Seminar

2008-2009 Seminars

Fall 2009

Sept 04: Organizational meeting

Sept 11: Vahid Anvari (York University, Toronto, Canada)

Mathematical modeling in epidemiology provides understanding of the underlying mechanisms that influence the process of spread of disease; it suggests prevention- and control- strategy to health policy makers. In fact, models often identify behaviors that are unclear in experimental data since data are non-reproducible, limited and subject to errors in measurement. In order to demonstrate the contribution of mathematics to epidemiology, the talk starts with a brief review on different types of mathematical models used for analysis of spread of diseases in both individual and population levels, followed by two case studies: the deterministic compartmental model to investigate the treatment policy for Fluroquinolone drug resistance of Gonorrhea and the probabilistic model to do risk analysis assessment for blood-borne transmission of pandemic influenza. Finally, an application of qualitative possibility theory to study the maturation process of Dendritic cells will be discussed.

Sept 18: No seminar today

Sept 25: Huseyin Yuce (New York City College of Technology-CUNY)

Perturbation Methods for Vibration of Moderately Elliptical Plates

Fundamental frequencies of vibrating moderately elliptic plates with a concentric circular core have been determined analytically. A boundary perturbation method is developed to extract the fundamental eigenvalue of the governing biharmonic boundary value problem. The method is then applied to moderately elliptic plates with clamped and simply supported outer boundary conditions. Clamped, simply supported, and free circular cores are considered. Approximate analytical formulations of the fundamental frequency for such plates with core are obtained.

Oct 02: Jan Vecer (Columbia University, New York, NY)

Change of Numeraire with Perspective Mapping

Typical contingent claims such as options are written on two or more underlying assets. Each of the underlying assets can be chosen as a numeraire for the purposes of pricing and hedging as long as the price of such asset is positive. This leads to at least two alternative formulations of the pricing problem, depending on the number of available reference assets with a positive price that enter a given contract. We show that the prices when expressed under different numeraires are connected by a functional relationship known as perspective mapping. This technique of computing prices under different reference assets is more general than simply computing the prices as expected discounted payoffs under the martingale measure associated with a given numeraire since it works also in situations when the reference asset does not have a corresponding martingale measure. For instance, an asset that represents the maximum price in the payoff of lookback options does not have a martingale measure, but the price of the contract with respect to the maximum can still be expressed using perspective mapping. This method applies for a general evolution of the price process. We give examples of the relationship of the pricing measures in the binomial model, the diffusion model, and the L\'evy jump model. We give two formulations of the pricing problem for European and American options, and three formulations of the problem for exotic options such as quantos, lookbacks, or Asians. In diffusion models, we obtain partial differential equations that correspond to the pricing problem.

Oct 09: Gerardo Hernandez-del-Valle (Columbia University, New York, NY)

Optimal Execution of Portfolio Transactions With Geometric Price Process

In this talk we derive the optimal execution trajectory for a trader who wishes to buy or sell a large position of shares which evolve as a geometric Brownian motion in contrast to the arithmetic model which prevails in the existing literature, and with a general temporary impact h. In general the problem may be viewed as a Stochastic Control Problem which alternatively leads to the Hamilton-Jacobi-Bellman partial differential equation. We provide a couple of examples which illustrate the results. We would like to stress the fact that in this paper we use understandable user-friendly techniques.

Oct 16: Bernard Mourrain (INRIA Sophia Antipolis, Cedex, France)

Robust and Efficient Algebraic-Geometric Computing

Geometric modeling plays an important role in many domains, linked with computer science, such as computer aided design, robotics, molecular biology, signal processing, physics simulation. To compute effectively with shapes on a computer, several approaches exist. One which we will discuss is based on semi-algebraic representation of shapes.

These piecewise non-linear models piecewise includes algebraic (or semi-algebraic) representations of shapes such representations based on b-spline (rational) parametrization or piecewise implicit representations. They provide precise, efficient, compact and powerful descriptions of the geometry, but require specific tools that we will describe.

Performing geometric operations on these models involves on one hand topological computation and on the other hand numerical approximation. Their treatment require dedicated methods that we will briefly review. The methods that we will describe are following a subdivision approach which deduced the topological structure from information on the boundary of regions. The robustness and efficiency of the methods relies on tools to isolate real roots of polynomials.

In particular, we will describe tools to compute points of intersection on curves and surfaces, which involve approximate, control or certified computation. We will describe methods for computing with certification the topology of curves in the plane or in higher dimensional space, or for meshing algebraic surfaces even in the presence of singularity. Extension to arrangement computations and Constraint Solid Geometry operations will also be described. Their use will be illustrated on specific geometric applications.

Oct 23: Tim S.T. Leung (Johns Hopkins University, Baltimore, MD)

Pricing American Option on Non-traded Asset in a Regime-Switching Market

Standard option pricing models assume continuous trading of the underlying asset. In many situations, however, the underlying asset is not traded. Instead, the option buyer or seller trades a correlated asset as a proxy to the underlying to manage risk exposure. With this setting, we consider the valuation of American options in a regime-switching market, where asset prices follow Markov-modulated dynamics. We adopt a utility maximization approach, in which the investor optimizes his/her investment strategy with respect to a time-consistent exponential utility function. This leads to the study of a system of coupled nonlinear PDEs or free boundary problems. We also provide an alternative interpretation for the option holder's indifference price in terms of relative entropy penalization with optimal stopping. A finite-difference numerical scheme is developed to solve for the optimal hedging and exercising strategies under different market regimes. Finally, we examine the impact of various factors, such as risk aversion and regime parameters, on option prices and investment strategies.

Oct 30: Tamay Ozgokmen (University of Miami, Miami, FL)

Large Eddy Simulations of Mixed Layer Instabilities and Sampling Strategies

The ocean's surface mixed layer is notoriously complex due to high spatial and temporal gradients of density and velocity fields. The surface mixed layer also exhibits sub-mesoscale instabilities which are challenging to observe due to their small scale and fast temporal evolution. Nevertheless, the small and sub-mesoscales represent the range of scale of naval operations and thus anomalous currents and perturbations in the acoustic and optical environment that can affect a variety of naval operations. Understanding the motion in this range of scales is therefore critical to help improve the predictive

capability of the existing ocean models.

In this preliminary study, large eddy simulations of an idealized mixed-layer problem are conducted using a spectral element model. Characteristics of the different phases of the evolution of a mixed-layer front, as well as the sensitivity of the solution to model parameters are described. The fields are then sampled using tracers and Lagrangian particles, and relative dispersion statistics are discussed.

Nov 06, 4:00-4:15 PM

Hongzhong Zhang (The Graduate Center of CUNY)

Formulas for the Laplace Transform of Stopping Times Based on Drawdowns and Drawups

The drawdown process is defined as the drop of the underlying asset from its running maximum. The first time to a given level of the drawdown process is usually referred to as the drawdown of that level, which we denote by TD. This time is clearly related to the maximum drawdown, a widely used risk measure in finance. Similarly a rally process is defined as the rise of the underlying asset from its running minimum and a rally is defined as the first hitting time of the rally process. Mathematicians H. Taylor and J. Lehoczky first derived the distributional properties of TD for any diffusion process. In this work, we extend their results by imposing a further condition on the order of occurrence of a drawdown and a rally. We finally discuss applications of this result in financial risk management.

Nov 06, 4:15-5:15 PM

Libor Pospisil (Columbia University, New York, NY)

Pricing Financial Derivatives with Jump-Diffusion Processes,

the corresponding partial integro-differential equations and application to maximum drawdown

In this talk, we assume that the price of an asset can be modeled as a diffusion process plus a compound Poisson process. Subsequently, we address the question of pricing contracts involving maximum drawdown of the asset. Given the complexity of the underlying model, the most suitable method is deriving the partial integro-differential equations and solving them numerically. The special feature of the equations is the presence of the running maximum and the running maximum drawdown, which may be discontinuous due to the jumps in the asset price. We will also discuss the question of hedging.

Nov 13: Carlo Lancellotti (College of Staten Island and Graduate Center of CUNY)

The Master Equation Approach in Kinetic Theory

I will discuss how certain linear and non-linear Fokker-Planck equations arise as the infinite-particle limit from appropriate N-particle systems. In particular, I will focus on a solvable case which was recently discovered by M. Kiessling and myself. In this case the kinetic equation is very similar to the standard linear equation for a Brownian particle, but arises instead as the kinetic limit for an isolated N-particle system.