Malliavin Calculus for Lévy Processes with Applications to Finance

While the original works on Malliavin calculus aimed to study the smoothness of densities of solutions to stochastic differential equations, this book has another goal. It portrays the most important and innovative applications in stochastic control and finance, such as hedging in complete and incomplete markets, optimisation in the presence of asymmetric information and also pricing and sensitivity analysis. In a self-contained fashion, both the Malliavin calculus with respect to Brownian motion and general Lévy type of noise are treated.

Besides, forward integration is included and indeed extended to general Lévy processes. The forward integration is a recent development within anticipative stochastic calculus that, together with the Malliavin calculus, provides new methods for the study of insider trading problems.

To allow more flexibility in the treatment of the mathematical tools, the generalization of Malliavin calculus to the white noise framework is also discussed.

This book is a valuable resource for graduate students, lecturers in stochastic analysis and applied researchers.

Malliavin Calculus is presented for both Brownian noise and Lévy type of noise Presents applications to mathematical finance New development of anticipating calculus Includes supplementary material: sn.pub/extras

Auteur

Giulia Di Nunno, Bernt Øksendal and Frank Proske are professors at the Department of Mathematics, University of Oslo, Norway. The three scholars are active in the fields of stochastic analysis, mathematical and quantitative finance.

Résumé

This book is an introduction to Malliavin calculus as a generalization of the classical non-anticipating Ito calculus to an anticipating setting. It presents the development of the theory and its use in new fields of application.

Contenu
The Continuous Case: Brownian Motion.- The WienerItô Chaos Expansion.- The Skorohod Integral.- Malliavin Derivative via Chaos Expansion.- Integral Representations and the ClarkOcone formula.- White Noise, the Wick Product, and Stochastic Integration.- The HidaMalliavin Derivative on the Space ? = S?(?).- The Donsker Delta Function and Applications.- The Forward Integral and Applications.- The Discontinuous Case: Pure Jump Lévy Processes.- A Short Introduction to Lévy Processes.- The WienerItô Chaos Expansion.- Skorohod Integrals.- The Malliavin Derivative.- Lévy White Noise and Stochastic Distributions.- The Donsker Delta Function of a Lévy Process and Applications.- The Forward Integral.- Applications to Stochastic Control: Partial and Inside Information.- Regularity of Solutions of SDEs Driven by Lévy Processes.- Absolute Continuity of Probability Laws.