New Research


2009/01/01

Statistical inference for stochastic processes

For the development in the field of financial data analysis

Associate professors Masayuki Uchida and Etsuo Kumagai

Statistical inference and its application to financial data analysis are mainly studied in Uchida and Kumagai laboratory. In the field of statistical inference, it is very important to construct a statistical model based on the data and to predict future behavior. In terms of globalization and development of the financial database, it is easy to get the financial data on the Internet. Furthermore, thanks to the development of computer, it is possible to do numerical analysis for enormous financial data. Under these situations, statistical inference for stochastic processes plays an important role in the development of financial data analysis.

In fitting financial data to a continuous time stochastic process, stochastic differential equation (SDE) is one of natural statistical models. However, the diffusion process defined by the SDE does not generally have an explicit transition density function (likelihood function). Because of it, the likelihood analysis can not be applied for discretely observed diffusion models. In order to overcome this difficulty, we propose the quasi-maximum likelihood estimator which maximizes the quasi-likelihood function (approximate likelihood function) and research its asymptotic properties. The financial data analysis based on the SDE model is one of interesting attempts in the field of mathematically statistical sciences.

Moreover, we study the Kullback-Leibler information, the Fisher information, Akaike’s information criterion, the Fisher information loss, Efron’s statistical curvature, Amari’s information geometry and information criteria in model selection for the following models: the standard autoregressive moving average (ARMA) model, autoregressive integrated moving average (ARIMA) model, seasonal ARIMA (SARIMA) model which are based on ARMA model, generalized autoregressive conditional heteroscedasticity (GARCH) model, stochastic volatility (SV) model which are based on heteroscedasticities, and nonlinear state space model which is a generalization of GARCH model and SV model.


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