1. No category

DEPARTAMENTO DE PROBABILIDAD
Y ESTADÍSTICA, IIMAS-UNAM
On a Harris process with
exchangeable increments to model
stochastic volatility
M. en C. Michelle Anzarut Chacalo
(Estudiante del Posgrado de Ciencias Matemáticas, IIMAS-UNAM)
There has been an increasing use of Lévy processes in applied sciences. However, such processes are spatially
homogeneous, a property that does not always characterize empirical data. In this work we explore a
one-dimensional, strongly-stationary Feller process. Some of its properties make it attractive for applications. In
particular, it has a spatially dependent structure and arbitrary, but fixed, invariant distribution, which can be chosen to
fit different scenarios. We develop procedures for the Bayesian estimation of the process, running rigid tests to prove
their convergence and accuracy. In addition, the proposal is employed to define a simple, yet useful, stochastic
volatility model, providing an alternative to models based on independent-increment or Ornestein-Uhlenbeck-type
processes.
Change-Point Detection on
Dependent Data, a
Random-Partition Approach
M. en C. Asael Fabian Martínez Martínez
(Estudiante del Posgrado de Ciencias Matemáticas, IIMAS-UNAM)
Change-point detection models aim to find abrupt changes in a given sample indexed on an ordered set. We tackle
this problem under a clustering approach restricted by such indexing set. Our proposed methodology is based on
exchangeable partition probability functions, specifically on Pitman's sampling formula, also known as two-parameter
Poisson Dirichlet process. Moreover, and due to the model-based nature of the problem, emphasis will be given to the
case where data inside each cluster is modelled by a Markovian process, in particular we will concentrate on discretely
observed Ornstein-Uhlenbeck diffusion processes. Some properties of the resulting model are explained and
posterior results are obtained via a novel Markov chain Monte Carlo algorithm.
Ruin probabilities for Bayesian
exchangeable claims processes
M. en C. Arrigo Coen Coria
(Estudiante del Posgrado de Ciencias Matemáticas, IIMAS-UNAM)
Among the driving assumptions in classical collective risk models, the independence among claims is frequently
violated by real applications. Therefore, there is an evident need of models that relax such a restriction. We undertake
the exchangeable claims platform and obtain some results for the infinite time ruin probability. The main result is that
the ruin probability under the exchangeable claims model can be represented as the expected value of the ruin
probabilities corresponding to certain independent claims cases. This allows us to extend some classical results to this
dependent claims scenario. The main tool is based on the de Finetti’s representation theorem for exchangeable
random variables, and as a consequence a natural Bayesian modelling feature for risk processes becomes available. In
particular, an interesting redefinition of the net profit condition is necessary.
28 de mayo
Salón 302, Edificio Anexo del IIMAS
13:00 horas
Circuito Escolar S/N, Ciudad Universitaria