Malliavin Calculus for L�vy Processes with Applications to Finance

by Giulia Di Nunno, Bernt ïksendal, Frank Proske

Overview

While the original works on Malliavin calculus aimed to study the smoothness of densities of solutions to shastic differential equations, this book has another goal. It portrays the most important and innovative applications in shastic 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 shastic 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 shastic analysis and applied researchers.

Product Details

ISBN-13:	9783540785712
Publisher:	Springer Berlin Heidelberg
Publication date:	12/12/2008
Series:	Universitext
Edition description:	2009
Pages:	418
Product dimensions:	6.10(w) x 9.30(h) x 1.00(d)

About the Author

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 shastic analysis, mathematical and quantitative finance.

Table of Contents

The Continuous Case: Brownian Motion.- The Wiener—Itô Chaos Expansion.- The Skorohod Integral.- Malliavin Derivative via Chaos Expansion.- Integral Representations and the Clark—Ocone formula.- White Noise, the Wick Product, and Shastic Integration.- The Hida—Malliavin 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 Wiener—Itô Chaos Expansion.- Skorohod Integrals.- The Malliavin Derivative.- Lévy White Noise and Shastic Distributions.- The Donsker Delta Function of a Lévy Process and Applications.- The Forward Integral.- Applications to Shastic Control: Partial and Inside Information.- Regularity of Solutions of SDEs Driven by Lévy Processes.- Absolute Continuity of Probability Laws.

Editorial Reviews

From the reviews: “The book under review gives a quite complete description of the Malliavin and white noise approaches to shastic 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)

From the Publisher