Malliavin Calculus for Lévy Processes with Applications to Finance

1. Front Matter

   Pages i-xvi

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2. The Continuous Case: Brownian Motion

   1. The Wiener—Itô Chaos Expansion

      Pages 7-18

   2. The Skorohod Integral

      Pages 19-26

   3. Malliavin Derivative via Chaos Expansion

      Pages 27-41

   4. Integral Representations and the Clark—Ocone formula

      Pages 43-62

   5. White Noise, the Wick Product, and Stochastic Integration

      Pages 63-85

   6. The Hida—Malliavin Derivative on the Space Ω = S′(ℝ)

      Pages 87-110

   7. The Donsker Delta Function and Applications

      Pages 111-130

   8. The Forward Integral and Applications

      Pages 131-158

3. The Discontinuous Case: Pure Jump Lévy Processes

   1. A Short Introduction to Lévy Processes

      Pages 161-175

   2. The Wiener—Itô Chaos Expansion

      Pages 177-182

   3. Skorohod Integrals

      Pages 183-186

   4. The Malliavin Derivative

      Pages 187-214

   5. Lévy White Noise and Stochastic Distributions

      Pages 215-242

   6. The Donsker Delta Function of a Lévy Process and Applications

      Pages 243-265

   7. The Forward Integral

      Pages 267-274

   8. Applications to Stochastic Control: Partial and Inside Information

      Pages 275-335

   9. Regularity of Solutions of SDEs Driven by Lévy Processes

      Pages 337-343

   10. Absolute Continuity of Probability Laws

       Pages 345-351

4. Back Matter

   Pages 353-417

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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.

• Brownian motion
• Levy processes
• Lévy process
• Malliavin calculus
• Stochastic Differential Equations
• Stochastic calculus
• asymmetric information
• calculus
• optimization
• stochastic control
• white noise
• 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)

• Department of Mathematics, University of Oslo, Oslo, Norway

  Giulia Di Nunno, Bernt Øksendal, Frank Proske

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.

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