The Black-Scholes Equation: Analysis, Discretization and Computational Implementation

Speaker: Cristian Amador Loli Prudencio, San Marcos e UFF.

Date: 04 nov 2022, 11h.

Place: Room 407, Bloco H, Campus Gragoatá, UFF.

Abstract: In the present work we analyze the Model of a Stochastic Differential Equation of Black-Scholes associated with a Financial behavior, which we transform by stochastic methods into a PDE, we do the analysis, the discretization and we finish with the computational simulation using Matlab.

The Black-Scholes model is given in stochastic form by the equation

dS(t)=μS(t)dt+σS(t)dW(t), 0≤t≤T

S(0)=S0

which, by stochastic and deterministic processes, we lead to a retrograde parabolic partial differential equation, we study the equation, we discretize it by Finite Differences, we analyze the convergence and stability, and finally we do the implementation and computational simulation.

Keywords: Black-Scholes, Brownian Motion, Itô, Finite Differences, Convergence.

 

 

Grupo de Trabalho: Análise Topológica de Dados (Session #6)

Speaker: Wilson Calmon, UFF.

Date: 05 nov 2020, 16h.

Place: contact organizer at O endereço de e-mail address está sendo protegido de spambots. Você precisa ativar o JavaScript enabled para vê-lo..

Abstract: Neste grupo de trabalho nos reuniremos para estudar trabalhos relevantes sobre Análise Topológica de Dados, tanto na parte teórica como aplicada.

Nesta conversa vamos falar sobre os métodos propostos em Fasy et al 2014 que visam a construção de intervalos de confiança para diagramas de persistência. De imediato, tais intervalos viabilizam um método estatístico formal para separar componentes significativas (sinais) de ruídos em diagramas obtidos a partir de dados reais. Em particular, discutiremos sobre: (i) intervalos de confiança do ponto de vista da inferência estatística, (ii) a factibilidade dos métodos apresentados no artigo e (iii) outras análises que podem ser feitas a partir dos mesmos.

Reference: Brittany Terese Fasy, Fabrizio Lecci, Alessandro Rinaldo, Larry Wasserman, Sivaraman Balakrishnan, and Aarti Singh. "Confidence sets for persistence diagrams". The Annals of Statistics. 2014, Vol. 42, No. 6, 2301–2339.

Artigo disponível (open access): https://projecteuclid.org/euclid.aos/1413810729"

 

Nash equilibria for quasi-linear parabolic problem 2D

Speaker: Orlando Romero, UFF.

Date: 23 sep 2022, 11h.

Place: Room 409, Bloco H, Campus Gragoatá, UFF.

Abstract: In this work, we present a study of Nash Equilibria for quasi-linear parabolic problem 2D via Fixed-Point.

 

Weapons of Math Destruction

How Big Data Increases Inequality and Threatens Democracy

Speaker: Cathy O'Neil, Mathbabe.org.

Date: 06 nov 2020, 11h Brasilia Time (09h EST).

Place: Google meet at meet.google.com/ncp-sfun-zzg.

Abstract: We live in the age of the algorithm. Increasingly, the decisions that affect our lives—where we go to school, whether we get a car loan, how much we pay for health insurance—are being made not by humans, but by mathematical models. In theory, this should lead to greater fairness: everyone is judged according to the same rules, and bias is eliminated.

But, often, the opposite is true. Many models used today are opaque, unregulated, and uncontestable, even when they’re wrong. Most troubling, they reinforce discrimination: if a poor student can’t get a loan because a lending model deems him too risky (by virtue of his zip code), he’s then cut off from the kind of education that could pull him out of poverty, and a vicious spiral ensues. Models are propping up the lucky and punishing the downtrodden, creating a “toxic cocktail for democracy.” Such “weapons of math destruction” score teachers and students, sort résumés, grant (or deny) loans, evaluate workers, target voters, set parole, and monitor our health. Welcome to the dark side of Big Data.

In this talk based on her award winning book, Cathy O’Neil calls on modelers to take more responsibility for their algorithms and on policy makers to regulate their use. But in the end, it’s up to us to become more savvy about the models that govern our lives.

Cathy O'Neil (Ph.D. Harvard 1999) is a mathematician turned quant data scientist. While working as a hedge fund analyst and later in the design of targeted ads, she realized the dangers of algorithmic bias. In 2011, she started the blog "Mathbabe", whose subjects span Math, Women in Math, Science Education, Data Science, Travel, and Cyborg Sex, among others. In 2016, she published the book "Weapons of Math Destruction", winner of the MAA's Euler Book Prize. More recently, you might have seen her in the documentary "The Social Dilemma".

 

On total coloring the direct product of complete graphs

Speaker: Caroline Patrão, COPPE-UFRJ.

Date: 26 oct 2020, 16h.

Place: Google meet at meet.google.com/sth-aebc-yxw.

Abstract: A k-total coloring of a graph G is an assignment of k colors to the elements (vertices and edges) of G so that adjacent or incident elements have different colors.  The total chromatic number is the smallest integer k for which G has a k-total coloring.  The well known Total Coloring Conjecture states that the total chromatic number of a graph is either ∆(G) + 1 or ∆(G) + 2, where ∆(G) is the maximum degree of G.  In this work, we consider the direct product of complete graphs Km×Kn.  It is known that if at least one of integers m or n is even then, Km×Kn has total chromatic number equal to ∆(Km×Kn) + 1, except when m=n=2.  We prove that the graph Km×Kn has a total chromatic number equal to ∆(Km×Kn) + 1 when both m and n are odd integers, ensuring in this way that all graphs Km×Kn have total chromatic number equal to ∆(Km×Kn) + 1, except when m=n= 2.

This is joint work with Celina M. H. de Figueiredo (COPPE-Universidade Federal do Rio de Janeiro), D. Sasaki (IME-Universidade Estadual do Rio de Janeiro), Luís Antonio Brasil Kowada (IC- Universidade Federal Fluminense), Diane Castonguay (INF-Universidade Federal de Goiás) and M. Valencia-Pabon (LIPN-Université Sorbonne Paris Nord).

 

Grupo de Trabalho: Análise Topológica de Dados (Session #5)

Speaker: Paulo Gusmão, UFF.

Date: 22 oct 2020, 16h.

Place: contact organizer at O endereço de e-mail address está sendo protegido de spambots. Você precisa ativar o JavaScript enabled para vê-lo..

Abstract: Neste grupo de trabalho nos reuniremos para estudar trabalhos relevantes sobre Análise Topológica de Dados, tanto na parte teórica como aplicada.

Neste encontro, continuaremos discutindo o artigo de Facundo Mémoli e Osman Berat Okutan intitulado "Quantitative Simplification of Filtered Simplicial Complexes" https://arxiv.org/abs/1801.02812. Nesse trabalho é apresentado um novo invariante definido nos vértices de um dado complexo simplicial filtrado, chamado condensidade, que controla o impacto de se remover vértices em homologia persistente. O trabalho se concentra na filtração de Vietoris-Rips.

 

Grupo de Trabalho: Análise Topológica de Dados (Session #4)

Speaker: Paulo Gusmão, UFF.

Date: 16 oct 2020, 16h.

Place: contact organizer at O endereço de e-mail address está sendo protegido de spambots. Você precisa ativar o JavaScript enabled para vê-lo..

Abstract: Neste grupo de trabalho nos reuniremos para estudar trabalhos relevantes sobre Análise Topológica de Dados, tanto na parte teórica como aplicada.

Neste encontro, discutiremos o artigo de Facundo Mémoli e Osman Berat Okutan intitulado "Quantitative Simplification of Filtered Simplicial Complexes" https://arxiv.org/abs/1801.02812. Nesse trabalho é apresentado um novo invariante definido nos vértices de um dado complexo simplicial filtrado, chamado condensidade, que controla o impacto de se remover vértices em homologia persistente. O trabalho se concentra na filtração de Vietoris-Rips.

 

Grupo de Trabalho: Análise Topológica de Dados (Session #3)

Speaker: Jones Colombo, UFF.

Date: 08 oct 2020, 16h.

Place: contact organizer at O endereço de e-mail address está sendo protegido de spambots. Você precisa ativar o JavaScript enabled para vê-lo..

Abstract: Neste grupo de trabalho nos reuniremos para estudar trabalhos relevantes sobre Análise Topológica de Dados, tanto na parte teórica como aplicada.

Neste encontro, vamos estudar outra forma de representar a um Diagrama de Persistência num espaço vetorial - com isso abrimos muitas possibilidades de estudo.

Reference: "H. Adams, T. Emerson et Al. Persistence images: a stable vector representation of persistent homology, The Journal of Machine Learning Research, 18(1) (2017) https://dl.acm.org/doi/abs/10.5555/3122009.3122017"

 

Grupo de Trabalho: Análise Topológica de Dados (Session #2)

Organizer: Carlos Meniño Cotón, UFF.

Date: 01 oct 2020, 16h.

Place: contact organizer at O endereço de e-mail address está sendo protegido de spambots. Você precisa ativar o JavaScript enabled para vê-lo..

Abstract: Neste grupo de trabalho nos reuniremos para estudar trabalhos relevantes sobre Análise Topológica de Dados, tanto na parte teórica como aplicada.

Neste segundo encontro, continuaremos com o estudo dos trabalhos de Bubenik e Y. Umeda et Al. sobre paisagens persistentes e classificação de séries temporais.

P. Bubenik. Statistical Topological Data Analysis using Persistence Landscapes. Journal of Machine Learning Research 16 (2015) 77-102

Y. Umeda, J. Kaneko, H. Kikuchi. Topological Data Analysis and Its Application to Time-Series Data Analysis. FUJITSU Sci. Tech. J., Vol. 55, No. 2 (2019).

 

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