In this talk I will present a stochastic model for the excitability of a neuron in a network. The neuron described by an Hodgkin-Huxley type model receives from the network a random input which is a perturbation of a periodic deterministic signal. For such a model we study ergodicity properties. Then, we prove limit theorems in order to be able to estimate characteristics of the sequence of spiking times. This talk is based on a joint work with R. Hoepfner (Univ. Mainz) and E. Loecherbach (Univ. Cergy-Pontoise).

Hodgkin-Huxley model - ergodicity - limit theorems - estimation[-]

Les processus de fragmentation sont des modèles aléatoires pour décrire l'évolution d'objets (particules, masses) sujets à des fragmentations successives au cours du temps. L'étude de tels modèles remonte à Kolmogorov, en 1941, et ils ont depuis fait l'objet de nombreuses recherches. Ceci s'explique à la fois par de multiples motivations (le champs d'applications est vaste : biologie et génétique des populations, formation de planètes, polymérisation, aérosols, industrie minière, informatique, etc.) et par la mise en place de modèles mathématiques riches et liés à d'autres domaines bien développés en Probabilités, comme les marches aléatoires branchantes, les processus de Lévy et les arbres aléatoires. L'objet de ce mini-cours est de présenter les processus de fragmentation auto-similaires, tels qu'introduits par Bertoin au début des années 2000s. Ce sont des processus markoviens, dont la dynamique est caractérisée par une propriété de branchement (différents objets évoluent indépendamment) et une propriété d'auto-similarité (un objet se fragmente à un taux proportionnel à une certaine puissance fixée de sa masse). Nous discuterons la construction de ces processus (qui incluent des modèles avec fragmentations spontanées, plus délicats à construire) et ferons un tour d'horizon de leurs principales propriétés.[-]

Les processus de fragmentation sont des modèles aléatoires pour décrire l'évolution d'objets (particules, masses) sujets à des fragmentations successives au cours du temps. L'étude de tels modèles remonte à Kolmogorov, en 1941, et ils ont depuis fait l'objet de nombreuses recherches. Ceci s'explique à la fois par de multiples motivations (le champs d'applications est vaste : biologie et génétique des populations, formation de planètes, polymérisation, aérosols, industrie minière, informatique, etc.) et par la mise en place de modèles mathématiques riches et liés à d'autres domaines bien développés en Probabilités, comme les marches aléatoires branchantes, les processus de Lévy et les arbres aléatoires. L'objet de ce mini-cours est de présenter les processus de fragmentation auto-similaires, tels qu'introduits par Bertoin au début des années 2000s. Ce sont des processus markoviens, dont la dynamique est caractérisée par une propriété de branchement (différents objets évoluent indépendamment) et une propriété d'auto-similarité (un objet se fragmente à un taux proportionnel à une certaine puissance fixée de sa masse). Nous discuterons la construction de ces processus (qui incluent des modèles avec fragmentations spontanées, plus délicats à construire) et ferons un tour d'horizon de leurs principales propriétés.[-]

We represent Hawkes process and their Volterra long term limits, which have recently been used as rough variance processes, as functionals of infinite dimensional affine Markov processes. The representations lead to several new views on affine Volterra processes considered by Abi-Jaber, Larsson and Pulido. We also discuss possible extensions to rough covariance modeling via Volterra Wishart processes.
The talk is based on joint work with Josef Teichmann.[-]

Consider random conductances that allow long range jumps. In particular we consider conductances $C_{xy} = w_{xy}|x − y|^{−d−\alpha}$ for distinct $x, y \in Z^d$ and $0 < \alpha < 2$, where $\lbrace w_{xy} = w_{yx} : x, y \in Z^d\rbrace$ are non-negative independent random variables with mean 1. We prove that under some moment conditions for $w$, suitably rescaled Markov chains among the random conductances converge to a rotationally symmetric $\alpha$-stable process almost surely w.r.t. the randomness of the environments. The proof is a combination of analytic and probabilistic methods based on the recently established de Giorgi-Nash-Moser theory for processes with long range jumps. If time permits, we also discuss quenched heat kernel estimates as well. This is a joint work with Xin Chen (Shanghai) and Jian Wang (Fuzhou).[-]

In this talk, we derive a novel non-reversible, continuous-time Markov chain Monte Carlo (MCMC) sampler, called Coordinate Sampler, based on a piecewise deterministic Markov process (PDMP), which can be seen as a variant of the Zigzag sampler. In addition to proving a theoretical validation for this new sampling algorithm, we show that the Markov chain it induces exhibits geometrical ergodicity convergence, for distributions whose tails decay at least as fast as an exponential distribution and at most as fast as a Gaussian distribution. Several numerical examples highlight that our coordinate sampler is more efficient than the Zigzag sampler, in terms of effective sample size.
[This is joint work with Wu Changye, ref. arXiv:1809.03388][-]

This talk will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential equations. Unfortunately, only particular parameter constellations allow for closed form solutions such that in general one needs to rely on numerical methods. Instead of studying these associated integro-differential equations, we adapt the problem in a way that allows us to apply deterministic numerical integration algorithms such as QMC rules.[-]

This talk will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential equations. Unfortunately, only particular parameter constellations allow for closed form solutions such that in general one needs to rely on numerical methods. Instead of studying these associated integro-differential equations, we adapt the problem in a way that allows us to apply deterministic numerical integration algorithms such as QMC rules.[-]

This talk will give an overview on the usage of piecewise deterministic Markov processes for risk theoretic modeling and the application of QMC integration in this framework. This class of processes includes several common risk models and their generalizations. In this field, many objects of interest such as ruin probabilities, penalty functions or expected dividend payments are typically studied by means of associated integro-differential equations. Unfortunately, only particular parameter constellations allow for closed form solutions such that in general one needs to rely on numerical methods. Instead of studying these associated integro-differential equations, we adapt the problem in a way that allows us to apply deterministic numerical integration algorithms such as QMC rules.[-]