Numerical Solution of SDEs on Matrix Lie Groups with Financial Applications
My name is Michelle Muniz and I am a second year PhD student at the Bergische Universität Wuppertal (BUW), where I have also received both my Bachelor’s and Master’s degree in mathematics. In my Bachelor’s and Master’s thesis I worked on geometric numerical methods for matrix ordinary differential equations (ODEs) on Lie groups with application in Lattice quantum chromodynamics (QCD). Trying to solve the problem of constructing numerical methods of high convergence order but at the same time having to preserve all the geometric requirements got me intrigued, such that I decided to keep on working on geometric numerical integration methods for my PhD thesis. Since I took financial economics as a minor subject during my studies it seemed naturally to me to focus my current research on numerical methods for stochastic differential equations (SDEs) on Lie groups with application in financial mathematics.
Why I decided to pursue a PhD
As I have not gathered any practical experience outside the university so far, I was not so sure about whether I should accept the PhD position at BUW, that was offered to me in the last semester of my Master’s studies. Therefore, I applied to several Consulting and Banking Groups. After doing some job interviews and getting job offers, I realized that I was not ready yet to give up on learning and researching about the mathematical interconnections behind the formulas applied in those kinds of companies. So, I decided to accept the offered PhD position at the Applied Mathematics and Numerical Analysis (AMNA) Group, where I have already been a research assistant since the last year of my Bachelor’s studies.
What I will do during my PhD
SDEs on Matrix Lie groups occur for instance in the description of robotic movements under the consideration of some uncertainty. The solution of differential equations on Lie groups can also be used to solve different equations on other manifolds. Considering the space of covariance matrices, occurring for example in portfolio optimization and other fields of finance, as such a manifold gives a new approach to approximating covariance matrices while respecting all geometric features as well as their stochastic behavior. My goal is to construct a class of numerical methods to solve these SDEs on Lie groups in order to solve differential equations on manifolds related to financial mathematics.