Stochastic calculus of variations in mathematical finance /
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| Author / Creator: | Malliavin, Paul, 1925-2010 |
|---|---|
| Imprint: | Berlin ; New York, NY : Springer, c2006. |
| Description: | 1 online resource (xi, 142 p.) : ill. |
| Language: | English |
| Series: | Springer finance Springer finance. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/8877451 |
Table of Contents:
- 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. Ito'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 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 R[superscript N]
- 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
- 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
- 8.4. Mean-Variance Minimal Hedging and Clark-Ocone Formula
- A. Volatility Estimation by Fourier Expansion
- A.1. Fourier Transform of the Volatility Functor
- A.2. Numerical Implementation of the Method
- B. Strong Monte-Carlo Approximation of an Elliptic Market
- B.1. Definition of the Scheme [characters not reproducible]
- B.2. The Milstein Scheme
- B.3. Horizontal Parametrization
- B.4. Reconstruction of the Scheme [characters not reproducible]
- C. Numerical Implementation of the Price-Volatility Feedback Rate
- References
- Index