レヴィ過程のためのマリアバン解析とファイナンスへの応用(テキスト) Malliavin Calculus for Levy Processes with Applications to Finance (Universitext)

DiNunno, Giulia/ Oksendal, Bernt/ Proske, Frank

Springer（2008発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 350 p.
• 商品コード 9783540785712

基本説明

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.

Full Description

There are already several excellent books on Malliavin calculus. However, most of them deal only with the theory of Malliavin calculus for Brownian motion, with [35] as an honorable exception. Moreover, most of them discuss only the applicationto regularityresults for solutions ofSDEs, as this wasthe original motivation when Paul Malliavin introduced the in?nite-dimensional calculus in 1978 in [158]. In the recent years, Malliavin calculus has found many applications in stochastic control and within ?nance. At the same time, L' evy processes have become important in ?nancial modeling. In view of this, we have seen the need for a book that deals with Malliavin calculus for L' evy processesin general,not just Brownianmotion, and that presentssome of the most important and recent applications to ?nance. It is the purpose of this book to try to ?ll this need. In this monograph we present a general Malliavin calculus for L' evy processes, covering both the Brownianmotioncaseand the purejump martingalecasevia Poissonrandom measures,and also some combination of the two.

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 Stochastic 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 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.