金融數(shù)學(xué)中的隨機(jī)變分法

內(nèi)容概要

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．

書籍目錄

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  ItS's Theory of Stochastic Integrals  1.6  Differential and Integral Calculus in Chaos Expansion  1.7  Monte-Carlo Computation of Divergence2 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 Option3 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 Rate4 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 Conditioning5 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 Market6 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 Lunches7 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 Scheme8 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 ExpansionB  Strong Monte-Carlo ApproximationC  Numerical ImplementationReferencesIndex

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《金融數(shù)學(xué)中的隨機(jī)變分法(英文版)》是一部金融數(shù)學(xué)名著，書中論述了隨機(jī)分析理論及其與金融數(shù)學(xué)的關(guān)聯(lián)性。目次如下：Gaussian隨機(jī)變分；Greeks計算與分布積分公式；市場均衡與價格-揮發(fā)度反饋率；多元條件與分布律的正則化；非橢圓市場與HJM模型的不穩(wěn)定性內(nèi)部貿(mào)易；漸近展開與弱收斂跳躍市場的隨機(jī)變分。附錄：利用Fourier展式進(jìn)行揮發(fā)評估；橢圓市場的Monte-Carlo強(qiáng)逼近；價格-揮發(fā)度反饋率的數(shù)值執(zhí)行。

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