Inference for Stochastic Processes and Applications

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The long-term goal of my research program is to develop novel methods and models to study inference for stochastic processes. The work proposed here involves stochastic processes such as nonlinear time series models, recently proposed random coefficient count time series models, circular count time series models, volatility models, infinite variance stochastic processes and semimartingales. The unified method of regularized combined estimating function theory in a Bayesian set-up as well as in a non-Bayesian set-up will be used in the context of discrete time models and semimartingales to obtain regularized filtering algorithms. Regularized filtering is also useful in dynamic data science for networks with time varying variance-covariance matrix. (a) Regularized time series methods allow data scientists and risk managers to enhance the forecasting power of a model and to improve the quality of the risk forecasts. While regularized estimates have been shown to perform well in many applications, their use in obtaining financial risk forecasts/filters has not yet been studied. Regularized data-driven adaptive forecasting algorithms for volatility and value at risk will be studied in detail and will be applied in automated trading systems. In this proposal, using an estimating function approach, regularized filtering for random coefficient time series models, state space models, stochastic volatility models and semimartingales will be studied. (b) There has been a growing interest in stochastic processes with infinite variance. For example Fama (2013 Nobel Prize winner for econometric modelling) studied estimation and prediction for infinite variance regression models. For time series models with infinite variance stable errors, closed form expressions for the density are not available and hence the maximum likelihood estimate cannot readily be obtained. We have used combined sine and cosine estimating functions to study estimation. I have developed a maximum information recursive estimation method and applied it to financial data. In this proposal, I will study maximum information filtered estimates for infinite variance models using sine and cosine estimating functions. (c) In further work, I will investigate de-biased estimates based on combined estimating functions. I will apply the proposed methodology to the study of data driven portfolio optimization and network inference. The proposed research will provide opportunities for training of highly qualified personnel at all levels. They will learn statistical theories and algorithms/R coding in regularized filtering, regularized risk forecasting and optimal portfolio selection. The regularized forecasts and regularized filters will provide practical tools to applied researchers in finance, computer science and AgriEconomics.