Functional data analysis has gained a lot of attention in the past few years. One very promising methodological paradigm is given by wavelet analysis. This methods have some general properties which are particularly interesting for FDA: parsimony, asymptotics, and computational performance.Wavelets are ideally suited for the formulation of computationally and statistically efficient methodologies on Functional data Analysis. We refer to Donoho and Johnstone (1998); Abramovich et al. (2004); Fan and Koo (2002); Klemelä (2006); and Morettin, Pinheiro and Vidakovi (2016) for details.Applications of statistical models for functional data is large and continuously growing. There are several major open questions which we hope to answer theoretically, such as:(i) Asymptotic properties of wavelet estimators, and its relative performance against other paradigms.(ii) Parsimonious wavelet representation.(iii) Optimal wavelet-based tests.(iv) Aletrnative underlying generating stochastic process to the ubiquotous Brownian motion.We will develop theoretical models based on Stochastic Differential Equations which generalize the current models in two main aspects:(A) Generating stochastic process, specially by CTARMA processes and Fractional Brownian motions.(B) Stochastic volatility and/or non-stationary models.Three main areas of application we will pursue are:(i) High frequency financial data and volatility models.(ii) Multidimensional models and high resolution satellite time series.(iii) Genomincs and proteinomics.
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