Oral Thesis Defense: "Bayesian Posterior Inference and LAN for Lévy Models Under High-frequency Data"

Abstract: Parameter estimation and inference for Lévy models under high-frequency data has been an interesting task in the field of financial mathematics. In this presentation, I will first talk about the Bayesian posterior inference on volatility in presence of infinite jump activity and microstructure noise. A ``purposely misspecified" posterior of the volatility is proposed by ignoring the jump-component of the process. It is further corrected by a simple estimate of the location shift and re-scaling of the log likelihood. The 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.
As one of the conditions of the Bernstein-von Mises theorem, local asymptotic normality (LAN) is also a key concept in the classical theory of asymptotic analysis. In the second part of the talk, LAN property is discussed for the Brownian motion time-changed by a generalized gamma convolution subordinator. A semiparametric model is also considered to include subordinator-related parameters. The results show that the LAN derived for the parametric part of the model also regulates the estimation behavior of the semiparametric model.

Host: Jose Figueroa-Lopez

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