Many problems in Statistics and Econometrics can be formulated in terms of the Bayesian
model. In fact, this can be interpreted as a penalized goodness-of-fit or more generic
decision or optimization problem. Most of these formulations need to resort to approximations and often suffer from the lack of efficient solution tools. Although usually complicated, simulation techniques can offer the exact solutions to these problems.
In fact, one of the interesting problems is statistical inference for the parameters defined on non-Euclidean spaces. An example is time series cointegration where linear transformation of multivariate process is stationary. Since the stationarity property is invariant with respect to this transformation, the objective of inference is a basis of the suitable vector space. In particular, in this thesis we focus on two aspects: 1) Markov Chain Monte Carlo
methods for cointegrated time series models; 2) application of Sequential Monte
Carlo for the optimization problems in automatic control.
In Chapter 3, we developed generic Hybrid Monte Carlo samplers allowing for efficient
Bayesian inference in the presence of cointegration. The proposed methods perform on
par with the existent state-of-the-art methods under Gaussian assumptions. We also show
they can be applied in more elaborate model formulations wherein other existent methods
are not available.
In Chapter 4, we proposed a cointegration model with dynamic factors and developed
a novel sampler enabling efficient inference. The model was developed specifically for
modelling spreads in commodities markets and we perform a thorough data analysis on
the soybean crush spread.
In Chapter 5, we change direction and look at stochastic regulation when formulated as an optimal control problem. For a multiplicative cost we show that the optimization part can be solved using filtering techniques and SMC. The method performance is investigated
in case studies with conditionally linear and non-linear state space models.