Wildfires have become more destructive and are seriously threatening societies and our ecosystems throughout the world. Once a wildfire escapes from its initial suppression attack, it can easily develop into a destructive huge fire that can result in significant loss of lives and resources. Some human-caused wildfires may be prevented; however, most nature-caused wildfires cannot. Consequently, wildfire suppression and contain- ment becomes fundamentally important; but suppressing and containing wildfires is costly. Since the budget and resources for wildfire management are constrained in reality, it is imperative to make important decisions such that the total cost and damage associated with the wildfire is minimized while wildfire containment effectiveness is maximized. To achieve this objective, wildfire attack-bases should be optimally located such that any wildfire is suppressed within the effective attack range from some bases. In addition, the optimal fire-fighting resources should be deployed to the wildfire location such that it is efficiently suppressed from an economic perspective. The two main uncertain/stochastic factors in wildfire management problems are fire occurrence frequency and fire growth characteristics. In this thesis two models for wildfire management planning are proposed. The first model is a strategic model for the optimal location of wildfire-attack bases under uncertainty in fire occurrence. The second model is a tactical model for the optimal deployment of fire-fighting resources under uncertainty in fire growth. A stochastic mixed-integer programming approach is proposed in order to take into account the uncertainty in the problem data and to allow for robust wildfire management decisions under uncertainty. For computational results, the tactical decision model is numerically experimented by two different approaches to provide the more efficient method for solving the model.

The problem of robotic mapping, also known as simultaneous localization and mapping (SLAM), by a mobile agent for large distributed environments is addressed in this dissertation. This has sometimes been referred to as the holy grail in the robotics community, and is the stepping stone towards making a robot completely autonomous. A hybrid solution to the SLAM problem is proposed based on "first localize then map" principle. It is provably consistent and has great potential for real time application. It provides significant improvements over state-of-the-art Bayesian approaches by reducing the computational complexity of the SLAM problem without sacrificing consistency. The localization is achieved using a feature based extended Kalman filter (EKF) which utilizes a sparse set of reliable features. The common issues of data association, loop closure and computational cost of EKF based methods are kept tractable owing to the sparsity of the feature set. A novel frequentist mapping technique is proposed for estimating the dense part of the environment using the sensor observations. Given the pose estimate of the robot, this technique can consistently map the surrounding environment. The technique has linear time complexity in map components and for the case of bounded sensor noise, it is shown that the frequentist mapping technique has constant time complexity which makes it capable of estimating large distributed environments in real time. The frequentist mapping technique is a stochastic approximation algorithm and is shown to converge to the true map probabilities almost surely. The Hybrid SLAM software is developed in the C-language and is capable of handling real experimental data as well as simulations. The Hybrid SLAM technique is shown to perform well in simulations, experiments with an iRobot Create, and on standard datasets from the Robotics Data Set Repository, known as Radish. It is demonstrated that the Hybrid SLAM technique can successfully map large complex data sets in an order of magnitude less time than the time taken by the robot to acquire the data. It has low system requirements and has the potential to run on-board a robot to estimate large distributed environments in real time.

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We consider the stochastic volatility model with smooth transition and persistent la- tent factors. We argue that this model has advantages over the conventional stochastic model for the persistent volatility factor. Though the linear filtering is widely used in the state space model, the simulation result, as well as theory, shows that it does not work in our model. So we apply the density-based filtering method; in particular, we develop two methods to get solutions. One is the conventional approach using the Maximum Likelihood estimation and the other is the Bayesian approach using Gibbs sampling. We do a simulation study to explore their characteristics, and we apply both methods to actual macroeconomic data to extract the volatility generating process and to compare macro fundamentals with them. Next we extend our model into multivariate model extracting common and id- iosyncratic volatility for multivariate processes. We think it is interesting to apply this multivariate model into measuring time-varying uncertainty of macroeconomic variables and studying the links to market returns via a consumption-based asset pric- ing model. Motivated by Bansal and Yaron (2004), we extract a common volatility factor using consumption and dividend growth, and we find that this factor predicts post-war business cycle recessions quite well. Then, we estimate a long-run risk model of asset prices incorporating this macroeconomic uncertainty. We find that both risk aversion and the intertemporal elasticity of substitution are estimated to be around two, and our simulation results show that the model can match the first and second moments of market return and risk-free rate, hence the equity premium.

A stochastic differential equation with vanishing martingale term is studied. Specifically, given a domain D, the asymptotic scaling properties of both the exit time from the domain and the exit distribution are considered under the additional (non-standard) hypothesis that the initial condition also has a scaling limit. Methods from dynamical systems are applied to get more complete estimates than the ones obtained by the probabilistic large deviation theory. Two situations are completely analyzed. When there is a unique critical saddle point of the deterministic system (the system without random effects), and when the unperturbed system escapes the domain D in finite time. Applications to these results are in order. In particular, the study of 2-dimensional heteroclinic networks is closed with these results and shows the existence of possible asymmetries. Also, 1-dimensional diffusions conditioned to rare events are further studied using these results as building blocks. The approach tries to mimic the well known linear situation. The original equation is smoothly transformed into a very specific non-linear equation that is treated as a singular perturbation of the original equation. The transformation provides a classification to all 2-dimensional systems with initial conditions close to a saddle point of the flow generated by the drift vector field. The proof then proceeds by estimates that propagate the small noise nature of the system through the non-linearity. Some proofs are based on geometrical arguments and stochastic pathwise expansions in noise intensity series.

Thesis (Ph. D.)--Lehigh University, 2000. / Includes vita. Includes bibliographical references (leaves 125-135).

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--Ohio State University, 1967. / Includes bibliographical references. Available online via OhioLINK's ETD Center

Thesis (Ph. D.)--Ohio State University, 1968. / Includes bibliographical references (leaves 76-77). Available online via OhioLINK's ETD Center

Thesis (M. Phil.)--University of Hong Kong, 1982.