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Probability
Org: M.-E. Caballero (UNAM), V. Pérez-Abreu (UAM-Cuajimalpa) and T. Salisbury (York)
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- MICHELLE BOUÉ, Trent University, 1600 West Bank Drive, Peterborough, Ontario
K9J 7B8, Canada
Critical values for an epidemic model with moving particles
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Kesten and Sidoravicius have recently introduced a spatial model for
the spread of epidemics that takes into account the movement of
individuals. We will discuss the phase transitions of this model and
of some extensions that incorporate immunization and traps.
- MARÍA-EMILIA CABALLERO, Universidad Nacional Autónoma de México, Instituto de
Matemáticas, Circuito Exterior, Ciudad Universitaria, 04510
D.F. México
Conditioned stable Lévy processes and the ruin problem
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In order to construct several interesting examples of Lévy
processes, we work with the Lamperti transformation between Lévy
processes and postive self-similar Markov processes.
By killing a stable Lévy process when it enters the positive half
line, or by conditioning it to stay positive, or by conditioning it to
hit 0 continuously, we obtain different positive self-similar Markov
processes. We compute the infinitesimal generator of each of them and
we also obtain, using the Lamperti's transformation, the corresponding
Lévy processes and the characteristics of them.
As an application we obtain explicitly the law of the minimum before
and independent exponential time, for some of these Lévy processes.
This provides the explicit form of the spatial Wiener-Hofp
factorization at a particular point and the value or the ruin
probability for these processes.
This is based on a joint paper (to appear) with Loic Chaumont.
- DONALD DAWSON, Carleton University
Catalytic branching processes
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A catalytic branching diffusion processes is a continuous state
branching process in which the branching rate depends on the presence
of a catalyst. A catalytic branching network corresponds to a
multitype system in which some types serve as catalysts for other
types. Catalytic branching networks in which there are closed cycles
of catalytic types, perturbations of these, and catalytic systems
distributed in space pose a number of challenging mathematical
problems. In this lecture we
discuss some aspects of these
problems from the viewpoint of the hierarchical mean-field limit.
- LUIS GOROSTIZA, CINVESTAV, Departamento de Matemáticas, Av. IPN No. 2508,
San Pedro Zacatenco, 07300 México D.F.
Some questions on occupation times of branching systems
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We consider occupation time fluctuations of (d,a,b)-branching systems. The long time rescaling limit processes
have different properties in intermediate, critical and large
dimensions (e.g., long-range dependence vs. independent
increments, path continuity vs. jumps), and questions arise on
interpretation of results in terms of the systems.
- DANIEL HERNÁNDEZ-HERNÁNDEZ, Centro de Investigación en Matemáticas
Pricing, hedging and PDE's
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In this talk we consider the mean-variance hedging problem when the
market is incomplete. More specifically, we consider a stochastic
volatility model, and study the problem using dynamic programming
techniques. The nonlinear PDE involved in the solution is a parabolic
quasi-linear equation with quadratic growth. Existence and uniqueness
of classical solutions within a suitable class of smooth functions is
obtained as well as relations with backward stochastic differential
equations. Using these results an optimal hedging strategy is
derived.
- GAIL IVANOFF, University of Ottawa, Ottawa, ON, K1N 6N5, Canada
Filtering of set-indexed stochastic processes
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Multiparameter and set-indexed stochastic processes have many
important applications in the natural sciences and engineering. The
concept of stopping, which plays a fundamental role for processes
indexed by the real line, is less well understood in this more general
framework. We introduce the concept of adapted filtering as an
appropriate generalization of stopping. Applications in areas such as
multivariate survival analysis and multivariate precedence tests will
be discussed.
- J. ALFREDO LÓPEZ-MIMBELA, Department of Probability and Statistics, CIMAT, Apartado
Postal 402, 36000 Guanajuato, México
Symmetric steady states of a semilinear equation with
fractional Laplacian
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We study a semilinear PDE equation whose evolution operator is the sum
of a fractional power of the Laplacian and a convex non-linearity. By
extending the method of moving planes to fractional powers of the
Laplacian we prove that all positive steady states of the
corresponding equation in a finite ball are radially symmetric.
- ANA MEDA, Facultad de Ciencias, UNAM
Estimates for the Value at Risk and ruin probabilities of
diffusion processes with jumps
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We have estimates for the tail distribution of X*t = sup0 £ s £ t Xs, where Xs is a diffusion process with jumps which
satisfies Xs = m + ò0s su dBu + ò0s bu du +ò0s gu- d[(N)\tilde]u, where B is a Brownian
motion; [(N)\tilde] a compound Poisson process independent of B; b
is an adapted integrable process; s and g are only
assumed to be predictable-hence random, which encompasses all the
stochastic volatility models. We discuss some applications to the
estimation of a Dynamic Value at Risk and to the Ruin Probability of a
risk process with stochastic investment.
- VICTOR PÉREZ-ABREU, Universidad Autónoma Metropolitana Cuajimalpa
Representation of Infinitely Divisible Distributions on
Cones
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In this talk we present a probabilistic characterization of cones in
Fréchet spaces. Specifically, we show that a normal cone K in a
Fréchet space is regular if and only if every infinitely divisible
probability measure concentrated on K has the regular
Lévy-Khintchine representation on cone.
This is joint work with Jan Rosinski.
- EDWIN PERKINS, Math. Dept., UBC, Vancouver, BC, V6T 1V2, Canada
Pathwise uniqueness for parabolic stochastic PDE's
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Consider the SPDE: du/dt = u"+g(u)dW/dtdx where dW/dtdx is
space-time white noise and g is Holder continuous of index h. It
is shown that if 2h3-h > 3/4 then pathwise uniqueness holds. The
proof is an infinite dimensional extension of the Yamada-Watanabe
Theorem.
This work is joint with Leonid Mytnik.
- BRUNO RÉMILLARD, HEC Montreal
On signed-measure solutions of stochastic differential
equations
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We study existence and uniqueness of signed-measure solutions of a
class of stochastic differential equations with respect to Wiener
sheet, including as particular cases the two-dimensional
Navier-Stokes equations in vorticity forms introduced by Kotelenez.
This is a joint work with Jean Vaillancourt.
- VICTOR RIVERO, Centro de Investigación en Matemáticas (CIMAT), Calle
Jalisco s/n Col. Valenciana, CP 36240 Guanajuato, Guanajuato
Recurrent extensions of positive self-similar Markov
processes and Cramer's condition
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Let (X,P) be a positive self-similar Markov process that
dies at its first hitting time of 0. In this work we study the
existence and characterization of all positive valued self-similar
Markov processes, [(X)\tilde], that behave like (X,P)
before its first hitting time of 0 and for which the state 0 is a
regular and recurrent state. A such process [(X)\tilde] is called
a recurrent extension of (X,P). Our main result establishes
that (X,P) admits a self-similar recurrent extension that
leaves 0 continuously if and only if the underlying Lévy process
satisfies Cramer's condition.
- ELIANE RODRIGUES, Instituto de Matemáticas, Universidad Nacional Autónoma
de México, Area de la Investigación Científica,
Circuito exterior, Ciudad Universitaria, México, DF 04510,
México
A non-homogeneous Poisson model to estimate the number of
ozone peaks in Mexico City
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In this talk we consider the problem of estimating the number of
exceedances of an air quality standard in a given period of time. A
non-homogeneous Poisson model is proposed to analyse this issue. The
rate at which the Poisson events occur is given by a rate function
l(t), t ³ 0. This rate function also depends on some
parameters that need to be estimated. Two forms for l(t), t ³ 0 are considered: Weibull and exponential-Weibull with
parameters a ³ 0, b ³ 0 and s ³ 0, that
will be estimated using a Bayesian formulation as well as a Gibbs
sampling algorithm. The model is applied to the ozone data provided
by the Mexico City monitoring network.
This is part of a joint work with Jorge A. Achcar from the University
of São Paulo, Brazil, and A. A. Fernández-Bremauntz and
G. Tzintzun both from the National Institute of Ecology of the
Ministry of Environment, México.
- BYRON SCHMULAND, University of Alberta, Edmonton, Alberta, Canada
Some recurrence sequences
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We consider some generalizations of the renewal theorem for Markov
chains. We will discuss both analytic and probabilistic approaches to
finding the asymptotic behaviour of the solution of a renewal
equation.
- BÁLINT VIRÁG, University of Toronto
Scaling limits of random matrices
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The sine and Airy point processes arising from random matrix
eigenvalues play a fundamental role in probability theory, partly due
to their connection to Riemann zeta zeros and random permutations.
I will describe recent work on the Stochastic Airy and Stochastic sine
differential equations, which are shown to describe these point
processes and can be thought of as scaling limits of random matrices.
This new approach resolves some open problems, e.g. it
generalizes these point processes for all values of the parameter
beta.
- JOHN WALSH, University of British Columbia, Vancouver, BC
The Rate of Convergence of Numberical Solutions of SPDEs
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Numerical solutions of stochastic differential equations are more
often used to simulate the solutions than to find them, so the rate of
convergence in distribution of numerical solutions is especially
interesting. We will talk about the rates of convergence, both almost
sure and in distribution, of various schemes, with emphasis on the
stochastic wave equation.
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