Malliavin calculus for Lévy processes with applications to finance

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Malliavin calculus for Lévy processes with applications to finance / Giulia Di Nunno, Bernt Øksendal, Frank Proske

版:

Corr. 2nd print

資料種別:

図書

出版情報:

Heidelberg : Springer, c2009

形態:

xiii, 417 p. ; 24 cm

シリーズ名:

Universitext <BA00021004>

著者名:

ISBN:

9783540785712 [354078571X]

書誌ID:

BB04624177

子書誌情報

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書誌詳細

注記:

Includes bibliographical references (p. 399-410) and index

主題:

Lévy processes; Malliavin calculus

言語:

英語

目次情報:

Introduction

The Continuous Case: Brownian Motion / Part I：

The Wiene-Ito Chaos Expansion / 1：

Iterated Ito Integrals / 1.1：

The Wiener-Ito Chaos Expansion / 1.2：

Exercises / 1.3：

The Skorohod Integral / 2：

Some Basic Properties of the Skorohod Integral / 2.1：

The Skorohod Integral as an Extension of the Ito Integral / 2.3：

Malliavin Derivative via Chaos Expansion / 2.4：

The Malliavin Derivative / 3.1：

Computation and Properties of the Malliavin Derivative / 3.2：

Chain Rules for Malliavin Derivative / 3.2.1：

Malliavin Derivative and Conditional Expectation / 3.2.2：

Malliavin Derivative and Skorohod Integral / 3.3：

Skorohod Integral as Adjoint Operator to the Malliavin Derivative / 3.3.1：

An Integration by Parts Formula and Closability of the Skorohod Integral / 3.3.2：

A Fundamental Theorem of Calculus / 3.3.3：

Integral Representations and the Clark-Ocone Formula / 3.4：

The Clark-Ocone Formula / 4.1：

The Clark-Ocone Formula under Change of Measure / 4.2：

Application to Finance: Portfolio Selection / 4.3：

Application to Sensitivity Analysis and Computation of the "Greeks" in Finance / 4.4：

White Noise, the Wick Product, and Stochastic Integration / 4.5：

White Noise Probability Space / 5.1：

The Wiener-Ito Chaos Expansion Revisited / 5.2：

The Wick Product and the Hermite Transform / 5.3：

Some Basic Properties of the Wick Product / 5.3.1：

Hermite Transform and Characterization Theorem for (S)* / 5.3.2：

The Spaces G and G* / 5.3.3：

The Wick Product in Terms of Iterated Ito Integrals / 5.3.4：

Wick Products and Skorohod Integration / 5.3.5：

The Hida-Malliavin Derivative on the Space [Omega] = S'(R) / 5.4：

A New Definition of the Stochastic Gradient and a Generalized Chain Rule / 6.1：

Calculus of the Hida-Malliavin Derivative and Skorohod Integral / 6.2：

Wick Product vs. Ordinary Product / 6.2.1：

Closability of the Hida-Malliavin Derivative / 6.2.2：

Wick Chain Rule / 6.2.3：

Integration by Parts, Duality Formula, and Skorohod Isometry / 6.2.4：

Conditional Expectation on (S)* / 6.3：

Conditional Expectation on G* / 6.4：

A Generalized Clark-Ocone Theorem / 6.5：

The Donsker Delta Function and Applications / 6.6：

Motivation: An Application of the Donsker Delta Function to Hedging / 7.1：

The Donsker Delta Function / 7.2：

The Multidimensional Case / 7.3：

The Forward Integral and Applications / 7.4：

A Motivating Example / 8.1：

The Forward Integral / 8.2：

Ito Formula for Forward Integrals / 8.3：

Relation Between the forward Integral and Skorohod Integral / 8.4：

Ito Formula for Skorohod Integrals / 8.5：

Application to Insider Trading Modeling / 8.6：

Markets with No Friction / 8.6.1：

Markets with Friction / 8.6.2：

The Discontinuous Case: Pure Jummp Levy Processes / 8.7：

A Short Introduction to Levy Processes / 9：

Basics on Levy Processes / 9.1：

The Ito Formula / 9.2：

The Ito Representation Theorem for Pure Jump Levy Processes / 9.3：

Application to Finance: Replicability / 9.4：

Skorohod Integrals / 9.5：

Definition and Basic Properties / 11.1：

Integration by Parts and Closability of the Skorohod Integral / 12.2：

Fundamental Theorem of Calculus / 12.3.3：

A Combination of Gaussian and Pure Jump Levy Noises / 12.4：

Application of Minimal Variance Hedging with Partial Information / 12.6：

Computation of "Greeks" in the Case of Jump Diffusions / 12.7：

The Barndorff-Nielsen and Shephard Model / 12.7.1：

Malliavin Weights for "Greeks" / 12.7.2：

Levy White Noise and Stochastic Distributions / 12.8：

The White Noise Probability Space / 13.1：

An Alternative Chaos Expansion and the White Noise / 13.2：

The Wick Product / 13.3：

Definition and Properties / 13.3.1：

Wick Product and Skorohod Integral / 13.3.2：

Levy-Hermite Transform / 13.3.3：

Spaces of Smooth and Generalized Random Variables: G and G* / 13.4：

The Malliavin Derivative on G* / 13.5：

A Generalization of the Clark-Ocone Theorem / 13.6：

A Combination of Gaussian and Pure Jump Levy Noises in the White Noise Setting / 13.7：

Generalized Chain Rules for the Malliavin Derivative / 13.8：

The Donsker Delta Function of a Levy Process and Applications / 13.9：

The Donsker Delta Function of a Pure Jump Levy Process / 14.1：

An Explicit Formula for the Donsker Delta Function / 14.2：

Chaos Expansion of Local Time for Levy Processes / 14.3：

Application to Hedging in Incomplete Markets / 14.4：

A Sensitivity Result for Jump Diffusions / 14.5：

A Representation Theorem for Functions of a Class of Jump Diffusions / 14.5.1：

Application: Computation of the "Greeks" / 14.5.2：

Definition of Forward Integral and its Relation with the Skorohod Integral / 14.6：

Ito Formula for Forward and Skorohod Integrals / 15.2：

Applications to Stochastic Control: Partial and Inside Information / 15.3：

The Importance of Information in Portfolio Optimization / 16.1：

Optimal Portfolio Problem under Partial Information / 16.2：

Formalization of the Optimization Problem: General Utility Function / 16.2.1：

Characterization of an Optimal Portfolio Under Partial Information / 16.2.2：

Examples / 16.2.3：

Optimal Portfolio under Partial Information in an Anticipating Environment / 16.3：

The Continuous Case: Logarithmic Utility / 16.3.1：

The Pure Jump Case: Logarithmic Utility / 16.3.2：

A Universal Optimal Consumption Rate for an Insider / 16.4：

Formalization of a General Optimal Consumption Problem / 16.4.1：

Characterization of an Optimal Consumption Rate / 16.4.2：

Optimal Consumption and Portfolio / 16.4.3：

Optimal Portfolio Problem under Inside Information / 16.5：

Characterization of an Optimal Portfolio under Inside Information / 16.5.1：

Examples: General Utility and Enlargement of Filtration / 16.5.3：

Optimal Portfolio Problem under Inside Information: Logarithmic Utility / 16.6：

The Pure Jump Case / 16.6.1：

A Mixed Market Case / 16.6.2：

Examples: Enlargement of Filtration / 16.6.3：

Regularity of Solutions of SDEs Driven by Levy Processes / 16.7：

The General Case / 17.1：

Absolute Continuity of Probability Laws / 17.3：

Existence of Densities / 18.1：

Smooth Densities of Solutions to SDE's Driven by Levy Processes / 18.2：

Malliavin Calculus on the Wiener Space / 18.3：

Preliminary Basic Concepts / A.1：

Wiener Space, Cameron-Martin Space, and Stochastic Derivative / A.2：

Malliavin Derivative via Chaos Expansions / A.3：

Solutions

References

Notation and Symbols

Index

Introduction

The Continuous Case: Brownian Motion / Part I：

The Wiene-Ito Chaos Expansion / 1：

Iterated Ito Integrals / 1.1：

The Wiener-Ito Chaos Expansion / 1.2：

Exercises / 1.3：

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