Stochastic Processes and Financial Mathematics

✍ Scribed by Ludger Rüschendorf

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 310
• Series: Mathematics Study Resources, 1
• Category: Library

✦ Synopsis

The book provides an introduction to advanced topics in stochastic processes and related stochastic analysis, and combines them with a sound presentation of the fundamentals of financial mathematics. It is wide-ranging in content, while at the same time placing much emphasis on good readability, motivation, and explanation of the issues covered.

Financial mathematical topics are first introduced in the context of discrete time processes and then transferred to continuous-time models. The basic construction of the stochastic integral and the associated martingale theory provide fundamental methods of the theory of stochastic processes for the construction of suitable stochastic models of financial mathematics, e.g. using stochastic differential equations. Central results of stochastic analysis such as the Itô formula, Girsanov's theorem and martingale representation theorems are of fundamental importance in financial mathematics, e.g. for the risk-neutral valuation formula (Black-Scholes formula) or the question of the hedgeability of options and the completeness of market models. Chapters on the valuation of options in complete and incomplete markets and on the determination of optimal hedging strategies conclude the range of topics.

Advanced knowledge of probability theory is assumed, in particular of discrete-time processes (martingales, Markov chains) and continuous-time processes (Brownian motion, Lévy processes, processes with independent increments, Markov processes). The book is thus suitable for advanced students as a companion reading and for instructors as a basis for their own courses.

This book is a translation of the original German 1st edition Stochastische Prozesse und Finanzmathematik by Ludger Rüschendorf, published by Springer-Verlag GmbH Germany

✦ Table of Contents

Contents
Introduction
1 Option Pricing in Discrete Time Models
2 Skorohod's Embedding Theorem and Donsker's Theorem
2.1 Skorohod's Embedding Theorem
2.2 Functional Limit Theorem
3 Stochastic Integration
3.1 Martingales and Predictable Processes
3.2 Itô Integral for the Brownian Motion
3.2.1 Extension of the Integral to L2 -Integrands
3.2.2 Construction of the Integral for L0 (B)
3.3 Quadratic Variation of Continuous Local Martingales
3.4 Stochastic Integral of Continuous Local Martingales
3.4.1 Stochastic Integral for Continuous L2-Martingales
3.4.2 Extension to the Set of Continuous Local Martingales
3.4.3 Extension to the Case of Continuous Semimartingales
3.5 Integration of Semimartingales
3.5.1 Decomposition Theorems
3.5.2 Stochastic Integral for Mloc2
3.5.3 Stochastic Integral for Semimartingales
4 Elements of Stochastic Analysis
4.1 Itô Formula
4.2 Martingale Representation
4.3 Measure Change, Theorem of Girsanov
4.3.1 Applications of the Girsanov Theorem
4.3.1.1 Theorem of Cameron–Martin
4.3.2 The Clark Formula
4.4 Stochastic Differential Equations
4.4.1 Strong Solution—Weak Solution of Stochastic Differential Equations
4.5 Semigroup, PDE, and SDE Approach to Diffusion Processes
5 Option Prices in Complete and Incomplete Markets
5.1 The Black–Scholes Model and Risk-Neutral Valuation
5.1.1 Risk-Neutral Valuation of Options
5.1.2 Discussion of the Black–Scholes Formula
5.1.3 Hedging Strategies and Partial Differential Equations
5.2 Complete and Incomplete Markets
6 Utility Optimization, Minimum Distance Martingale Measures, and Utility Indifference Pricing
6.1 Utility Optimization and Utility Indifference Pricing
6.2 Minimum Distance Martingale Measures
6.3 Duality Results
6.3.1 Minimum Distance Martingale Measures and Minimax Measures
6.3.2 Relationship to Portfolio Optimization
6.4 Utility-Based Hedging
6.5 Examples in Exponential Lévy Models
6.6 Properties of the Utility Indifference Price
7 Variance-Minimal Hedging
7.1 Hedging in the Martingale Case
7.2 Hedging in the Semimartingale Case
7.2.1 Föllmer–Schweizer Decomposition and Optimality Equation
7.2.2 Minimal Martingale Measures and Optimal Strategies
Bibliography
Index

✦ Subjects

Stochastic Integration, Stochastic Analysis, Mathematical Finance, Option Prices, Utility Optimization, Hedging