Research Statement 1 Coefficient Polynomial Expansions 1.1 Parabolic Pdes

Fundamentally, financial mathematics is concerned with modeling the random movements of underlyers (e.g., stocks, indexes, interest rates, etc.) and developing the computational tools that are needed to price and hedge the risks associated with these movements. One of the principal modeling challenges is that different financial underlyers exhibit an array of different behaviors (e.g., stochastic volatility, mean-reversion, jumps, etc.). Moreover, for any given underlyer, there is no consensus on how to properly capture its dynamics either under the physical measure or the risk-neutral pricing measure. With this in mind, my research focuses on developing computational methods that are applicable to large classes of models. Speaking broadly, my work can be divided into three categories (i) coefficient polynomial expansions, (ii) multiscale expansions, and (iii) semi-parametric methods. Below, I highlight some of my key contributions in these three areas.