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