Department of Mathematicshttp://hdl.handle.net/1957/137372015-10-13T04:19:03Z2015-10-13T04:19:03ZRuin Problems with Risky InvestmentsLoke, Sooie-Hoehttp://hdl.handle.net/1957/572632015-09-25T16:31:51Z2015-09-10T00:00:00ZRuin 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
2015-09-10T00:00:00ZSymmetry breaking and uniqueness for the incompressible Navier-Stokes equationsDascaliuc, RaduMichalowski, NicholasThomann, EnriqueWaymire, Edward C.http://hdl.handle.net/1957/571102015-09-02T16:45:42Z2015-07-01T00:00:00ZSymmetry breaking and uniqueness for the incompressible Navier-Stokes equations
Dascaliuc, Radu; Michalowski, Nicholas; Thomann, Enrique; Waymire, Edward C.
The present article establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. While the uniqueness problem itself remains unresolved, these connections provide interesting problems and possible methods for investigating symmetry breaking and the uniqueness problem for Navier-Stokes equations. In particular, new branching Markov chains, including a dilogarithmic branching random walk on the multiplicative group (0, ∞), naturally arise as a result of this investigation.
Copyright 2015 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.; The following article appeared in Chaos: An Interdisciplinary Journal of Nonlinear Science and may be found at http://scitation.aip.org/content/aip/journal/chaos
2015-07-01T00:00:00ZJoint optimization of well placement and control for nonconventional well typesHumphries, T. D.Haynes, R. D.http://hdl.handle.net/1957/569852015-08-28T19:54:53Z2015-02-01T00:00:00ZJoint optimization of well placement and control for nonconventional well types
Humphries, T. D.; Haynes, R. D.
Optimal well placement and optimal well control are two important areas of study in
oilfield development. Although the two problems differ in several respects, both are important
considerations in optimizing total oilfield production, and so recent work in the field
has considered the problem of addressing both problems jointly. Two general approaches
to addressing the joint problem are a simultaneous approach, where all parameters are optimized
at the same time, or a sequential approach, where a distinction between placement
and control parameters is maintained by separating the optimization problem into two (or
more) stages, some of which consider only a subset of the total number of variables. This
latter approach divides the problem into smaller ones which are easier to solve, but may not
explore search space as fully as a simultaneous approach.
In this paper we combine a stochastic global algorithm (Particle Swarm Optimization)
and a local search (Mesh Adaptive Direct Search) to compare several simultaneous and sequential
approaches to the joint placement and control problem. In particular, we study
how increasing the complexity of well models (requiring more variables to describe the well’s
location and path) affects the respective performances of the two approaches. The results of
several experiments with synthetic reservoir models suggest that the sequential approaches
are better able to deal with increasingly complex well parameterizations than the simultaneous
approaches.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by Elsevier and can be found at: http://www.journals.elsevier.com/journal-of-petroleum-science-and-engineering/
2015-02-01T00:00:00ZTechnical Note: Convergence analysis of a polyenergetic SART algorithmHumphries, T.http://hdl.handle.net/1957/569842015-08-28T19:50:39Z2015-07-01T00:00:00ZTechnical Note: Convergence analysis of a polyenergetic SART algorithm
Humphries, T.
PURPOSE: We analyze a recently proposed polyenergetic version of the simultaneous algebraic
reconstruction technique (SART). This algorithm, denoted pSART, replaces the monoenergetic
forward projection operation used by SART with a post-log, polyenergetic forward projection, while
leaving the rest of the algorithm unchanged. While the proposed algorithm provides good results
empirically, convergence of the algorithm was not established mathematically in the original paper.
METHODS: We analyze pSART as a nonlinear fixed point iteration by explicitly computing the Jacobian of the iteration. A necessary condition for convergence is that the spectral radius of the Jacobian, evaluated at the fixed point, is less than one. A short proof of convergence for SART is also provided as a basis for comparison..
RESULTS: We show that the pSART algorithm is not guaranteed to converge, in general. The Jacobian of the iteration depends on several factors, including the system matrix and how one models the energy dependence of the linear attenuation coefficient. The authors provide a simple numerical example that shows that the spectral radius of the Jacobian matrix is not guaranteed to be less than one. A second set of numerical experiments using realistic CT system matrices, however, indicates that conditions for convergence are likely to be satisfied in practice
CONCLUSION: Although pSART is not mathematically guaranteed to converge, our numerical
experiments indicate that it will tend to converge at roughly the same rate as SART for system
matrices of the type encountered in CT imaging. Thus we conclude that the algorithm is still a
useful method for reconstruction of polyenergetic CT data.
This is an author's peer-reviewed final manuscript, as accepted by the publisher. The published article is copyrighted by American Association of Physicists in Medicine and can be found at: http://www.medphys.org/
2015-07-01T00:00:00Z