Date of Award
Doctor of Philosophy (PhD)
Parameter estimation and inference for L̩vy models under high-frequency data has been an exciting and important task in the field of financial mathematics and has been found practically useful when analyzing real financial data. One feature of L̩vy models is the allowance of jumps to model the abrupt changes sometimes observed in the market. In this thesis, we discuss some problems related to the statistical inference of L̩vy models based on high-frequency data emphasizing on the presence of the jumps. The first problem we consider focuses on the estimation of the volatility, which is critical to measure and control the risk of financial assets accurately. In order to accommodate the high-frequency data, infinite jump activity and microstructure noise are considered to be present. We propose a "purposely misspecified" posterior of the volatility obtained by ignoring the jump-component of the process. The misspecified posterior is further corrected by a simple estimate of the location shift and re-scaling of the log-likelihood. Our main result establishes a Bernstein-von Mises theorem, which states that the proposed adjusted posterior is asymptotically Gaussian, centered at a consistent estimator, and with variance equal to the inverse of the Fisher information. In the absence of microstructure noise, our approach can be extended to inferences of the integrated variance of general It̫ semimartingales. This method simplifies the complete Bayesian inference of the volatility, which would require the construction of the joint posterior. Thus, it avoids the requirement of a prior distribution on the jump process, the difficulties in specifying the likelihood function, and the implementation of computationally-expensive methods to sample from the joint posterior. Simulations are provided to demonstrate the accuracy of the resulting credible intervals, and the frequentist properties of the approximate Bayesian inference based on the adjusted posterior. As one of the conditions of the Bernstein-von Mises theorem, the local asymptotic normality (LAN) is a key concept in the classical theory of asymptotic analysis. We consider a class of jump processes, which can be represented as a Brownian motion time-changed by a driftless generalized gamma convolution subordinator. Sufficient conditions for LAN property to hold are first summarized. Then, LAN property is proved uniformly for the whole class. We also consider a semiparametric model, including any subordinator-related parameters. Our results show that the LAN derived for the parametric part of the model also regulates the estimation behavior of the semiparametric model. The estimation behavior includes the nonexistence of an estimator, whose convergence rate is faster than the rate imposed by the LAN property, and also an asymptotic risk lower bound of all estimators with the optimal rate.
Chair and Committee
Jos̩ E. Figueroa-L̟pez
Likai Chen, Soumendra Lahiri, Michael Landis, Syring Nicholas,
Wang, Qi, "Bayesian Posterior Inference and LAN for L̩vy Models Under High-frequency Data" (2020). Arts & Sciences Electronic Theses and Dissertations. 2253.