Monte Carlo methods in nonlinear filtering theory. - Alexsandro Machado Jacob (2006)
This thesis is focused on two basic aspects of the Control Problem: the stochastic modelling of physical systems, and Monte Carlo-based numerical approximation of the nonlinear filtering problem solution. In the first topic this thesis concerns about clarifying some issues in the mathematical modeling of continuous-time systems with Brownian motion. The hypothesis that physical systems should be modelled in continuous-time approach is defended, once the main results in Physics provide solutions for dynamic systems via continuous-time differential equations. It was shown, recalling a main result from the 1960's that a physical system is represented by Fisk-Stratonovich stochastic differential equation, though Ito approach is better to manipulate the mathematical operations. The required conditions for implementing these equations in computers were also studied by using Euler-Maruyama and Milstein schemes of discretization. In the second topic a unified treatment of the available Monte Carlo methods solving the nonlinear filtering problem for continuous and discrete-time modelling is presented with sufficient emphasis on basic applications enabling the engineer to use results provided by the theory. This topic is branched in the study of the theory of nonlinear filtering problem in continuous and discrete-time approaches, and in the investigation of the aspects of Monte Carlo-based numerical solutions approximating unnormalized conditional expectations, as those given by the classical Kallianpur-Striebel formula and its derived robust representation. Investigations showed that the estimates obtained via numerical approximations of the robust representation, or pathwise filter, might accumulate errors when the observation makes this filter alternative equation unstable, a limitation of the method. Another result of this thesis refers to the implementation of Monte Carlo filters using Bayesian representation for discretized models. Although Monte Carlo methods are attractive due to their facility of parallelization, their main drawback is the degeneracy phenomenon of the particles. The traditional resampling scheme solves the problem, but it difficulties the parallelization of the algorithm. The restoration method was then proposed to move the particles towards higher regions in the likelihood function, given information about the model parameters. This open method, in some sense, might decrease the particles degeneracy.