This project has three parts. The first part is to use the efficient frontier and find the tangency portfolio to form our optimal portfolio. We built our portfolio using the Interactive Brokers software and rebalanced every week for 4 holding periods to see the relationship between our projected returns and actual market returns. In the second part we considered the Capital Asset Pricing Model (CAPM) and ran linear regressions on the stocks we chose in the first part of the project. This process is based on our idea of finding the systematic risk in each stock to improve our stock choosing ability. In the last part we introduce the concept of factor models and add more factors into our original CAPM model. Via a back-testing method, we test the reasonability of our factors and give advice to further improve our portfolio optimization project.

In this thesis I develop a mathematical model for an apartment building during an earthquake. The movement of the building is restricted to a plane and twisting motions have been assumed negligible. A control system for the building is developed using optimal control techniques. For a quadratic objective functional, the existence of an optimal control is determined and numerical results are generated that show that the controller significantly lowers the chaotic oscillations in the building. The numerical work was done with the Miser3 package for Matlab. Relaxation of different constraints are considered, including multiple controls, varying stories, and different objective functionals.

<p>The main achievement of this work is the development of a duality theory for optimal control problems with stochastic differential equations. Incipient with the Hamilton-Jacobi-Bellman equation we established a dual problem to a given stochastic control problem and were also able to generalise the assembled theory.</p>

The main achievement of this work is the development of a duality theory for optimal control problems with stochastic differential equations. Incipient with the Hamilton-Jacobi-Bellman equation we established a dual problem to a given stochastic control problem and were also able to generalise the assembled theory.

Mathematical models often involve unknown parameters that must be fit to experimental data. These so-called parameter estimation problems have many applications that may involve differential equations, optimization, and control theory. EASY-FIT and SOCS are two software packages that solve parameter estimation problems. In this thesis, we discuss the design and implementation of a source-to-source translator called EFtoSOCS used to translate EASY FIT input into SOCS input. This makes it possible to test SOCS on a large number of parameter estimation problems available in the EASY-FIT problem database that vary both in size and difficulty.<p>Parameter estimation problems typically have many locally optimal solutions, and the solution obtained often depends critically on the initial guess for the solution. A 3-stage approach is followed to enhance the convergence of solutions in SOCS. The stages are designed to use an initial guess that is progressively closer to the optimal solution found by EASY-FIT. Using this approach we run EFtoSOCS on all translatable problems (691) from the EASY-FIT database. We find that all but 7 problems produce converged solutions in SOCS. We describe the reasons that SOCS was not able solve these problems, compare the solutions found by SOCS and EASY-FIT, and suggest possible improvements to both EFtoSOCS and SOCS.

Farm size economies size measure the relationship between the size of operation and the average cost of production. Along with increasing farm size, the average cost of production per unit may decline. One reason farms have been growing in size is that larger sized farms tend to have more recent and advanced machines capable of covering more land with less labor. However, it is still questionable how farm size affects on input costs and field operation costs in Saskatchewan. The major objective of this study was to examine the issues related to size economies in larger crop farms in Saskatchewan. The project has taken a different approach than is traditionally done in economies of size research where various forms of statistical data are analyzed. <p>First, the study analyzed several different operating and investment costs to see whether they are decreasing or staying the same as a result of increasing farm size. Next the study determined the probabilities of available field workdays using conditional probability equations derived from the Markov Chain method. The analysis was carried out for the West central and East central Saskatchewan regions to determine spring and fall field workability. Based on the field workdays estimation, the optimal area of combine for larger farms were analysed using a least-cost machinery size approach. The last part of this study analysed farm operational costs per unit for larger crop farms in order to determine how machinery efficiency and farm size have an effect on the farm production costs. <p>The study found that however there were certain combine costs that increase with farm size in Saskatchewan. In addition, soil types, weather conditions and field efficiency can strongly affect combine cost per acre. <p>The results of this research provide a reference for policy makers in designing policy recommendations. In addition, the results may offer useful information for farmers in designing management plans to control farm operation costs.

Quota-share and stop-loss/excess-of-loss reinsurances are two important reinsurance strategies. An important question, both in theory and in application, is to determine optimal retentions for these reinsurances. In this thesis, we study the optimal retentions of quota-share and stop-loss/excess-of-loss reinsurances under ruin-related optimization criteria. We attempt to balance the interest for a ceding company and a reinsurance company and employ an optimization criterion that considers the interests of both a cedent and a reinsurer. We also examine the influence of interest, dividend, commission, expense, and diffusion on reinsurance retentions.

Quota-share and stop-loss/excess-of-loss reinsurances are two important reinsurance strategies. An important question, both in theory and in application, is to determine optimal retentions for these reinsurances. In this thesis, we study the optimal retentions of quota-share and stop-loss/excess-of-loss reinsurances under ruin-related optimization criteria. We attempt to balance the interest for a ceding company and a reinsurance company and employ an optimization criterion that considers the interests of both a cedent and a reinsurer. We also examine the influence of interest, dividend, commission, expense, and diffusion on reinsurance retentions.

Farm size economies size measure the relationship between the size of operation and the average cost of production. Along with increasing farm size, the average cost of production per unit may decline. One reason farms have been growing in size is that larger sized farms tend to have more recent and advanced machines capable of covering more land with less labor. However, it is still questionable how farm size affects on input costs and field operation costs in Saskatchewan. The major objective of this study was to examine the issues related to size economies in larger crop farms in Saskatchewan. The project has taken a different approach than is traditionally done in economies of size research where various forms of statistical data are analyzed. <p>First, the study analyzed several different operating and investment costs to see whether they are decreasing or staying the same as a result of increasing farm size. Next the study determined the probabilities of available field workdays using conditional probability equations derived from the Markov Chain method. The analysis was carried out for the West central and East central Saskatchewan regions to determine spring and fall field workability. Based on the field workdays estimation, the optimal area of combine for larger farms were analysed using a least-cost machinery size approach. The last part of this study analysed farm operational costs per unit for larger crop farms in order to determine how machinery efficiency and farm size have an effect on the farm production costs. <p>The study found that however there were certain combine costs that increase with farm size in Saskatchewan. In addition, soil types, weather conditions and field efficiency can strongly affect combine cost per acre. <p>The results of this research provide a reference for policy makers in designing policy recommendations. In addition, the results may offer useful information for farmers in designing management plans to control farm operation costs.

Mathematical models often involve unknown parameters that must be fit to experimental data. These so-called parameter estimation problems have many applications that may involve differential equations, optimization, and control theory. EASY-FIT and SOCS are two software packages that solve parameter estimation problems. In this thesis, we discuss the design and implementation of a source-to-source translator called EFtoSOCS used to translate EASY FIT input into SOCS input. This makes it possible to test SOCS on a large number of parameter estimation problems available in the EASY-FIT problem database that vary both in size and difficulty.<p>Parameter estimation problems typically have many locally optimal solutions, and the solution obtained often depends critically on the initial guess for the solution. A 3-stage approach is followed to enhance the convergence of solutions in SOCS. The stages are designed to use an initial guess that is progressively closer to the optimal solution found by EASY-FIT. Using this approach we run EFtoSOCS on all translatable problems (691) from the EASY-FIT database. We find that all but 7 problems produce converged solutions in SOCS. We describe the reasons that SOCS was not able solve these problems, compare the solutions found by SOCS and EASY-FIT, and suggest possible improvements to both EFtoSOCS and SOCS.