Project Description
This project seeks to extend the concept of integration from regular Riemann integration to Riemann-Stieltjes (R-S) integration with functions as the integrators. R-S integration will first be used in the context of probability and statistics to provide a unified approach to finding expectations for both discrete and continuous random variables (rvs) as well as rvs which have mixed distributions. Next R-S integration will be used with stochastic processes as the integrators to develop more general models of processes in continuous time whose paths have instantaneous jumps. These so-called stochastic integrals will then be used to model real-world processes having this type of behavior. A topical example would be the modelling of the spread and containment of an infection using an SEIR (susceptible-exposed-infected-recovered) model. The first semester will primarily concentrate on the theory of Riemann-Stieltjes integration and its application to random variables. The second semester will establish the theory of stochastic integrals in this context and use these new models for the dynamics of an infection over time. It is anticipated that this effort will rely on statistical techniques to estimate parameters and that the model will be used to predict the length of time the infection persists in the population.
Tasks and Responsibilites
The student will begin by reading the theory of Riemann-Stieltjes integration, with an emphasis on integration in time, and applying it to some straightforward examples. Through a careful choice of examples, the student will understand the care needed in defining the integration when both integrand and integrator have discontinuities at the same time. By the end of the first semester, the student will produce a careful exposition of this work. Then, they will focus on integration with stochastic processes as both integrands and integrators. The student will then apply stochastic integration to an infection, probably COVID-19 since data is widely and freely available. They will develop an SEIR model and fit the model parameters to the data using statistical techniques.
Desired Qualifications
None Listed.