MSRI-UP 2011: Mathematical Finance
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Prof. Talithia Williams, Harvey Mudd College
An Incidence Estimation Model For Multi-Stage Diseases With Differential Mortality
Prevalence and incidence are two important measures of the impact of a disease. For many diseases, incidence is the most useful measure for response planning. However, the longitudinal studies needed to calculate incidence are resource-intensive, so prevalence estimates are often more readily available. In 1986, Podgor and Leske developed a model to estimate incidence of a single disease from one survey of age-specific prevalence, even where the presence of the disease increases the mortality rate of patients. Here, we extend their model to the case of progressive diseases, where the incidence of all disease stages is desired. As an example, we consider the case of cataract disease in Africa, where ophthalmologists wish to distinguish between unilateral and bilateral cataract incidence in order to plan the number of cataract surgeries needed. Our method has successfully provided cataract incidence estimates based on prevalence data from new Rapid Assessment of Avoidable Blindness surveys in Africa. In this talk, we provide a more general form of the model in order to promote its applicability to other diseases.
Prof. Duane Cooper, Morehouse College
Analysis of Cumulative Voting's Potential to Yield Fair Representation
For representative bodies, the election method of cumulative voting replaces the democratic principle of "one person, one vote" with "one person, n votes", where n is the number of representatives to be elected in the jurisdiction. We describe results on the potential of cumulative voting to yield fair representation to minority populations by omparison to apportionment methods. We extend this consideration beyond measures of fairness to population subgroups to consideration of fairness to individual voters via spatial modeling.
Prof. Ricardo Cortez, Tulane University
The Gambler's Ruin is a Random Walk
The "Gambler's Ruin" is a game in which two players exchange money by flipping a coin. If the coin lands heads, the gambler pays the opponent $1. If it lands tails, the opponent pays the gambler $1. The game goes on until one of them has no money. An interesting question is: What is the probability that the gambler will lose? A one-dimensional "Random Walk" is a process in which a person flips a coin and moves one step to the left if it lands tails; the person moves one step to the right if it lands heads. Where does the person end up after repeating this process N times? I will make a connection between these two games using ideas from probability and differential equations. Bring a coin!
Prof. Joseph Teran, University of California Los Angeles
Virtual Surgery: Scientific Computing in Real Time
As a general rule, scientific computing for solid and fluid mechanics is regarded an offline task, often requiring days of CPU time to complete. However, it is now evident that future microprocessors will be highly parallel, incorporating a large number of cores with multi-threading and vector processing capabilities. This revolution in architecture will afford future chips the computational capacity found in today's massive clusters. Unfortunately, realization of this potential revolution in computing power is contingent upon the ability of numerical algorithms to successfully leverage the raw capacity of these parallel multiprocessors. This task is non-trivial given the nascent state of the architecture. Although the computing environment will resemble traditional high-performance computing, multi-core hardware will be sufficiently different to prevent simple porting of existing techniques from parallel computing. Novel approaches are needed that leverage the mathematical nuances of the various governing equations to meet the memory and scalability constraints of the hardware. I will discuss ongoing challenges developing such techniques and the potentially revolutionary applications they will admit.
Prof. Myron Scholes, Emeritus, Stanford University
A Conversation with Prof. Myron Scholes
Myron Scholes is the Frank E. Buck Professor of Finance, Emeritus, at the Stanford Graduate School of Business, Nobel laureate in Economic Sciences, and co-originator of the Black-Scholes options pricing model. Scholes was awarded the Nobel Prize in 1997 for his new method of determining the value of derivatives.