter E. Kloeden, Eckhard Platen

Backmatter

Titel: Numerical Solution of Stochastic Differential Equations
verfasst von: Peter E. Kloeden
Eckhard Platen
Copyright-Jahr: 1992
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-12616-5
Print ISBN: 978-3-642-08107-1
DOI: https://doi.org/10.1007/978-3-662-12616-5