Ph.D. Theses (Mathematics)
http://hdl.handle.net/1957/15739
2015-09-28T00:14:28ZRuin Problems with Risky Investments
http://hdl.handle.net/1957/57263
Ruin Problems with Risky Investments
Loke, Sooie-Hoe
In this dissertation, we study two risk models. First, we consider the dual risk process which models the surplus of a company that incurs expenses at a constant rate and earns random positive gains at random times. When the surplus is invested in a risky asset following a geometric Brownian motion, we show that the ruin probability decays algebraically for small volatility and that ruin is certain for large volatility. We use numerical methods to approximate the ruin probability when the surplus is invested in a risk-free asset. When there are no investments, we recover the exact expression for the ruin probability via Wiener-Hopf factorization. Second, we are concerned with incurred but not reported (IBNR) claims, modeled by delaying the settlement of each claim by a random time. When the investments follow a geometric Brownian motion, we derive a parabolic integro-partial-differential equation (IPDE) for the ultimate ruin probability with final value condition given by the ruin probability under risky investments with no delay. Assuming that the delay times are bounded by a constant, we obtain an existence theorem of the final value IPDE in the space of bounded functions, and a uniqueness theorem in the space of square integrable functions. When the delay times are deterministic, we show that delaying the settlement of claims does not reduce the probability of ruin when the volatility is large.
Graduation date: 2016
2015-09-10T00:00:00ZThe Variable Speed Wave Equation and Perfectly Matched Layers
http://hdl.handle.net/1957/56308
The Variable Speed Wave Equation and Perfectly Matched Layers
Kim, Dojin
A perfectly matched layer (PML) is widely used to model many different types of wave propagation in different media. It has been found that a PML is often very effective and also easy to set, but still many questions remain.
We introduce a new formulation from regularizing the classical Un-Split PML of the acoustic wave equation and show the well-posedness and numerical efficiency. A PML is designed to absorb incident waves traveling perpendicular to the PML, but there is no effective absorption of waves traveling with large incident angles. We suggest one method to deal with this problem and show well-posedness of the system, and some numerical experiments. For the 1-d wave equation with a constant speed equipped a PML, stability and the exponential decay rate of energy has been proved, but the question for variable sound speed equation remained open. We show that the energy decays exponentially in the 1-d PML wave equation with variable sound speed.
Most PML wave equations appear as a first-order hyperbolic system with as a zero-order perturbation. We introduce a general formulation and show well-posedness and stability of the system. Furthermore we develop a discontinuous Galerkin method and analyze both the semi-discrete and fully discretized system and provide a priori error estimations.
Graduation date: 2016
2015-06-04T00:00:00Z3D cone beam reconstruction formulas for the transverse-ray transform with source points on a curve
http://hdl.handle.net/1957/51754
3D cone beam reconstruction formulas for the transverse-ray transform with source points on a curve
Wongsason, Patcharee
3D vector tomography has been explored and results have been achieved in the last few decades. Among these was a reconstruction formula for the solenoidal part of a vector field from its Doppler transform with sources on a curve. The Doppler transform of a vector field is the line integral of the component parallel to the line. In this work, we shall study the transverse ray transform of a vector field, which instead integrates over lines the component of the vector field perpendicular to the line. We provide a reconstruction procedure for the transverse ray transform of a vector field with sources on a curve fulfilling Tuy’s condition of order 3. We shall recover both the potential and solenoidal parts. We present two steps for the reconstruction. The first one is to reconstruct the solenoidal part and the techniques we use are inspired by work of Katsevich and Schuster. A procedure for recovering the potential part will be the second step. The main ingredient is the difference between the measured data and the reprojection of the solenoidal part. We also provide a variation of the Radon inversion formula for the vector part of a quaternionic-valued function (or vector field) and an inversion formula in cone-beam setting with sources on the sphere.
Graduation date: 2015
2014-07-23T00:00:00ZMathematical treatment and simulation of methane hydrates and adsorption models
http://hdl.handle.net/1957/50424
Mathematical treatment and simulation of methane hydrates and adsorption models
Medina, Francis Patricia
In this dissertation we develop mathematical treatment for two important applications: (i) evolution of methane in coalbeds with the associated phenomena of adsorption, and (ii) formation of methane hydrates in seabed. We use simplified models for (i) and (ii) since we are more interested in qualitative properties of the solutions rather than direct applications to engineering.
For methane hydrates we focus on a scalar problem with diffusion only, and we discuss it as a nonlinear parabolic problem in a single variable with monotone operators. We show how the problem can be cast in the framework of a free boundary problem. The particular nonlinearity that we deal with comes from a constraint on one of the variables. For the simplified model of methane hydrates, we establish well-posedness of the problem in an abstract weak setting. We also perform simulations with a novel approach based on semismooth Newton methods. We demonstrate convergence rates of the numerical approximation which are similar to those for Stefan free boundary value problem.
On the other hand, for adsorption problems, we focus on their structure as systems of conservation laws, with equilibrium and non-equilibrium type nonlinearities, where the latter are associated with microscale diffusion. We also work with an unusual type isotherm called Ideal Adsorbate Solution, which is defined implicitly. For the IAS adsorption system, we show sufficient conditions that render the system hyperbolic. We also construct numerical approximations for equilibrium and nonequilibrium models.
Graduation date: 2015
2014-05-13T00:00:00Z