Stochastic Processes for Finance

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 theoremPreliminaries Girsanov theorem Application Stochastic differential equations Existence and unicity of solutions A specific case: linear equations Bibliography Index