Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2018 | Book

Scaling Limits of Markov-Branching Trees and Applications Read chapter Read first chapter

XII Symposium of Probability and Stochastic Processes

Merida, Mexico, November 16–20, 2015

Editors: Daniel Hernández-Hernández, Juan Carlos Pardo, Prof. Victor Rivero

Publisher: Springer International Publishing

Book Series : Progress in Probability

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

This volume contains the proceedings of the XII Symposium of Probability and Stochastic Processes which took place at Universidad Autonoma de Yucatan in Merida, Mexico, on November 16–20, 2015. This meeting was the twelfth meeting in a series of ongoing biannual meetings aimed at showcasing the research of Mexican probabilists as well as promote new collaborations between the participants.

The book features articles drawn from different research areas in probability and stochastic processes, such as: risk theory, limit theorems, stochastic partial differential equations, random trees, stochastic differential games, stochastic control, and coalescence. Two of the main manuscripts survey recent developments on stochastic control and scaling limits of Markov-branching trees, written by Kazutoshi Yamasaki and Bénédicte Haas, respectively. The research-oriented manuscripts provide new advances in active research fields in Mexico.

The wide selection of topics makes the book accessible to advanced graduate students and researchers in probability and stochastic processes.

Table of Contents

Frontmatter

Courses

Frontmatter
Scaling Limits of Markov-Branching Trees and Applications
Lecture Notes of the XII Simposio de Probabilidad y Procesos Estocásticos 16–20 Novembre 2015, Mérida, Yucatán
Abstract
The goal of these lecture notes is to survey some of the recent progress on the description of large-scale structure of random trees. We use the framework of Markov-Branching sequences of trees and discuss several applications.
Bénédicte Haas
Optimality of Two-Parameter Strategies in Stochastic Control
Abstract
In this note, we study a class of stochastic control problems where the optimal strategies are described by two parameters. These include a subset of singular control, impulse control, and two-player stochastic games. The parameters are first chosen by the two continuous/smooth fit conditions, and then the optimality of the corresponding strategy is shown by verification arguments. Under the setting driven by a spectrally one-sided Lévy process, these procedures can be efficiently performed owing to the recent developments of scale functions. In this note, we illustrate these techniques using several examples where the optimal strategy and the value function can be concisely expressed via scale functions.
Kazutoshi Yamazaki

Research Articles

Frontmatter
Asymptotic Results for the Severity and Surplus Before Ruin for a Class of Lévy Insurance Processes
Abstract
We investigate a classical two-sided jumps risk process perturbed by a spectrally negative α-stable process, in which the gain size distribution has a rational Laplace transform. We consider three classes of light- and heavy-tailed claim size distributions. We obtain the asymptotic behaviors of the ruin probability and of the joint tail of the surplus prior to ruin and the severity of ruin, for large values of the initial capital. We also show that our asymptotic results are sharp. This extends our previous work (Kolkovska and Martín-González, Gerber-Shiu functionals for classical risk processes perturbed by an α-stable motion. Insur Math Econ 66:22–28, 2016).
Ekaterina T. Kolkovska, Ehyter M. Martín-González
Characterization of the Minimal Penalty of a Convex Risk Measure with Applications to Robust Utility Maximization for Lévy Models
Abstract
The minimality of the penalty function associated with a convex risk measure is analyzed in this paper. First, in a general static framework, we provide necessary and sufficient conditions for a penalty function defined in a convex and closed subset of the absolutely continuous measures with respect to some reference measure \(\mathbb {P}\) to be minimal on this set. When the probability space supports a Lévy process, we establish results that guarantee the minimality property of a penalty function described in terms of the coefficients associated with the density processes. These results are applied in the solution of the robust utility maximization problem for a market model based on Lévy processes.
Daniel Hernández-Hernández, Leonel Pérez-Hernández
Blackwell-Nash Equilibria in Zero-Sum Stochastic Differential Games
Abstract
Advanced-type equilibria for a general class of zero-sum stochastic differential games have been studied in part by Escobedo-Trujillo et al. (J Optim Theory Appl 153:662–687, 2012), in which a comprehensive study of the so-named bias and overtaking equilibria was provided. On the other hand, a complete analysis of advanced optimality criteria in the context of optimal control theory such as bias, overtaking, sensitive discount, and Blackwell optimality was developed independently by Jasso-Fuentes and Hernández-Lerma (Appl Math Optim 57:349–369, 2008; J Appl Probab 46:372–391, 2009; Stoch Anal Appl 27:363–385, 2009). In this work we try to fill out the gap between the aforementioned references. Namely, the aim is to analyze Blackwell-Nash equilibria for a general class of zero-sum stochastic differential games. Our approach is based on the use of dynamic programming, the Laurent series and the study of sensitive discount optimality.
Beatris Adriana Escobedo-Trujillo, Héctor Jasso-Fuentes, José Daniel López-Barrientos
A Note on Γ-Convergence of Monotone Functionals
Abstract
In this note we present a criterion under which a functional defined on vectors of non-decreasing functions is the Γ-limit of a functional defined on vectors of continuous non-decreasing functions. To this end, we present a separation principle in which a weakly converging sequence of continuous non-decreasing functions is decomposed in two parts, one converging to a non-decreasing function with a finite number of jumps and the other to the complementary jumps.
Erick Treviño Aguilar
A Criterion for Blow Up in Finite Time of a System of 1-Dimensional Reaction-Diffusion Equations
Abstract
We give a criterion for blow up in finite time of the system of semilinear partial differential equations \(\frac {\partial u_{i}(t,x)}{\partial t}=\frac {1}{2}\frac {\partial ^{2}u_{i}\left (t,x\right )}{\partial x^{2}}+\frac {\varphi ^{\prime }_{i}\left (x\right )} {\varphi _{i}\left (x\right )}\frac {\partial u_{i}\left (t,x\right )}{\partial x}+u_{j}^{1+\beta _{i}}\left (t,x\right )\), t > 0, \(x\in \mathbb {R}\), with initial values of the form \(u_{i}\left (0,x\right )={h_{i}\left (x\right )}/{\varphi _{i}\left (x\right )}\), where \(0<\varphi _{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )\cap C^{2}\left (\mathbb {R}\right )\), \(0\leq h_{i}\in L^{2}\left (\mathbb {R},\mathrm {d} x\right )\), β i  > 0 and i = 1, 2, j = 3 − i. Moreover, we find an upper bound T ∗ for the blowup time of such system which depends both on the initial values f 1, f 2, and the measures \(\mu _i(\mathrm {d} x)=\varphi _i^2(x)\,\mathrm {d} x\), i = 1, 2.
Eugenio Guerrero, José Alfredo López-Mimbela
A Note on the Small-Time Behaviour of the Largest Block Size of Beta n-Coalescents
Abstract
We study the largest block size of Beta n-coalescents at small times as n tends to infinity, using the paintbox construction of Beta-coalescents and the link between continuous-state branching processes and Beta-coalescents established in Birkner et al. (Electron J Probab 10(9):303–325, 2005) and Berestycki et al. (Ann Inst H Poincaré Probab Stat 44(2):214–238, 2008). As a corollary, a limit result on the largest block size at the coalescence time of the individual/block {1} is provided.
Arno Siri-Jégousse, Linglong Yuan
Metadata
Title
XII Symposium of Probability and Stochastic Processes
Editors
Daniel Hernández-Hernández
Juan Carlos Pardo
Prof. Victor Rivero
Copyright Year
2018
Electronic ISBN
978-3-319-77643-9
Print ISBN
978-3-319-77642-2
DOI
https://doi.org/10.1007/978-3-319-77643-9