本书是一部金融数学名著，书中论述了随机分析理论及其与金融数学的关联性。目次如下：Gaussian随机变分；Greeks计算与分布积分公式；市场均衡与价格-挥发度反馈率；多元条件与分布律的正则化；非椭圆市场与HJM模型的不稳定性内部贸易；渐近展开与弱收敛跳跃市场的随机变分。附录：利用Fourier展式进行挥发评估；椭圆市场的Monte-Carlo强逼近；价格-挥发度反馈率的数值执行。

1 Gaussian Stochastic Calculus of Variations

 1.1 Finite-Dimensional Gaussian Spaces,Hermite Expansion

 1.2 Wiener Spaceas Limit of its Dyadic Filtration

 1.3 Stroock-Sobolev Spaces of Fnctionals on Wiener Space

 1.4 Divergence of Vector Fields,Integration by Parts

 1.5 Itб'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 Feedback 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 Transformand Multivariate Conditioning

 4.6 Example of the Univariate Conditioning

5 Non-Elliptic Marketsand 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 Matrixand Hбrmander Bracket Conditions

 5.4 Regularity by Predictable Smearing

 5.5 Forward Regularity by an Infinite-Dimensional Heat Equation

 5.6 Instability of Hedging Digital Optionsin HJM Models

 5.7 Econometric Observation of an Interest RateMarket

6 Insider Trading

 6.1 A Toy Model:the Brownian Bridge

 6.2 Information Driftand 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

8 Stochastic Calculus of Variations for Market swith 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

 8.4 Mean-Variance Minimal Hedging and Clark-Ocone Formula

A　Volatility Estimation by Fourier Expansion

B Strong Monte-Carlo Approximation

C Numerical Implementation of the Price-Volatility Feedback Rate

References

Index