Stochastic Processes for Finance

This book is an extension of “Probability for Finance” to multi-period financial models, either in the discrete or continuous-time framework. It describes the most important stochastic processes used in finance in a pedagogical way, especially Markov chains, Brownian motion and martingales. It also shows how mathematical tools like filtrations, Itô’s lemma or Girsanov theorem should be understood in the framework of financial models. It also provides many illustrations coming from the financial literature.

Introduction

1 Discrete-time stochastic processes
1.1 Introduction
1.2 The general framework
1.3 Information revelation over time
1.3.1 Filtration on a probability space
1.3.2 Adapted and predictable processes
1.4 Markov chains
1.4.1 Introduction
1.4.2 Definition and transition probabilities
1.4.3 Chapman-Kolmogorov equations
1.4.4 Classification of states
1.4.5 Stationary distribution of a Markov chain
1.5 Martingales
1.5.1 Doob decomposition of an adapted process
1.5.2 Martingales and self-financing strategies
1.5.3 Investment strategies and stopping times
1.5.4 Stopping times and American options

2 Continuous-time stochastic processes
2.1 Introduction
2.2 General framework
2.2.1 Filtrations, adapted and predictable processes
2.2.2 Markov and diffusion processes
2.2.3 Martingales
2.3 The Brownian motion
2.3.1 Intuitive presentation
2.3.2 The assumptions
2.3.3 Definition and general properties
2.3.4 Usual transformations of the Wiener process
2.3.5 The general Wiener process
2.3.6 Stopping times
2.3.7 Properties of the Brownian motion paths

3 Stochastic integral and Itô’s lemma
3.1 Introduction
3.2 The stochastic integral
3.2.1 An intuitive approach
3.2.2 Counter-example
3.2.3 Definition and properties of the stochastic integral
3.2.4 Calculation rules
3.3 Itô’s lemma
3.3.1 Taylor’s formula, an intuitive approach to Itô’s lemma
3.3.2 Itô’s lemma
3.3.3 Applications
3.4 The Girsanov theorem
3.4.1 Preliminaries
3.4.2 Girsanov theorem
3.4.3 Application
3.5 Stochastic differential equations
3.5.1 Existence and unicity of solutions
3.5.2 A specific case: linear equations

Bibliography

Index