Malliavin Calculus for Lévy Processes with Applications to Finance
Authors: Di Nunno, Giulia, Øksendal, Bernt, Proske, Frank

Springer-Verlag Berlin Heidelberg 2009


Covers Malliavin Calculus for both Brownian noise and Lévy type of noise
Presents applications to mathematical finance
Examines a new development of anticipating calculus

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

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.

From the reviews: “The book under review gives a quite complete description of the Malliavin and white noise approaches to stochastic analysis on both the Wiener and Poisson spaces with applications to mathematical finance. … In addition each chapter is accompanied with exercises and their solutions. … The technical requirements of the book are kept at a reasonable level and its organisation into short chapters not only facilitates the reading but also provides several alternative study plans making it a valuable learning and reference tool.” (Nicolas Privault, Mathematical Reviews, Issue 2010 f)