Stochastic calculus for fractional brownian motion and related processes /

Stochastic calculus for fractional brownian motion and related processes /

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Bibliographic Details
Author / Creator:Mishura, Yuliya.
Imprint:Berlin ; New York : Springer-Verlag, c2008.
Description:xv, 393 p. ; 24 cm.
Language:English
Series:Lecture notes in mathematics, 0075-8434 ; 1929
Lecture notes in mathematics (Springer-Verlag) ; 1929.
Subject:
Stochastic analysis.
Brownian motion processes -- Mathematical models.
Brownian movements -- Mathematical models.
Stochastic analysis.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/6655783
Hidden Bibliographic Details
ISBN:3540758720
9783540758723
Notes:Includes bibliographical references (p. [369]-389) and index.
Table of Contents:
  • 1. Wiener Integration with Respect to Fractional Brownian Motion
  • 1.1. The Elements of Fractional Calculus
  • 1.2. Fractional Brownian Motion: Definition and Elementary Properties
  • 1.3. Mandelbrot-van Ness Representation of fBm
  • 1.4. Fractional Brownian Motion with H ¿ (1/2, 1) on the White Noise Space
  • 1.5. Fractional Noise on White Noise Space
  • 1.6. Wiener Integration with Respect to fBm
  • 1.7. The Space of Gaussian Variables Generated by fBm
  • 1.8. Representation of fBm via the Wiener Process on a Finite Interval
  • 1.9. The Inequalities for the Moments of the Wiener Integrals with Respect to fBm
  • 1.10. Maximal Inequalities for the Moments of Wiener Integrals with Respect to fBm
  • 1.11. The Conditions of Continuity of Wiener Integrals with Respect to fBm
  • 1.12. The Estimates of Moments of the Solution of Simple Stochastic Differential Equations Involving fBm
  • 1.13. Stochastic Fubini Theorem for the Wiener Integrals w.r.t fBm
  • 1.14. Martingale Transforms and Girsanov Theorem for Long-memory Gaussian Processes
  • 1.15. Nonsemimartingale Properties of fBm; How to Approximate Them by Semimartingales
  • 1.15.1. Approximation of fBm by Continuous Processes of Bounded Variation
  • 1.15.2. Convergence B 2+H,ß → B 2+H in Besov Space W 2+¿ [a,b]
  • 1.15.3. Weak Convergence to fBm in the Schemes of Series
  • 1.16. Holder Properties of the Trajectories of fBm and of Wiener Integrals w.r.t. fBm
  • 1.17. Estimates for Fractional Derivatives of fBm and of Wiener Integrals w.r.t. Wiener Process via the Garsia-Rodemich-Rumsey Inequality
  • 1.18. Power Variations of fBm and of Wiener Integrals w.r.t fBm
  • 1.19. Levy Theorem for fBm
  • 1.20. Multi-parameter Fractional Brownian Motion
  • 1.20.1. The Main Definition
  • 1.20.2. Holder Properties of Two-parameter fBm
  • 1.20.3. Fractional Integrals and Fractional Derivatives of Two-parameter Functions
  • 2. Stochastic Integration with Respect to fBm and Related Topics
  • 2.1. Pathwise Stochastic Integration
  • 2.1.1. Pathwise Stochastic Integration in the Fractional Sobolev-type Spaces
  • 2.1.2. Pathwise Stochastic Integration in Fractional Besov-type Spaces
  • 2.2. Pathwise Stochastic Integration w.r.t Multi-parameter fBm
  • 2.2.1. Some Additional Properties of Two-parameter Fractional Integrals and Derivatives
  • 2.2.2. Generalized Two-parameter Lebesgue-Stieltjes Integrals
  • 2.2.3. Generalized Integrals of Two-parameter fBm in the Case of the Integrand Depending on fBm
  • 2.2.4. Pathwise Integration in Two-parameter Besov Spaces
  • 2.2.5. The Existence of the Integrals of the Second Kind of a Two-parameter fBm
  • 2.3. Wick Integration with Respect to fBm with H ¿ [1/2,1) as S* -integration
  • 2.3.1. Wick Products and S* -integration
  • 2.3.2. Comparison of Wick and Pathwise Integrals for "Markov" Integrands
  • 2.3.3. Comparison of Wick and Stratonovich Integrals for "General" Integrands
  • 2.3.4. Reduction of Wick Integration w.r.t. Fractional Noise to the Integration w.r.t. White Noise
  • 2.4. Skorohod, Forward, Backward and Symmetric Integration w.r.t. fBm. Two Approaches to Skorohod Integration
  • 2.5. Isometric Approach to Stochastic Integration with Respect to fBm
  • 2.5.1. The Basic Idea
  • 2.5.2. First- and Higher-order Integrals with Respect to X
  • 2.5.3. Generalized Integrals with Respect to fBm
  • 2.6. Stochastic Fubini Theorem for Stochastic Integrals w.r.t. Fractional Brownian Motion
  • 2.7. The Ito Formula for Fractional Brownian Motion
  • 2.7.1. The Simplest Version
  • 2.7.2. Ito Formula for Linear Combination of Fractional Brownian Motions with H i ¿ [1/2,1] in Terms of Pathwise Integrals and Itô Integral
  • 2.7.3. The Itô Formula in Terms of Wick Integrals
  • 2.7.4. The Itô Formula for H ¿ (0,1/2)
  • 2.7.5. Itô Formula for Fractional Brownian Fields
  • 2.7.6. The Itô Formula for H ¿ (0,1) in Terms of Isometric Integrals, and Its Applications
  • 2.8. The Girsanov Theorem for fBm and Its Applications
  • 2.8.1. The Girsanov Theorem for fBm
  • 2.8.2. When the Conditions of the Girsanov Theorem Are Fulfilled? Differentiability of the Fractional Integrals
  • 3. Stochastic Differential Equations Involving Fractional Brownian Motion
  • 3.1. Stochastic Differential Equations Driven by Fractional Brownian Motion with Pathwise Integrals
  • 3.1.1. Existence and Uniqueness of Solutions: the Results of Nualart and R&acaron;şcanu
  • 3.1.2. Norm and Moment Estimates of Solution
  • 3.1.3. Some Other Results on Existence and Uniqueness of Solution of SDE Involving Processes Related to fBm with (H ¿ (1/2,1))
  • 3.1.4. Some Properties of the Stochastic Differential Equations with Stationary Coefficients
  • 3.1.5. Semilinear Stochastic Differential Equations Involving Forward Integral w.r.t. fBm
  • 3.1.6. Existence and Uniqueness of Solutions of SDE with Two-Parameter Fractional Brownian Field
  • 3.2. The Mixed SDE Involving Both the Wiener Process and fBm
  • 3.2.1. The Existence and Uniqueness of the Solution of the Mixed Semilinear SDE
  • 3.2.2. The Existence and Uniqueness of the Solution of the Mixed SDE for fBm with H ¿ (3/4,1)
  • 3.2.3. The Girsanov Theorem and the Measure Transformation for the Mixed Semilinear SDE
  • 3.3. Stochastic Differential Equations with Fractional White Noise
  • 3.3.1. The Lipschitz and the Growth Conditions on the Negative Norms of Coefficients
  • 3.3.2. Quasilinear SDE with Fractional Noise
  • 3.4. The Rate of Convergence of Euler Approximations of Solutions of SDE Involving fBm
  • 3.4.1. Approximation of Pathwise Equations
  • 3.4.2. Approximation of Quasilinear Skorohod-type Equations
  • 3.5. SDE with the Additive Wiener Integral w.r.t. Fractional Noise
  • 3.5.1. Existence of a Weak Solution for Regular Coefficients
  • 3.5.2. Existence of a Weak Solution for SDE with Discontinuous Drift
  • 3.5.3. Uniqueness in Law and Pathwise Uniqueness for Regular Coefficients
  • 3.5.4. Existence of a Strong solution for the Regular Case
  • 3.5.5. Existence of a Strong Solution for Discontinuous Drift
  • 3.5.6. Estimates of Moments of Solutions for Regular Case and H ¿ (0,1/2)
  • 3.5.7. The Estimates of the Norms of the Solution in the Orlicz Spaces
  • 3.5.8. The Distribution of the Supremum of the Process X on [0,T]
  • 3.5.9. Modulus of Continuity of Solution of Equation Involving Fractional Brownian Motion
  • 4. Filtering in Systems with Fractional Brownian Noise
  • 4.1. Optimal Filtering of a Mixed Brownian-Fractional-Brownian Model with Fractional Brownian Observation Noise
  • 4.2. Optimal Filtering in Conditionally Gaussian Linear Systems with Mixed Signal and Fractional Brownian Observation Noise
  • 4.3. Optimal Filtering in Systems with Polynomial Fractional Brownian Noise
  • 5. Financial Applications of Fractional Brownian Motion
  • 5.1. Discussion of the Arbitrage Problem
  • 5.1.1. Long-range Dependence in Economics and Finance
  • 5.1.2. Arbitrage in "Pure" Fractional Brownian Model. The Original Rogers Approach
  • 5.1.3. Arbitrage in the "Pure" Fractional Model. Results of Shiryaev and Dasgupta
  • 5.1.4. Mixed Brownian-Fractional-Brownian Model: Absence of Arbitrage and Related Topics
  • 5.1.5. Equilibrium of Financial Market. The Fractional Burgers Equation
  • 5.2. The Different Forms of the Black-Scholes Equation
  • 5.2.1. The Black-Scholes Equation for the Mixed Brownian-Fractional-Brownian Model
  • 5.2.2. Discussion of the Place of Wick Products and Wick-Ito-Skorohod Integral in the Problems of Arbitrage and Replication in the Fractional Black-Scholes Pricing Model
  • 6. Statistical Inference with Fractional Brownian Motion
  • 6.1. Testing Problems for the Density Process for fBm with Different Drifts
  • 6.1.1. Observations Based on the Whole Trajectory with ¿ and H Known
  • 6.1.2. Discretely Observed Trajectory and ¿ Unknown
  • 6.2. Goodness-of-fit Test
  • 6.2.1. Introduction
  • 6.2.2. The Whole Trajectory Is Observed and the Parameters ¿ and ¿ Are Known
  • 6.2.3. Goodness-of-fit Tests with Discrete Observations
  • 6.2.4. On Volatility Estimation
  • 6.2.5. Goodness-of-fit Test with Unknown ¿ and ¿
  • 6.3. Parameter Estimates in the Models Involving fBm
  • 6.3.1. Consistency of the Drift Parameter Estimates in the Pure Fractional Brownian Diffusion Model
  • 6.3.2. Consistency of the Drift Parameter Estimates in the Mixed Brownian-fractional-Brownian Diffusion Model with "Linearly" Dependent W t and B 2+H t
  • 6.3.3. The Properties of Maximum Likelihood Estimates in Diffusion Brownian-Fractional-Brownian Models with Independent Components
  • A. Mandelbrot-van Ness Representation: Some Related Calculations
  • B. Approximation of Beta Integrals and Estimation of Kernels
  • References
  • Index
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