## A virtual and online seminar on Stochastic Processes and their Applications

The Working Group on «Stochastic Processes and their Applications» organizes a virtual and online seminar, which usually takes place within the last week of the month.

The SPA Webinar Series is aimed at the scientific community interested in stochastic processes and their applications, and hosts talks by well-recognized experts and young researchers, as well as training courses for pre-doctoral students.

The talks are announced at least two weeks in advance and can be attended by sending an email to the Coordinators (mruiz@ugr.es and cminuesaa@unex.es). You will receive an email with the link to access the seminar a few hours before the seminar starts.

### Upcoming talks

To be announced soon.

For information about how to access the virtual room, please, contact mruiz (at) ugr (dot) es or cminuesaa (at) unex (dot) es

### Past talks and training courses

**March 14, 2024, Thursday, at 16:30 p.m. (UTC+1)
Title:** Estabilidad de la solución en ecuaciones diferenciales estocásticas semilineales con condición inicial anticipante

**Josep Vives, Universitat de Barcelona**

Speaker:

Speaker:

**Abstract: **En este seminario presentaremos el artículo [1], un trabajo conjunto con Jorge A. León del CINVESTAV-IPN (Ciudad de México) y David Márquez-Carreras (UB). En este artículo estudiamos diferentes tipos de estabilidad de la solución de una ecuación diferencial estocástica anticipante semilineal, dirigida por un movimiento browniano, con una variable aleatoria como condición inicial. La integral estocástica involucrada es la de Skorohod. Siendo la condición inicial aleatoria, es necesario redefinir los conceptos de estabilidad. Los nuevos criterios de estabilidad dependen de la derivada de la condición inicial en el sentido del cálculo de Malliavin.

Reference: [1] J. A. León, D. Márquez-Carreras and J. Vives (2023) Stability of Some Anticipating Semilinear Stochastic Differential Equations of Skorohod Type. Journal of Dynamics and Differential Equations. 11-10-2023. https://doi.org/10.1007/s10884-023-10312-z

**January 18, 2024, Thursday, at 16:00 p.m. (UTC+1)
Title:** Enlargements of filtrations and applications to portfolio optimization problems

**Bernardo D’Auria, Università degli Studi di Padova**

Speaker:

Speaker:

**Abstract: **Enlargements of filtration constitute a powerful concept in stochastic analysis, particularly in the realm of financial mathematics. It allows for the incorporation of additional information into the information flow available to market participants, and it proves useful to model situations in which different agents have asymmetric information about the underlying stochastic processes. By employing techniques from enlargements of filtration, one can determine the intrinsic value of a given type of information.We apply the technique of enlargements of filtration to model the introduction of anticipating information in simple financial market models, in which the dynamics of a risky asset are driven by a Brownian motion and a Poisson process. Then, under the logarithmic utility, we compute the additional expected value made by the anticipating information by using standard Itô as well as Malliavin calculus techniques.

Organized jointly with the Departamento de Matemáticas Fundamentales, Universidad Nacional de Educación a Distancia.

**December 19, 2023, Tuesday, at 16:00 p.m. (UTC+1)
Title:** Performance paradox in stochastic dynamic matching models

**Jocu Doncel Vicente, Universidad del País Vasco/Euskal Herriko Unibertsitatea**

Speaker:

Speaker:

**Abstract:** In this presentation, we will analyze the stochastic dynamic matching model. In this model, the items of different classes arrive according to independent Poisson processes, and compatibilities between items are described by an undirected graph. We investigate the existence of a performance paradox in dynamic matching models. More precisely, we analyze the impact on the expected number of unmatched items when we add an edge to the compatibility graph, i.e., whether increasing flexibility to the compatibility graph leads to a larger expected number of unmatched items.

(This is an activity organized within * A Series of Invited Talks*.)

**November 23, 2023, Tuesday, at 16:00 p.m. (UTC+1)
Title:** Dispersión óptima de poblaciones: un enfoque con procesos de decisión markovianos

**Tomás Prieto Rumeau, Universidad Nacional de Educación a Distancia**

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Speaker:

**Abstract:** Abordamos un problema que surge en ecología matemática con la siguiente motivación: pensemos en una población de animales, con una dinámica aleatoria, que es separada, tras la construcción de una carretera o de una vía de tren, en dos grupos independientes que tendrán ahora, cada uno, su propia dinámica. ¿Cómo conectar de manera óptima estas dos poblaciones para maximizar la población total? Plantearemos y resolveremos este problema como un proceso de decisión markoviano. En esta charla, haremos una breve presentación de la herramienta de los procesos de decisión markovianos y mostraremos cómo se pueden aplicar estos modelos para estudiar el problema de dispersión óptima de dos poblaciones. Presentaremos los resultados teóricos obtenidos, algunas simulaciones numéricas y se discutirán posibles extensiones del problema.

(This is an activity organized within * A Series of Invited Talks*.)

**October 19, 2023, Thursday, 12:00-13:30 p.m. and 16:00-17:30 p.m. (UTC+1)
Title:** Birth-death processes and their applications

**Antonio Di Crescenzo, Università degli Studi di Salerno**

Speaker:

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**Abstract:** The seminar consists of two parts. In the first part, the focus is on birth-death processes, growth models and diffusion approximations. A summary is as follows:

Birth-death processes constitute the continuous-time analog of random walks and are largely adopted as a tool for stochastic modeling. Indeed, the richness of the birth and death rates allows modeling a variety of phenomena, ranging for instance from evolutionary dynamics and neuronal modeling, to queueing and reliability theory. Various methods of analysis have been developed for determining quantities of interest, such as stationary distributions and first-passage-time distributions. This first part is aimed at providing a review of some recent results on growth-evolution models characterized by time-dependent growth rates and their stochastic counterpart described by birth-death processes. The analysis focuses on generalizations of the Gompertz and logistic growth models, and related birth-death processes having linear and quadratic rates. A diffusion approximation leading to a time-inhomogeneous geometric Brownian motion is also treated. Some applications will be outlined as well.

In the second part, the interest is in birth-death processes and related diffusions on a star graph for multi-type evolution models. An abstract is as follows:

Recent advances in the theory of birth-death processes have been oriented to describe continuous-time random walks on graphs. In this second part, we aim to discuss certain generalizations involving extended birth-death processes on a star graph defined as a lattice formed by the integers of semiaxes joined at the origin. The study is oriented to: (i) the analysis of the transient and asymptotic behavior of a multispecies birth-death-immigration process and of a continuous-time multi-type Ehrenfest model; and (ii) the construction and the study of suitable diffusion approximations for the considered models, leading to two processes belonging to the class of Pearson diffusions on the spider, i.e. a domain formed by semiaxis joined at the origin. Special attention is devoted to: the diffusion approximations involving a Feller process and the Ornstein-Uhlenbeck process; the stationary distribution based on the switching rules among the semiaxis; and the goodness of the diffusion approximation.

(This is an activity organized within * A Week of Research*.)

**October 18, 2023, Wednesday, 12:00-13:30 p.m. and 16:00-17:30 p.m. (UTC+1)
Title:** Branching processes in the Markovian framework

**Miguel González Velasco, Universidad de Extremadura**

Speaker:

Speaker:

**Abstract:** Branching processes are useful probabilistic models, essentially Markovian, to describe population dynamics in the broadest sense. The notion of branching has been relevant in the development of theoretical approaches to problems in fields as diverse and applied as population growth and extinction, biology, epidemiology, cell proliferation kinetics, cancer, genetics, nuclear physics and algorithm and data structures. The seminar reviews the state of the art with regarding recent advances in the theory and applications of these models.

(This is an activity organized within * A Week of Research*.)

**October 17, 2023, Tuesday, 12:00-13:30 p.m. and 16:00-17:30 p.m. (UTC+1)
Title:** Epidemiological models and human behavior: using game theory to model the effects of voluntary vaccination

**Fabio A.C.C. Chalub, Universidade Nova de Lisboa**

Speaker:

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**Abstract:** There are many different ways to model the spread of a disease in a population, as, for example, ordinary differential equations (deterministic models in homogeneous populations), partial differential equations (populations with space or age structure, for example), Markov process (to include stochasticity, which is important when modeling the dynamics in small populations).

One particularly important challenge is how to include human behavior into the model, which is traditionally made with the use of game theory.

In the first part, we will discuss the basic concepts of epidemiological models, based on ordinary differential equations, and game theory, with a particular interest in studying the effects of voluntary vaccinations on disease dynamics.

In the second part, we will discuss how to model the same problem using the Markov process, discuss the similarities and the differences between both models and present some challenges that can be of interest in the near future.

(This is an activity organized within * A Week of Research*.)

**September 26, 2023, Tuesday, at 16:00 p.m. (UTC+2)
Title:** Spatial population synchrony in two-species stochastic dynamics models

**Javier Jarillo Díaz, Universidad Complutense de Madrid**

Speaker:

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**Abstract:** Markovian processes have been widely applied in stochastic dynamics models in Ecology, to analyze the effects of both demographic and environmental variability in the species population dynamics and stability. Such stochasticity is usually modeled through Itô Stochastic Differential Equations, which include Wiener processes. However, when trying to study spatial models, as for example it could be the dynamics of a species in a multi-patched habitat, we must generalize these Wiener processes to include the possible existence of spatial correlations of the environmental variable conditions. For example, P.A.P. Moran already proved that the existence of such a spatial correlation (or synchrony) can be transferred from the environmental variability to the dynamics of the population size of the species, which have consequences for the species resistance against extinction events. To model this spatially correlated environment, multivariate random fields with the required spatial covariance should be employed. We have then analyzed the spatial community dynamics in two-species ecological systems with these spatially correlated random fields. First, we have seen that inter-species competition may increase the population synchrony of the species, reinforcing the previously analyzed synchronization effect of dispersal. Then, we have also obtained how, in predator-prey systems, predators tend to be synchronized at longer distances than their prey. And more generally, we have obtained that the typical synchrony distance tends to increase from the species more directly affected by the environmental random fields, to other species of the community linked via species interactions.

(This is an activity organized within * A Series of Invited Talks*.)

**March 13, 2023, Monday, at 17:00 p.m. (UTC+1)
Title:** Escape probability [0,b]. Particular cases

**Juan Antonio Vega Coso**

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Speaker:

**Abstract:** Given a compound renewal process with drift *{X_t}_{t>=0}*, we will study the probability that the first escape from the interval *[0,b]* is through the upper barrier. We will deduce the integral equations that satisfy this probability and solve them in some interesting particular cases.

**January 31, 2022, Monday, at 16:30 p.m. (UTC+1)**

**Title: **A new stochastic model for growth curves based on the random telegraph process, with application to listeriosis **
Speaker:** Michael Peter Wiper, Universidad Carlos III de Madrid

**Abstract:** In this talk, we introduce a new, stochastic model for bacteria growth using an integrated, time stretched, random telegraph process. The process is designed so that the mean function is a regular bacterial growth curve such as the Gompertz curve. The random telegraph process is a Markov process with two states, and we also illustrate that our model can capture symmetry around the mean curve (when the rates of the Markov process are equal) or asymmetry (when the rates are different). It is complex to calculate an exact likelihood function for this model and therefore, we consider alternative approaches to inference for the symmetric model, based on (constrained), non-linear, least squares estimation for the mean curve and using the method of moments for the rate of the Markov process. We also show that an approximate likelihood function can be calculated. Finally, we consider approaches to selection of the symmetric or asymmetric model. Our technique is applied to data from petri dish experiments on listeria growth.

**September 30, 2021, Thursday, at 16:30 p.m. (UTC+2)**

**Title:** Brownian motion with regeneration at Poisson epochs

**Speaker:** Javier Villarroel, Universidad de Salamanca

**Abstract:** Brownian motion is the paradigmatic example of purely diffusive random motion where the position has a Gaussian distribution with a variance that grows linearly in time. Here we consider a continuous-time Brownian motion in one space dimension, and we suppose that an external random mechanism is operating, triggered via an independent Poisson process. At the Poisson epochs the process is «regenerated», i.e. the system is restarted to the initial position to start a new so the «future» process is independent of the past. Given two fixed levels a<0<b we describe the probability that starting from 0 the Brownian particle escapes (a,b) via the upper barrier b. We then study the distribution of the escape time and optimal reset mechanisms appropriate to search problems. In the last part we show how to incorporate a drift by means of the Girsanov-Radon-Nikodym theorem.

**May-June, 2021**

**Online course:** Introduction to Malliavin Calculus and Applications

**Instructor:** Carlos Escudero Liébana, Universidad Nacional de Educación a Distancia

**Abstract:** This course will give a short introduction to the Malliavin calculus or stochastic calculus of variations. It is well known as powerful framework that allows for the construction of a pure probabilistic proof of the Hörmander theorem and for an extension of the Itô theory of stochastic integration. We will give a relatively accessible introduction to the subject and to one of its financial applications, the computation of hedging portfolios by means of pure probabilistic tools. General prerequisites for the course: the Itô theory of stochastic integration. Specific prerequisites for the financial part: the Black-Scholes theory of option pricing.

**Content:** Double Wiener-Itô integrals; Chaos expansions; Multiple Wiener-Itô integrals; The Wiener-Itô theorem; Malliavin derivatives; The Itô and martingale representation theorems; The Clark-Ocone formula; Computation of hedging portfolios

**Course structure:** The teaching will be delivered using eight one-hour-and-a-half recorded lectures and four live seminars with the instructor, as follows:

- Week 1: Lectures 1 and 2 (May 17-20); seminar 1 (May 21)
- Week 2: Lectures 3 and 4 (May 24-27); seminar 2 (May 28)
- Week 3: Lectures 5 and 6 (June 7-10); seminar 3 (June 11)
- Week 4: Lectures 7 and 8 (June 14-17); seminar 4 (June 18)

**April 29, 2021, Thursday, at 16:30 p.m. (UTC+2)**

**Title:** Near-maxima and near-records

**Speaker:** F. Javier López Lorente, Universidad de Zaragoza

**Abstract:** There are several definitions of observations near the maximum in the literature. These definitions can be roughly classified in two types, depending on whether observations are recorded sequentially or not. Examples of the first type are near-records and delta-records while near-maxima and epsilon-repetitions of a record are examples of the second type. In this talk we give an overview of the main probabilistic properties of these objects. Near-maxima were introduced by Pakes and Steutel (1997) extending the idea of Eisenberg, Stengle and Strang (1993) on the number of winners in a discrete sample. The main problem is to establish the asymptotic behaviour of the number of near-maxima when the sample size tends to infinity. Near-records, which can be seen as the sequential version of near-maxima, were introduced by Balakrishnan, Pakes and Stepanov (2005). Here, the results in the literature focus both on the number of near-records in a random sequence and on their values. The tools used in the proofs are extreme value theory, point processes and martingales, among others. We also discuss some statistical applications of near-maxima and near-records, such as the estimation of tail-related quantities, maximum likelihood estimation of parameters, Bayesian inference or nonparametric estimation of the hazard function.

**March 24, 2021, Wednesday, at 16:30 p.m. (UTC+1)**

**Title:** Some structural aspects related to spatial deformation, threshold exceedance and risk in random fields

**Speaker:** José Miguel Angulo, Universidad de Granada

**Abstract:** In this talk I intend to give a synthetic view on some significant aspects related to the effect of deformation transformations of the spatial support of random fields. In particular, I will refer to the assessment regarding the implications derived on structural characteristics of excursion sets defined by threshold exceedance, related risk indicators, and risk measures. Among other approaches, information measures constitute useful tools for quantitative analysis in this context.