
Contents
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3.1 Estimating Integrated Volatility in Simple Cases 3.1 Estimating Integrated Volatility in Simple Cases
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3.1.1 Constant Volatility 3.1.1 Constant Volatility
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3.1.2 Deterministic Time-Varying Volatility 3.1.2 Deterministic Time-Varying Volatility
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3.1.3 Stochastic Volatility Independent of the Driving Brownian Motion W 3.1.3 Stochastic Volatility Independent of the Driving Brownian Motion W
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3.1.4 From Independence to Dependence for the Stochastic Volatility 3.1.4 From Independence to Dependence for the Stochastic Volatility
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3.2 Stable Convergence in Law 3.2 Stable Convergence in Law
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3.3 Convergence for Stochastic Processes 3.3 Convergence for Stochastic Processes
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3.4 General Stochastic Volatility 3.4 General Stochastic Volatility
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3.5 What If the Process Jumps? 3.5 What If the Process Jumps?
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3 Introduction to Asymptotic Theory: Volatility Estimation for a Continuous Process
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Published:July 2014
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Abstract
This chapter presents the simple situation of a one-dimensional continuous martingale to “estimate” its integrated volatility. For pedagogical reasons, it demonstrates nearly every statement beyond classical probability theory, without appealing to outside results: “demonstrate” is in fact too ambitious a word, since complete proofs are quite intricate in the most general situation, but a precise scheme is given for all proofs, emphasizing methods and ideas. These schematic proofs can serve as a template for the other situations encountered in the book, whose proofs will typically invoke general central limit theorems that are only stated in Appendix A, or elsewhere in the literature.
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