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Table of Contents
Intro; Preface; Introduction and Synopsis; Volatility and Intermittency; Ambit Sets; Synopsis; Part I: Purely Temporal Case; Part II: Spatio-Temporal Case; Part III: Applications; Reading Guide; Conclusion; Contents; Part I The Purely Temporal Case; 1 Volatility Modulated Volterra Processes; 1.1 Motivation; 1.1.1 Notation; 1.2 Infinite Divisibility; 1.3 Lévy Processes; 1.3.1 Definition and First Properties; 1.3.2 Subordinators; 1.4 Volatility Modulated Volterra Processes; 1.4.1 Definition and First Properties; 1.4.1.1 Integrability Conditions for a VMLV Process Without Drift
1.4.1.2 Integrability Conditions for X21.4.1.3 Integrability Conditions for X1; 1.4.1.4 Integrability Conditions for X; 1.4.1.5 Integrability Conditions for the Drift Term; 1.4.1.6 Square Integrability; 1.4.2 A Note on Stationarity; 1.4.3 Characteristic Function and Functional for Integrals w.r.t. Lévy Processes; 1.4.3.1 Cumulant Functional; 1.4.4 Second Order Structure; 1.4.5 Remarks; 1.5 Brownian and Lévy Semistationary Processes; 1.5.1 Definition and First Properties; 1.5.2 Second Order Structure; 1.5.2.1 Link Between the Autocorrelation Function and the Kernel Function
1.6 Semimartingale and Non-semimartingale Settings in the Lévy Semistationary Case1.6.1 Semimartingale Condition; 1.7 Examples of LSS Processes; 1.7.1 The Ornstein-Uhlenbeck Process; 1.7.2 CAR and CARMA Processes; 1.7.3 Complex-Valued LSS Processes; 1.7.4 Ratio Kernel Function; 1.7.5 BSS Processes with Gamma Kernel; 1.8 Examples for the Stochastic Volatility/Intermittency Process; 1.9 BSS Processes with Generalised Hyperbolic Marginal Law; 1.10 Path Properties of LSS Processes; 1.10.1 Kernels of Power-Type Behaviour; 1.10.2 The Special Case of a BSS Process with GammaKernel
1.11 Volatility Modelling Through Amplitude and/or Intensity Modulation1.11.1 Definition and First Properties; 1.11.1.1 The Time-Change Process; 1.11.1.2 The Brownian Case; 1.11.2 Combining Amplitude and Intensity Modulation; 1.12 Further Reading; 2 Simulation; 2.1 Numerical Integration; 2.1.1 Error Bounds; 2.2 A Representation of LSS Processes in Terms of (Complex-Valued) Volatility Modulated Ornstein-Uhlenbeck Processes; 2.2.1 A Fourier-Based Representation; 2.2.2 A Laplace-Based Representation; 2.3 A Stepwise Simulation Scheme Based on the Laplace Representation
2.4 A Stepwise Simulation Scheme Based on the Fourier Representation2.4.1 The Complex-Valued Volatility Modulated Ornstein-Uhlenbeck Process; 2.4.2 Numerical Integration and Simulation Algorithm; 2.4.3 Error Analysis; 2.5 Simulation Based on Numerically Solving a Stochastic Partial Differential Equation; 2.6 Proofs of Some Results; 2.7 Further Reading; 3 Asymptotic Theory for Power Variation of LSS Processes; 3.1 Convergence Concept; 3.2 Asymptotic Theory in the Semimartingale Setting; 3.2.1 The Brownian Case; 3.2.2 The Pure-Jump Case; 3.3 Asymptotic Theory in the Non-semimartingale Setting
1.4.1.2 Integrability Conditions for X21.4.1.3 Integrability Conditions for X1; 1.4.1.4 Integrability Conditions for X; 1.4.1.5 Integrability Conditions for the Drift Term; 1.4.1.6 Square Integrability; 1.4.2 A Note on Stationarity; 1.4.3 Characteristic Function and Functional for Integrals w.r.t. Lévy Processes; 1.4.3.1 Cumulant Functional; 1.4.4 Second Order Structure; 1.4.5 Remarks; 1.5 Brownian and Lévy Semistationary Processes; 1.5.1 Definition and First Properties; 1.5.2 Second Order Structure; 1.5.2.1 Link Between the Autocorrelation Function and the Kernel Function
1.6 Semimartingale and Non-semimartingale Settings in the Lévy Semistationary Case1.6.1 Semimartingale Condition; 1.7 Examples of LSS Processes; 1.7.1 The Ornstein-Uhlenbeck Process; 1.7.2 CAR and CARMA Processes; 1.7.3 Complex-Valued LSS Processes; 1.7.4 Ratio Kernel Function; 1.7.5 BSS Processes with Gamma Kernel; 1.8 Examples for the Stochastic Volatility/Intermittency Process; 1.9 BSS Processes with Generalised Hyperbolic Marginal Law; 1.10 Path Properties of LSS Processes; 1.10.1 Kernels of Power-Type Behaviour; 1.10.2 The Special Case of a BSS Process with GammaKernel
1.11 Volatility Modelling Through Amplitude and/or Intensity Modulation1.11.1 Definition and First Properties; 1.11.1.1 The Time-Change Process; 1.11.1.2 The Brownian Case; 1.11.2 Combining Amplitude and Intensity Modulation; 1.12 Further Reading; 2 Simulation; 2.1 Numerical Integration; 2.1.1 Error Bounds; 2.2 A Representation of LSS Processes in Terms of (Complex-Valued) Volatility Modulated Ornstein-Uhlenbeck Processes; 2.2.1 A Fourier-Based Representation; 2.2.2 A Laplace-Based Representation; 2.3 A Stepwise Simulation Scheme Based on the Laplace Representation
2.4 A Stepwise Simulation Scheme Based on the Fourier Representation2.4.1 The Complex-Valued Volatility Modulated Ornstein-Uhlenbeck Process; 2.4.2 Numerical Integration and Simulation Algorithm; 2.4.3 Error Analysis; 2.5 Simulation Based on Numerically Solving a Stochastic Partial Differential Equation; 2.6 Proofs of Some Results; 2.7 Further Reading; 3 Asymptotic Theory for Power Variation of LSS Processes; 3.1 Convergence Concept; 3.2 Asymptotic Theory in the Semimartingale Setting; 3.2.1 The Brownian Case; 3.2.2 The Pure-Jump Case; 3.3 Asymptotic Theory in the Non-semimartingale Setting