### Fall 2017

 Date/Time/Location: September 14, 2017 11:00-11:50 a.m. @ ECOT 226 Speaker: Arash Fahim (Department of Mathematics, Florida State University) Title: A Monte Carlo scheme for a singular control problem Abstract: We provide a numerical solution of the nonlinear parabolic double obstacle problem arising from a finite portfolio selection with proportional  transaction costs. The problem is mainly governed by a time-dependent Hamilton-Jacobi-Bellman equation with gradient constraints due to the singularity of the control. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the high-dimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties from theoretical results.

 Date/Time/Location: September 21, 2017 11:00-11:50 a.m. @ ECOT 226 Speaker: Matt Lorig (Department of Applied Mathematics, University of Washington) Title: Semiparametric Pricing and Hedging of Volatility and Hybrid Derivatives on Jump Diffusions Abstract: We show how to price and replicate a variety of claims written on the log price X of a stock and its realized quadratic variation [X] when the underling stock price exhibits both jumps and stochastic volatility.  A key feature of our approach is that the volatility process need not be specified parametrically in order for the pricing and replication results to hold.  As a result, our methods are robust to misspecification of the volatility process.

 Date/Time/Location: October 19, 2017 11:00-11:50 a.m. @ ECOT 226 Speaker: Oleksii Mostovyi (Department of Mathematics, University of Connecticut) Title: Sensitivity Analysis of the Utility Maximization Problem with Respect to Model Perturbations Abstract: We study the sensitivity of the expected utility maximization problem in a continuous semimartingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled with a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and construct trading strategies that match the indirect utility function up to the second order. If a risk-tolerance wealth process exists, using it as a numeraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita-Watanabe decomposition. (Joint work with Mihai Sirbu).

 Date/Time/Location: October 26, 2017 11:00-11:50 a.m. @ ECOT 226 Speaker: Gu Wang (Department of Mathematical Sciences, Worcester Polytechnic Institute) Title: High-water Mark Fees with Stochastic Benchmark Abstract: A hedge fund manager invests the fund in a constant investment opportunity, and receives high-water mark fees when the fund reaches a new maximum relative to a stochastic benchmark, aiming to maximize the expected power utility from fees in the long run. The manager's optimal portfolio includes a Merton component with the risk aversion parameter shifted towards one, and a hedging component against the risk in the benchmark, both of which depend on how the fund investment opportunity compares to the benchmark. A stochastic benchmark in the calculation of high-water mark fees introduces the moral hazard of risky fund investment with hedging motives, but also provides a possibility of regulating the manager's risk taking with a carefully chosen benchmark.

 Date/Time/Location: November 16, 2017 11:00-11:50 a.m. @ ECOT 226 Speaker: Manuel Lladser (Department of Applied Mathematics, University of Colorado at Boulder) Title: Mixtures with Uncharacterized Components Abstract: This talk addresses probabilistic and statistical considerations related to the problem of estimating the weights of components in a mixture with uncharacterized components. For simplicity, it will focus on the case of a single characterized component when this component, as well as the whole mixture, can be observed indirectly through data. Time permitting, some initial results when data characterize more than one component will also be discussed. This work is in collaboration with my student, A. Pearson (APPM/IQ Biology), and the Smith Lab (USC).

 Date/Time/Location: November 30, 2017 11:00-11:50 a.m. @ ECOT 226 Speaker: Sean O'Rourke (Department of Mathematics, University of Colorado at Boulder) Title: Eigenvectors of random matrices and graphs Abstract: Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. For instance, many properties of a graph can be deduced or estimated from its eigenvectors.  The goal of this talk is to present several properties of the eigenvectors when the matrix (or graph) is random.  I will also discuss several applications of these results to control theory and community detection.  The talk is based on joint work with Behrouz Touri, Van Vu, and Ke Wang.

 Date/Time/Location: December 14, 2017 11:00-11:50 a.m. @ ECOT 226 (Cancelled; will be rescheduled to Spring 2018). Speaker: Antony Pearson (Department of Applied Mathematics, University of Colorado at Boulder) Title: TBA Abstract: TBA

### Spring 2018

 Date/Time/Location: February 22, 2018 11:00-11:50 a.m. @ ECCR 257 Speaker: Yuchong Zhang (Department of Statistics, Columbia University) Title: Large tournament games Abstract: We consider a stochastic tournament game in which each player works toward accomplishing her goal and is rewarded based on her rank in terms of the time to completion. We prove existence, uniqueness and stability of the game with infinitely many players, and existence of approximate equilibrium with finitely many players. When players are homogeneous, the equilibrium has an explicit characterization. We find that the welfare may be increasing in cost of effort in its low range, as the cost reduces players’ eagerness to work too hard. The reward function that minimizes the expected time until a given fraction $\alpha$ of the population has reached the target, as well as the aggregate welfare, only depends on whether the rank is above or below $\alpha$. However, that is no longer true when maximizing a function of the completion time. Numerical examples are also provided when players are inhomogeneous. (Joint work with Erhan Bayraktar and Jaksa Cvitanic).

 Date/Time/Location: March 1, 2018 11:00-11:50 a.m. @ ECCR 257 Speaker: Ibrahim Ekren (Department of Mathematics, University of Michigan) Title: A Dynamic Equilibrium Model for Brokerage Fees Abstract: We develop a dynamic equilibrium model for market liquidity. To wit, we solve for the equilibrium prices at which liquidity takers' demands are absorbed by liquidity providers, who can in turn gradually transfer these positions to a group of end users. We also find the optimal strategy of a liquidity taker in such a market and compute the equilibrium price dynamics. This is joint work in progress with Peter Bank and Johannes Muhle-Karbe.

 Date/Time/Location: April 5, 2018 11:00-11:50 a.m. @ ECCR 257 Speaker: Leonard Wong (Department of Mathematics, University of Southern California) Title: Porfolios, optimal transport and information geometry Abstract: Can we outperform a market index in the presence of volatility? What is the optimal frequency to rebalance a portfolio? We show that these questions can be analyzed using modern ideas in probability and information geometry (geometry in information theory). We quantify market volatility by a logarithmic divergence which is a distance-like quantity analogous to the relative entropy, and in this context portfolio selection (such as the universal portfolio) has a lot in common with nonparametric statistics. Mathematically, the divergence is intimately related to exponentially concave functions and the solution of an optimal transport problem with a logarithmic cost. It induces a rich differential geometric structure with numerous applications. In particular, a dualistic Pythagorean theorem gives insight into the optimal frequency of discrete rebalancing.

 Date/Time/Location: April 26, 2018 11:00-11:50 a.m. @ ECCR 257 Speaker: Justin Sirignano (Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign) Title: TBA Abstract: TBA

 Date/Time/Location: May 3, 2018 11:00-11:50 a.m. @ ECCR 257 Speaker: Xingtan Zhang (Leeds Business School, CU Boulder) Title: The Value of Scattered Information Abstract: We analyze a model in which the value of a security is comprised of multiple distinct parts and private information about these pieces is scattered among investors. We show that as information is scattered into smaller, distinctively informative pieces, endogenous information acquisition activity can increase, even if the acquisition cost does not decrease. Our paper generalizes Grossman-Stiglitz (1980) for an arbitrary number of distinct pieces of information and demonstrates that when information is scattered among investors, information free-riding can be alleviated. Our model generates new insights and testable predictions about financial information markets, segmentation of firm-specific information, and informed trading. Link: https://ssrn.com/abstract=2846482