Research
In my Ph.D. dissertation and later work, I studied differential equations (DEs) that govern the motion of nonlinearly suspended structures, such as suspension bridges. More recently, my research has focused on qualitative and quantitative properties of solutions to DEs and free- boundary problems that arise in Actuarial and Financial Mathematics (AFM).
Motivation for my current research
There is growing concern among Americans about financial ruin during retirement. These concerns are justified as a significant financial crisis is looming; it is projected that retired Americans’ living expenses will exceed their financial resources by $400 billion over the ten- year period 2020-2030. This shortfall is driven by demographic trends, the increased longevity of our aging population, changes in Social Security, inadequate private retirement savings, and the continuing trend toward defined contribution plans such as 401(k)s, under which the individual, not the employer, assumes all investment and longevity risk.
A life annuity is a financial instrument that pays a fixed amount periodically throughout the life of the recipient; the payments are contingent on the recipient’s survival. Since they provide guaranteed, periodic income, life annuities are instrumental in helping individuals sustain a given level of consumption. Recent proposed legislation giving tax incentives for individuals who purchase these instruments testifies to their potential effectiveness in preventing poverty in retirement. Though life annuities provide income security in retirement, something retirees describe as “very important,” few retirees choose a life annuity over a lump sum.
Motivated by these observations, I have worked on problems in AFM that fall into the following broad categories:
- pricing and design of equity-indexed annuities (EIAs) and
- optimal strategies for investors who seek to achieve a goal, such as maximizing the utility of wealth or minimizing the probability of lifetime ruin. A utility function is a measure of an individual’s subjective attitudes toward risk and wealth. The probability of lifetime ruin is the probability that an individual runs out of money before she dies.
The mathematical tools for studying these problems come from stochastic optimal control (including optimal stopping and singular control) and analysis of the related Hamilton-Jacobi- Bellman (HJB) equations. I also employ numerical techniques for approximating solutions to boundary value problems and variational inequalities.