U. Shaked, P.R. Kumar. / Bibliography: p. 30. / "May, 1984" / "ECS-8304435" "DAAG29-84-K-0005"

Bernard C. Levy, John N. Tsitsiklis. / "October, 1981" / Bibliography: p. 39-41. / "This work was supported by the National Science Foundation under Grant ECS-80-12608."

by Robert R. Tenney. / "November, 1982." / Bibliography: p. 38-39. / ONR contract N00014-77-0532C (N041-519)

by Milton B. Adams, Alan S. Willsky, Bernard C. Levy. / Caption title. "March 1983." / Bibliography: leaf 8. / NSF grant ECS-8012668

W.J. Runggaldier. / "October 1985." / Bibliography: p. 20-21.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2008. / Includes bibliographical references (leaves 79-80).

This dissertation addresses various aspects of estimation and inference for multivariate stochastic processes with jumps. The first chapter develops an unbiased Monte Carlo estimator of the transition density of a multivariate jump-diffusion process. The drift, volatility, jump intensity, and jump magnitude are allowed to be state-dependent and non-affine. The density estimator proposed enables efficient parametric estimation of multivariate jump-diffusion models based on discretely observed data. Under mild conditions, the resulting parameter estimates have the same asymptotic behavior as maximum likelihood estimators as the number of data points grows, even when the sampling frequency of the data is fixed. In a numerical case study of practical relevance, the density and parameter estimators are shown to be highly accurate and computationally efficient. In the second chapter, I examine continuous-time stochastic volatility models with jumps in returns and volatility in which the parameters governing the jumps are allowed to switch according to a Markov chain. I estimate the parameters and the latent processes using the S&P 500 and Nasdaq indices from 1990 to 2014. The Markov-switching parameters characterize well the periods of market stress, such as those in 1997-1998, 2001 and 2007-2010. Several statistical tests favor the model with Markov-switching jump parameters. These results provide empirical evidence about the state-dependent and time-varying nature of asset price jumps, a feature of asset prices that has recently been documented using high-frequency data. The third chapter considers applying Markov-switching affine stochastic volatility models with jumps in returns and volatility, where the jump parameters are not regime-switching. The estimation is performed via Markov Chain Monte Carlo methods, allowing to obtain the latent processes induced by the structure of the models. Furthermore, I propose some misspecification tests and develop a Markov-switching test based on the odds ratios. The parameters and the latent processes are estimated using the S&P 500 index from 1970 to 2014. I show that the S&P 500 stochastic volatility exhibits a Markov-switching behavior, and that most of the high volatility regimes coincide with the recessions identified ex-post by the National Bureau of Economic Research.

Title: Multivariate stochastic dominance and its application in portfolio optimization Problems Author: Barbora Petrová Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis discusses the concept of multivariate stochastic dominance, which serves as a tool for ordering random vectors, and its possible usage in dynamic portfolio optimization problems. We strictly focus on different types of the first-order multivariate stochastic dominance for which we describe their generators in the sense of von Neumann-Morgenstern utility functions. The first one, called strong multivariate stochastic dominance, is generated by all nondecreasing multivariate utility functions. The second one, called weak multivariate stochastic dominance, is defined by relation between survival functions, and the last one, called the first-order linear multivariate stochastic dominance, applies the first-order univariate stochastic dominance notion to linear combinations of marginals. We focus on the main characteristics of these types of stochastic dominance, their relationships as well as their relation to the cumulative and marginal distribution functions of considered random vectors. Formulated...

This thesis introduces three new model-free tests for isotropy, equal distribution and random superposition in non-rectangular windows respectively. For isotropy, a bootstrap-type test is proposed. The corresponding test statistic assesses the discrepancy between the uniform distribution and the empirical normalised reduced second-order moment measure of a sector of fixed radius with increasing central angle. The null distribution of the discrepancy is then estimated by stochastic reconstruction, which generates bootstrap-type samples of point patterns that resemble the spatial structure of the given pattern. The new test is applicable for small sample sizes and is shown to have more robust powers to different choices of user-chosen parameter when compared with the asymptotic chi-squared test by Guan et al. (2006) in our simulation. For equal distribution, a model-free asymptotic test is introduced. The proposed test statistic compares the discrepancy between the empirical second-order product densities of the observed point patterns at some pre-chosen lag vectors. Under certain mild moment conditions and a weak dependence assumption, the limiting null distribution of the test statistic is the chi-squared distribution. Simulation results show that the new test is more powerful than the permutation test by Hahn (2012) for comparing point patterns with similar structures but different distributions. The new test for random superposition is a modification of the toroidal shift test by Lotwick and Silverman (1982). The idea is to extrapolate the pattern observed in a non-rectangular window to a larger rectangular region by the stochastic reconstruction so that the toroidal shift test can be applied. Simulation results show that the powers of the test applied to patterns with extrapolated points are remarkably higher than those of the test applied to the largest inscribed rectangular windows, with only slightly increased type I error rates. Real data sets are used to illustrate the advantages of the tests developed in this thesis over the existing tests in the literature.

This dissertation discusses the construction of some stochastic models for population dynamics with a variety of birth and death rate functions. A general model is constructed considering a fundamental growth rate function of the population while allowing random births and deaths in the population. Four stochastic discrete delay models and two non-delay models using the infinitesimal mean and variance given by birth and death rate functions have been produced and analyzed. In these constructions drift terms are in the form of logistic growth or logistic growth with delay. Logistic growth models are well known to biologists and economists. For each model, the existence and uniqueness of the global solution, non-negativeness of the solution is discussed, and for some models, boundedness of the path is also given. Persistence of the population and the boundary behavior have also been discussed through the hitting times. Here, a new method to analyze the hitting times for a specific class of stochastic delay models is presented. This work is related to and also extends the work of Edward Allen, Linda Allen and Bernt Oksendal in population dynamics.