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Examining Committee

Supervisory Committee
Dr. Yu-Ting Chen, Department of Mathematics and Statistics, 呱呱爆料 (Supervisor)
Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Member)
Dr. Kristan Jensen, Department of Physics and Astronomy, UVic (Outside Member)

External Examiner
Dr. Cl茅ment Cosco, Centre de recherche en math茅matiques de la decision, Universit茅 Paris-Dauphine

Chair of Oral Examination
Dr. Falk Herwig, Department of Physics and Astronomy, UVic

Abstract
This thesis investigates the mollified versions of the stochastic heat equation (SHE) and the Kardar鈥揚arisi鈥揨hang (KPZ) equation in high dimensions (d 鈮� 3), focusing on their limiting behaviors as the mollification is removed. These limiting behaviors are closely related to a key parameter, known as the coupling constant, which characterizes the strength of the driving noise.

34 The first part of this article analyzes the space-time fluctuations of the mollified SHE and mollified KPZ equation around their stationary states. These rescaled fluctuations, previously considered in [23, 24], provide a new viewpoint for describing the convergence of the above mollified equations. For every coupling constant that is strictly less than a specific critical threshold, called the L2-critical point, we establish Gaussian limits, possibly with additional random perturbations, for these rescaled fluctuations.

The second part of this article investigates the limiting higher moments for the mollified SHE in high dimensions. Motivated by a recent result [21] in two dimensions, a natural question is whether the higher moments also converge at the L2-critical point in high dimensions. Our main theorem gives a negative answer: in high dimensions, the limiting higher moments diverge for all coupling constants in a nontrivial interval containing the L2-critical point. As an application, we derive sharp estimates for quantities, which are believed to be closely related to probability distributions of the limiting partition function of the continuous directed polymer.

As a follow-up investigation, the third part of this article analyzes the above conjecture on the higher moments in three dimensions by considering a specific version of the limiting higher moment at the L2-critical point, referred to as the sub-limiting higher moment. This special limiting higher moment can be regarded as a three-dimensional analogue of the limiting higher moment in two dimensions at the corresponding L2-critical point, established in [21]. Our main result proves the spatial pointwise divergence of this limiting object. This divergence can be related to a well-known phenomenon in quantum physics known as the Efimov effect.

Examining Committee

Supervisory Committee
Dr. Xuekui Zhang, Department of Mathematics and Statistics, 呱呱爆料 (Supervisor)
Dr. Ke Xu, Department of Economics, UVic (Outside Member)

External Examiner
Dr. Min Tsao, Department of Mathematics and Statistics, UVic

Chair of Oral Examination
Prof. Merrie Klazek, School of Music, UVic

Abstract
This study investigates the pricing efficiency of Chinese convertible bonds and presents evidence of systematic mispricing. To support this analysis, we develop a pricing framework based on the Least Squares Monte Carlo (LSM) method, tailored to reflect contractual features unique to the Chinese market. Using this model, we simulate fair values over the full lifespan of 154 convertible bonds issued between 2015 and 2019 and compare them to observed market prices. The model-predicted price curves generally align well with observed price patterns, demonstrating the robustness and practical value of our approach. However, we also find that trading prices occasionally deviate from model-implied values by more than 10%, with these deviations exhibiting consistent patterns rather than random fluctuations. Furthermore, we demonstrate that simple trading strategies鈥攂oth at the individual bond level and at the portfolio level鈥攃an exploit these discrepancies to generate substantial excess returns. These findings suggest that the Chinese convertible bond market is only partially efficient and highlight persistent arbitrage opportunities, underscoring the importance of market-specific valuation models in emerging financial markets.

Programme

The Final Oral Examination
for the Degree of
Master of Science

(Department of Mathematics and Statistics)

Seyedeh Shaghayegh AhooeiNejad
Shahid Beheshti University, B.Sc. in Statistics (2022)

Breeding Patterns of Ancient Murrelet: A Multievent Model Approach

April 15, 2025
10:30 am
Zoom:

Supervisory Committee:
Dr. Laura Cowen, Department of Mathematics and Statistics, UVic (Supervisor)
Dr. Simon Bonner, Department of Mathematics and Statistics, UVic (Member)

Chair of Oral Examination:
Dr. Peter Dukes, Department of Mathematics and Statistics, UVic

See announcement and abstract (PDF).

See

Notice of the Final Oral Examination for the Degree of Doctor of Philosophy of

MANTING WANG

MSc (Donghua University, 2020)
BSc (Huaibei Normal University, 2017)

鈥淒eterministic and Stochastic Modelling of Infectious Diseases in the Early Stages鈥�

Department of Mathematics and Statistics
Friday, April 11, 2025
10:00 A.M.
Clearihue Building
Room B019

Supervisory Committee:
Dr. Junling Ma, Department of Mathematics and Statistics, 呱呱爆料 (Co-Supervisor)
Dr. Pauline van den Driessche, Department of Mathematics and Statistics, UVic (Co-Supervisor)
Dr. Dean Karlen, Department of Physics and Astronomy, UVic (Outside Member)

External Examiner:
Dr. Michael Li, Department of Mathematical and Statistical Sciences, University of Alberta

Chair of Oral Examination:
Dr. Mihai Sima, Department of Electrical and Computer Engineering, UVic

Abstract
During the early stages of an epidemic, case counts typically grow exponentially, influenced by disease transmissibility, contact patterns, and implemented control measures. Understanding this exponential growth and disentangling the effects of various interventions are critical for public health decision-making. This dissertation investigates the dynamics of the early stages of an epidemic under control measures, addressing two key topics: evaluating the effectiveness of contact tracing and estimating the exponential growth rate of cases.

Contact tracing is a key public health measure to reduce disease transmission. However, due to limited public health capacity, it is mostly effective during the early stage when the case counts are low. In Chapter 2, I develop a novel modelling framework to track contacts in a randomly mixed population. This approach borrows the idea of edge dynamics from network models to track contacts included in a compartmental SIR model for an epidemic spreading. Using COVID-19 as a case study, I evaluate the effectiveness of contact tracing during the early stage when multiple control measures were implemented in Chapter 3. I conduct a simulation study to determine the necessary dataset for parameter estimation. I find that new case counts, cases identified through contact tracing (or voluntary testing), and symptomatic onset counts are necessary for parameter identification. Finally, I apply our models to the early stages of the COVID-19 pandemic in Ontario, Canada.

Chapters 4 and 5 focus on reliably estimating the exponential growth rate during the early stages of an outbreak, a key measure of the speed of disease spread. To establish a suitable likelihood function for accurate growth rate estimation, I derive the probability generating function for new cases using a linear stochastic SEIR model and obtain formulas for its mean and variance in Chapter 4. Numerical simulations show that the binomial or negative binomial distribution closely approximates the distribution of new cases. To determine the most appropriate method for estimating the growth rate, I compare the performance of the negative binomial regression model and the hidden Markov model (HMM) in Chapter 5. My results show that the 95% credible intervals produced by the HMM have a higher probability of covering the true growth rate.