Mathematics & Statistics

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Open Access
Efficient Numerical Methods for Reynolds Averaged Navier-Stokes Equations of Flow over Topography and Application
(2014-07-09) Yu, Xiao; Liang, Dong
Recently, wind energy has been used widely as a complement to the common used energy resources such as oil, coal, and natural gas. Fossil fuels can generate heavy pollution and release greenhouse gases, which are recognized as the main cause of the global warming. As a result, green and renewable energy technologies, such as wind energy and solar energy, are highly recommended nowadays. In order to build the wind farms and make the wind energy assessments, wind flow over topography has been studied intensively in wind energy industry. In my thesis, we first improve an under-relaxed iteration scheme for the steady-state RANS equations of neutrally stratified airflow over complex topography. The NLMSFD scheme failed on predicting flow over terrains with a relatively high slope and we improve this iteration scheme to a much higher maximum slope. In the second part, we develop the efficient characteristic finite volume method (CFV) to solve the time-dependent RANS equations of flow over topography with various surface roughnesses. In viscous flow, the convective term plays a more important role than the diffusive term, especially for the turbulent flow with a high Reynolds number. The CFV scheme is developed by combining the characteristic method and the finite volume method. It treats the convective term efficiently. Numerical experiments of solving the time-dependent RANS equations with k-epsilon closure show the advantages of the accuracy, efficiency and stability of the method. In the last part, the CFV method is further applied to model wind flow and turbine wakes of large wind farms. We simulate the wind turbine wakes behind a cluster of wind farms which take into account the roughness change on the topography. We propose to consider RANS models with the Coriolis effect in modelling wind flows under a large scale due to the rotation of Earth. The wind flows within and downwind of the wind farms are predicted numerically. Simulation results on the Horns Rev wind farm are compared with field measurements.
Open Access
The Impact of Stochastic Interest and Mortality Rates on Ruin Probability and Annuitization Decisions Faced by Retirees
(2014-07-09) Wang, Jinlian; Huang, Huaxiong; Milevsky, Moshe
This dissertation focuses on two issues in retirement planning. The first issue, annuitization problem, provides insight on how interest rates may affect annuitization decisions for retirees under an all-or-nothing framework. The second issue, ruin probability, studies the probability for a retired individual who might run out of money, under a fixed consumption strategy before the end of his/her life under stochastic hazard rates. These two financial problems have been very important in personal finance for both retirees and financial advisors throughout the world, especially in the developed countries as the baby boom generation nears retirement. They are the direct results of both longevity risk and demise of Defined Benefit (DB) pension plans. The existing literature of the annuitization problem, such as Richard (1975), concludes that it is always optimal to annuitize with no bequest motives under a constant interest rate. To see the effect of stochastic interest rates on the annuitization decisions under a constrained consumption strategy without bequest motives, we present two life cycle models. They investigate the optimal annuitization strategy for a retired individual whose objective is to maximize his/her lifetime utility under a variety of institutional restrictions, in an all-or-nothing framework. The individual is required to annuitize all his/her wealth in a lump sum at some time at retirement. The first life cycle model we have presented assumes full consumption after annuity purchasing. A free boundary exists in this case upon the assumption of constant spread between the expected return of the risky asset and the riskless interest rate. The second life cycle model applies the optimal consumption strategy after annuitization, and numerical analysis shows that it is always optimal to annuitize no matter what the current interest rate is. This conclusion is based on the assumption of constant risk premium, no loads and no bequest motives. Historical data show that mortality rates for human beings behave stochastically. Motivated by this, we study the ruin probability for a retired individual who withdraws $1 per annum with various initial wealth for log-normal mortality with constant drift and volatility, which is a special form of the most widely accepted Lee-Carter model. This problem is converted to a Partial Differential Equation (PDE) and solved numerically by the Alternative Direction Implicit (ADI) method. For any given initial wealth, ruin probability can be obtained for various initial hazard rates. The correlation between the wealth process and the mortality process slightly affects the ruin probability at time zero.
Open Access
Finite Pseudo-Differential Operators, Localization Operators for Curvelet and Ridgelet Transforms
(2014-07-09) Li, Jiawei; Wong, Man Wah
Pseudo-differential operators can be built from the Fourier transform. However, besides the difficult problems in proving convergence and L^2-boundedness, the problem of finding eigenvalues is notoriously difficult. Finite analogs of pseudo-differential operators are desirable and indeed are constructed in this dissertation. Energized by the success of the Fourier transform and wavelet transforms, the last two decades saw the rapid developments of new tools in time-frequency analysis, such as ridgelet transforms and curvelet transforms, to deal with higher dimensional signals. Both curvelet transforms and ridgelet transforms give the time/position-frequency representations of signals that involve the interactions of translation, rotation and dilation, and they can be ideally used to represent signals and images with discontinuities lying on a curve such as images with edges. Given the resolution of the identity formulas for these two transforms, localization operators on them are constructed. The later part of this dissertation is to investigate the L^2-boundedness of the localization operators for curvelet transforms and ridgelet transforms, as well as their trace properties.
Open Access
Modeling, Dynamics and Optimal Control of West Nile Virus with Seasonality
(2015-01-26) Abdelrazec, Ahmed Hassan Manaa; Zhu, Huaiping
West Nile virus (WNv) is a mosquito-borne disease which arrived in Canada in 2001. It has kept spreading across the country and still remains a threat to public health. In this dissertation, we formulate dynamical models and apply theory of dynamical systems to investigate the behavior of the transmission of WNv in the mosquito-bird cycle and humans. In the first part, we propose a system of ordinary differential equations to model the role of corvids and non-corvids birds in the transmission of WNv in the mosquito-bird cycle in a single season and proved the existence of backward bifurcation in the model. In the second part, we consider another deterministic model to study the impact of seasonal variations of the mosquito population on the transmission dynamics of WNv. We prove the existence of periodic solutions under specific conditions. As for the third part, the latter model is extended to assess the impact of some anti-WNv control measures; by re-formulating the model as an optimal control problem. For mosquito-borne diseases, it is essential to access and forcast the virus risk. Therefore in the final part, we generalize the risk index, minimum infection rate (MIR) by using a compartment model for WNv, to define a dynamical minimum infection rate (DMIR) for assessing risk of WNv. By using the data from Peel region, we test and forecast the weekly risk of WNv which can help identify the optimal mitigation strategies.
Open Access
Statistical Modeling to Information Retrieval for Searching from Big Text Data and Higher Order Inference for Reliability
(2015-01-26) Zhou, Xiaofeng; Wong, Augustine Chi Mou
This thesis examined two research projects: probabilistic information retrieval modeling and third-order inference on reliability. In the first part of this dissertation, two research topics in the information retrieval are carried out and experimented on large-scale text data set. First, we conduct an in-depth study of relationship between information of document length and document relevance to user need. Two statistical methods are proposed which incorporates document length as a substantial weighting factor to achieve higher retrieval performance. Second, we utilize the property of survival function to propose a cost-based re-ranking method to promote ranking diversity for biomedical information retrieval, and to model the proximity between query terms to improve retrieval performance. Through extensive experiments on standard TREC collections, our proposed models perform significantly better than the classical probabilistic information retrieval models. In the second part of this dissertation, a small sample asymptotic method is proposed for higher order inference in the stress-strength reliability model, R=P(Y
Open Access
Pricing and Hedging Options in Discrete Time with Liquidity Risk
(2015-01-26) Sorokin, Yegor; Ku, Hyejin
Different derivative securities, including European options, are very popular and widely used in forms of exchange-traded instruments or over-the-counter products. For practical purposes the European options are often priced using analytic solution to the Black-Scholes formula. Hedging, according to the Black-Scholes model, is accomplished via the construction of dynamically rebalanced replicating portfolio. However, the model makes several critical assumptions. I extend the Black-Scholes model by relaxing the assumption of no trading costs and considering the market liquidity risk for the underlying asset. Liquidity risk is understood as the effect of the trade size on the price of the underlying asset. I use stochastic supply curve to model liquidity risk. The problem is to hedge a European option in the presence of the market liquidity risk for an underlying asset. One hedges with the underlying, as the option price depends on the price of the underlying asset. The underlying asset has market liquidity risk; thus, studying the impact of market liquidity risk is important for devising more effective and efficient option hedging algorithms. The main contributions of the thesis arise from the investigation of mathematical techniques for hedging and pricing of European options in discrete time with liquidity risk. First, I study delta hedging in Chapter 3. I show L2 convergence of the replicating trading strategy payoff to the option payoff. In other words, the optimal strategy minimizes the mean squared replication error. I also show that for European put and call options with varying trading times the recommendation is to trade closer to expiry as the spot price of the underlying asset deviates from the strike price. Then I apply the local risk-minimizing hedge in Chapter 4. This time the optimal strategy minimizes the conditional mean squared hedging error. I prove the existence of the local risk-minimizing trading strategy and characterize its structure.
Open Access
Convergence Rate Analysis of Markov Chains
(2015-01-26) Jovanovski, Oliver; Madras, Neal
We consider a number of Markov chains and derive bounds for the rate at which convergence to equilibrium occurs. For our main problem, we establish results for the rate of convergence in total variation of a Gibbs sampler to its equilibrium distribution. This sampler is motivated by a hierarchical Bayesian inference construction for a gamma random variable. The Bayesian hierarchical method involves statistical models that incorporate prior beliefs about the likelihood of observed data to arrive at posterior interpretations, and appears in applications for information technology, statistical genetics, market research and others. Our results apply to a wide range of parameter values in the case that the hierarchical depth is 3 or 4, and are more restrictive for depth greater than 4. Our method involves showing a relationship between the total variation of two ordered copies of our chain and the maximum of the ratios of their respective co-ordinates. We construct auxiliary stochastic processes to show that this ratio does converge to 1 at a geometric rate. In addition, we also consider a stochastic image restoration model proposed by A. Gibbs, and give an upper bound on the time it takes for a Markov chain defined by this model to be arbitrarily close in total variation to equilibrium. We use Gibbs' result for convergence in the Wasserstein metric to arrive at our result. Our bound for the time to equilibrium is of similar order to that of Gibbs.
Open Access
Algebraic-Delay Differential Systems: Co - Extendable Banach Manifolds and Linearization
(2015-01-26) Kosovalic, Nemanja; Wu, Jianhong
Consider a population of individuals occupying some habitat, and assume that the population is structured by age. Suppose that there are two distinct life stages, the immature stage and the mature stage. Suppose that the mature and immature population are not competing in the sense that they are consuming different resources. A natural question is ``What determines the age of maturity?" A subsequent natural question is ``How does the answer to the latter question affect the population dynamics?" In many biological contexts, including those from plant and insect populations, the age of maturity is not merely constant but is more accurately determined by whether or not the food concentration reaches a prescribed threshold. We consider a model for such a population in terms of a nonlinear transport equation with nonlocal boundary conditions. The variable age of maturity gives rise to an implicit state-dependent delay in the system of first order partial differential equations. We explain the relevance of this problem and provide a mechanistic derivation of the model equations. We address the existence, positivity, and continuity of the solution semiflow arising from the model equations, and then we discuss the differentiability of the semiflow with respect to initial data, in a suitable weak sense. The problem of the differentiability of the solution semiflow arising from even ordinary differential equations containing state-dependent delays was a long standing open problem for some time. Prior to this work, there were no results which addressed the linearization of the solution semiflow corresponding to a partial differential equation having a state-dependent delay.
Open Access
Operator Algebras and Abstract Classification
(2015-08-28) Lupini, Martino; Farah, Ilijas
This dissertation is dedicated to the study of operator spaces, operator algebras, and their automorphisms using methods from logic, particularly descriptive set theory and model theory. The material is divided into three main themes. The first one concerns the notion of Polish groupoids and functorial complexity. Such a study is motivated by the fact that the categories of Elliott-classifiable algebras, Elliott invariants, abelian separable C*-algebras, and arbitrary separable C*-algebras have the same complexity according to the usual notion of Borel complexity. The goal is to provide a functorial refinement of Borel complexity, able to capture the complexity of classifying the objects in a functorial way. Our main result is that functorial Borel complexity provides a finer distinction of the complexity of functorial classification problems. The second main theme concerns the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory. Our results show that, for any non-elementary simple separable C*-algebra, the problem of classifying its automorphisms up to unitary equivalence transcends countable structures. Furthermore we prove that in the unital case the relation of unitary equivalence obeys the following dichotomy: it is either smooth, when the algebra has continuous trace, or not classifiable by countable structures. The last theme concerns applications of model theory to the study and construction of interesting operator spaces and operator systems. Specifically we show that the Gurarij operator space introduced by Oikhberg can be characterized as the Fraisse limit of the class of finite-dimensional 1-exact operator spaces. This proves that the Gurarij operator space is unique, homogeneous, and universal among separable 1-exact operator spaces. Moreover we prove that, while being 1-exact, the Gurarij operator space does not embed into any exact C*-algebra. Furthermore the ternary ring of operators generated by the Gurarij operator space is canonical, and does not depend on the concrete representation chosen. We also construct the operator system analog of the Gurarij operator space, and prove that it has analogous properties.
Open Access
Theoretical and Computational Analysis of Credit and Liquidity Risk with Multiple Defaults
(2015-08-28) Zhu, Hongmei; Huang, Haohan
Since the 2008 global financial crisis, regulators have been paying considerable attention to the credit and liquidity risks. Two such concepts (related to credit and liquidity risks) that have been repeatedly mentioned in the regulatory announcements are the credit value adjustment (CVA) and the Incremental Risk Charge (IRC). The CVA is an adjustment to the previous trade price when the counterparty risk has been added. The IRC is a new type of risk charge defined in Basel II which covers the major exposures of the counterparty and liquidity risk in the trading book. The current models on CVA and IRC have specific shortcomings. The CVA is currently calculated on a one-period model with restriction on the number of defaults. The IRC is computed using the time consuming Monte Carlo simulations. In this dissertation, we have made significant contributions to risk analysis by solving CVA in both two-default and full model without the restriction on the number of defaults as well as providing an analytical method for calculating IRC. Our research can be considered as a major step forward in expanding the current credit and liquidity risks models. Compared to the current one-default CVA calculations, our two-default and full calculations offer the distinct advantages of more accurate and practical CVA results. On top of that, our PDE method provides the speed and accuracy which allows us to finish a thorough risk exposure analysis and identify the conditions when the first default CVA overestimates or underestimates the counterparty risk. As opposed to the current numerical approach of calculating IRC, we offer an analytical method which provides an approximation of VaR on the two-period model and exact value of VaR on the infinite-default model. This is the first analytic solution in the literature on the multi-period capital model and may impact the view of current measure of risk controls in the banks. Thus credit risk control can be greatly improved if this new analytic solution can be applied in financial industry. Combined together, the work in this dissertation makes significant improvements in credit risk analysis in the multi-period credit and liquidity risks models.
Open Access
Machine Learning and its Application in Automatic Change Detection in Medical Images
(2015-08-28) Nika, Varvara; Hongmei Zhu, Hongmei; Babyn, Paul
Change detection is a fundamental problem in various fields, such as image surveillance, remote sensing, medical imaging, etc. The challenge of change detection in medical images is to detect disease-related changes while rejecting changes caused by noise, patient position change, and imaging acquisition artifacts such as field inhomogeneity. In this thesis, first, we overview the existing change detection methods, their underlying mathematical frameworks and limitations. Second, we present our contributions in solving the problem. We design optimal subspaces to approximate the background image in more efficient fashion. This is based on our structure principal component analysis, aiming to capture the structural similarity between scans in the context of change detection. We theoretically and numerically discuss the proper choices of norms used in the subspace approximation. The mathematical frameworks developed in this thesis consist of: (i) a new mathematical model to change detection by defining it as an optimization problem involving a cost function, input and output image sets, projection onto a subspace, and a similarity measure; (ii) development and implementation of numerical pipelines to compute the clinical changes by designing four mathematical algorithms; (iii) refining our algorithms by introducing the co-registration step utilizing the local dictionaries; and (iv) two new structure subspace learning models that are robust to outliers and noise, reduce the dimensionality of the dataset, and computationally efficient. We defined the co-registration step as a minimization problem involving a cost function, input and output image sets, a set of transform functions, projection onto a subspace, and a similarity measure. Based on the mathematical frameworks discussed above, numerical schemes are developed to automatically filter out clinically unrelated changes and identify true structure changes that may be of clinical importance. Our approaches are data-driven and utilize the knowledge of machine learning. We quantitatively analyze the performance of these algorithms using both synthetic and real human data. Our work has the potential to be used in computer aided diagnosis.
Open Access
Extensions of the Formlet Model of Planar Shape
(2015-08-28) Yakubovich, Alexander; Elder, James
This thesis addresses the problem of shape representation using the GRID/formlet theory, a system based on localized diffeomorphisms. While this framework enjoys many desirable properties, it suffers from several limitations: it converges slowly for shapes with elongated parts, and it can be sensitive to parameterization as well as grossly ill-conditioned. Several innovations are proposed to address these problems: 1) The formlet basis is generalized to include oriented deformation, improving convergence for elongated parts. 2) A recent contour remapping method is applied in order to eliminate problems due to drift of the model parameterization during matching pursuit. 3) A regularizing term is introduced in order to limit redundancy in formlet parameters and improve the model’s identifiability. Finally, an algorithm is proposed to hierarchically cluster formlets, and is shown to induce a partial ordering on the representation.
Open Access
Combining Test Statistics and Information Criteria for High Dimensional Data Integration
(2015-08-28) Xu, Yawen; Gao, Xin; Wang, Xiaogang
This research is focused on high dimensional data integration by combing test statistics or information criteria. Our research contains four projects. Firstly, an integration method is developed to perform hypothesis testing and biomarkers selection based on multi-platform data sets observed from normal and diseased populations. Secondly, non-parametric method is developed to cluster continuous data mixed with categorical data, where modified Chi-squared tests are used to detect of cluster patterns on the product space. Thirdly, weighted integrative AICs criterion is developed to be used for model selection across multiple data sets. Finally, Linhart's and Shimodaria's test statistics are extended onto composite likelihood function to perform model comparison test for correlated data.
Open Access
Strongly Summable Ultrafilters: Some Properties and Generalizations
(2015-08-28) Breton, David Jose Fernandez; Steprans, Juris
This dissertation focuses on strongly summable ultrafilters, which are ultrafilters that are related to Hindman’s theorem in much the same way that Ramsey ultrafilters are related to Ramsey’s theorem. Recall that Hindman’s theorem states that whenever we partition the set of natural numbers into two (or any finite number of) cells, one of the cells must entirely contain a set of the form FS(X) for some infinite set X (here FS(X) is the collection of all nonrepeating sums of finitely many elements of X). A nonprincipal ultrafilter on the set of natural numbers is said to be strongly summable if it has a base of sets of the form FS(X), this is, if for every element A of p, there exists an infinite X such that FS(X) is both a subset of A and an element of p. These ultrafilters were first introduced by Hindman, and subsequently studied by people such as Blass, Eisworth, Hindman, Krautzberger, Matet, Protasov and others. Now, from the viewpoint of the definitions, there is nothing special about the semigroup of natural numbers, and analogous definitions for FS(X) and strongly summable ultrafilter can be considered for any semigroup (in the non-abelian case, one must first fix an ordering for X on order-type omega). It is not immediate, however, that the results that hold for strongly summable ultrafilters on the semigroup of natural numbers are still satisfied in general. Some of the main results of this dissertation are generalizations of these properties for all abelian groups and some non-abelian cases as well. Notably among these, a strongly summable ultrafilter p on an abelian group G has the so-called trivial sums property: whenever q, r are ultrafilters on G such that q+r=p, it must be the case that for some element g of G, q=p+g and r=−g+p (this is all in the context of the right-topological semigroup of all ultrafilters on G). The other significant result from this dissertation is a consistency result. It has long been known that the existence of strongly summable ultrafilters (on any abelian group) is not provable from the ZFC axioms, for it implies the existence of P-points. It is also known, however, that (at least on the semigroup of natural numbers) the existence of strongly summable ultrafilters follows from the restriction of Martin’s axiom to countable forcing notions. We prove here that there exist models of ZFC that satisfy the failure of this restriction of Martin's axiom, while at the same time in these models there exist strongly summable ultrafilters on all abelian groups. This can be done using iterations, both with finite or with countable support, of σ-centred forcing notions which resemble Mathias’s or Laver’s forcing.
Open Access
High Order Energy - Conserved Splitting FDTD Methods for Maxwell's Equations
(2015-08-28) Yuan, Qiang; Liang, Dong
Computation of Maxwell's equations has been playing an important role in many applications, such as the radio-frequencies, microwave antennas, aircraft radar, integrated optical circuits, wireless engineering and materials, etc. It is of particular importance to develop numerical methods to solve the equations effectively and accurately. During the propagation of electromagnetic waves in lossless media without sources, the energies keep constant for all time, which explains the physical feature of electromagnetic energy conservations in long term behaviors. Preserving the invariance of energies is an important issue for efficient numerical schemes for solving Maxwell's equations. In my thesis, we first develop and analyze the spatial fourth order energy-conserved splitting FDTD scheme for Maxwell's equations in two dimensions. For each time stage, while the spatial fourth-order difference operators are used to approximate the spatial derivatives on strict interior nodes, the important feature is that on the near boundary nodes, we propose a new type of fourth-order boundary difference operators to approximate the derivatives for ensuring energy conservative. The proposed EC-S-FDTD-(2,4) scheme is proved to be energy-conserved, unconditionally stable and of fourth order convergence in space. Secondly, we develop and analyze a new time fourth order EC-S-FDTD scheme. At each stage, we construct a time fourth-order scheme for each-stage splitting equations by converting the third-order correctional temporal derivatives to the spatial third-order differential terms approximated further by the three central difference combination operators. The developed EC-S-FDTD-(4,4) scheme preserves energies in the discrete form and in the discrete variation forms and has both time and spatial fourth-order convergence and super-convergence. Thirdly, for the three dimensional Maxwell's equations, we develop high order energy-conserved splitting FDTD scheme by combining the symplectic splitting with the spatial high order near boundary difference operators and interior difference operators. Theoretical analyses including energy conservations, unconditional stability, error estimates and super-convergence are established for the three dimensional problems. Finally, an efficient Euler-based S-FDTD scheme is developed and analyzed to solve a very important application of Maxwell's equations in Cole-Cole dispersive medium. Numerical experiments are presented in all four parts to confirm our theoretical results.
Open Access
Analytical Methods For Levy Processes With Applications To Finance
(2015-08-28) Hackmann, Daniel; Kuznetsov, Alexey
This dissertation is divided into two parts: the first part is a literature review and the second describes three new contributions to the literature. The literature review aims to provide a self-contained introduction to some popular Levy models and to two key objects from the theory of Levy processes: the Wiener-Hopf factors and the exponential functional. We pay special attention to techniques and results associated with two “analytically tractable” families of processes known as the meromorphic and hyper-exponential families. We also demonstrate some important numerical techniques for working with these families and for solving numerical integration and rational approximation problems. In the second part of the dissertation we prove that the exponential functional of a meromorphic Levy process is distributed like an infinite product of independent Beta random variables. We also identify the Mellin transform of the exponential functional, and then, under the assumption that the log-stock price follows a meromorphic process, we use this to develop a fast and accurate algorithm for pricing continuously monitored, fixed strike Asian call options. Next, we answer an open question about the density of the supremum of an alpha-stable process. We find that the density has a conditionally convergent double series representation when alpha is an irrational number. Lastly, we develop an effective and simple algorithm for approximating any process in the class of completely monotone processes –some members of this class include the popular variance gamma, CGMY, and normal inverse Gaussian processes – by a hyper-exponential process. Under the assumption that the log-stock price follows a variance gamma or CGMY process we use this approximation to price several exotic options such as Asian and barrier options. Our algorithms are easy to implement and produce accurate prices.
Open Access
Rigidity of Corona Algebras
(2015-08-28) Ghasemi, Saeed; Farah, Ilijas
In this thesis we use techniques from set theory and model theory to study the isomorphisms between certain classes of C*-algebras. In particular we look at the isomorphisms between corona algebras of the form $\prod\mathbb{M}_{k(n)}(\mathbb{C})/\bigoplus \mathbb{M}_{k(n)}(\mathbb{C})$ for sequences of natural numbers $\{k(n): n\in\mathbb{N}\}$. We will show that the question ``whether any isomorphism between these C*-algebras is trivial", is independent from the usual axioms of set theory (ZFC). We extend the classical Feferman-Vaught theorem to reduced products of metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent. We also use this to find examples of corona algebras of the form $\prod\mathbb{M}_{k(n)}(\mathbb{C}) / \bigoplus \mathbb{M}_{k(n)}(\mathbb{C})$ which are non-trivially isomorphic under the Continuum Hypothesis. This gives the first example of genuinely non-commutative structures with this property. In chapter 6 we show that $SAW^{*}$-algebras are not isomorphic to $\nu$-tensor products of two infinite dimensional C*-algebras, for any C*-norm $\nu$. This answers a question of S. Wassermann who asked whether the Calkin algebra has this property.
Open Access
Optimal Retirement Investment Strategies Under Health Shocks and Jump-Diffusion Processes
(2015-12-16) Cara, Mirela Elena; Huang, Huaxiong
The dissertation focuses on two problems applied to personal financial management for individuals, either before or after retirement. The first topic examines a lifetime ruin probability (LRP) model in which a jump-diffusion process drives the investment return of the agent. The value of the LRP is important to an investor who wants to find out the probability of running out of money, while maintaining a desired standard of living for the rest of his life. Our model leads to a partial-integro-differential equation (PIDE) which is solved by a numerical algorithm. Results are compared against diffusion-related LRP values that do not assume jumps by using calibrated parameters. Retirees are often exposed to large and unpredictable medical expenses due to health shocks. The second topic examines the effect of health shocks and mortality risk on the optimal medical insurance-consumption-allocation strategy. We also derived a solution for the optimal retirement-triggering wealth in a life-cycle framework. As in the first problem, we investigated model changes, for asset return rates which obey a jump-diffusion dynamics.
Open Access
Some Aspects of Statistical Volatility Analysis
(2015-12-16) Xu, Min; Wu, Yuehua
Volatility is the key of the option price in the stock market. Changes in volatility will dramatically lead to changes of the option price. One of the most important volatilities is historical volatility(HV). HV is essentially the annualized standard deviation of the first order difference of logarithm of the asset price. Therefore, changes in HV in finance may be detected by the variance change detection methods in statistics. We propose a weighted sum of powers of variances method to detect single change in HV. It is noted that this method only examines if there is one single change-point in the data sequence. In the second part of the dissertation, we propose the empirical Bayesian information criterion (emBIC) method to detect multiple change-points simultaneously. The empirical BIC method can not only detect change-points in HV, but also in mean, and mean-and-variance. Simulation study shows that both of the above methods perform very well. We also apply these methods to detect changes in HV by using real stock data. Another important volatility is the implied volatility (IV). IV is the volatility of asset implied by the market option price based on Black-Sholes model. The long term IV and HV have totally different behaviours. We find the optimal time range by using the emBIC method aforementioned above. We explain the long term IV behaviour by interest rate risk and capital charge in the last part of the dissertation.
Open Access
Imaginary Whittaker Modules For Extended Affine Lie Algebras
(2016-09-20) Shi, Song; Gao, Yun
We classify irreducible Whittaker modules for generalized Heisenberg Lie algebra t and irreducible Whittaker modules for Lie algebra t obtained by adjoining m degree derivations d1, d2, . . . , dm to t. Using these results, we construct imaginary. Whittaker modules for non-twisted extended affine Lie algebras and prove that the imaginary Whittaker modules of Z-independent level are always irreducible.