## Probability Seminar

**Date:** November 1, 2019

**Time:** 11:30AM - 12:30PM

**Location:** BLOC 628

**Speaker:** Quan Zhou, TAMU (Statistics)

**Title:** *Optimal detection of a drifting Brownian coordinate*

**Abstract:** Given a stochastic process X_t, how to find a stopping time \tau that maximizes the expectation of some reward function, say G(X_\tau), is known as an optimal stopping problem. Two famous examples are the secretary problem and the pricing of American options. Interestingly, continuous-time optimal stopping problems can often be converted to free-boundary PDE problems. The primary goal of this talk is to introduce the theory of optimal stopping using a class of problems which we refer to as “optimal detection of a drifting Brownian coordinate”: Imagine N independent Brownian motions. One of them has a nonzero drift while all the others are just standard Brownian motions. The question is how to find out which one is drifting as soon as possible. This problem can be formulated in many ways. In this talk we will focus on one particular formulation as an optimal stopping problem and solve the corresponding free boundary problem. Some other formulations will be briefly discussed as well. We will also mention applications of the optimal stopping theory (and more generally stochastic optimization) in statistics.