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1.

図書

図書
M.N. Feller
出版情報: Cambridge, UK ; New York : Cambridge University Press, 2005  vi, 153 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 166
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Introduction
The Lévy Laplacian / 1:
Lévy-Laplace operators / 2:
Symmetric Lévy-Laplace operators / 3:
Harmonic functions of infinitely many variables / 4:
Linear elliptic and parabolic equations with Lévy Laplacians / 5:
Quasilinear and nonlinear elliptic equation with Lévy Laplacians / 6:
Nonlinear parabolic equations with Lévy Laplacians / 7:
Appendix. LévyâÇôDirichlet forms and associated Markov processes / 8:
Bibliography
Index
Introduction
The Lévy Laplacian / 1:
Lévy-Laplace operators / 2:
2.

図書

図書
Jean Bertoin
出版情報: Cambridge [Eng.] : Cambridge University Press, 1996  x, 265 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 121
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3.

図書

図書
Horst Osswald
出版情報: Cambridge : Cambridge University Press, 2012  xix, 407 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 191
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The Fundamental Principles / Part I:
Preface / 1:
Martingales / 2:
Fourier and Laplace transformations / 3:
Abstract Wiener-Fréchet spaces / 4:
Two concepts of no-anticipation in time / 5:
Malliavin calculus on the space of real sequences / 6:
Introduction to poly-saturated models of mathematics / 7:
Extension of the real numbers and properties / 8:
Topology / 9:
Measure and integration on Loeb spaces / 10:
An Introduction to Finite- and Infinite-Dimensional Stochastic Analysis / Part II:
From finite- to infinite-dimensional Brownian motion / 11:
The Itô integral for infinite-dimensional Brownian motion / 12:
The iterated integral / 13:
Infinite-dimensional Ornstein-Uhlenbeck processes / 14:
Lindstrøm's construction of standard Lévy processes from discrete ones / 15:
Stochastic integration for Lévy processes / 16:
Malliavin Calculus / Part III:
Chaos decomposition / 17:
The Malliavin derivative / 18:
The Skorokhod integral / 19:
The interplay between derivative and integral / 20:
Skorokhod integral processes / 21:
Girsanov transformation / 22:
Malliavin calculus for Lévy processes / 23:
Poly-saturated models / Appendix A:
The existence of poly-saturated models / Appendix B:
References
Index
The Fundamental Principles / Part I:
Preface / 1:
Martingales / 2:
4.

図書

図書
Giulia Di Nunno, Bernt Øksendal, Frank Proske
出版情報: Heidelberg : Springer, c2009  xiii, 417 p. ; 24 cm
シリーズ名: Universitext
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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:
5.

図書

図書
Ming Liao
出版情報: Cambridge : Cambridge University Press, 2004  x, 266 p. ; 24 cm
シリーズ名: Cambridge tracts in mathematics ; 162
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