Mathematics
http://hdl.handle.net/1808/263
2021-09-08T11:11:08ZUnique Factorization Domains in Commutative Algebra
http://hdl.handle.net/1808/31857
Unique Factorization Domains in Commutative Algebra
Huang, Yongjian
In this project, we learn about unique factorization domains in commutative algebra. Most importantly, we explore the relation between unique factorization domains and regular local rings, and prove the main theorem: If R is a regular local ring, so is a unique factorization domain.
This project was submitted to the Mathematics department in partial fulfillment of the requirements for the degree of Master of Arts.
2021-05-20T00:00:00ZInitial-boundary value problems for a reaction-diffusion equation
http://hdl.handle.net/1808/31590
Initial-boundary value problems for a reaction-diffusion equation
Himonas, A. Alexandrou; Mantzavinos, Dionyssios; Yan, Fangchi
A novel approach that utilizes Fokas’s unified transform is employed for studying a reaction-diffusion equation with power nonlinearity formulated either on the half-line or on a finite interval with data in Sobolev spaces. This approach was recently introduced for initial-boundary value problems involving dispersive nonlinear equations such as the nonlinear Schrödinger and the Korteweg-de Vries equations. Thus, the present work extends the new approach from dispersive equations to diffusive ones, demonstrating the universality of the unified transform in the analysis of nonlinear evolution equations on domains with a boundary.
This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Math. Phys. 60, 081509 (2019); doi: 10.1063/1.5118767 and may be found at https://aip.scitation.org/doi/10.1063/1.5118767.
2019-08-27T00:00:00ZA General Stochastic Volatility Model on VIX Options
http://hdl.handle.net/1808/31502
A General Stochastic Volatility Model on VIX Options
Cui, Yanhao
Abstract In this dissertation, we study a general stochastic volatility model for the VIX options (Chicago Board Options Exchange) volatility index, which is a stochastic differential equation with 8 unknown parameters. It originated from a nested stochastic model based on several known models in the paper [7], stochastic volatility models and the Pricing of VIX Options. To estimate the parameters in these models from the real financial data a commonly used approach is the Generalized Method of Moments of Hansen (1982). We will study the model in more generality and we shall provide a completely different parameter estimation technique using the ergodic theory. Since our equation is more general and new and since our equation is singular in the sense it does not satisfy the global Lipschitz condition, we shall first study the existence, uniqueness and positivity of the solution of the SDE, in which Feller’s test will be used to calculate a criteria of all parameters such that the SDE has a unique and positive weak solution. The positivity property of the solution is crucial, since volatility is always positive. Then, we use the strong large law of numbers theorems given e.g. in [4] to give the region for the parameters to live in order that the model is ergodic. In important condition for the ergodicity is the positive recurrency. We give verifiable condition on the parameters so that process is positive recurrent. This results also provide ways to calculate the invariant distribution (limiting distribution). The next step is to provide a theoretical methodology of parameter estimation. Simulation process will be introduced with giving an example for each case. In the future study, I will work on testing the model using numerical schemes. Keywords: Stochastic volatility model, VIX options, Feller’s test, ergodicity, parameter estimation.
2019-12-31T00:00:00ZOn the Generation of Stable Kerr Frequency Combs in the Lugiato--Lefever Model of Periodic Optical Waveguides
http://hdl.handle.net/1808/31470
On the Generation of Stable Kerr Frequency Combs in the Lugiato--Lefever Model of Periodic Optical Waveguides
Hakkaev, Sevdzhan; Stanislavova, Milena; Stefanov, Atanas G.
2019-03-07T00:00:00Z