Handbook of Brownian Motion - Facts and Formulae | springerprofessional.de

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2002 | Buch | 2. Auflage

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Handbook of Brownian Motion - Facts and Formulae

verfasst von: Andrei N. Borodin, Paavo Salminen

Verlag: Birkhäuser Basel

Buchreihe : Probability and its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

There are two parts in this book. The first part is devoted mainly to the proper ties of linear diffusions in general and Brownian motion in particular. The second part consists of tables of distributions of functionals of Brownian motion and re lated processes. The primary aim of this book is to give an easy reference to a large number of facts and formulae associated to Brownian motion. We have tried to do this in a "handbook-style". By this we mean that results are given without proofs but are equipped with a reference where a proof or a derivation can be found. It is our belief and experience that such a material would be very much welcome by students and people working with applications of diffusions and Brownian motion. In discussions with many of our colleagues we have found that they share this point of view. Our original plan included more things than we were able to realize. It turned out very soon when trying to put the plan into practice that the material would be too wide to be published under one cover. Excursion theory, which most of the recent results concerning linear Brownian motion and diffusions can be classified as, is only touched upon slightly here, not to mention Brownian motion in several dimensions which enters only through the discussion of Bessel processes. On the other hand, much attention is given to the theory of local time.

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Inhaltsverzeichnis

Frontmatter

Chapter 1 Stochastic Processes in General

Andrei N. Borodin, Paavo Salminen

Chapter 2 Linear Diffusions

Andrei N. Borodin, Paavo Salminen

Chapter 3 Stochastic Calculus

Andrei N. Borodin, Paavo Salminen

Chapter 4 Brownian Motion

Andrei N. Borodin, Paavo Salminen

Chapter 5 Local Time as a Markov Process

Andrei N. Borodin, Paavo Salminen

Chapter 6 Differential Systems Associated to Brownian Motion

Andrei N. Borodin, Paavo Salminen

Introduction

Andrei N. Borodin, Paavo Salminen

1. Brownian Motion

Andrei N. Borodin, Paavo Salminen

2. Brownian Motion with Drift

Andrei N. Borodin, Paavo Salminen

3. Reflecting Brownian Motion

Andrei N. Borodin, Paavo Salminen

4. Bessel Process of Order ע

Andrei N. Borodin, Paavo Salminen

5. Bessel Process of Order 1/2

Andrei N. Borodin, Paavo Salminen

6. Bessel Process of Order Zero

Andrei N. Borodin, Paavo Salminen

7. Ornstein–Uhlenbeck Process

Andrei N. Borodin, Paavo Salminen

8. Radial Ornstein–Uhlenbeck Process

Andrei N. Borodin, Paavo Salminen

9. Geometric Brownian Motion

Andrei N. Borodin, Paavo Salminen

9. Geometric Brownian Motion

Abstract

1.0.5 \({P_x}\left( {{V_\tau } \in dz} \right) = \left\{ \begin{gathered} \frac{\lambda }{{z{\sigma ^2}\sqrt {{v^2} + 2\lambda /{\sigma ^2}} }}{\left( {\frac{x}{z}} \right)^{\sqrt {{v^2} + 2\lambda /{\sigma ^2}} - v}}dz, x \leqslant z \hfill \\ \frac{\lambda }{{z{\sigma ^2}\sqrt {{v^2} + 2\lambda /{\sigma ^2}} }}{\left( {\frac{z}{x}} \right)^{\sqrt {{v^2} + 2\lambda /{\sigma ^2}} - v}}dz, z \leqslant x \hfill \\ \end{gathered} \right. \)

Andrei N. Borodin, Paavo Salminen

Appendix 2. Special Functions

Abstract

sh \(x: = \frac{1}{2}\left( {{e^x} - {e^{ - x}}} \right) = \sum\limits_{k = 0}^\infty {\frac{{{x^{2k + 1}}}}{{^{\left( {2k + 1} \right)!}}}} \)

Andrei N. Borodin, Paavo Salminen

Appendix 3. Inverse Laplace Transforms

Abstract

General formulae 0. \(\mathcal{L}_\gamma ^{ - 1}\left( {F\left( \gamma \right)} \right) = :f(y), where F\left( \gamma \right) = \mathop \smallint \limits_0^\infty {e^{ - \gamma y}}f(y)dy, Re \gamma \geqslant 0 \)

Andrei N. Borodin, Paavo Salminen

Appendix 4. Differential Equations and Their Solutions

Abstract

Let \(w = \not \upsilon '\varphi - \not \upsilon \varphi ', \lambda > 0, \gamma > 0, p > 0, q > 0. \)

Andrei N. Borodin, Paavo Salminen

Appendix 5. Formulae for n-Fold Differentiation

Abstract

For the n times differentiable real valued function f introduce

$$ {\left( {f\left( x \right)} \right)^{(n)}}: = {f^{(n)}}(x): = \frac{{{d^n}}}{{d{x^n}}}f(x), n = 1,2,3, \ldots , {f^{(0)}}(x): = f(x). $$

Andrei N. Borodin, Paavo Salminen

Backmatter

Titel: Handbook of Brownian Motion - Facts and Formulae
verfasst von: Andrei N. Borodin
Paavo Salminen
Copyright-Jahr: 2002
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-0348-8163-0
Print ISBN: 978-3-7643-6705-3
DOI: https://doi.org/10.1007/978-3-0348-8163-0