Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

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Lévy Processes and Stochastic Calculus

This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation

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Quantitative Risk Management: Concepts, Techniques, and Tools

We study the implications of imperfect information for term structures of credit spreads on corporate bonds. We suppose that bond investors cannot observe the issuer's assets directly, and receive instead only periodic and imperfect accounting reports. For a setting in which the assets of the firm are a geometric Brownian motion until informed equityholders optimally liquidate, we derive the conditional distribution of the assets, given accounting data and survivorship. Contrary to the perfect-information case, there exists a default-arrival intensity process. That intensity is calculated in terms of the conditional distribution of assets. Credit yield spreads are characterized in terms of accounting information. Generalizations are provided.

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Term Structures of Credit Spreads with Incomplete Accounting Information

The main objective of Credit Risk: Modeling, Valuation and Hedging is to present a comprehensive survey of the past developments in the area of credit risk research, as well as to put forth the most recent advancements in this field. An important aspect of this text is that it attempts to bridge the gap between the mathematical theory of credit risk and the financial practice, which serves as the motivation for the mathematical modeling studied in the book. Mathematical developments are presented in a thorough manner and cover the structural (value-of-the-firm) and the reduced (intensity-based) approaches to credit risk modeling, applied both to single and to multiple defaults. In particular, the book offers a detailed study of various arbitrage-free models of defaultable term structures with several rating grades. This volume will serve as a valuable reference for financial analysts and traders involved with credit derivatives. Some aspects of the book may also be useful for market practitioners engaged in managing credit-risk sensitive portfolios. Graduate students and researchers in areas such as finance theory, mathematical finance, financial engineering and probability theory will benefit from the book as well. On the technical side, readers are assumed to be familiar with graduate level probability theory, theory of stochastic processes, and elements of stochastic analysis and PDEs some aquaintance with arbitrage pricing theory is also expected. A systematic exposition of mathematical techniques underlying the intensity-based approach is however provided.

https://www.springer.com/productFlyer_978-3-540-67593-8.pdf?SGWID=0-0-1297-2053538-0

Credit Risk: Modeling, Valuation and Hedging

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

Stochastic processes of common use in mathematical finance are presented throughout this book, which consists of eleven chapters, interlacing on the one hand financial concepts and instruments, such as arbitrage opportunities, admissible strategies, contingent claims, option pricing, default risk, ruin, and on the other hand, Brownian motion, diffusion processes, Levy processes, together with the basic properties of these processes. The first half of the book is devoted to continuous path processes whereas the second half deals with discontinuous processes.

Only basic knowledge of probability theory is assumed; the book is organized so that the mathematical facts pertaining to a given financial question are gathered close to the study of that question.

Mathematical Methods for Financial Markets

We rst discuss some mathematical tools used to compute the intensity of a single jump process, in its canonical ltration. In the second part, we try to clarify the meaning of default and the links between the default time, the asset’s ltration, and the intensity of the default time. We nally discuss some examples.

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On models of default risk

In this work, we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process. We use backward stochastic differential equations (BSDEs) and Snell envelopes. Finally, we give some numerical results.

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On the Starting and Stopping Problem: Application in Reversible Investments

We provide a concise exposition of theoretical results that appear in modeling default time as a random time, we study in details the invariance martingale property and we establish a representation theorem which leads, in a complete market setting, to the hedging portfolio of a vulnerable claim. Our main result is that, to hedge a defaultable claim one has to invest the value of this contingent claim in the defaultable zero-coupon.

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Hazard rate for credit risk and hedging defaultable contingent claims

https://www.maths.univ-evry.fr/pages_perso/jeanblanc/pubs/ejl_guaran.pdf

Monique Jeanblanc

Papers

Mathematical Methods for Financial Markets

On models of default risk

On the Starting and Stopping Problem: Application in Reversible Investments

Hazard rate for credit risk and hedging defaultable contingent claims

Optimal portfolio management with American capital guarantee