This thesis considers classical methods to solve stochastic control problems and valuation problems from financial mathematics numerically. To this end, (linear) partial differential equations (PDEs) in non-divergence form or the optimality conditions known as the (nonlinear) Hamilton-Jacobi-Bellman (HJB) equations are solved by means of finite differences, volumes and elements. We consider all of these three approaches in detail after a thorough introduction to stochastic control problems and discuss various solution terms including classical solutions, strong solutions, weak solutions and viscosity solutions. A particular role in this thesis play degenerate problems. Here, a new model for the optimal control of an energy storage facility is developed which extends the model introduced in [Chen, Forsyth (2007)]. This four-dimensional HJB equation is solved by the classical finite difference Kushner-Dupuis scheme [Kushner, Dupuis (2001)] and a semi-Lagrangian variant which are both discussed in detail. Additionally, a convergence proof of the standard scheme in the setting of parabolic HJB equations is given. Finite volume schemes are another classical method to solve partial differential equations numerically. Sharing similarities to both finite difference and finite element schemes we develop a vertex-centered dual finite volume scheme. We discuss convergence properties and apply the scheme to the solution of HJB equations, which has not been done in such a broad context, to the best of our knowledge. Astonishingly, this is one of the first times the finite volume approach is systematically discussed for the solution of HJB equations. Furthermore, we give many examples which show advantages and disadvantages of the approach. Finally, we investigate novel tailored non-conforming finite element approximations of second-order PDEs in non-divergence form, utilizing finite-element Hessian recovery strategies to approximate second derivatives in the equation. We study approximations with both continuous and discontinuous trial functions. Of particular interest are a-priori and a-posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature. We discuss implementations of all three approaches in MATLAB (finite differences and volumes) and FEniCS (finite elements) publicly available in GitHub repositories under https://github.com/janblechschmidt. Many numerical experiments show convergence properties as well as pros and cons of the respective approach. Additionally, a new postprocessing procedure for policies obtained from numerical solutions of HJB equations is developed which improves the accuracy of control laws and their incurred values.

BACKGROUND: Major Depressive Disorder (MDD) is a common mental health illness, characterized by persistent feelings of sadness, diminished interests, guilt, low-self esteem, and disturbances in sleep and appetite. A significant percentage of patients with MDD are treatment resistant. Electroconvulsive Therapy (ECT) is a biological procedure utilized for treatment resistant illnesses. Diagnosis and clinical conditions primarily dictate when ECT is the appropriate treatment modality for an individual. Circumstances requiring rapid clinical response, risks affiliated with alternative treatments, resistance to pharmacotherapy, and medical history are all factors that designate ECT as the treatment of choice. METHODS: The objective of this systematic review was to examine how different anesthetics or combinations of agents affect ECT’s therapeutic efficacy in depressed, adult patients. Electroencephalography (EEG) and motor seizure durations and Hamilton Depression Rating Scale (HDRS) scores were used as primary measures of clinical outcomes. Two rounds of literature searches were conducted in the PubMed, Web of Science, and Google Scholar databases to identify randomized controlled trials and crossover trials that examined the effects of different intravenous sedatives and hypnotic agents on ECT. Two reviewers independently evaluated the internal validity and quality of studies, extracted data, and analyzed statistics. Utilizing all relevant data, standardized mean differences (SMD) with 95% confidence intervals (CIs), and heterogeneity measures were calculated. Ten studies with 373 participants were included. RESULTS: Thiopental only anesthesia was associated with longer EEG seizure duration when compared to propofol only treatment. The pooled effect size from studies with propofol anesthesia also suggests that this agent is associated with shorter seizure durations. If assessed individually with thiopental, the combination of ketamine and thiopental is correlated with increased motor as well as EEG seizure durations. When pooled; however, studies with patient groups assigned to anesthesia consisting of ketamine and another primary agent do not show significant differences either in EEG or motor seizure durations. Additionally, no difference exists in HDRS score reductions between propofol and methohexital. Of note; however, ketamine combined with either propofol or thiopental had significantly greater decreases in HDRS scores. CONCLUSION: Choice of anesthetic should be determined based on anticipated clinical outcome, adverse effect profile, reemergence, and patient preference. If long seizures are preferred, thiopental may be a reasonable option. However, if significantly larger decreases in depression score are preferred, then the combinations of ketamine and propofol or ketamine and thiopental appear to be the therapies of choice. Small sample sizes and insufficient clinical data limit the interpretations of these variables that determine therapeutic efficacy. Larger randomized control trials and crossover trials would provide greater insight into the optimal use of intravenous anesthetic agents with minimal adverse effects.

The Hamilton Rating Scale for Depression (HRS), a 17-item observer-rated scale, was first developed in 1960 to assess the severity of depressive symptomatology in patients diagnosed as suffering from depression. The HRS has since demonstrated high inter-rater reliability (with coefficients ranging from .87 to .94), and has proven useful in measuring changes of severity following treatment.

京都大学 / 0048 / 新制・課程博士 / 博士(農学) / 甲第20006号 / 農博第2190号 / 新制||農||1045(附属図書館) / 学位論文||H28||N5015(農学部図書室) / 33102 / 京都大学大学院農学研究科地域環境科学専攻 / (主査)教授 藤原 正幸, 教授 村上 章, 准教授 宇波 耕一 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

Merton’s Optimal Investment-Consumption Problem is a classic optimization problem in finance. It aims to find the optimal controls for a portfolio with both risky and risk-less assets, inorder to maximize an investor’s utility function. One of the controls is the optimal allocationof wealth invested in a risky asset and the other control is the consumption rate. The problemis solved by using Dynamic Programming and the related Hamilton-Jacobi-Bellman equation.One of the disadvantages of the original problem is the consideration of constant volatility. Inthis thesis, we extend Merton’s problem considering the Grzelak-Oosterlee-Van Veeren modelthat describes the dynamics of a risky asset with stochastic volatility and stochastic interestrate. We derive the related Hamilton-Jacobi-Bellman for Merton’s problem considering theGrzelak-Oosterlee-Van Veeren model. We simulate the controls from Merton’s problem intwo different cases, one case where the volatility and interest rate are stochastic, following theGOVV-model. In the other case, the volatility and interest rate are assumed to be constant, asin Merton’s problem. The results obtained from simulations show that the case with stochasticvolatility and interest gave the same results as the case where the volatility and the interest ratewere assumed to be constant.

This thesis examines the relationship between childhood exposure to air pollution and long-term health. The research is based on an earlier study (1978-1986) that examined the relationship between exposure to air quality and respiratory health for a cohort of elementary school-aged children (n=3,202). These children resided in four distinct neighbourhoods in Hamilton, Ontario, which exhibited significant gradients in air pollution levels. Informed by the Life Course Health Development model, a survey was developed and administered on a reconstructed cohort (n = 395). The following objectives were addressed: 1) to determine the current health status of the reconstructed cohort; 2) to assess the potential relationship between childhood exposure to air pollution and adult respiratory health; and, 3) to explore factors mediating this relationship. Data was collected for a range of variables including residential and occupational histories, socio-demographic variables, and health outcomes. The dataset was merged with data from childhood on respiratory health, exposure to air pollution, and socio-demographic variables. Results indicate that a relatively high percentage of respondents had asthma in childhood (11%) compared to the original cohort (5.5%). In addition, prevalence rates of most health outcomes were higher than those of the Canadian population. Despite the gradient in air pollution levels, there were no significant differences in health status across neighborhoods. However, results of bi-variate and multi-variate analysis indicate possible significant associations between childhood exposure to S02 and hospital visits for asthma, asthma incidence in adulthood, and ever being diagnosed with asthma for females. For males, results suggest that childhood exposure to S02 is not linked to respiratory health. Factors related to the macro and micro environments also play significant roles in long-term health. This thesis made significant contributions to knowledge by suggesting that childhood exposure to S02 may impact long-term respiratory health for females, and may be linked to inflammatory diseases. / Thesis / Doctor of Philosophy (PhD)

The purpose of this manuscript is to explain Alexander Hamilton's idea that a national bank was essential for America's survival. Three key ingredients, clarified through the use of letters and documents, are used to understand the importance of Hamilton's objective: 1) Hamilton's relationship with George Washington, discussed in chapter one; 2) James Madison's and Thomas Jefferson's arguments against Hamilton's ideas, discussed in chapters two and three; and 3) Hamilton's proposal for the bank and his opinion in favor of its constitutionality, including the idea that the necessary and proper clause expands the authority of government, discussed in chapter four. The prosperity and stability America experienced after the national bank threw open its door is proof that Hamilton's initiatives were needed. While James Madison is considered the Father of the Constitution, Hamilton is most definitely it foremost interpreter.

This thesis concerns the approximation of optimally controlled partial differential equations for applications in optimal design and reconstruction. Such optimal control problems are often ill-posed and need to be regularized to obtain good approximations. We here use the theory of the corresponding Hamilton-Jacobi-Bellman equations to construct regularizations and derive error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method where the first, analytical, step is to regularize the Hamiltonian. Next its stationary Hamiltonian system, a nonlinear partial differential equation, is computed efficiently with the Newton method using a sparse Jacobian. An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L2 projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems. In the thesis we present solutions to applications such as optimal design and reconstruction of conducting materials and elastic structures. / QC 20101110

Although Paul Hamilton Hayne was the acknowledged poet laureate of the South at the time of his death in 1886, he and his poetry have virtually disappeared from the recent American literary histories and anthologies. Even the literary histories and anthologies of Southern literature tend to down play his role as a man of letters and poet of consequence. This diminution of Hayne's literary reputation has taken place despite the respect for his poetry and criticism that came from leading poets and critics in the United States and England during the mid to late nineteenth century. In this thesis, I analyze the neglect of Hayne's work by first outlining his reception history as a poet. Certain trends are evident, specifically a movement in the United States away from the Anglo-American tradition to a new style of poetry, best represented by Walt Whitman. This change in what was fashionable in poetry has had the effect of undermining Hayne's reputation as a poet. Moreover, Hayne's literary reputation became more tenuous after the War Between the States given his strong affiliation with the conquered Confederacy. To bolster my argument that Hayne's reputation ought not be left to the whims of literary fashion, I conduct a preliminary examination of Hayne's poetry by analyzing several poems. I conclude, after examining the evidence, that Hayne deserves inclusion in the literary canons of American and Southern literature as an important representative nineteenth-century Southern poet writing within the Anglo-American tradition. / Master of Arts

<p> This study determines the potential for expansion within Third Sector Employment Enterprises: the recycling operation in Hamilton-Wentworth. Although the company has been in existence since 1977, it has not yet reached the level of viability achieved by other firms established at that time . This study involves a comparison of Third Sector with Halton,s Recycled Resources: the recycling firm in the Region of Halton. The comparison attempts to determine what factors within Third Sector and Hamilton-Wentworth are retarding the growth of the company. A quantitative comparison was used to contrast the net revenues of the two companies, and descriptive data was used to explain and qualify the quantitative findings. The analysis reveals that despite the operational differences between the two companies, it is community participation that makes Halton's Recycled Resources more viable than Third Sector. The concluding remarks make some suggestions as to how community participation, and thus Third Sector's operation can be improved. </p> / Thesis / Bachelor of Arts (BA)