Approximate Bayesian inference methods for stochastic state space models

This thesis collects together research results obtained during my doctoral studies related to approximate Bayesian inference in stochastic state-space models. The published research spans a variety of topics including 1) application of Gaussian filtering in satellite orbit prediction, 2) outlier robust linear regression using variational Bayes (VB) approximation, 3) filtering and smoothing in continuous-discrete Gaussian models using VB approximation and 4) parameter estimation using twisted particle filters. The main goal of the introductory part of the thesis is to connect the results to the general framework of estimation of state and model parameters and present them in a unified manner.Bayesian inference for non-linear state space models generally requires use of approximations, since the exact posterior distribution is readily available only for a few special cases. The approximation methods can be roughly classified into to groups: deterministic methods, where the intractable posterior distribution is approximated from a family of more tractable distributions (e.g. Gaussian and VB approximations), and stochastic sampling based methods (e.g. particle filters). Gaussian approximation refers to directly approximating the posterior with a Gaussian distribution, and can be readily applied for models with Gaussian process and measurement noise. Well known examples are the extended Kalman filter and sigma-point based unscented Kalman filter. The VB method is based on minimizing the Kullback-Leibler divergence of the true posterior with respect to the approximate distribution, chosen from a family of more tractable simpler distributions.The first main contribution of the thesis is the development of a VB approximation for linear regression problems with outlier robust measurement distributions. A broad family of outlier robust distributions can be presented as an infinite mixture of Gaussians, called Gaussian scale mixture models, and include e.g. the t-distribution, the Laplace distribution and the contaminated normal distribution. The VB approximation for the regression problem can be readily extended to the estimation of state space models and is presented in the introductory part.VB approximations can be also used for approximate inference in continuous-discrete Gaussian models, where the dynamics are modeled with stochastic differential equations and measurements are obtained at discrete time instants. The second main contribution is the presentation of a VB approximation for these models and the explanation of how the resulting algorithm connects to the Gaussian filtering and smoothing framework.The third contribution of the thesis is the development of parameter estimation using particle Markov Chain Monte Carlo (PMCMC) method and twisted particle filters. Twisted particle filters are obtained from standard particle filters by applying a special weighting to the sampling law of the filter. The weighting is chosen to minimize the variance of the marginal likelihood estimate, and the resulting particle filter is more efficient than conventional PMCMC algorithms. The exact optimal weighting is generally not available, but can be approximated using the Gaussian filtering and smoothing framework

Financial Implications of Engineering Decisions

When society fails to effectively integrate natural and constructed environments, one of the cataclysmic byproducts of this disconnect is an increased risk of natural disasters. On top of the devastation that is the aftermath of such disasters, poor planning and engineering decisions have detrimental effects on communities as they attempt to recover and rebuild. While there is an inherent difficulty in the quantification of the cost of human life, interruption in business operations, and damage to the properties, it is critical to develop plans and mitigation strategies to promote fast recovery. Traditionally insurance and reinsurance products have been used as a mitigation strategy for financing post-disaster recovery. However, there are number of problems associated with these models such as lack of liquidity, defaults, long litigation process, etc. In light of these problems, new Alternative Risk Transfer (ART) methods are introduced. The pricing of these risk mitigating instruments, however, has been mostly associated with the hazard frequency and intensity; and little recognition is made of the riskiness of the structure to be indemnified. This study proposes valuation models for catastrophe-linked ART products and insurance contracts in which the risks and value can be linked to the characteristics of the insured portfolio of constructed assets. The results show that the supply side ? structural parameters are as important as the demand ? hazard frequency, and are in a highly nonlinear relationship with financial parameters such as risk premiums and spreads

Pricing and hedging of a portfolio of options in the presence of stochastic volatility

Dopo aver fatto pricing di un basket di opzioni sul S&P500 sia con black-scholes che Heston, vengono effettuate diverse strategie di hedging dinamico (Delta, Delta-Gamma, Delta-Gamma-Vega

Stochastic Optimization; Proceedings of the International Conference, Kiev, USSR, September 1984

The purpose of this conference, which was attended by 240 scientists from 20 countries, was to survey the latest developments in the field of controlled stochastic processes, stochastic programming, control under incomplete information and applications of stochastic optimization techniques to problems in economics, engineering, modeling of energy systems, etc. The conference reflected a number of recent important developments in the field, notably new results in control theory with incomplete information, stochastic maximum principle, new numerical techniques for stochastic programming and related software, application of probabilistic methods to the modeling of the economy. The contributions to this book are divided into three categories: (1) Controlled stochastic processes; (2) Stochastic extremal problems; and (3) Stochastic optimization problems with incomplete information

Girsanov's transformation based variance reduced Monte Carlo simulation schemes for reliability estimation in nonlinear stochastic dynamics

The study considers the problem of simulation based time variant reliability analysis of nonlinear randomly excited dynamical systems. Attention is focused on importance sampling strategies based on the application of Girsanov's transformation method. Controls which minimize the distance function, as in the first order reliability method (FORM), are shown to minimize a bound on the sampling variance of the estimator for the probability of failure. Two schemes based on the application of calculus of variations for selecting control signals are proposed: the first obtains the control force as the solution of a two point nonlinear boundary value problem, and, the second explores the application of the Volterra series in characterizing the controls. The relative merits of these schemes, visa-vis the method based on ideas from the FORM, are discussed. Illustrative examples, involving archetypal single degree of freedom (dof) nonlinear oscillators, and a multi degree of freedom nonlinear dynamical system, are presented. The credentials of the proposed procedures are established by comparing the solutions with pertinent results from direct Monte Carlo simulations. (C) 2017 Elsevier Inc. All rights reserved