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.

This volume is an original collection of articles by 44 leading mathematicians on the theme of the future of the discipline. The contributions range from musings on the future of specific fields, to analyses of the history of the discipline, to discussions of open problems and conjectures, including first solutions of unresolved problems. Interestingly, the topics do not cover all of mathematics, but only those deemed most worthy to reflect on for future generations. These topics encompass the most active parts of pure and applied mathematics, including algebraic geometry, probability, logic, optimization, finance, topology, partial differential equations, category theory, number theory, differential geometry, dynamical systems, artificial intelligence, theory of groups, mathematical physics and statistics.

Modelling with the Ito integral or stochastic differential equations has become increasingly important in various applied fields, including physics, biology, chemistry and finance. However, stochastic calculus is based on a deep mathematical theory. This book is suitable for the reader without a deep mathematical background. It gives an elementary introduction to that area of probability theory, without burdening the reader with a great deal of measure theory. Applications are taken from stochastic finance. In particular, the Black -- Scholes option pricing formula is derived. The book can serve as a text for a course on stochastic calculus for non-mathematicians or as elementary reading material for anyone who wants to learn about Ito calculus and/or stochastic finance.

This collection of selected, revised and extended contributions resulted from a Workshop on BSDEs, SPDEs and their Applications that took place in Edinburgh, Scotland, July 2017 and included the 8th World Symposium on BSDEs. The volume addresses recent advances involving backward stochastic differential equations (BSDEs) and stochastic partial differential equations (SPDEs). These equations are of fundamental importance in modelling of biological, physical and economic systems, and underpin many problems in control of random systems, mathematical finance, stochastic filtering and data assimilation. The papers in this volume seek to understand these equations, and to use them to build our understanding in other areas of mathematics. This volume will be of interest to those working at the forefront of modern probability theory, both established researchers and graduate students.

This volume includes selected technical papers presented at the Forum “Math-for-Industry” 2018. The papers written by eminent researchers and academics working in the area of industrial mathematics from the viewpoint of financial mathematics, machine learning, neural networks, inverse problems, stochastic modelling, etc., discuss how the ingenuity of science, technology, engineering and mathematics are and will be expected to be utilized. This volume focuses on the role that mathematics-for-industry can play in interdisciplinary research to develop new methods. The contents are useful for researchers both in academia and industry working in interdisciplinary sectors.

Financial Markets in Practice: From Post-Crisis Intermediation to FinTechs delivers an overview of the development of risk-transformation undertaken by the financial services industry from the perspective of quantitative finance. It provides an instructional and comprehensive explanation of the structure of the financial system as a network of risk suppliers and risk consumers, where different categories of market participants buy, transform, net, and re-sell different kinds of risks. This risk-transformation oriented view is supported by the changes that followed the last global financial crisis: consumers of financial products asked for less complex risk transformations, regulators demanded limiting risks inside financial institutions to the maximum extent possible, and market participants turned to run mass market-like businesses and away from bespoke 'haute couture'-like businesses.This book portrays the network of intermediaries that compose the financial system, describes their most common business models, explains the exact role of each kind of market participant, and underlines the interaction between them. It seeks to reveal the potential disintermediation that could occur inside the financial sector, led by FinTechs and Artificial Intelligence-based innovations.Readers are invited to reconsider the role of market participants in the post-crisis world and are prepared for the next wave of changes driven by data science, AI, and blockchain. Amid these innovations, quantitative finance will be increasingly involved in all aspects of the financial system. This handy resource helps practitioners from both the buy-side and sell-side gain insights to, and provides an overview of, business models in the financial system from an intermediation perspective, and guides students to comprehensively understand the complex ecosystem in which they will evolve.

This milestone 50th volume of the "Séminaire de Probabilités" pays tribute with a series of memorial texts to one of its former editors, Jacques Azéma, who passed away in January. The founders of the "Séminaire de Strasbourg", which included Jacques Azéma, probably had no idea of the possible longevity and success of the process they initiated in 1967. Continuing in this long tradition, this volume contains contributions on state-of-art research on Brownian filtrations, stochastic differential equations and their applications, regularity structures, quantum diffusion, interlacing diffusions, mod-Ø convergence, Markov soup, stochastic billiards and other current streams of research.