Stochastic Analysis

verfasst von: Prof. Dr. Shigeo Kusuoka

Verlag: Springer Singapore

Buchreihe : Monographs in Mathematical Economics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is intended for university seniors and graduate students majoring in probability theory or mathematical finance. In the first chapter, results in probability theory are reviewed. Then, it follows a discussion of discrete-time martingales, continuous time square integrable martingales (particularly, continuous martingales of continuous paths), stochastic integrations with respect to continuous local martingales, and stochastic differential equations driven by Brownian motions. In the final chapter, applications to mathematical finance are given. The preliminary knowledge needed by the reader is linear algebra and measure theory. Rigorous proofs are provided for theorems, propositions, and lemmas.

In this book, the definition of conditional expectations is slightly different than what is usually found in other textbooks. For the Doob–Meyer decomposition theorem, only square integrable submartingales are considered, and only elementary facts of the square integrable functions are used in the proof. In stochastic differential equations, the Euler–Maruyama approximation is used mainly to prove the uniqueness of martingale problems and the smoothness of solutions of stochastic differential equations.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preparations from Probability Theory

Abstract

We review fundamental results in probability theory including notation in this section.

Shigeo Kusuoka

Chapter 2. Martingale with Discrete Parameter

Abstract

Let \(\mathbf{T}\) be an ordered set. We only consider the case that \(\mathbf{T}\) is a subset of \([-\infty , \infty ]\) in this book. In almost all cases, \(\mathbf{T}\) is \(\{ 0,1, 2,\ldots ,m\} ,\) \(m\geqq 1,\) or \(\mathbf{Z}_{\geqq 0}\) \(=\{ 0, 1,2, \ldots \} ,\) or [0, T], \(T>0,\) or \([0,\infty ),\) although we consider the case that \(\mathbf{T}\) is \( \mathbf{Z}_{\leqq 0}\) \(=\{ n \in \mathbf{Z}; \; n \leqq 0\} .\)

Shigeo Kusuoka

Chapter 3. Martingale with Continuous Parameter

Abstract

Let \((\Omega ,{\mathcal {F}}, P)\) be a complete probability space, that is, any subset B of \(\Omega \) for which there is an \(A\in {\mathcal {F}}\) such that \(B\subset A\) and \(P(A)=0\) belongs to \({\mathcal {F}}.\)

Shigeo Kusuoka

Chapter 4. Stochastic Integral

Abstract

Let \((\Omega ,{\mathcal {F}},P, \{ {\mathcal {F}}_t\}_{t\in [0,\infty )})\) be a standard filtered probability space throughout this section.

Shigeo Kusuoka

Chapter 5. Applications of Stochastic Integral

Abstract

We assume that \((\Omega ,{\mathcal {F}},P, \{ {\mathcal {F}}_t\}_{t\in [0,\infty )})\) is a standard filtered probability space throughout this section.

Shigeo Kusuoka

Chapter 6. Stochastic Differential Equation

Abstract

The definition of stochastic differential equations is not unique, since many kinds of definitions are used for each application. In this book we consider Markov type stochastic differential equations only.

Shigeo Kusuoka

Chapter 7. Application to Finance

Abstract

In this section, we discuss option pricing theory in the following situation.

Shigeo Kusuoka

Chapter 8. Appendices

Abstract

We prove Proposition 1.1.6 in this section. We say that a family \({\mathcal {C}}\) of subsets of a set S is a \(\pi \)-system, if \( A\cap B\in {\mathcal {C}}\) for any \(A,B\in {\mathcal {C}}.\)

Shigeo Kusuoka

Backmatter

Titel: Stochastic Analysis
verfasst von: Prof. Dr. Shigeo Kusuoka
Verlag: Springer Singapore
Electronic ISBN: 978-981-15-8864-8
Print ISBN: 978-981-15-8863-1
DOI: https://doi.org/10.1007/978-981-15-8864-8