John Wiley & Sons Inc., 2010. – 104 p. – ISBN: 9788776816667This 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.Contents:
Introduction
Discrete-time stochastic processes
Introduction
The general framework
Information revelation over time
Filtration on a probability space
Adapted and predictable processes
Markov chains
Introduction
Definition and transition probabilities
Chapman-Kolmogorov equations
Classification of states
Stationary distribution of a Markov
Martingales
Doob decomposition of an adapted
Martingales and self-financing strategies
Investment strategies and stopping times
Stopping times and American options
Continuous-time stochastic processes
Introduction
General framework
Filtrations, adapted and predictable processes
Markov and diffusion processes
Martingales
The Brownian motion
Intuitive presentation
The assumptions
Definition and general properties
Usual transformations of the Wiener process
The general Wiener process
Stopping times
Properties of the Brownian motion paths
Stochastic integral and Ito’s lemma
Introduction
The stochastic integral
An intuitive approach
Counter-example
Definition and properties of the stochastic integral
Calculation rules
Ito’s lemma
Taylor’s formula, an intuitive approach to Ito’s lemma
Ito’s lemma
Applications
The Girsanov theorem
Preliminaries
Girsanov theorem
Application
Stochastic differential equations
Existence and unicity of solutions
A specific case: linear equations
Bibliography
Index
