Stochastic Calculus of Variations in Mathematical Financ数学财务随机变分法
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作者： Paul Malliavin，Anton Thalmaier著
出 版 社： 湖南文艺出版社
出版时间： 2005-11-1字数：版次： 1页数： 142印刷时间： 2005/11/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540434313包装： 精装内容简介
Malliavin calculus provides an infinite-dimensional differential calculus in the context of continuous paths stochastic processes. The calculus includes formulae of integration by parts and Sobolev spaces of differentiable functions defined on a probability space. This new book, demonstrating the relevance of Malliavin calculus for Mathematical Finance, starts with an exposition from scratch of this theory. Greeks (price sensitivities) are reinterpreted in terms of Malliavin calculus. Integration by parts formulae provide stable Monte Carlo schemes for numerical valuation of digital options. Finite-dimensional projections of infinite-dimensional Sobolev spaces lead to Monte Carlo computations of conditional expectations useful for computing American options. The discretization error of the Euler scheme for a stochastic differential equation is expressed as a generalized Watanabe distribution on the Wiener space. Insider information is expressed as an infinite-dimensional drift. The last chapter gives an introduction to the same objects in the context of jump processes where incomplete markets appear.
目录
1 Gaussian Stochastic Calculus of Variations
1.1 Finite-Dimensional Gaussian Spaces, Hermite Expansion
1.2 Wiener Space as Limit of its Dyadic Filtration
1.3 Stroock-Sobolev Spaces of Functionals on Wiener Space
1.4 Divergence of Vector Fields, Integration by Parts
1.5 It6's Theory of Stochastic Integrals
1.6 Differential and Integral Calculus in Chaos Expansion
1.7 Monte-Carlo Computation of Divergence
2 Computation of Greeks and Integration by Parts Formulae
2.1 PDE Option Pricing; PDEs Governing the Evolution of Greeks
2.2 Stochastic Flow of Diffeomorphisms; Ocone-Karatzas Hedging
2.3 Principle of Equivalence of Instantaneous Derivatives
2.4 Pathwise Smearing for European Options
2.5 Examples of Computing Pathwise Weights
2.6 Pathwise Smearing for Barrier Option
3 Market Equilibrium and Price-Volatility Feedack Rate
3.1 Natural Metric Associated to Pathwise Smearing
3.2 Price-Volatility Feedback Rate
3.3 Measurement of the Price-Volatility Feedback Rate
3.4 Market Ergodicity and Price-Volatility Feedback Rate
4 Multivariate Conditioning and Regularity of Law
4.1 Non-Degenerate Maps
4.2 Divergences
4.3 Regularity of the Law of a Non-Degenerate Map
4.4 Multivariate Conditioning
4.5 Riesz Transform and Multivariate Conditioning
4.6 Example of the Univariate Conditioning
5 Non-Elliptic Markets and Instability in HJM Models
5.1 Notation for Diffusions on RN
5.2 The Malliavin Covariance Matrix of a Hypoelliptic Diffusion
5.3 Malliavin Covariance Matrix and Hormander Bracket Conditions
5.4 Regularity by Predictable Smearing
5.5 Forward Regularity by an Infinite-Dimensional Heat Equation
5.6 Instability of Hedging Digital Options in HJM Models
5.7 Econometric Observation of an Interest Rate Market
6 Insider Trading
6.1 A Toy Model: the Brownian Bridge
6.2 Information Drift and Stochastic Calculus of Variations
6.3 Integral Representation of Measure-Valued Martingales
6.4 Insider Additional Utility
6.5 An Example of an Insider Getting Free Lunches
7 Asymptotic Expansion and Weak Convergence
7.1 Asymptotic Expansion of SDEs Depending on a Parameter
7.2 Watanabe Distributions and Descent Principle
7.3 Strong Functional Convergence of the Euler Scheme
7.4 Weak Convergence of the Euler Scheme Stochastic Calculus of Variations for Markets with Jumps
8.1 Probability Spaces of Finite Type Jump Processes
8.2 Stochastic Calculus of Variations for Exponential Variables
8.3 Stochastic Calculus of Variations for Poisson Processes
……
A Volatility Estimation by Fourier Expansion
B Strong Monte-Carlo Approximation of an Elliptic Market
C Numerical Implementation of the Price-Volatillity Feedback Rate
References
Index
