PHD, Kent State University, 2008, College of Arts and Sciences / Department of Mathematical Sciences

We use CUSUM procedure to analyze trading the line strategy. Closed form expressions concerning probabilistic characteristics of the CUSUM stopping time and stopped process were obtained in discrete time setting for a wide class of processes. This class of discrete processes was recently defined by K. M. Khan and R. A. Khan. In continuous time, the CUSUM procedure applied to the processes driven by a particular stochastic differential equation was studied. As a result the joint Laplace transform of the maximum process and CUSUM stopping time was derived. Finally, the trading the line strategy was studied for the process driven by the fractional Brownian motion. As in regular Brownian motion case, the Laplace transform was linked to the partial differential equation. Although the lack of optional sampling theorem in this case prevents us from getting a closed form expression, the structure of the Laplace transform is derived. By using these results we point some of the subtle features of the trading the line strategy.

Committee: Kazim Khan (Committee Chair); Sergey Anokhin (Committee Member); Hassan Allouba (Committee Member); Oana Mocioalca (Committee Member); Doug Delahanty (Committee Co-Chair) Subjects: Mathematics

Doctor of Philosophy (Ph.D.), Bowling Green State University, 2008, Mathematics/Probability and Statistics

One of the primary issues in mathematical finance is the ability to construct portfolios that are optimal with respect to the risk. The stock price is subject to stochastic variability so the risk an investor encounters is due to the stock prices. A commonly used measure of risk is the expected maximum loss of a stock, in other words, how much one can lose. It can be defined informally as the largest drop from a stock peak to a stock nadir. Over a certain fixed length of time, a reasonably low expected maximum loss is as crucial to the success of any fund asa high maximum gain or maximum profit. The correlation coefficient of the maximum loss and the maximum gain indicates the relation between the gain and the risk using measures which are functions of the Sharpe ratio. The price of one share of the risky asset, the stock, is modeled by geometric Brownian motion. By taking the log of geometric Brownian motion, Brownian motion can be used as basis of the calculations related to the geometric Brownian motion. In this dissertation work, we present analytical results related to the joint distribution of the maximum loss and maximum gain of a Brownian motion and the correlation of them, and detailed explanation of this theoretical result which requires a review of standard but difficult literature. We have given an analytical expression for the correlation of the supremum and the infimum of standard Brownian motion up to an independent exponential time, we have shown convexity of the maximum gain and the maximum loss, and we have calculated some bounds for the expected values of maximum gain and maximum loss. We also search for a relation between the Sharpe ratio and the correlation coefficient for Brownian motion with drift and geometric Brownian motion with drift. Using the scaling property, we have shown that the correlation coefficient does not depend on the diffusion coefficient for Brownian motion. And finally, using real-life data, we have presented the correlatio (open full item for complete abstract)

Committee: Gabor Szekely (Advisor); Craig Zirbel (Advisor); Bullerjahn George (Committee Member); Rizzo Maria (Committee Member); Chen John (Committee Member) Subjects: Mathematics

Doctor of Philosophy, Case Western Reserve University, 2020, Macromolecular Science and Engineering

A new form factor model is introduced to describe small-angle neutron scattering measurements of star polymers that explicitly includes an excluded volume parameter and Flory interaction parameter, χ. Using 3-, 4-, and 6-arm poly(N-isopropylacrylamide), the stretching predicted by the Daoud-Cotton model was found in addition to significant effects of end-group chemistries on χ. These star polymers and their deviations from the ideal conformation were too small to investigate with resistive pulse sensing (RPS). Using RPS of the translocation events of phytoglycogen nanoparticles, dendrimers with uniform density, were found to permeable by using a “hardness” parameter to explain the current blockade, which is confirmed with recent neutron scattering results in the literature. The current blockade response of the viral nanoparticle Qβ is more complicated, suggesting a divergence between ionic and hydrodynamic permeability of the capsid with the ionic permeability of Qβ increasing with salt concentration. The timescale of translocation dwell times for both phytoglycogen and Qβ was found to obey the Stokes-Einstein relation. A Strouhal framework was then used to define the parameter space of RPS experimental conditions in terms of a Strouhal number, Sr. When Sr >> 1, nanoparticle motion is dominated by thermal energy, kBT, and diffusion coefficients derived from dwell times correspond to values derived from dynamic light scattering measurements. For 1 < Sr << 1, the electrophoretic force, QE, applied by the electric field decreases dwell times from Brownian diffusion times. A new model is introduced to relate the rate of this decrease to the hydrodynamic radius, Rh, electrophoretic mobility, μe, electric field, E, and length of the nanochannel, lp. This relationship finds the electrodes must be very close together, at least 1 mm for the low-charge particles measured in this work, for Sr to be in this range. Further application of the model developed in this work will all (open full item for complete abstract)

Committee: Michael Hore (Committee Chair); LaShanda Korley (Committee Member); Svetlana Morozova (Committee Member); Horst von Recum (Committee Member) Subjects: Engineering; Experiments; Materials Science; Nanoscience; Nanotechnology; Physical Chemistry; Physics; Polymer Chemistry; Polymers

PhD, University of Cincinnati, 2022, Arts and Sciences: Mathematical Sciences

Joint modeling of noisily measured biomarkers alongside time-to-event outcomes, such as those used for evaluating disease progression over time and dynamic prediction of survival, has revolutionized statistical science. It is also becoming increasingly clear with the advent of non-stationary Gaussian processes applied to medical monitoring studies that typical random effects within a longitudinal sub-model do not properly reflect complex fluctuations in biological processes. In this body of work, we propose a novel, flexible five-component longitudinal sub-model for univariate and multivariate joint modeling frameworks. The most noteworthy development is our introduction in these modeling frameworks of scaled integrated fractional Brownian Brownian motion (IFBM), a more generalized version of the integrated Brownian motion stochastic process that has been demonstrated to accurately represent biological processes observed with uncertainty. While integrated Brownian motion has been investigated as a method for developing target functions capable of properly predicting threshold changes in disease progression markers, no applications have used joint longitudinal-survival modeling or assessed the utility of IFBM for such purposes. As the event sub-model, the Cox proportional hazards model is used. For Bayesian posterior calculation and inference, we employ Markov chain Monte Carlo (MCMC) techniques. Using data from national patient registries, we use this novel method to assess lung function trajectories and mortality in two distinct rare lung diseases-lymphangioleiomyomatosis and cystic fibrosis. We investigate clinically important target functions with the goal of predicting rapid lung function decline in each disease, including environmental exposures and community characteristics in the application of cystic fibrosis. Each application includes a comparison of IFBM, with literature-based integrated Ornstein-Uhlenbeck (IOU), and random intercepts and slopes modeling. (open full item for complete abstract)

Committee: Seongho Song Ph.D. (Committee Member); Xuan Cao (Committee Member); Xia Wang Ph.D. (Committee Member); Rhonda Szczesniak Ph.D. (Committee Member); Siva Sivaganesan Ph.D. (Committee Member) Subjects: Statistics

Doctor of Philosophy, The Ohio State University, 2022, Statistics

Stochastic differential equations (SDEs) are used to model random phenomena which evolve over time and space. Applications of SDEs range from finance (modelling the evolution of stock prices, interest rates and complex derivatives) to biology (modelling growth of populations, spread of disease) to weather systems (modelling changes in temperature and precipitation over a region) and many others. Solutions of stochastic differential equations are stochastic processes. The realizations of such stochastic processes are referred to as sample paths. A closed form solution of a given SDE is generally not available and in general, SDEs are not solvable. However, the existence of a solution is guaranteed when the SDE satisfies certain conditions which are easily verifiable. Numerical methods which provide approximate solutions for a given SDE have been widely used as a compromise due to the lack of a closed form solution or existence of exact simulation methods. Such methods have a tremendous advantage due to the simplicity of their implementation. However, the inexact nature of approximate solutions introduce unknown bias and corrective measures come at a larger computational cost. Exact simulation methods for a class of one-dimensional time-homogeneous SDEs emerged in mid 2000s. The classic rejection sampling method for random variables could be extended to random elements whose realizations are continuous functions over an interval. Sample paths are proposed from the Wiener measure and accepted or rejected without requiring knowledge about the entire sample path. The simplicity of the early exact methods led to a thrust in developing new algorithms for a wider class of SDEs. In this context I began my research to develop exact simulation methods for a large class of one-dimensional time-homogeneous SDEs. In pursuit of this objective, I first developed new algorithms to perform exact joint simulation of the maximum and the time of maximum for constrained Brownian motio (open full item for complete abstract)

Committee: Radu Herbei (Advisor) Subjects: Statistics

PHD, Kent State University, 2020, College of Arts and Sciences / Department of Mathematical Sciences

Ever since J. Burgers introduced the famous equation bearing his name, it has been used by so many authors in both the deterministic as well as the stochastic settings as a model for turbulence and as a simplification for the notorious Navier-Stokes equations. On the other hand, the theory of time-fractional PDEs/SPDEs and their equivalent high order equations with memory is currently garnering intense interest from many authors. We use Allouba's Brownian-time Brownian motion (BTBM) approach to formulate and establish detailed dimension-dependent Lp regularity and local well posedness results for a new class of time-fractional Burgers type equations that are on one hand related to slow diffusion, diffusion in material with memory, and viscoelasticity; and on the other hand, they are related to turbulent flow in viscous fluid.

Committee: Hassan Allouba (Advisor); Lothar Reichel (Committee Member); Gang Yu (Committee Member); Austin Melton (Committee Member); Spyridon Margetis (Committee Member) Subjects: Mathematics

Doctor of Philosophy, The Ohio State University, 1981, Graduate School

Committee: Not Provided (Other) Subjects: Mathematics

Master of Science, The Ohio State University, 2011, Mechanical Engineering

Brownian motion is an important phenomena demonstrated by sub-micron sized particles in fluids. The objective of the current work is to develop a general-purpose numerical model to simulate Brownian motion and explore it for various applications of practical interest. To accomplish this goal, a model based on the Langevin equation was developed and implemented. The resulting stochastic ordinary differential equations were solved using the Newton-Raphson method after backward Euler discretization. In order to track the particles through the background fluid, a three-dimensional (3-D) tracking algorithm was developed. The solver is applicable to arbitrary two-dimensional (2-D) and 3-D geometries discretized using unstructured mesh topology. The Brownian motion solver was fully coupled to the carrier fluid flow solver, enabling unsteady flows to be simulated. In order to apply the model to various applications, viscous drag, Brownian forces, lift, and thermophoretic forces were considered. These forces all contribute to the particle's velocity and resulting motion. The Brownian motion model was first validated against analytical results and then explored for various applications where Brownian motion is important. From the simulations, it is shown that Brownian motion is increased by decreasing the particle size, increasing the fluid temperature, or decreasing the fluid viscosity. One application where Brownian motion is taken advantage of is in micro-scale filters. By simulating micro-scale filters, they can be optimized so that capture efficiencies can be improved. It is shown by performing simulations under different operating conditions, that the capture efficiency of micro-filters can be increased for a given particle diameter by increasing fluid temperature or decreasing fluid viscosity. Another application investigated in this work is the chemical vapor deposition (CVD) of Aluminum Nitride (AlN). During this process, it has been experimentally observed that part (open full item for complete abstract)

Committee: Sandip Mazumder Dr. (Advisor); Shaurta Prakash Dr (Committee Member) Subjects: Fluid Dynamics

Master of Science, The Ohio State University, 2011, Geological Sciences

The viscosity of water is a first-order constraint on the transport of material from a subducting plate to the mantle wedge. The viscosity of fluids that are released during the dehydration of hydrous minerals during subduction can vary by more than 9 orders of magnitude between the limits of pure liquid water and silicate melts. Accurate determination of low viscosities (<1 mPa·s) for liquids at simultaneous high pressures (>1 GPa) and high temperatures (>373 K) is hindered by the geometry and sample size of high-pressure devices. Here the viscosity of water at pressures representative of the deep crust and upper mantle through use of Brownian motion in the hydrothermal diamond anvil cell (HDAC) is reported. By tracking the Brownian motion of 2.8 and 3.1 micron polystyrene spheres suspended in H2O, the viscosity of the water at high pressure and high temperature can be determined in situ using Einstein's relation. Accuracies of 3-10% are achieved and measurements are extended to pressures relevant to fluid release from subducting slabs and temperatures up to 150% of the melting temperature. Unhampered by wall effects of previous methods, the results from this study are consistent with a homologous temperature dependence of water viscosity in which the viscosity is a function of the ratio of the temperature to the melting temperature at a given pressure. Based on the homologous temperature dependence of water, transport times for fluids released from subducted plates inferred from geochemical proxies are too short for transport via porous flow alone, and suggest transport through a combination of channel-flow and porous flow implying hydrofracturing at 50-150 km depth.

Committee: Wendy Panero (Advisor); Michael Barton (Committee Member); David Cole (Committee Member) Subjects: Geophysics

Doctor of Philosophy, The Ohio State University, 2009, Mechanical Engineering

Due to its capabilities of three-dimensional (3D) non-contact manipulation and measurement with sub-picoNewton force resolution, optical trapping is a modern technique that has been particularly important for studying biological systems under physiological conditions. An optical trapping system, composed of an FPGA-based digital controller, 3D high-speed laser measurement, and 3D rapid laser steering, is developed. The 3D steering actuators consist of a deformable mirror enabling axial actuation and a two-axis acousto-optic deflector for lateral steering. The actuation range is designed and calibrated to be over 20μm along the two lateral axes and over 10μm along the axial direction. The actuation bandwidth along lateral axes is over 50 kHz and the associated resolution is 0.016nm (1σ). The axial resolution is 0.16nm, while the bandwidth is enhanced to over 3 kHz by model cancellation method. To enhance the manipulation resolution of the developed system, Brownian motion control is theoretically and experimentally investigated. A 1st-order ARMAX model describing the Brownian motion of an optically trapped probe is derived for controller design and analysis. The derived model is experimentally validated by proportional control results. An optimal controller based on minimum variance control theory is then designed and implemented. The theoretical analysis is validated by both experiment and simulation to illustrate the performance envelope of active control. Moreover, adaptive minimum variance control is implemented and experimentally verified to be capable of maintaining the optimal control performance in a time-varying environment. Adaptive estimation is developed to enhance the system's dynamic force probing capability. An adaptive observer is designed using the augmented system model that includes the dynamics of the external interaction and trapping system variation. It recursively estimates the external force and the system parameter from the noisy motion of t (open full item for complete abstract)

Committee: Chia-Hsiang Menq PhD (Advisor); Walter Lempert PhD (Committee Member); Andrea Serrani PhD (Committee Member); Krishnaswamy Srinivasan PhD (Committee Member) Subjects: Biophysics; Electrical Engineering; Engineering; Mechanical Engineering; Optics; Systems Design

Doctor of Philosophy, The Ohio State University, 2009, Mechanical Engineering

This dissertation presents the development of quadrupole magnetic tweezers, which are capable of actuating, measuring, and controlling biological samples at nanometer scale. Magnetic force, with the advantages of biocompatibility and specificity, is employed as the actuation force for biological manipulation. Quadrupole magnetic tweezers are designed and implemented to realize force generation in arbitrary two-dimensional (2D) directions. To characterize the relationship between the applied currents to the coils and the resulting magnetic force on the magnetic probe, a lumped parameter model with magnetic monopole approximation is employed to describe the magnetic field generated by the magnetic poles. The magnetic force model is then developed based on this approximation. According to the force model, the magnetic force exerted on the magnetic probe is nonlinear with respect to the applied currents to the coils and is position dependent.Three-dimensional (3D) particle tracking algorithm based on microscope off-focus images is developed to measure the motion of the magnetic probe. Subnanometer resolution in all three axes at 400 Hz sampling rate is achieved using a high speed CMOS camera in bright-field illumination. At each sampling, the lateral position of the particle is first estimated by the centroid method. The axial position is then estimated by comparing the radius vector, which is converted from the off-focus 2D image of the probe with no information loss, with an object-specific model, calibrated automatically prior to each experiment. By normalizing the radius vectors, the algorithm becomes a shape-based method, thus invariant to image intensity change and robust to photobleaching. The algorithm is therefore updated and utilized to measure the 3D position of fluorescent particles by analyzing the fluorescent images acquired by a high sensitivity CCD camera. Furthermore, according to a detailed analysis of measurement noise, variance equalization and corre (open full item for complete abstract)

Committee: Chia-Hsiang Menq (Advisor); Krishnaswamy Srinivasan (Committee Member); Marcelo Dapino (Committee Member); Junmin Wang (Committee Member) Subjects: Electrical Engineering; Mechanical Engineering

MS, Kent State University, 2008, College of Arts and Sciences / Department of Computer Science

Several simulation using Brownian motion to model neuromuscular junctions have been published in the past. Information published includes algorithm details, data files, simulation results and binary code, but source code was not made publicly available. A new open source implementation of the algorithms presented was created, to use as a platform for future research, and to attempt to validate the results of previous simulations.

Committee: Arden Ruttan PhD (Advisor); Gwenn Volkert PhD (Committee Member); Ye Zhao PhD (Committee Member) Subjects: Computer Science