| Περίληψη : | The present thesis deals with the problem of developing statistical methodology for modellingand inference of high dimensional financial data. The motivation of our research wasthe identification of infrequent and extreme movements, which are called jumps, in the pricesof the 600 stocks of Euro STOXX index. This is known in the financial and statistical literatureas the problem of separating jumps from the volatility of the underlying process whichis assumed for the evolution of the stock prices.The main contribution of the thesis is the modelling and the development of methodsfor inference on the characteristics of the jumps across multiple stocks, as well as across thetime horizon. Following the Bayesian paradigm we use prior information in order to modela known characteristic of financial crises, which is that jumps in stock prices tend to occurclustered in time and to a↵ect several markets within a sort period of time. An improvementin the prediction of future stock prices has been achieved.The proposed model combines the stochastic volatility (SV) model with a multivariatejump process and belongs to the very broad class of latent Gaussian models. Bayesian inferencefor latent Gaussian models relies on a Markov chain Monte Carlo (MCMC) algorithmwhich alternates sampling from the distribution of the latent states of the model conditionalon the parameters and the observations, and sampling from the distribution of the parametersof the model conditional on the latent states and the observations. In the case of SVmodels with jumps, sampling the latent volatility process of the model is not a new problem.Over the last few years several methods have been proposed for separating the jumps fromthe volatility process but there is not a satisfactory solution yet, since sampling from a highdimensional nonlinear and non-Gaussian distribution is required. In the present thesis wepropose a Metropolis-Hastings algorithm in which we sample the whole path of the volatilityprocess of the model without using any approximation. We compare the resulting MCMCalgorithm with existing algorithms. We apply our proposed methodology on univariate SVwith jumps models in order to identify jumps in the stock prices of the real dataset thatmotivated our research.To model the propagation of the jumps across stocks and across time we combine the SVmodel with a doubly stochastic Poisson process, also known as Cox process. The intensityof the jumps in the Poisson process is modelled using a dynamic factor model. Furthermore,we develop an MCMC algorithm to conduct Bayesian inference for the parameters and thelatent states of the proposed model. We test the proposed methods on simulated data and weapplied them on our real dataset. We compare the prediction of future stock prices using theproposed model with the predictions obtained using existing models. The proposed modelprovides better predictions of future stock prices and this is an indication for a predictablepart of the jump process of SV models.IIIThe MCMC algorithm that is implemented in order to conduct Bayesian inference forthe aforementioned models is also employed on a demographic application. More precisely,within the context of latent Gaussian models we present a novel approach to model andpredict mortality rates of individuals.
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