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

The Working Group on «Stochastic Processes and their Aplications» 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

**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:

Speaker:

**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*.)

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

### Past talks and training courses

**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.