Mathematics of the Bond Market A Lévy Processes Approach

Mathematical models of bond markets are of interest to researchers working in applied mathematics, especially in mathematical finance. This book concerns bond market models in which random elements are represented by Lévy processes. These are more flexible than classical models and are well suited to describing prices quoted in a discontinuous fashion. The book's key aims are to characterize bond markets that are free of arbitrage and to analyze their completeness. Nonlinear stochastic partial differential equations (SPDEs) are an important tool in the analysis. The authors begin with a relatively elementary analysis in discrete time, suitable for readers who are not familiar with finance or continuous time stochastic analysis. The book should be of interest to mathematicians, in particular to probabilists, who wish to learn the theory of the bond market and to be exposed to attractive open mathematical problems.

Introduction; Part I. Bond Market in Discrete Time: 1. Elements of the bond market; 2. Arbitrage-free bond markets; 3. Completeness; Part II. Fundamentals of Stochastic Analysis: 4. Stochastic preliminaries; 5. Lévy processes; 6. Martingale representation and Girsanov's theorems; Part III. Bond Market in Continuous Tme: 7. Fundamentals; 8. Arbitrage-free HJM markets; 9. Arbitrage-free factor forward curves models; 10. Arbitrage-free affine term structure; 11. Completeness; Part IV. Stochastic Equations in the Bond Market: 12. Stochastic equations for forward rates; 13. Analysis of the HJMM equation; 14. Analysis of Morton's equation; 15. Analysis of the Morton–Musiela equation; Appendix A. Martingale representation for jump Lévy processes; Appendix B. Semigroups and generators; Appendix C. General evolution equations; References; Index.

Michal Barski is Professor of Mathematics at the University of Warsaw. His interests include mathematical finance, especially bond market and risk measures. In the years 2011–2016 he held the position of Junior-Professor in Stochastic Processes and their Applications in Finance at the University of Leipzig.
Jerzy Zabczyk is Professor Emeritus in the Institute of Mathematics at the Polish Academy of Sciences. His research interests include stochastic processes, evolution equations, control theory and mathematical finance. He published over ninety research papers. He is the author or co-author of seven books including Stochastic Equations in Infinite Dimensions (Cambridge, 1992, 2008, 2014), Stochastic Partial Differential Equations with Lévy Noise (Cambridge, 2007) and Mathematical Control Theory: An Introduction (1992, 1996, 2020).