Delving into the interplay between mathematics and the concept of error, this book explores how errors influence various fields, from science to philosophy. It examines the historical development of error analysis and its significance in mathematical theories and real-world applications. Through engaging examples and thought-provoking discussions, readers gain insight into the nature of uncertainty and the role it plays in shaping knowledge and understanding. The work challenges conventional perceptions of accuracy and invites a deeper reflection on the mathematics of imperfection.
This richly illustrated book is an exploration of how chance and risk, on the one hand, and meaning or significance on the other, compete for the limelight in art, in philosophy, and in science. In modern society, prudence and probability calculation permeate our daily lives. Yet it is clear for all to see that neither cautious bank regulations nor mathematics have prevented economic crises from occurring time and again. Nicolas Bouleau argues that it is the meaning we assign to an event that determines the perceived risk, and that we generally turn a blind eye to this important fact, because the word „meaning“ is itself awkward to explain. He tackles this fundamental question through examples taken from cultural fields ranging from painting, architecture, and music, to poetry, biology, and astronomy. This enables the reader to view overwhelming risks in a different light. Bouleau clarifies that the most important thing in a time of uncertainty is to think of prudence on a higher level, one that truly addresses the various subjective interpretations of the world.
Is it really possible to make money on the financial markets? This is just one of the questions posed in this practical and thought-provoking book, winner, in the original French version, of the "Best financial economics book" prize 1999 from the Institute de Haute Finance, and the "Prix FNAC-Arthur Anderson du meilleur livre d'entreprise 2000" Starting with games of chance, from which probability theory was born, Nicolas Bouleau explains how the financial markets operate, and demonstrates how the application of mathematics has turned finance into a high-tech business, as well as a formidable and efficient tool. The human side of finance is also considered, with a look at the influence of the trader and the working relationships that are woven into the market rooms. Concise and accessible, with no previous knowledge of finance or mathematics required, the aim of this book is simply to articulate the main ideas and put them into perspective, leading readers to a fresh understanding of this complex area.
Many recent advances in modelling within the applied sciences and engineering have focused on the increasing importance of sensitivity analyses. For a given physical, financial or environmental model, increased emphasis is now placed on assessing the consequences of changes in model outputs that result from small changes or errors in both the hypotheses and parameters. The approach proposed in this book is entirely new and features two main characteristics. Even when extremely small, errors possess biases and variances. The methods presented here are able, thanks to a specific differential calculus, to provide information about the correlation between errors in different parameters of the model, as well as information about the biases introduced by non-linearity. The approach makes use of very powerful mathematical tools (Dirichlet forms), which allow one to deal with errors in infinite dimensional spaces, such as spaces of functions or stochastic processes. The method is therefore applicable to non-elementary models along the lines of those encountered in modern physics and finance. This text has been drawn from presentations of research done over the past ten years and that is still ongoing. The work was presented in conjunction with a course taught jointly at the Universities of Paris 1 and Paris 6. The book is intended for students, researchers and engineers with good knowledge in probability theory.
This book analyzes Wiener space using Dirichlet forms and Malliavin calculus, offering distinct viewpoints compared to existing literature. The authors begin with a review of Dirichlet forms, focusing on functional analytic, potential theoretical, and algebraic properties, intentionally omitting connections to Markov processes or stochastic calculus typically found in other texts. Instead of the Beuring-Deny formula, they delve into "carré du champ" operators introduced by Meyer and Bakry, discussing their existence under challenging conditions, which they later verify for the Ornstein-Uhlenbeck operator in Wiener space. Notably, the existence of the "carré du champ" operator can be demonstrated more easily using Shigekawa’s H-derivative. In the Malliavin calculus section, the authors concentrate on the absolute continuity of the probability law of Wiener functionals, addressing the limitations of Dirichlet forms, which correspond only to first derivatives, thus focusing on the first step of Malliavin calculus. They tackle intricate issues, such as the absolute continuity of solutions to stochastic differential equations with Lipschitz continuous coefficients and the domain of stochastic integrals. While the book emphasizes the abstract structure of Dirichlet forms and Malliavin calculus over applications, it includes numerous exercises and references to aid readers in exploring related topics.
