Filtering or estimation is of great importance to virtually all disciplines of engineering and science that need inference, learning, information fusion, and knowledge discovery of dynamical systems. The filtering problem is to recursively determine the states and/or parameters of a dynamical system from a sequence of noisy measurements made on the system. The theory and practice of optimal estimation of linear Gaussian dynamical systems have been well established and successful, but optimal estimation of nonlinear and non-Gaussian dynamical systems is much more challenging and in general requires solving partial differential equations and intractable high-dimensional integrations. Hence, Gaussian approximation filters are widely used. In this dissertation, three innovative point-based Gaussian approximation filters including sparse Gauss-Hermite quadrature filter, sparse-grid quadrature filter, and the anisotropic sparse-grid quadrature filter are proposed. The relationship between the proposed filters and conventional Gaussian approximation filters is analyzed. In particular, it is proven that the popular unscented Kalman filter and the cubature Kalman filter are subset of the proposed sparse-grid filters. The sparse-grid filters are employed in three aerospace applications including spacecraft attitude estimation, orbit determination, and relative navigation. The results show that the proposed filters can achieve better estimation accuracy than the conventional Gaussian approximation filters, such as the extended Kalman filter, the cubature Kalman filter, the unscented Kalman filter, and is computationally more efficient than the Gauss-Hermite quadrature filter.

The system of sparse gridpoints was used to propagate uncertainty forward in time through orbital mechanics simulations. Propagation of initial uncertainty through a nonlinear dynamic model is examined in regards to the uncertainty of orbit estimation. The necessary underlying mechanics of orbital mechanics, probability, and nonlinear estimation theory are reviewed to allow greater understanding of the problem. The sparse grid method itself and its implementation is covered in detail, along with the necessary properties and how to best it to a given problem based on inputs and desired outputs. Three test cases were run in the form of a restricted two-body problem, a perturbed two-body problem, and a three-body problem in which the orbiting body is positioned at a Lagrange point. It is shown that the sparse grid method shows sufficient accuracy for all mean calculations in the given problems and that higher accuracy levels allow for accurate estimation of higher moments such as the covariance.

Efficiently pricing multi-asset options is a challenging problem in computational finance. Although classical Fourier methods are extremely fast in pricing single asset options, maintaining the tractability of Fourier techniques for multi-asset option pricing is still an area of active research. Fourier methods rely on explicit knowledge of the characteristic function of the suitably stochastic price process, allowing for calculation of the option price by evaluation of multidimensional integral in the Fourier domain. The high smoothness of the integrand in the Fourier space motivates the exploration of deterministic quadrature methods that are highly efficient under certain regularity assumptions, such as, adaptive sparse grids quadrature (ASGQ), and Randomized Quasi-Monte Carlo (RQMC). However, when designing a numerical quadrature method for most of the existing Fourier pricing approaches, two key factors affecting the complexity should be carefully controlled, (i) the choice of the vector of damping parameters that ensure the Fourier-integrability and control the regularity class of the integrand, (ii) the high-dimensionality of the integration problem. To address these challenges, in the first part of this thesis we propose a rule for choosing the damping parameters, resulting in smoother integrands. Moreover, we explore the effect of sparsification and dimension-adaptivity in alleviating the curse of dimensionality. Despite the efficiency of ASGQ, the error estimates are very hard to compute. In cases where error quantification is of high priority, in the second part of this thesis, we design an RQMC-based method for the (inverse) Fourier integral computation. RQMC integration is known to be highly efficient for high-dimensional integration problems of sufficiently regular integrands, and it further allows for computation of probabilistic estimates. Nonetheless, using RQMC requires an appropriate domain transformation of the unbounded integration domain to the hypercube, which may originate in a transformed integrand with singularities at the boundaries, and consequently deteriorate the rate of convergence. To preserve the nice properties of the transformed integrand,we propose a model-dependent domain transformation to avoid these corner singularities and retain the optimal efficiency of RQMC. The effectiveness of the proposed optimal damping rule, the designed domain transformation procedure, and their combination with ASGQ and RQMC are demonstrated via several numerical experiments and computational comparisons to the MC approach and the COS method.

Compuesta por tres capítulos: El primero es un estudio sobre la implementación the Sparse Grid métodos para es el estudio de modelos económicos con muchas dimensiones. Llevado a cabo mediante aplicaciones noveles del método de Smolyak con el objetivo de favorecer la tratabilidad y obtener resultados preciso. Los resultados muestran mejoras en la eficiencia de la implementación de modelos con múltiples agentes. El segundo capítulo introduce una nueva metodología para la evaluación de políticas económicas, llamada Synthetic Control with Statistical Learning, todo ello aplicado a políticas particulares: a) reducción del número de horas laborales en Portugal en 1996 y b) reducción del coste del despido en España en 2010. La metodología funciona y se erige como alternativa a previos métodos. En términos empíricos se muestra que tras la implementación de la política se produjo una reducción efectiva del desempleo y en el caso de España un incremento del mismo. El tercer capítulo utiliza la metodología utiliza en el segundo capítulo y la aplica para evaluar la implementación del Tercer Programa Europeo para la Seguridad Vial (Third European Road Safety Action Program) entre otras metodologías. Los resultados muestran que la coordinación a nivel europeo de la seguridad vial a supuesto una ayuda complementaria. En el año 2010 se estima una reducción de víctimas mortales de entre 13900 y 19400 personal en toda Europa.