This volume presents classical results of the theory of enlargement of filtration. The focus is on the behavior of martingales with respect to the enlarged filtration and related objects. The study is conducted in various contexts including immersion, progressive enlargement with a random time and initial enlargement with a random variable.
The aim of this book is to collect the main mathematical results (with proofs) previously spread among numerous papers, great part of which is only available in French. Many examples and applications to finance, in particular to credit risk modelling and the study of asymmetric information, are provided to illustrate the theory. A detailed summary of further connections and applications is given in bibliographic notes which enables to deepen study of the topic.
This book fills a gap in the literature and serves as a guide for graduate students and researchers interested in the role of information in financial mathematics and in econometric science. A basic knowledge of the general theory of stochastic processes is assumed as a prerequisite.
Theory of Stochastic Processes.- Semimartingales.- Change of probability and Girsanov's Theorem.- Projections and Dual Projections.- Exercises .-Bibliographic.- Compensators of Random .- Compensator of a Default Indicator in its own Filtration.- Compensator of the Default Process in a General Setting .- Cox Processes and Extensions.- Study of Azéma's supermartingale in general setting.- Exercices .- Bibliographic Notes.-Immersion Property.- Immersion of Immersion in a Progressive Enlargement of Filtration.- Multidefaults Setting.-Exercices .- Bibliographic.- Initial Enlargement.- Brownian and Poisson Bridges.- Insider Trading.- Enlargement of Filtration setting.- Yor's Method.-Jacod's Absolute Continuity Condition.- Jacod's Equivalence Condition.- List of examples in the Literature.- Bibliographic Notes.- Progressive Enlargement.- G-semimartingale decomposition of F-martingales before t.- Honest Times.- (E)-times.- 5.4 Pseudo-stopping Times.- Predictable Representation property.-Enlargement with the filtration generated by a continuous process .- Arbitrages in a progressive Enlargement.- Applications of (E)-times to Finance.- Exercises.- Bibliographic Notes.- Solutions to some exercises.- Indexes.