Mathematics Dissertations and Theses http://hdl.handle.net/1808/14149 2021-09-25T12:50:59Z A 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 , 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  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. 2019-12-31T00:00:00Z Optimal Energy Decay for the Damped Klein-Gordon Equation http://hdl.handle.net/1808/31358 Optimal Energy Decay for the Damped Klein-Gordon Equation Malhi, Satbir Singh In this dissertation we study the long time dynamics of damped Klein-Gordon and damped fractional Klein-Gordon equations using $C_0$- Semigroup theory and its application. The $C_0$-semigroups are used to solve a large class of problems commonly known as evolution equations. Such models arise from delay differential equations and partial differential equations in many disciplines including physics, chemistry, biology, engineering, and economics. Water waves, sound waves and simple harmonic motion of strings are few important models of evolution equations. The Klein-Gordon equation is a relativistic version of the Schr\"odinger equation. It was named after Oskar Klein and Walter Gordon who proposed it to describe quantum particles in the framework of relativity. It describes the motion of spinless composite particles. Indeed, one of the most fundamental questions that should be asked when studying these equations is whether the solution (if it exist) goes to equilibrium (stable) state or behaves erratically as time evolves. Understanding these properties can help determine how robust a system is, as well as provides insight on the characteristics of the corresponding phenomena it is modeling. In the first part we consider a one dimensional damped Klein-Gordon equation on the real line. It is well known fact that if there is no external force (i.e damping) acting in the system, the wave will oscillate forever in time since the energy is conserved in the system. An interesting question to ask is at what rate the energy starts leaving the system when we introduce damping force? This question was intensely studied in the last ten years. In this direction, Burq and Joly have proved that the energy decays at exponential rate if the damping force $\gamma(x)$ satisfies the geometric control condition (GCC) in a sense that there exist $T$, $\epsilon>0$, such that $\int_0^T\gamma(x(t))dt\geq \epsilon$ along every straight line unit speed trajectory. However, GCC does not provide an optimal condition to ensure exponential rate of energy decay. We address this problem in chapter 2 and provide optimal conditions on the damping coefficient $\gamma$ under which the exponential decay holds in one-dimensional setting. In addition, we derive simple to verify necessary and sufficient conditions for such exponential rate of decay. In the second part we relate the energy decay rate for the fractional damped wave equation to the order of its fractional derivative. In fact we prove that the energy decays at a polynomial rate if the order of derivative lies between $00$, such that $\int_0^T\gamma(x(t))dt\geq \epsilon$ along every straight line unit speed trajectory. However, GCC does not provide an optimal condition to ensure exponential rate of energy decay. We address this problem in chapter 2 and provide optimal conditions on the damping coefficient $\gamma$ under which the exponential decay holds in one-dimensional setting. In addition, we derive simple to verify necessary and sufficient conditions for such exponential rate of decay. In the second part we relate the energy decay rate for the fractional damped wave equation to the order of its fractional derivative. In fact we prove that the energy decays at a polynomial rate if the order of derivative lies between $0<s<2$ and at an exponential rate when $s\geq 2$ provided the damping coefficient is non-zero and periodic. An important ingredient of the proof is the derivation of a new observability estimate for the fractional Laplacain. Such important estimate has potential applications in control theory. 2019-08-31T00:00:00Z Sharp time asymptotics for the quasi-geostrophic equation, the Boussinesq system and near plane waves of reaction-diffusion models http://hdl.handle.net/1808/31318 Sharp time asymptotics for the quasi-geostrophic equation, the Boussinesq system and near plane waves of reaction-diffusion models Hadadifard, Fazel Through this dissertation we present the sharp time decay rates for three equations, namely quasi--geostrophic equation (SQG), Boussinesq system (BSQ) and plane wave of general reaction-diffusion models. In addition, in each case, we provide the dominant part of the solution which leads to the long term asymptotic profiles of each model. The first two equations, arising in fluid dynamics, model some aspect of the shallow waters with horizontal and vertical structures. Indeed, quasi--geostrophis equation models the horizontal inertia forces of a flow. As a result of that, atmospheric and oceanographic flows which take place over horizontal length scales, which are very large compare to their vertical length scales, are studied by SQG equation. On the other hand BSQ system models some vertical aspect of the flow, namely the speed, pressure and the temperature of the flow. In coastal engineering, BSQ type equations have a vast application in computer modeling. Lastly, a plane wave is a constant-frequency wave whose wavefronts (surfaces of constant phase) are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector. In order to study these equations, we made some developments in the "scaling variable" methods, so that it fits over models. In particular, we now have a good understanding of this method when it is applied to the equations with fractional dissipations. 2019-05-31T00:00:00Z A canonical form for the differential equations of curves in n-dimensional space http://hdl.handle.net/1808/30710 A canonical form for the differential equations of curves in n-dimensional space Smith, Ronald Gibson Dissertation (Ph.D.)--University of Kansas, Mathematics, 1930. 1930-05-31T00:00:00Z