金融數學中的隨機變分法-金融數學名著（簡體書）

stochaLstic Calculus of Variations(or Malliavin Calculus)consists，in brief，in constructing and exploiting natural differentiable structures on abstract Drobability spaces；in other words，Stochastic Calculus of Variations proceeds from a merging of differential calculus and probability theory．As optimization under a random environment iS at the heart of mathemat’ical finance，and as differential calculus iS of paramount importance for the search of extrema，it is not surprising that Stochastic Calculus of Variations appears in mathematical finance．The computation of price sensitivities(orGreeksl obviously belongs to the realm of differential calculus．
Nevertheless，Stochastic Calculus of Variations Was introduced relatively late in the mathematical finance literature：first in 1991 with the Ocone-Karatzas hedging formula，and soon after that，many other applications alDeared in various other branches of mathematical finance；in 1999 a new irapetus came from the works of P．L．Lions and his associates．

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 Fnctionals on Wiener Space
1.4 Divergence of Vector Fields, Integration by Parts
1.5 ItSs 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 Smearin
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 HSrmander 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
8 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
C Numerical Implementation
References
Index