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We study the asymptotic theory of misspecified models for diffusion processeswith noisy nonsynchronous observations. Unlike with correctly specified models,the original maximum-likelihood-type estimator has an asymptotic bias under themisspecified setting and fails to achieve an optimal rate of convergence. Toaddress this, we consider a new quasi-likelihood function that arrowsconstructing a maximum-likelihood-type estimator that achieves the optimal rateof convergence. Study of misspecified models enables us to applymachine-learning techniques to the maximum-likelihood approach. With thesetechniques, we can efficiently study the microstructure of a stock market byusing rich information of high-frequency data. Neural networks haveparticularly good compatibility with the maximum-likelihood approach, so wewill consider an example of using a neural network for simulation studies andempirical analysis of high-frequency data from the Tokyo Stock Exchange. Wedemonstrate that the neural network outperforms polynomial models in volatilitypredictions for major stocks in Tokyo Stock Exchange.

Misspecified diffusion models with high-frequency observations and an application to neural networks