## 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

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

**Juan Antonio Vega Coso**

Speaker:

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.

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

### Past talks and training courses

**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.