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Title: CONSTRUCTIONS IN NON-ADAPTIVE GROUP TESTING STEINER SYSTEMS AND LATIN SQUARES
Creator: Balint, Gergely `greg' T.
Date: 2014, 2014-05
Description: This thesis explores and introduces new constructions for non-adaptive group testing which are particulary important for the parameter range...
Show moreThis thesis explores and introduces new constructions for non-adaptive group testing which are particulary important for the parameter range we encounter in real life problems. After a summary of existing results, the rst part of this thesis introduces our own constructions, the Latin Square Construction and the Column Augmented Concatenation. Both of these constructions take existing good group testing matrices to create test matrices of larger dimensions. These new matrices are easy to nd for the practical small parameter range we are most interested in. We also address and prove asymptotic results of our Latin Square Construction. In case of the Column Augmented Concatenation the asymptotic results depend greatly on the codes used for the construction. The second part of our work is to address possible ways of augmentation of the Latin Square Construction. Here we explore the di erence in augmentation based on the properties of the starting matrix. In the appendices we give tables of best matrices coming from our constructions with xed, small column weights. We also give a list of the known best 2-disjunct matrices for small row numbers.
PH.D in Applied Mathematics, May 2014
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Title: SPECTRALLY ACCURATE BOUNDARY INTEGRAL METHOD FOR FREE SURFACES IN STOKES FLOW
Creator: Kuang, Yin
Date: 2011-04-18, 2011-05
Description: spectrally accurate boundary integral method is developed for solving the velocity of the film flow with suspended particles down an inclined...
Show morespectrally accurate boundary integral method is developed for solving the velocity of the film flow with suspended particles down an inclined plane in Stokes flow. The problem is a two-dimensional gravity-driven film flow with a rigid particle flowing down an inclined plane. We present the governing equations and the numerical methods for solving it with the help of periodic Green’s function. To obtain the system of discretized equations, we discuss the smoothness of each integrand appearing in the boundary integral formulation and use the composite trapezoidal rule for smooth periodic integrands to achieve the spectral accuracy. For the weakly singular integral, we approximate it by a special spectrally accurate quadrature. The Krylov subspace iterative method GMRES is employed to solve the resulting linear system. This method can be also applied to compute the velocity of interface in some other cases of film flow.
M.S. in Applied Mathematics, May 2011
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Title: Quantitative Tools for Stochastic Dynamical Systems: Invariant Structures and Escape Probabilities
Creator: Kan, Xingye
Date: 2012-07-16, 2012-07
Description: Three types of quantitative structures, stochastic inertial manifolds, random invariant foliations, and escape probabilities, are investigated...
Show moreThree types of quantitative structures, stochastic inertial manifolds, random invariant foliations, and escape probabilities, are investigated to study stochastic dynamical systems. Invariant structures for stochastic dynamical systems are reviewed and detailed techniques for their simulation, approximation and construction are presented with several illustrative examples. First, a numerical approach for the simulation of inertial manifolds of stochastic evolutionary equations with multiplicative noise is presented and illustrated. After splitting the stochastic evolutionary equations into a backward and a forward part, a numerical scheme is devised for solving this backward-forward stochastic system, and an ensemble of graphs representing the inertial manifold is consequently obtained. This numerical approach is tested in two illustrative examples: one is for a stochastic differential equation and the other is for a stochastic partial differential equation. Second, invariant foliations for dynamical systems with small white noisy perturbation are approximated via asymptotic analysis. In other words, random invariant foliations are represented as a perturbation of the corresponding deterministic invariant foliations, with deviation errors estimated. The escape probability is a deterministic concept making methods of partial differential equations theory attainable to stochastic dynamics. Finally, the escape probability p(x) for dynamical systems driven by non-Gaussian L´evy motions, especially symmetric α-stable L´evy motions, is considered and characterized. More precisely, it is represented as the solution of the Balayage-Dirichlet problem of a certain partial differential-integral equation. This issue has been investigated previously for dynamical systems driven by Wiener process. Differences between escape probabilities for dynamical systems driven by Gaussian and non-Gaussian noises are highlighted.
Ph.D. in Applied Mathematics, July 2012
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Title: TOPICS IN COUNTERPARTY RISK AND DYNAMIC CONIC FINANCE
Creator: Iyigunler, Ismail
Date: 2012-11-02, 2012-12
Description: This thesis consists of three essays about modeling counterparty risk and pricing derivative securities. In the rst essay, we analyze the...
Show moreThis thesis consists of three essays about modeling counterparty risk and pricing derivative securities. In the rst essay, we analyze the counterparty risk embedded in CDS contracts, in presence of a bilateral margin agreement. We focus on the pricing of collateralized counterparty risk, and we derive the bilateral Credit Valuation Adjustment (CVA), unilateral Credit Valuation Adjustment (UCVA), and Debt Valuation Adjustment (DVA). We propose a model for the collateral by incorporating all related factors such as the thresholds, haircuts and margin period of risk. We derive the dynamics of the bilateral CVA in a general form with related jump martingales. Counterparty risky and the counterparty risk-free spread dynamics are derived and the dynamics of the Spread Value Adjustment (SVA) is found as a consequence. We nally employ a Markovian copula model for default intensities and illustrate our ndings with numerical results. In the second essay we address the issue of computation of the bilateral CVA under rating triggers in presence of ratings-linked margin agreements. We consider collateralized OTC contracts, that are subject to rating triggers, between two parties { an investor and a counterparty. Moreover, we model the margin process as a function of the credit ratings of the counterparty and the investor. We employ a Markovian approach for modeling of the rating transitions and of the default probabilities of the counterparties. In this framework, we derive the representation for bilateral CVA. We also introduce a new component in the decomposition of the counterparty risky price: namely the rating valuation adjustment (RVA) that accounts for the rating triggers. We consider several dynamic collateralization schemes where the margin thresholds are linked to the credit ratings of the counterparties. We account for the rehypothecation risk in the presence of independent amounts. Our results are ix illustrated in terms of a CDS contract and an IRS contract. In the third essay, we study the problem of pricing in incomplete markets with risk measures and acceptability indices. We propose a model for nding the dynamic ask and bid prices of derivative securities using Dynamic Coherent Acceptability Indices (DCAI) in the presence of transaction costs. In this framework, we de ne and prove a representation theorem for dynamic bid ask prices. We show that our prices can be computed using dynamic Gain-Loss Ratio (dGLR), which is a DCAI. To illustrate our results, we provide several numerical examples, by pricing barrier options with dGLR.
PH.D in Applied Mathematics, December 2012
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Title: MULTILEVEL ALGORITHMS FOR PHASE RETRIEVAL
Creator: Tanoue, Cullen
Date: 2015, 2015-05
Description: Phase retrieval is an important optimization problem that arises in di raction imaging, where the original structure of an object needs to be...
Show morePhase retrieval is an important optimization problem that arises in di raction imaging, where the original structure of an object needs to be reconstructed from its measured di raction data that does not have information concerning the phase of the object. Multilevel algorithms can be used to compute solutions to the standard phase retrieval optimization problem by constructing a hierarchy of problems using a series of restriction and prolongation operations. The coarser problems have a quarter of the variables as the ner problems, and hence, there are much less linear algebra requirements for solving the coarser problems. Further, the prolongation of the solutions computed for the coarser problems yield good starting points for the ner problems. We can also use an approach that alternates between solving the coarse and ne problem. Parameters for these methods include the number of levels, prolongation and restriction operations, and the number of iterations to perform at each level. We study the solutions to the standard phase retrieval optimization problem that result from exploring these parameters and compare them to the results obtained from using single-level methods.
M.S. in Applied Mathematics, December 2014
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Title: NONLINEAR SIMULATIONS OF MULTI-VESICLE DYNAMICS
Creator: Hamiilton, Caleb
Date: 2015, 2015-07
Description: Vesicles in biology are closed forms of membranes. The dimensions of vesicles can vary in terms of surface area and enclosed volume. Examples...
Show moreVesicles in biology are closed forms of membranes. The dimensions of vesicles can vary in terms of surface area and enclosed volume. Examples range from small organelles to large cell bodies which all play a variety of resource transportation roles in biological systems. Research from the fields of chemistry and physics helps mathematical modeling by providing the mechanisms behind certain observed morphologies. Mathematical models and methods for simulating vesicle dynamics have produced accurate numerical solutions to verify experimental data and can be used to design new experiments that lead to more discoveries. The most researched case has been a single vesicle under shear flow. However, recent numerical and experimental results consider extensional flows on a single vesicle and hydrodynamic interactions among multiple vesicles. This thesis extends work on hydrodynamic interactions between vesicles in viscous fluid. We investigate numerically cases with multiple vesicles relaxing in asymmetrical configurations, time-dependent flow with more oscillation, and stochastic dynamics. Subjecting vesicles to these various cases reveals sensitivity to initial conditions such as distance and relative orientation. The effects from adding more vesicles are: increased time before equilibrium for the relaxation tests, and distributive wrinkling dynamics for the extensional flow tests. In stochastic cases, there are similar wrinkling distributions. However, initial conditions like distance and orientation have less important effects when competing with influence from thermal fluctuations. Additionally, in the presence of other vesicles under extensional flow, a vesicle may change the number and amplitude of wrinkles it would have experienced alone.
M.S. in Applied Mathematics, July 2015
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Title: MEAN-VARIANCE HEDGING WITH TIME CHANGED LEVY PROCESS
Creator: Liu, J Ingran
Date: 2012-11-17, 2012-12
Description: The goal of this thesis is to consider asset pricing model which driven by an exponential time changed process: Brownian motion with time...
Show moreThe goal of this thesis is to consider asset pricing model which driven by an exponential time changed process: Brownian motion with time changing process{ Poisson process. We rst present the characteristic function of the time change exponential Brown motion and its ltration. Second we exhibit the explicit European call pricing formula then discuss the mean-variance hedging method in this thesis.
M.S. in Applied Mathematics, December 2012
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Title: COMPUTATIONAL COST OF SIMULATING MEAN EXIT TIME USING STOCHASTIC DIFFERENTIAL EQUATIONS
Creator: Liu, Fanjing
Date: 2016, 2016-05
Description: Stochastic di erential equations play an important role in modern science, including engineering, physics, computer science and nance. It has...
Show moreStochastic di erential equations play an important role in modern science, including engineering, physics, computer science and nance. It has been shown that numerically solving stochastic di erential equation is a productive methodto deal with such problems. In this work, we try to analyze the procedure of numerically computing the mean exit time of some stochastic processes from a given boundary using Monte Carlo simulations. The two methods, including the Euler-Maruyama Method and Milstein's higher order method, will be explained and used extensively when we simulate paths of the random process. The simulated processes generated through the methods will then be used to identify the exit times. Later we use the average of the exit times as a numerical solution of Mean Exit Time. We compare the e ciency of the above two methods by evaluating their computational complexity and CPU cost of reaching the same level of accuracy.
M.S. in Applied Mathematics, May 2016
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Title: A STUDY OF HIGH FREQUENCY TRADING IN LIMIT ORDER BOOKS
Creator: Jiang, Yuan
Date: 2013, 2013-12
Description: In the thesis we study the high frequency trading and its applications in limit order books. We discuss the basic concepts and review the...
Show moreIn the thesis we study the high frequency trading and its applications in limit order books. We discuss the basic concepts and review the models in the limit order books. The review section focuses on the queues in the limit order books, optimal trading strategies, short-term volatilities and multi-agent problems in the scenario of limit order markets. Discussions on the shortage of some prevalent models of limit order books are addressed thereafter. For the main results of the thesis, market data are calibrated to facilitate the comparison between a theoretical model and the empirical behaviors in terms of order flows, price changes and diffusion limit of prices.
M.S. in Applied Mathematics, December 2013
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Title: CONTRIBUTIONS TO ALGORITHMIC MATROID PROBLEMS
Creator: Huang, Jinyu
Date: 2015, 2015-07
Description: In this thesis, we obtain several algorithms for problems related to matroids, a structure that generalizes the concept of linear independence...
Show moreIn this thesis, we obtain several algorithms for problems related to matroids, a structure that generalizes the concept of linear independence in a vector space and an acyclic subgraph structure in a graph. Matroids have been widely applied in combinatorial optimization, graph theory, coding theory and so forth. Specifically, our results include: In this thesis, we present a constant-competitive online algorithm of the matroid secretary problem for the partition matroids without information of the partition and for the paving matroids. We also introduce the multi-objective matroid secretary problem that extends the matroid secretary problem, in which the weight function is a k-vector w = [w1, · · · , wk]. We show a constant competitive algorithm of the multiobjective matroid secretary problem for the uniform matroids and for the paving matroids. Since bases of a matroid generalize many important combinatorial structures, many counting problems can be expressed as a problem that counts the number of bases of a matroid. An efficient approximate counting algorithm can be designed provided that a rapidly-mixing Markov chain that samples bases of a matroid can be constructed. Let Φ(G) be the conductance of the base-exchange graph G. Matroid- Expansion Conjecture (1989, Mihai and Vazirani) states that Φ(G) ≥ 1 for any base-exchange graph G of a matroid, which implies an FPRAS (fully-polynomial randomized approximation scheme) for counting the number of bases of a matroid. We use λ2, the second smallest eigenvalue of L, the discrete Laplacian matrix of G, to prove the Matroid-Expansion Conjecture for any paving matroid, for any balanced matroid, and for the direct sum of a paving matroid with a balanced matroid. A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. Finally, we prove that if there is a black-box NC algorithm for PIT (Polynomial Identity Testing), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).
Ph.D. in Applied Mathematics, July 2015
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Title: MOTION OF BUBBLY FLUID IN A TANK
Creator: Langman, Michael
Date: 2014, 2014-07
Description: Computational uid dynamics is the numerical study of the motion of uids. In this thesis, an introduction to uid mechanics is presented and the...
Show moreComputational uid dynamics is the numerical study of the motion of uids. In this thesis, an introduction to uid mechanics is presented and the governing equations of uid mechanics are derived. The open-source computational uid dynamics library OpenFOAM is then used to simulate uid dynamics and to model the formation and movement of bubbles in a tank.
M.S. in Applied Mathematics, July 2014
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Title: MODELS AND SIMULATIONS OF SPROUTING ANGIOGENESIS
Creator: Langman, Catherine
Date: 2016, 2016-05
Description: All living mammalian cells need to consume oxygen and nutrients for cellular processes and need a way to remove waste from those cellular...
Show moreAll living mammalian cells need to consume oxygen and nutrients for cellular processes and need a way to remove waste from those cellular processes. Capillary networks provide places for such exchanges to occur. The process of creating new capillaries from existing blood vessels is called angiogenesis. Understanding angiogenesis is critical to the advancement of knowledge in the life sciences, as well as in medical applications where blood vessels play an important role. Angiogenesis is a complex process composed of many subprocesses which are not yet fully understood and take place over varying temporal and spatial scales. Mathematically modeling and simulating angiogenesis, and evaluating the capillary networks that result from angiogenesis, can help further understanding of angiogenesis and improve therapeutic treatments. This thesis examines mathematical models and simulations of sprouting angiogenesis and proposes two generic models of sprouting angiogenesis based on descriptions found in educational and scientific literature. Future research opportunities for scientific study and educational study using these models as a starting place are discussed.
M.S. in Applied Mathematics, May 2016
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Title: DYNAMICS OF VESICLES IN VISCOUS FLUID
Creator: Liu, Kai
Date: 2014, 2014-12
Description: Modeling vesicle dynamics involves a complicated moving boundary problem where uids, thermal uctuations, and vesicle morphology are intimately...
Show moreModeling vesicle dynamics involves a complicated moving boundary problem where uids, thermal uctuations, and vesicle morphology are intimately coupled. In this thesis, we study the dynamics of a two-dimensional membrane in linear viscous ows. In the asymptotic analysis section, we derive deterministic and stochastic equations describing the motion of a slightly perturbed membrane interface. Using a 2nd order Runge-Kutta method, we solve these equations numerically, and explain the formation and development of wrinkling patterns. We then develop a boundary integral method and an immersed boundary method for simulating the nonlinear wrinkling dynamics of a homogenous vesicle in viscous ows. The nonlinear results agree with the asymptotic theory for a nearly circular vesicle, and also agree with experimental results for an elongated vesicle. Using a stochastic immersed boundary method, we investigate the e ects of thermal uctuations in vesicle dynamics. Comparing with the deterministic results, thermal uctuation can lead to the development of odd modes and asymmetric wrinkles. Finally, we investigate the nonlinear wrinkling dynamics of a multi-component vesicle. The model includes a 4th order Cahn-Hilliard type equation describing the phase transitions on the vesicle surface. We nd that for an elongated vesicle with large excess arc length, the inhomogeneous bending introduces nontrivial asymmetric wrinkling and buckling dynamics.
Ph.D. in Applied Mathematics, December 2014
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Title: AN ENERGY-PRESERVING SCHEME FOR THE POISSON-NERNST-PLANCK EQUATIONS
Creator: Kabre, Julienne
Date: 2017, 2017-07
Description: Transport of ionic particles is ubiquitous in all biology. The Poisson-Nernst- Planck (PNP) equations have recently been used to describe the...
Show moreTransport of ionic particles is ubiquitous in all biology. The Poisson-Nernst- Planck (PNP) equations have recently been used to describe the dynamics of ion transport through biological ion channels (besides being widely employed in semiconductor industry). This dissertation is about the design of a numerical scheme to solve the PNP equations that preserves exactly (up to roundoff error) a discretized form of the energy dynamics of the system. The proposed finite difference scheme is of second-order accurate in both space and time. Comparisons are made between this energy dynamics preserving scheme and a standard finite difference scheme, showing a difference in satisfying the energy law. Numerical results are presented for validating the orders of convergence in both time and space of the new scheme for the PNP system. The energy preserving scheme presented here is one dimensional in space. A highlight of an extension to the multi-dimensional case is shown.
Ph.D. in Applied Mathematics, July 2017
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Title: THE SIMPLE EQUAL FLOW PROBLEM ON GENERALIZED NETWORKS
Creator: Fidler, Mary E.
Date: 2011-07, 2011-07
Description: We study algorithms for the simple equal ow problem on generalized networks. Network ows problems are concerned with optimization of the ow of...
Show moreWe study algorithms for the simple equal ow problem on generalized networks. Network ows problems are concerned with optimization of the ow of commodities over a network, a directed graph. In a network, the amount of ow that leaves a node equals the ow that arrives at the destination node. However, generalized networks have arc multipliers which change the rate of ow on each arc. A classical network ow problem is the min cost ow problem which asks for minimum cost required for the ow of a commodity that satis es individual commodity requirements of each node in a network. The simple equal ow problem considers the min cost ow problem with an additional non-network constraint that requires certain arcs to have equal ow. Ahuja et al. [2] developed a combinatorial parametric algorithm, binary search algorithm, and capacity scaling algorithm for the simple equal ow problem. In this thesis, we extend the rst two algorithms to generalized networks. To do so, we must rst reformulate the simple equal ow problem on generalized networks to parameterize the equal ow arcs. The resulting linear program creates a piecewise linear convex curve as a function of the parameter. Then, we exploit the simplex algorithm derived combinatorial basis of generalized networks to determine the distance between breakpoints of the piecewise parametric linear convex curve of optimal solutions, which helps to determine the appropriate termination condition for the algorithms. This allows us to formulate the modi ed combinatorial parametric algorithm and the modi ed binary search algorithm, and their running times.
M.S. in Applied Mathematics, July 2011
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Title: GUARANTEED ADAPTIVE MONTE CARLO METHODS FOR ESTIMATING MEANS OF RANDOM VARIABLES
Creator: Jiang, Lan
Date: 2016, 2016-05
Description: Monte Carlo is a versatile computational method that may be used to approximate the means, μ, of random variables, Y , whose distributions are...
Show moreMonte Carlo is a versatile computational method that may be used to approximate the means, μ, of random variables, Y , whose distributions are not known explicitly. This thesis investigates how to reliably construct fixed width confidence intervals for μ with some prescribed absolute error tolerance, "a, relative error tolerance, "r or some generalized error criterion. To facilitate this, it is assumed that the kurtosis, , of the random variable, Y , does not exceed a user specified bound max. The key idea is to confidently estimate the variance of Y by applying Cantelli’s Inequality. A Berry-Esseen Inequality makes it possible to determine the sample size required to construct such a confidence interval. When relative error is involved, this requires an iterative process. This idea for computing μ = E(Y ) can be used to develop a numerical integration method by writing the integral as μ = E(f(x)) = RRd f(x)⇢(x)dx, where x is a d dimensional random vector with probability density function ⇢. A similar idea is used to develop an algorithm for computing p = E(Y) where Y is a Bernoulli random variable. All of the algorithms have been implemented in the Guaranteed Automatic Integration Library (GAIL).
Ph.D. in Applied Mathematics, May 2016
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Title: ANALYSIS OF THE APPLICATION OF THE LIAR MACHINE TO THE Q-ARY PATHOLOGICAL LIAR GAME WITH A FOCUS ON LOWER DISCREPANCY BOUNDS
Creator: Williamson, James W
Date: 2011-12-12, 2011-12
Description: The binary pathological liar game, as described by Ellis and Yan in [Ellis and Yan, 2004], is a variation of the original liar game, as...
Show moreThe binary pathological liar game, as described by Ellis and Yan in [Ellis and Yan, 2004], is a variation of the original liar game, as described by Berlekamp, R enyi, and Ulam in [Berlekamp, 1964], [R enyi, 1961], and [Ulam, 1976]. This two person, questioner/ responder, game is played for n rounds for a set of M messages. The game begins by the responder selecting a message from the set M. Each round the questioner partitions the messages into two distinct subsets. The responder selects one subset, and elements not in the selected subset each accumulate a lie. Elements accumulating more than e lies are eliminated. The questioner wins the original game provided after the completion of n rounds there is at most one surviving message. The questioner wins the pathological game provided there is at least one surviving message. The focus here will be to generalize the pathological game from two subsets to q subsets with a focus on providing a winning condition for the questioner. The q-ary variant of the pathological liar game has been studied, with rst results in [Ellis and Nyman, 2009]. We let the number of rounds the game is played go to in nity, with e a linear fraction of n, and present an upper bound on the number of messages required by the questioner to win the q-ary Pathological Liar Game. The liar machine and linear machine as discussed by Cooper and Ellis in [Cooper and Ellis, 2010] have been adapted to t this generalization and are used to track the approximate progression of the game. We provide an upper bound on the initial number of chips by bounding the discrepancy between the actual progression of the game and the approximate progression of the game as described by the linear and liar machines respectively. A similar upper bound can be found in [Tietzer, 2011], with di erent elements in the argument. Using methods similar to those found in [Cooper and Ellis, 2010], we provide a partial order argument to show that the winning condition bound for one response strategy by the questioner transfers to all possible response strategies.
M.S. in Applied Mathematics, December 2011
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Title: The Relationship Between Default and Volatility and Its Impact on Counterparty Credit Risk
Creator: Yang, Jiarui
Date: 2012-07-16, 2012-07
Description: This thesis presents a uni ed framework for studying the impact of the correlation between interest rate volatility and counterparty default...
Show moreThis thesis presents a uni ed framework for studying the impact of the correlation between interest rate volatility and counterparty default probability on the credit risk of collateralized interest-rate derivative contracts. A defaultable term structure model is proposed in which the default risk is correlated with interest rate volatility. In particular, an existence and uniqueness theorem of this model is proved. The pricing formula of credit derivatives under the proposed model is derived and the stochastic interest rate model and credit model are calibrated together . Finally, given all the parameters calibrated by the unscented Kalman lter, a sensitivity analysis of the impact of the correlation between interest rate volatility and a counterparty's default probability on the credit risk of collateralized interest-rate derivative contracts is presented.
Ph.D. in Applied Mathematics, July 2012
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Title: APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH NON-GAUSSIAN NOISE AND APPLICATION TO A VOLATILITY MODEL
Creator: Jianhua, Wang
Date: 2015, 2015-05
Description: In recent decades, stochastic processes with non-Gaussian noise are widely utilized in financial models. The a-stable Levy motion, one type of...
Show moreIn recent decades, stochastic processes with non-Gaussian noise are widely utilized in financial models. The a-stable Levy motion, one type of non-Gaussian noise processes, provides robust data ts and events simulations in financial world. Due to "heavy" tails and path jumps property, the a-stable Levy motion modeling becomes extremely popular among financial decision makers and risk hedgers. The a-stable Levy motion, however, usually has neither closed form of probability density function nor the higher moments, which raises implement obstacles. We exhibited distributions of a-stable random variables by different values. In contrast to the Gaussian distribution, the a-stable distribution illustrated the "heavy" tails and shape skews with various parameters. We analyzed jump behaviors along with calculating tails probabilities. We exploited scenario simulation method to solve stochastic differential equations with a-stable Levy motions. Except Euler scheme, we derived two strong convergence 1.0 order numerical schemes via the Wagner-Platen expansion. After we executed the schemes on the Merton Jump-Diffusion model, we roughly proved the convergence order of the schemes. We successfully applied the derived schemes to simulate a sophisticated stochastic volatility model with skewed a-stable Levy motions. With the approximated underlying asset process, we priced an european call option value and visualized implied volatility curve. As the result, we concluded the logarithm of underlying asset follows a skewed distribution rather than a symmetric one.
M.S. in Applied Mathematics, May 2015
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Title: GRAPH PARTITIONING WITH EIGENVECTORS
Creator: Panek, James
Date: 2017, 2017-05
Description: The Cheeger constant of a graph quantities how well a graph can be cut yield- ing two (typically) large vertex sets by a small edge cut. Lower...
Show moreThe Cheeger constant of a graph quantities how well a graph can be cut yield- ing two (typically) large vertex sets by a small edge cut. Lower and upper bounds have been developed using the eigenvalues and eigenvectors of the normalized Laplacian matrix of the graph. Here a classic sweep algorithm is studied using linear combinations of eigenvectors, specifically the columns of approximate discrete Green's functions. It is then shown, statistically on certain families of random graphs following a stochastic block model, that it is enough to use two eigenvalues and vectors to improve this classic algorithm's upper bound in most cases.
M.S. in Applied Mathematics, May 2017
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