Title: Langevin diffusions and Monte Carlo methodology.

 Abstract: Gradient descent is a method of iterative optimization to minimize an objective function f(x). When a noise is added and a step size converges to zero, this iterative algorithm is viewed as a Langevin diffusion. It eventually reaches equilibrium in distribution proportional to exp(-f(x)). In this talk we introduce a notion of ``stochastic flow'' of sample path and ``duality'' of stochastic processes. A dual process, called an ``intertwining dual,'' will play a pivotal role in determining a stopping time (of algorithm) so that we can sample ``exactly'' from the stationary distribution in a finite steps. Thus, it will enable us to perform Monte Carlo simulations. [Demonstration code (bm.R) for this talk and more can be found at vps63.heliohost.us/e-math/MCMC]

Title: Feynman-Kac Formulation of Black-Scholes Option Pricing.

Abstract:  Financial institutions have the ability to sell to buyers derivatives and options. Typical contracts that sell to the buyer the right to purchase stock for a strike price at a future time regardless of the stocks value at that time are called European call options. These institutions rely on the Black-Scholes option pricing to set contract price for European call options. The primary result we investigate is what the Black-Scholes model is and how it can be derived.  We will explore its foundation to understand stochastic differential equations starting with probability theory and Ito calculus. From there, infinitesimal generators are introduced along with related results to deduce the Feynman-Kac formula in order to solve the Black-Scholes option pricing problem. Once this has been examined, we move onto relating the Feynman-Kac formula to the partial differential equation associated to asset portfolio. The solution to this equation via the Feynman-Kac formula becomes the Black-Scholes option pricing.

Title:  Looking Ahead with Hope Amidst the Trials and Tribulations of Graduate
School in Mathematics

Abstract:  In this talk, I will address questions graduate students often pose to me when I am either
in attendance at a conference or when I am teaching a class. The questions center around
themes such as survival in graduate school, developing good research writing skills, being
ready for life after graduate school, and establishing collaborations. At conferences or in
classes, my focus is often on the delivery of the mathematics and my goal in this talk, is
to elaborate on some of the tricky non-mathematical issues that come along as we focus
on the mathematics.