由纳努著作的《Lévy过程的Malliavin分析及其在金融学中的应用》旨在描述其在随机控制和金融中最重要和新具有创造性的应用，例如在完全市场和和不完全市场中的对冲、非对称信息出现的优化和定价与敏感分析中。《levy过程的malliavin分析及其在金融学中的应用》自称体系，malliavin 微积分的布朗运动和一般的噪音levy型都讲述的比较清楚；另外，包含了向前积分和延伸到一般levy过程。向前积分是在预期随机微积分中发展起来的，和malliavin 微积分一起提供了研究内部贸易问题的新方法。为了让数学工具的处理更具有弹性，讨论了从malliavin微积分到白噪音框架的所有问题。本书既是研究生学习的很好的资料、也是随机分析和应用研究人员很好的讲义。

Introduction

Part I The Continuous Case：Brownian Motion

1 The Wiener-It5 Chaos Expansion

1．1 Iterated It6 Integrals

1．2 The Wiener—It6 Chaos Expansion

1．3 Exercises

2 The Skorohod Integral

2．1 The Skorohod Integral

2．2 Some Basic Properties of the Skorohod Integral

2．3 The Skorohod Integral as an Extension of the It6 Integral

2．4 Exercises

3 Malliavin Derivative via Chaos Expansion

3．1 The Malliavin Derivative

3．2 Computation and Properties of the Malliavin Derivative

3．2．1 Chain Rules for Malliavin Derivative

3．2．2 Malliavin Derivative and Conditional Expectation

3．3 Malliavin Derivative and Skorohod Integral

3．3．1 Skorohod Integral as Adjoint Operator to the

Malliavin Derivative

3．3．2 An Integration by Parts Formula and Closability

of the Skorohod Integral

3．3．3 A Fundamental Theorem of Calculus

 3．4 Exercises

4 Integral Representations and the Clark-Ocone Formula

4．1 The Clark—Ocone Formula

4．2 The Clark-Ocone Formula under Change of Measure

4．3 Application to Finance：Portfolio Selection

4．4 Application to Sensitivity Analysis and Computation

of the“Greeks”in Finance

 4．5 Exercises

5 White Noise，the Wick Product，and Stochastic

 Integration

 5．1 White Noise Probability Space

 5．2 The Wiener-It6 Chaos Expansion Revisited

 5．3 The Wick Product and the Hermite Transform

5．3．1 Some Basic Properties ofthe Wick Product

5．3．2 Hermite Transform and Characterization

Theorem for for(S)

5．3．3 The Spaces 9 and 9

5．3．4 The Wick Product in Terms of Iterated It6 Integrals

5．3．5 Wick Products and Skorohod Integration

5．4 Exercises

6 The Hida—Malliavin Derivative on the Space

6．1 A New Definition ofthe Stochastic Gradient and a Generalized

Chain Rule

6．2 Calculus of the Hida—Malliavin Derivative

and Skorohod Integral

6．2．1 Wick Product VS．Ordinary Product

6．2．2 Closability of the Hida—Malliavin Derivative

6．2．3 Wick Chain Rule

6．2．4 Integration by Parts，Duality Formula

and Skorohod Isometry

6．3 Conditional Expectation on(S)

6．4 Conditional Expectation on g

6．5 A Generalized Clark-Ocone Theorem

6．6 Exercises

7 The Donsker Delta Function and Applications

7．1 Motivation：An Application of the Donsker Delta Function

tO Hedging

7．2 The Donsker Delta Function

7．3 The Multidimensional Case

7．4 Exercises

8 The Forward Integral and Applications

8．1 A Motivating Example

8．2 The Forward Integral

8．3 It6 Formula for Forward Integrals

8．4 Relation Between the Forward Integral

and the Skorohod Integral

8．5 It5 Formula for Skorohod Integrals

8．6 Application to Insider Trading Modeling

8．6．1 Markets W|th NO Friction

8．6．2 Markets With Friction

 8．7 Exercises

Part II The Discontinuous Case：Pure Jump L6vy Processes

9 A Short Introduction to L6vy Processes

9．1 Basics on L6vy Processes．

9．2 The It6 Formula

9．3 The It6 Representation Theorem for Pure Jump

L6vv Processes

 9．4 Application to Finance：Replicability

 9．5 Exercises

10 The Wiener-It6 Chaos Expansion

10．1 Iterated It6 Integrals

10．2 The Wiener—It6 Chaos Expansion

10．3 Exercises

11 Skorohod Integrals

11．1 TIle Skorohod Integral

1 1．2 The Skorohod Integral aS an Extension of the It6 Integral

11．3 Exercises

12 The Malliavin Derivative

12．1 Definition and Basic Properties

12．2 Chain Rules for Malliavin Derivative

12．3 Malliavin Derivative and Skorohod Integral

12．3．1 Skorohod Integral as Adjoint Operator

to the Malliavin Derivative

12．3．2 Integration by Parts and Closability

of the Skorohod Integral

12．3．3 Fundamental Theorem of Calculus

12．4 The Clark_Ocone Formula

12．5 A Combination of Gaussian and Pure Jump L6vy Noises

12．6 Application to Minimal Variance Hedging with Partial

Information

12．7 Computation of“Gteeks”in the Case of Jump Diflusions

12．7．1 The Barndorff一Nielsen and Shephard Model

12．7．2 Malliavin Weights for“Greeks”

 12．8 Exercises

13 L@vy White Noise and Stochastic Distributions

13．1 The White Noise Probability Space

13．2 An Alternative Chaos Expansion and the White Noise

13．3 The Wick Product

13．3．1 Definition and Properties．

13．3．2 Wick Product and Skorohod Integral

13．3．3 Wick Product VS．Ordinary Product

13．3．4 Ldvy—Hermite Transform

13．4 Spaces of Smooth and Generalized Random Variables：

g and g

13．5 The Malliavin Derivative on g

13．6 A GeneraIization of the Clark—Ocone Theorem

13．7 A Combination of Gaussian and Pure Jump Ldvy Noises

in the White Noise Setting

1 3．8 Generalized Chain Rules for the Malliavin Derivative

13．9 Exercises

14 The Donsker Delta Function of a L@vy Process

and AppIications

14．1 The Donsker Delta Function of a Pure Jump L@vy Process

14．2 An Explicit Formula for the Donsker Delta Function

14．3 Chaos Expansion ofLocal Time for L6vv Processes

14．4 Application to Hedging in Incomplete Markets

14．5 A Sensitivity Resuit for Jump Diffusions

14．5．1 A Representation Theorem for Functions

of a Class of Jump Di肋sions

14．5．2 Application：Computation of the“Greeks”

 14．6 Exercises

15 The Forward Integral

15．1 Definition of Forward Integral and its Relation

with the Skorohod Integral

15．2 It6 Formula for Forward and Skorohod Integrals

15．3 Exercises

16 Applications to Stochastic Control：Partial

and Inside Information

1 6．1 The Importance of Information in Portfolio Optimization

16．2 Optimal Portfolio Problem under Partial：nformation

16．2．1 F0rmalization of the Optimization Problem：

General Utility Function

16．2．2 Characterization of an Optimal Portfolio

Under Partial Information

16．2．3 Examples

16．3 Optimal Portfolio under Partial Information

in an Anticipating Environment

16．3．1 The Continuous Cuse：Logarithmic Utility

16．3．2 The Pure Jump Cuse：Logarithmic Utility

16．4 A Universal Optimal Consumption Rate for an Insider

16．4．1 F0rmalization of a General Optimal

Consumption Problem

16．4．2 Characterization of an Optimal Consumption Rate

16．4．3 Optimal Consumption and Portfolio

16．5 Optimal Portfolio Problem under Inside Information

16．5．1 F0rmalization of the Optimization Problem：

General Utility Function

16．5．2 Characterization of an Optimal Portfolio

under Inside Information

16．5．3 Examples：General Utility and Enlargement

of Filtration

16．6 Optimal Portfolio Problem under Inside Information：

Logarithmic Utility

16．6．1 The Pure Jump Cuse

16．6．2 A Mixed Market Cuse

16．6．3 Examples：Enlargement ofFiltration．

 16．7 Exercises

1 7 Regularity of Solutions of SDEs Driven

by Ldvy Processes

17．1 The Pure Jump Case

17．2 The General Cuse

17．3 Exercises

18 Absolute Continuity of Probability Laws

18．1 Existence of Densities

18．2 Smooth Densities of Solutions to SDE’S Driven

by Ldvy Processes

 18．3 Exercises

Appendix A：Malliavin Calculus on the Wiener Space

A．1 Preliminary Busic Concepts

A．2 Wiener Space，Cameron—Martin Space

and Stochastic Derivative

A．3 Malliavin Derivative via Chaos Expansions

Solutions

References

Notation and Symbols

Index