Séminaire d’analyse – Gisèle Ruiz Goldstein et Jerome Goldstein (University of Memphis)

Séminaire d’analyse

Le vendredi 20 octobre 2023 à partir de 13 h 30
Salle : VCH-3870

Gisèle Ruiz Goldstein (13 h 30)
University of Memphis

Chaos and Deterministic PDEs of Mathematical Finance

Résumé: Major developments in mathematical Finance have come from the study of two deterministic parabolic partial differential equations, the Nobel Prize winning Black-Scholes equation for stock options, $$ \frac{\partial u}{\partial t}=\frac{\sigma^2}{2}x^2 \frac{\partial^2 u}{\partial x^2}+rx \frac{\partial u}{\partial x}-rx $$ and the Cox-Ingersoll-Ross equation for zero coupon bonds, $$ \frac{\partial u}{\partial t}=\frac{\sigma^2}{2}x \frac{\partial^2 u}{\partial x^2}+(\beta x+\gamma)\frac{\partial u}{\partial x}-xu, $$ where $(x,t) \in (0,\infty) \times [0,\infty)$. Each has a particular initial condition $u(x,0) = u_0(x)$ of relevance in economics. In both models \sigma is the volatility, r is an interest rate, \beta and \gamma are also parameters given by the economic modeling.

We study these problems in weighted sup norm Banach spaces whose functions are unbounded near infinity (and possibly also near 0). The Black-Scholes equation is governed by a semigroup that is strongly continuous, quasicontractive, and chaotic. Extentions to time dependent coe¢cients will be given for this model. Recent results embedding the chaotic behavior of the Black-Scholes equation into a class of equations which are of interest in mathematical finance will be discussed.

The Cox-Ingersoll-Ross equation is governed by a strongly continuous quasi- contractive semigroup, and the solution is given by a new type of Feynman-Kac formula. Extensions to more general potential terms will be explained as well as extensions to time dependent coefficients.

Jerome Goldstein (14 h 30)
University of Memphis

Equipartition of Energy, Old and New Results

Résumé: Equipartition of energy results for the wave equation on R^n (even with n = 1) could have been proved by Euler, but they were not discovered until the mid 1960s. For the usual wave equation on all of Euclidean space, the energy becomes half kinetic and half potential as t -> 1. This fails on bounded domains. The subject quickly evolved into a chapter in the theory of selfadjoint operators. We shall survey the history of the theory, some applications, and recent results. The newest results involve systems of wave equations with “cross friction” terms. These systems are ill posed and temporally inhomogeneous, yet equipartition of energy in a general context still makes sense and holds. The newest results comprise new joint work with Gisèle Goldstein, Sandra Lucente, and Silvia Romanelli.

Lien Zoom : https://ulaval.zoom.us/j/62595792267?pwd=Y1hZYlRKUVdBaXZSNE9RN0xYYXNUUT09