With the emergence of modern technology and availability of high frequency data, the study of functional time series has become popular in recent years. One of the main goals of time series is to predict the outcome of the future observations. Though methods of predicting a functional time series have been explored in the literature, not much work has been done to obtain prediction error estimates. The first part of the dissertation proposes several estimates of prediction errors in the functional time series context. Prediction errors are necessary inputs for construction of prediction bands. This dissertation introduces methods of getting prediction bands using the different functional prediction error estimates. The proposed methods are evaluated based on simulation studies as well as real data applications.

A second important application of functional time series can be found in high-frequency finance, viewing the intra-day price movements as functions. Financial time series are characterized by volatility clustering, implying that large price movements tend to be followed by further large price movements and small movements by small movements. The stochastic volatility model is a widely used multiplicative financial model originally introduced to capture this form of heteroscedastic behavior for univariate financial time series. Unlike the competitor GARCH models that depend on past volatility and past residuals, and also aim at capturing the clustering tendency, the stochastic volatility process depends on the product of an independent noise sequence with a latent volatility sequence. When the observations are scalars or vectors, stochastic volatility estimation is often performed within the state-space modeling framework. The second part of the dissertation introduces the functional stochastic volatility model along with a method to estimate the model parameters using the state-space modeling framework and evaluates the proposed methodology on simulated data.