Anticipating Stochastic Integrals and Related Linear Stochastic Differential Equations (original) (raw)

This dissertation results from a culmination of years of effort, help from many people, a robust support system, and love for knowledge. I would like to express heartfelt gratitude to my advisors-Professors Hui-Hsiung Kuo and Padmanabhan Sundar. Without their timely guidance, encouragement, patience, kindness, empathy, and unwavering support, I would not be where I am today. I shall strive to uphold the strong work ethic that I have witnessed in them. I would also like to thank Professors Ambar Sengupta and Gerry Knapp for serving on my advisory committee. I am honored that Pujan Shrestha chose to work with me throughout the doctoral journey. Having a colleague with whom I could work in tandem made me cherish my time here. I am thankful to him for being patient with my mathematical ramblings, research discussions, and bringing me out of procrastination loops. I am indebted to my master's thesis advisor Professor Fabio Antonelli. It is from him that I started to understand and appreciate mathematics for what it is. His teaching and advice motivated me to pursue research in stochastic analysis. I also want to thank Professors Davide Gabrielli, Ida Germana Minelli, and Prasanna Kumar for teaching me and supporting my endeavors. I am grateful to Professor Jeffrey Roland for helping me explore the foundations and philosophies of mathematics and to Professor James Oxley for helping me refine my communication techniques. I thank Professors Shipman, Adkins, Dasbach, and the other professors and graduate students for creating a lively and healthy environment conducive to research. None of this would have been possible without a sturdy support system. I am grateful to my family for their unconditional love and support. I would like to thank Bree, Kaju, Jessica, Rachel, Steven, Irfan, Liem, Huy, and all my friends who made living in Baton Rouge an enjoyable experience. My heartfelt gratitude to Ankan and Anirudh for lively discussions on mathematics, the universe, and everything. I am grateful to Tuppu for being a constant source of love, support, and inspiration. Finally, my work builds on the mathematical framework laid by mathematicians and philosophers over millennia of painstaking effort. Quoting Isaac Newton, "If I have seen further, it is by standing on the shoulders of giants." v