Parameter estimation and random number generation for student Lévy processes

Abstract

To address the challenges in estimating parameters of the widely applied Student-Lévy process, the study introduces two distinct methods: a likelihood-based approach and a data-driven approach. A two-step quasi-likelihood-based method is initially proposed, countering the non-closed nature of the Student-Lévy process's distribution function under convolution. This method utilizes the limiting properties observed in high-frequency data, offering estimations via a quasi-likelihood function characterized by asymptotic normality. Additionally, a novel neural-network-based parameter estimation technique is advanced, independent of high-frequency observation assumptions. Utilizing a CNN-LSTM framework, this method effectively processes sparse, local jump-related data, extracts deep features, and maps these to the parameter space using a fully connected neural network. This innovative approach ensures minimal assumption reliance, end-to-end processing, and high scalability, marking a significant advancement in parameter estimation techniques. The efficacy of both methods is substantiated through comprehensive numerical experiments, demonstrating their robust performance in diverse scenarios.