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Title: A Dynamic Model of Central Counterparty Risk and Liquidity Risk Measures
Creator: Feng, Shibi
Date: 2019
Description: The thesis consists of two major parts, and it contributes to two topics in risk models - a dynamic model of central counterparty risk and...
Show moreThe thesis consists of two major parts, and it contributes to two topics in risk models - a dynamic model of central counterparty risk and liquidity risk measures.Chapter 2 is devoted to the first part of the thesis, where we propose a dynamic model of central counterparty risk by introducing a dynamic model of the default waterfall of derivatives Central Counterparties (CCPs) and by designing a risk sensitive method for sizing the initial margin (IM), and the default fund (DF) and its allocation among clearing members. Using a Markovian structure model of joint credit migrations, our evaluation of DF takes into account the joint credit quality of clearing members as they evolve over time. Another important aspect of the proposed methodology is the use of the time consistent dynamic risk measures for computation of IM and DF. We carry out a comprehensive numerical study, where, in particular, we analyze the advantages of the proposed methodology and its comparison with the currently prevailing methods used in industry. The second part of the thesis is divided into four chapters, and the primary goal of this part is to develop a general framework for liquidity risk management in an order driven market. Chapter 3 describes the essential elements of an order driven market and introduces the notions that are of critical financial meaning, for instance, trading strategy and its corresponding value process. Moreover, we propose a model for the dynamics of the limit order book by using marked point process. Chapter 4 is devoted to the identification and measurement of liquidity risk. We describe the importance of demand for liquidity in measuring liquidity risk and we introduce the concept of liquidity provision. By considering a trader who is subject to liquidity provision only, we demonstrate that liquidity provision impacts the valuation of the portfolio through the trading costs of the foreseen transactions. Then, we propose two portfolio liquidity risk measures to account for the liquidity risk introduced by the liquidity provision. Besides measuring the liquidity risk of a portfolio, we also design a method to measure the liquidity provision adjusted risk for any contingent claim in the financial market established in Chapter 3. Chapter 5 attends to the hedging problem under liquidity provision. We prove the existence of an optimal hedging strategy in terms of minimizing the hedging error under liquidity provision. We demonstrate that the optimal hedging strategy can be solved in terms of associated Bellman equations.
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Title: Identity and Self-Efficacy Among Mathematically Successful African American Single Mothers in Urban Community College Contexts
Creator: Devi, Shavila
Date: 2019
Description: This dissertation is a phenomenological, multi-case study of 13 mathematically successful African American single mothers from two urban...
Show moreThis dissertation is a phenomenological, multi-case study of 13 mathematically successful African American single mothers from two urban community colleges in Chicago. While a number of recent studies have focused on Black girls and women in K-12 and university contexts, the community college context remains understudied despite the presence of large numbers of Black women. Moreover, there has been a tendency in mainstream research contexts to normalize failure, and focus on problematic aspects of being a Black single mother or being a Black mathematics learner. Bringing together considerations of identity (racial, mathematics, single mother) and mathematics self-efficacy, this study will be the first to focus on mathematically successful African American single mothers in the community college context. The following research questions guided the research for this dissertation:1. How do African American single mothers, who return to study mathematics at the community college and are successful in their courses, narrate their identities and life experiences around race, gender, mathematics learning, and being a mother?2. How do these women score on the Mathematics Self-Efficacy Scale (MSES) and what sources of and influences on their self-efficacy are reported by these women via interviews? 3. What other factors (intrapersonal and beyond) do these women report as being particularly salient in their mathematics success?Multiple forms of data–semi-structured interviews, pre-and-post responses to a widely-used mathematics self-efficacy survey, and mathematics artifacts–were collected to address the research questions. A cross-case analysis of the data revealed four themes that emerged across the 13 participants. Within-case analyses of three participants reveals how the themes play out in-depth for these women. The four themes are (1) strong counter-narratives of being a single mother that resisted dominant and deficit-oriented discourses; (2) education as a key tool and resource to manage and mitigate risks associated with single motherhood; (3) multifaceted stories of resilience to achieve success in mathematics and life; and (4) positive, success-oriented mathematics identities and positive math self-efficacy. This study contributes to an emerging success-oriented literature on Black women and mathematics, and a growing research literature on identity in mathematics education. In surfacing how the participants narrate and negotiate race, gender, and class, this dissertation also contributes to an emerging literature on intersectionality in mathematics education. Results from this study can inform community college administrators and faculty in crafting practice and policy to support African American single mothers in mathematics.
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Title: GUARANTEED, ADAPTIVE, AUTOMATIC ALGORITHMS FOR UNIVARIATE INTEGRATION: METHODS, COSTS AND IMPLEMENTATIONS
Creator: Zhang, Yizhi
Date: 2018
Description: This thesis investigates how to solve univariate integration problems using numerical methods, including the trapezoidal rule and the Simpson...
Show moreThis thesis investigates how to solve univariate integration problems using numerical methods, including the trapezoidal rule and the Simpson's rule. Most existing guaranteed algorithms are not adaptive and require too much a priori information. Most existing adaptive algorithms do not have valid justification for their results. The goal is to create adaptive algorithms utilizing the two above-mentioned methods with guarantees. The classes of integrands studied in this thesis are cones. The algorithms are analytically proved to be a success if the integrand lies in the cone. The algorithms are adaptive and automatically adjust the computational costs based on the integrand values. The lower and upper bounds on the computational costs for both algorithms are derived. The lower bounds on the complexity of the problems are derived as well. By comparing the upper bounds on the computational cost and the lower bounds on the complexity, our algorithms are shown to be asymptotically optimal. Numerical experiments are implemented.
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Title: Learning Stochastic Governing Laws from Noisy Data Using Normalizing Flows
Creator: McClure, William Jacob
Date: 2021
Description: With the increasing availability of massive collections of data, researchers in all sciences need tools to synthesize useful and pertinent...
Show moreWith the increasing availability of massive collections of data, researchers in all sciences need tools to synthesize useful and pertinent descriptors of the systems they study. Perhaps the most fundamental knowledge of a dynamical system is its governing laws, which describe its evolution through time and can be lever-aged for a number of analyses about its behavior. We present a novel technique for learning the infinitesimal generator of a Markovian stochastic process from large, noisy datasets generated by a stochastic system. Knowledge of the generator in turn allows us to find the governing laws for the process. This technique relies on normalizing flows, neural networks that estimate probability densities, to learn the density of time-dependent stochastic processes. We establish the efficacy of this technique on multiple systems with Brownian noise, and use our learned governing laws to perform analysis on one system by solving for its mean exit time. Our approach also allows us to learn other dynamical behaviors such as escape probability and most probable pathways in a system. The potential impact of this technique is far-reaching, since most stochastic processes in various fields are assumed to be Markovian, and the only restriction for applying our method is available data from a time near the beginning of an experiment or recording.
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Title: WIENER-HOPF FACTORIZATION FOR TIME-INHOMOGENEOUS MARKOV CHAINS AND BAYESIAN ESTIMATIONS FOR DIAGONALIZABLE BILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
Creator: Cheng, Ziteng
Date: 2021
Description: This thesis consists of two major parts, and contributes to two areas of research in stochastic analysis: (i) Wiener-Hopf factorization (WHf)...
Show moreThis thesis consists of two major parts, and contributes to two areas of research in stochastic analysis: (i) Wiener-Hopf factorization (WHf) for Markov Chains, (ii) statistical inference for Stochastic Partial Differential Equations (SPDEs).WHf for Markov chains is a methodology concerned with computation of expectation of some types of functionals of the underlying Markov chain. Most results in WHf for Markov chains are done in the framework of time-homogeneous Markov chains. The major contribution of this thesis in the area of WHf for Markov chains are: • We extend the classical theory to the framework of time-inhomogeneous Markov chains. • In particular, we establish the existence and uniqueness of solutions for a new class of operator Riccati equations. • We connect the solution of the Riccati equation to some expectations of interest related to a time-inhomogeneous Markov chain. Statistical inference for SPDEs regards estimating parameters of a SPDE based on available and relevant observations of the underlying phenomenon that is modeled by the given SPDE. We summarize the contribution of this thesis in the area statistical inference for SPDEs as follows: • We conduct the statistical inference for a diagonalizable SPDE driven by a multiplicative noise of special structure, using spectral approach. We show that the corresponding statistical model fits the classical uniform asymptotic normality (UAN) paradigm. • We prove a Bernstein-Von Mises type result that strengthens the existing results in the literature. • We prove the asymptotic consistency, asymptotic normality and asymptotic efficiency of two Bayesian type estimators.
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Title: IMPLEMENTING ASYNCHRONOUS DISCUSSION AS AN INSTRUCTIONAL STRATEGY IN THE DEVELOPMENTAL MATHEMATICS COURSES TO SUPPORT STUDENT LEARNING
Creator: Zenati, Lynda
Date: 2020
Description: Remedial known as developmental coursework are designed to get under-prepared students ready for college. Ninety one percent of colleges offer...
Show moreRemedial known as developmental coursework are designed to get under-prepared students ready for college. Ninety one percent of colleges offer remedial courses in mathematics and English (Seo, 2014). Evidence suggests that traditional teaching methods do not enable all students to engage with the types of academic literacy constitutive to higher education (Lea and Street, 2006). The popularity of online discussion has been made possible through their availability in most LMS which are widely used in higher education (Dahlstrom, Brooks, & Bichsel, 2014). This study aimed at examining the use of asynchronous discussion (AD) as an instructional strategy to help alleviate some of the difficulties developmental math students make in different topics. Participants were 15 students enrolled in Summer, 2019 semester at a Community College. Results showed that students’ performance increased from pretest to posttest for students’ who participated in AD. Comparison was made with two other sections of the same course at the same college taught by two different instructors. Controlling for prior academic ability, results showed a statistically significant difference between students’ performance in the posttest in the section that utilized the AD but not the other two sections. Content analysis of students' posts showed the use of AD at least temporarily corrected students’ misconceptions when they were active and Consistent. Results were mixed for the lurker and the passive students. Moreover, correlation analysis showed no relationship for the frequency of interaction; however, a significant relationship was found for the quality of participation and students’ performance as measured by the final exam. Furthermore; no relationship between the CoI presences and students’ performance. Students’ reflections indicated that students valued the online experience. Benefits were related to students’ engagement and collaborative learning. Obstacles included students’ behavior, timing and the structure of the AD. This may imply that using structured AD may help in building a community of learners. Also, instructor presence and facilitation were necessary to promote deep learning. Future research can build on this finding by replicating the study using a bigger sample size and a longer period to allow students to reflect and discuss any conflict with their peers.
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Title: NUMERICAL ANALYSIS ON MAXIMUM LIKELIHOOD ESTIMATORS FOR LINEAR PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
Creator: Zhang, Jun
Date: 2019
Description: The thesis contributes to the numerical analysis on statistical inference for stochastic partial differential equations (SPDEs). We study the...
Show moreThe thesis contributes to the numerical analysis on statistical inference for stochastic partial differential equations (SPDEs). We study the maximum likelihood estimation problem of the drift parameter for a large class of linear parabolic SPDEs. As in the existing literature on statistical inference for SPDEs, we take a spectral approach, and assume that one path of the first N Fourier modes is observed continuously in a fixed finite time interval [0, T]. We first provide a review of the asymptotic properties of the maximum likelihood estimator (MLE) of the drift parameter in the large number of Fourier modes regime, N  ∞, while the time horizon T > 0 is fixed. The main part of this thesis is dedicated to the numerical study of the asymptotic properties of the MLEs for two examples of linear parabolic SPDEs: the one-dimensional stochastic heat equation and a d-dimensional linear, diagonalizable, parabolic SPDE, where d  ℕ. For the one-dimensional stochastic heat equation, we perform the sensitivity analysis to assess the effect of changes in model parameters on the speed of convergence of the MLE. For the second linear parabolic SPDE, our simulations verify the theoretical results in the literature that both the consistency and asymptotic normality of the MLE hold for such equation only when d ≥ 2.
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Title: Mathematics of Civil Infrastructure Network Optimization
Creator: Rumpf, Adam Andrew
Date: 2020
Description: We consider a selection of problems from civil infrastructure network design that are of great importance in modern urban planning but have,...
Show moreWe consider a selection of problems from civil infrastructure network design that are of great importance in modern urban planning but have, until relatively recently, gone largely ignored in mathematical literature. Each of these problems is approached from the perspective of network optimization-based modeling, with a major focus placed on the development of efficient solution algorithms.We begin with a study of the phenomenon of interdependent civil infrastructure networks, wherein the functionality of one network (such as a telecommunications system) requires the input of resources from another network (such as the electrical power grid). We first consider a linear relaxation of an established binary interdependence minimum-cost network flows model, including its unique modeling applications and its use as part of a randomized rounding approximation algorithm for the mixed integer model. We also develop a generalized network simplex algorithm for the efficient solution of this generalized minimum-cost network flows problem. We then move on to consider a trilevel network interdiction game for use in planning the fortification of interdependent networks subject to targeted attacks. A variety of solution algorithms are developed for both the binary and the linear interdependence models, and the linear interdependence model is used to develop an approximation algorithm for the more computationally expensive binary model.We then develop a public transit network design model which incorporates a social access objective in addition to traditional operator cost and user cost objectives. The model is meant for use in planning minor modifications to a public transit network capable of improving equity of access to important services while guaranteeing that service levels remain within a specified tolerance of their initial values. A hybrid tabu search/simulated annealing algorithm is developed to solve this model, which is then applied to a test case based on the Chicago public transit network with the objective of improving equity of primary health care access across the city.
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Title: Applications of Optimal Contract Theory in Brokerage
Creator: Alonso Alvarez, Guillermo
Date: 2023
Description: In this thesis we study optimal brokerage problems in different scenarios. The thesisis structured in two parts:...
Show moreIn this thesis we study optimal brokerage problems in different scenarios. The thesisis structured in two parts: In the first part of this thesis, corresponding to Chapter 2 and 3, we construct optimal brokerage contracts for multiple (heterogeneous) clients trading a single asset whose price follows the Almgren-Chriss model. The distinctive features of this work are as follows: (i) the reservation values of the clients are determined endogenously, and (ii) the broker is allowed to not offer a contract to some of the potential clients, thus choosing her portfolio of clients strategically. We find a computationally tractable characterization of the optimal portfolios of clients (up to a digital optimization problem, which can be solved efficiently if the number of potential clients is small) and conduct numerical experiments which illustrate how these portfolios, as well as the equilibrium profits of all market participants, depend on the price impact coefficients. In the second part of this thesis, corresponding to Chapter 4, we establish existence of a solution to the optimal contract problem in models where the state process is given by a multidimensional diffusion with linearly controlled drift. Then, under certain concavity assumptions, we show that the optimal contracts in the relaxed formulation also solve the associated strong optimal contract problem. The main advantages of this approach, relative to the existing methods, are due to the fact that it allows (i) to obtain the existence of an optimal contract (as a limit point of epsilon-optimal ones), and (ii) to include various additional constraints on the associated control problems (e.g., state constraints, difference in filtrations of the agent and of the principal, etc.). Finally, we apply our results to the problem of brokerage fees when the agent has access to a larger filtration.
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Title: Phase field modeling and computation of vesicle growth or shrinkage
Creator: Tang, Xiaoxia
Date: 2023
Description: Lipid bilayers are the basic structural component of all biological cell membranes. It is a semipermeable barrier to most solutes, including...
Show moreLipid bilayers are the basic structural component of all biological cell membranes. It is a semipermeable barrier to most solutes, including ions, glucoses, proteins and other molecules. Vesicles formed by a bilayer lipid membrane are often used as a model system for studying fundamental physics underlying complicated biological systems such as cells and microcapsules. Mathematical modeling of membrane deformation has become an important topic in biological and industrial system for a long time. In this thesis, we develop a phase field model for vesicle growth or shrinkage based on osmotic pressure that arises due to a chemical potential gradient. This thesis consists of three main parts.In the first part, we establish a phase field model for vesicle growth or shrinkage without flow. It consists of an Allen-Cahn equation, which describes the evolution of the phase field parameter (the shape of the vesicle), and a Cahn-Hilliard-type equation, which simulates the evolution of the ionic fluid. The model is mass conserved and surface area constrained during the membrane deformation. Conditions for vesicle growth or shrinkage are analyzed via the common tangent construction. We develop the numerical computing in two-dimensional space using a nonlinear multigrid method which is a combination of nonlinear Gauss-Seidel relaxation operator and V-cycles multigrid solver, and perform convergence tests that suggest an $\mathcal{O}(t+h^2)$ accuracy. Numerical results demonstrate the growth and shrinkage effects graphically and numerically, which agree with the conditions analyzed via the common tangent construction.In the second part, we present a model for vesicle growth or shrinkage with flow. The dynamical equations considered are an Allen-Cahn equation, which describes the phase field evolution, a Cahn-Hilliard-type equation, which simulates the fluid concentration, and a Stokes-type equation, which models the flow. The numerical scheme in two-dimensional space includes a nonlinear multigrid method comprised of a standard FAS method for the Allen-Cahn and Cahn-Hilliard part, and the Vanka smoothing strategy for the Stokes part. Convergence tests imply an $\mathcal{O}(t+h^2)$ accuracy. Numerical results are demonstrated under zero velocity boundary condition and with boundary-driven shear flows, respectively.In the last part, we give an unconditionally energy stable and uniquely solvable finite difference scheme for the model established in the first part. The finite difference scheme is based on a convex splitting of the discrete energy and is semi-implicit. One key difficulty associated with the energy stability is due to the fact that some nonlinear energy functional terms in the expansion is neither convex nor concave. To overcome this subtle difficulty, we add auxiliary terms to make the combined term convex, which in turn yields a convex–concave decomposition of the physical energy. As a result, both the unique solvability and energy stability of the proposed numerical scheme are assured. In addition, we show the scheme is stable in the defined discrete norm.
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Title: KERNEL FREE BOUNDARY INTEGRAL METHOD AND ITS APPLICATIONS
Creator: Cao, Yue
Date: 2022
Description: We developed a kernel-free boundary integral method (KFBIM) for solving variable coefficients partial differential equations (PDEs) in a...
Show moreWe developed a kernel-free boundary integral method (KFBIM) for solving variable coefficients partial differential equations (PDEs) in a doubly-connected domain. We focus our study on boundary value problems (BVP) and interface problems. A unique feature of the KFBIM is that the method does not require an analytical form of the Green’s function for designing quadratures, but rather computes boundary or volume integrals by solving an equivalent interface problem on Cartesian mesh. We decompose the problem defined in a doubly-connected domain into two separate interface problems. Then we evaluate integrals using a Krylov subspace iterative method in a finite difference framework. The method has second-order accuracy in space, and its complexity is linearly proportional to the number of mesh points. Numerical examples demonstrate that the method is robust for variable coefficients PDEs, even for cases when diffusion coefficients ratio is large and when two interfaces are close. We also develop two methods to compute moving interface problems whose coefficients in governing equations are spatial functions. Variable coefficients could be a non-homogeneous viscosity in Hele-Shaw problem or an uptake rate in tumor growth problems. We apply the KFBIM to compute velocity of the interface which allows more flexible boundary condition in a restricted domain instead of free space domain. A semi-implicit and an implicit methods were developed to evolve the interface. Both methods have few restrictions on the time step regardless of numerical stiffness. Theyalso could be extended to multi-phase problem, e.g., annulus domain. The methods have second-order accuracy in both space and time. Machine learning techniques have achieved magnificent success in the past decade. We couple the KFBIM with supervised learning algorithms to improve efficiency. In the KFBIM, we apply a finite difference scheme to find dipole density of the boundary integral iteratively, which is quite costly. We train a linear model to replace the finite difference solver in GMRES iterations. The cost, measured in CPU time, is significantly reduced. We also developed an efficient data generator for training and derived an empirical rule for data set size. In the future work, the model could be expanded to moving interface problems. The linear model will be replaced by neural network models, e.g., physics-informed neural networks (PINNs).
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Title: Thermal Effects in Fluid Dynamics
Creator: Sulzbach, Jan-Eric
Date: 2021
Description: In this thesis we propose a mathematical framework modeling non-isothermal fluids.The framework is based on a coupling between non-equilibrium...
Show moreIn this thesis we propose a mathematical framework modeling non-isothermal fluids.The framework is based on a coupling between non-equilibrium thermodynamics and an energetic variational approach for the mechanical parts of the system. From this general model we derive and analyze three separate systems.The first application is the Brinkman-Fourier model. This is related to the ideal gas system, where the pressure and internal energy depend linearly on the product of density and temperature. This is a subsystem to the general Navier-Stokes-Fourier system. We prove the existence of local-in-time weak solutions via compensated compactness arguments.The next model we study is a non-isothermal diffusion system involving chemical reactions. For a system close to chemical equilibrium we show the well-posedness of classical solution using a fixed-point argument involving theory of homogeneous Besov spaces.The third application of the general theory is for another general diffusion system with a Cahn-Hilliard energy. In this framework, we study in detail how the temperature can affect the system on different scales, leading to different models. For the analysis, we focus on one case and show the well-posedness of classical solutions. The proof relies on methods from the theory of Besov spaces and paradifferential calculus.
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Title: Numerical Analysis and Deep Learning Solver of the Non-local Fokker-Planck Equations
Creator: Jiang, Senbao
Date: 2022
Description: This thesis is divided into three mutually connected parts. ...
Show moreThis thesis is divided into three mutually connected parts. In the first part, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the forms \begin{align*} I_{i,j} = \int_{\mathbb{R}^2}\phi(x)\frac{x_ix_j}{|x|^{2+\alpha}} \d x, \quad 0< \alpha < 2 \end{align*} where $i,j\in\{1,2\}$ and $\phi\in C_c^N$ for $N\geq 2$. This type of singular integrals and its quadrature rule appear in the numerical discretization of fractional Laplacian in non-local Fokker-Planck Equations in 2D. The quadrature rules are trapezoidal rules equipped with correction weights for points around singularity. We prove the order of convergence is $2p+4-\alpha$, where $p\in\mathbb{N}_{0}$ is associated with total number of correction weights. We present numerical experiments to validate the order of convergence of the proposed modified quadrature rules. In the second part, we propose and analyze a general arbitrarily high-order modified trapezoidal rule for a class of weakly singular integrals of the forms $I = \int_{\R^n}\phi(x)s(x)\d x$ in $n$ dimensions, where $\phi$ and $s$ is the regular and singular part respectively. The admissible class requires $s$ satisfies three hypotheses and is large enough to contain singular kernel of the form $P(x)/|x|^r,\ r > 0$ where $P(x)$ is any monomial with degree strictly less than $r$. The modified trapezoidal rule is the singularity-punctured trapezoidal rule plus correction terms involving the correction weights for grid points around singularity. Correction weights are determined by enforcing the quadrature rule to exactly evaluate some monomials and solving corresponding linear systems. A long-standing difficulty of these types of methods is establishing the non-singularity of the linear system, despite strong numerical evidence. By using an algebraic-combinatorial argument, we show the non-singularity always holds and prove the general order of convergence of the modified quadrature rule. We present numerical experiments to validate the order of convergence. In the final part, we propose \emph{trapz-PiNN}, a physics-informed neural network incorporated with a modified trapezoidal rule and solve the space-fractional Fokker-Planck equations in 2D and 3D. We verify the modified trapezoidal rule has the second-order accuracy for evaluating the fractional laplacian. We demonstrate trapz-PiNNs have high expressive power through predicting solutions with low $\mathcal{L}^2$ relative error on a variety of numerical examples. We also use local metrics such as point-wise absolute and relative errors to analyze where could be further improved. We present an effective method for improving performance of trapz-PiNN on local metrics, provided that physical observations of high-fidelity simulation of the true solution are available. Besides the usual advantages of the deep learning solvers such as adaptivity and mesh-independence, the trapz-PiNN is able to solve PDEs with fractional laplacian with arbitrary $\alpha\in (0,2)$ and specializes on rectangular domains. It also has potential to be generalized into higher dimensions.
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Title: Choice-Distinguishing Colorings of Cartesian Products of Graphs
Creator: Tomlins, Christian James
Date: 2022
Description: A coloring $f: V(G)\rightarrow \mathbb N$ of a graph $G$ is said to be \emph{distinguishing} if no non-identity automorphism preserves every...
Show moreA coloring $f: V(G)\rightarrow \mathbb N$ of a graph $G$ is said to be \emph{distinguishing} if no non-identity automorphism preserves every vertex color. The distinguishing number, $D(G)$, of a graph $G$ is the smallest positive integer $k$ such that there exists a distinguishing coloring $f: V(G)\rightarrow [k]$ and was introduced by Albertson and Collins in their paper ``Symmetry Breaking in Graphs.'' By restricting what kinds of colorings are considered, many variations of distinguishing numbers have been studied. In this paper, we study proper list-colorings of graphs which are also distinguishing and investigate the choice-distinguishing number $\text{ch}_D(G)$ of a graph $G$. Primarily, we focus on the choice-distinguishing number of Cartesian products of graphs. We determine the exact value of $\text{ch}_D(G)$ for lattice graphs and prism graphs and provide an upper bound on the choice-distinguishing number of the Cartesian products of two relatively prime graphs, assuming a sufficient condition is satisfied. We use this result to bound the choice distinguishing number of toroidal grids and the Cartesian product of a tree with a clique. We conclude with a discussion on how, depending on the graphs $G$ and $H$, we may weaken the sufficient condition needed to bound $\text{ch}_D(G\square H)$.
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Title: Stochastic dynamical systems with non-Gaussian and singular noises
Creator: Zhang, Qi
Date: 2022
Description: In order to describe stochastic fluctuations or random potentials arising from science and engineering, non-Gaussian or singular noises are...
Show moreIn order to describe stochastic fluctuations or random potentials arising from science and engineering, non-Gaussian or singular noises are introduced in stochastic dynamical systems. In this thesis we investigate stochastic differential equations with non-Gaussian Lévy noise, and the singular two-dimensional Anderson model equation with spatial white noise potential. This thesis consists of the following three main parts. In the first part, we establish a linear response theory for stochastic differential equations driven by an α-stable Lévy noise (1<α<2). We first prove the ergodic property of the stochastic differential equation and the regularity of the corresponding stationary Fokker-Planck equation. Then we establish the linear response theory. This result is a general fluctuation-dissipation relation between the response of the system to the external perturbations and the Lévy type fluctuations at a steady state.In the second part, we study the global well-posedness of the singular nonlinear parabolic Anderson model equation on a two-dimensional torus. This equation can be viewed as the nonlinear heat equation with a random potential. The method is based on paracontrolled distribution and renormalization. After splitting the original nonlinear parabolic Anderson model equation into two simpler equations, we prove the global existence by some a priori estimates and smooth approximations. Furthermore, we prove the uniqueness of the solution by classical energy estimates. This work improves the local well-posedness results in earlier works.In the third part, we investigate the variation problem associated with the elliptic Anderson model equation in a two-dimensional torus in the paracontrolled distribution framework. The energy functional in this variation problem is arising from the Anderson localization. We obtain the existence of minimizers by a direct method in the calculus of variations, and show that the Euler-Lagrange equation of the energy functional is an elliptic singular stochastic partial differential equation with the Anderson Hamiltonian. We further establish the L2 estimates and Schauder estimates for the minimizer as weak solution of the elliptic singular stochastic partial differential equation. Therefore, we uncover the natural connection between the variation problem and the singular stochastic partial differential equation in the paracontrolled distribution framework.Finally, we summarize our results and outline some research topics for future investigation.
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Title: Dynamic Risk and Dynamic Performance Measures Generated by Distortion Functions and Diversification Benefits Optimization
Creator: Liu, Hao
Date: 2023
Description: This thesis consists of two major parts, and it contributes to the fields of risk management and optimization.One contribution to risk...
Show moreThis thesis consists of two major parts, and it contributes to the fields of risk management and optimization.One contribution to risk management is made via developing dynamic risk measures and dynamic acceptability indices that can be characterized by distortion functions. In particular, we proved a representation theorem illustrating that the class of dynamic coherent risk measures generated by distortion functions coincides with a specific type of dynamic risk measures, the dynamic WV@R. We also investigate thoroughly various types of time consistencies for dynamic risk measures and dynamic acceptability indices in terms of distortion functions. Another contribution to risk management is proving strong consistency and asymptotic normality of two estimators of dynamic WV@R. In contrast to the exist- ing literature, our results do not rely on the assumptions of distribution of random variables. Instead, we investigate the asymptotic normality of estimators in terms of the generating distortion functions. Last but not least, we give counterexample to show that a sufficient condition of asymptotic normality is not necessary. The contribution to optimization is twofold. On the one hand, we formulate the (scalar) diversification optimization problem as a vector optimization problem (VOP), and show that a set-valued Bellman principle is satisfied by this VOP. On the other hand, we derive explicit policy gradient formula and implement the deep neural network to solve diversification optimization problem numerically. This deep learning technique allows to overcome computation difficulty caused by the non-convexity of VOP.
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