# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics
**Dr. Massimiliano
Tamborrino**, Institute for Stochastics, JKU Linz,
will give a public talk (followed by a discussion) on **Wed, Nov. 20, 2019**
at **15:15 o'clock** at **S3 047**
on the topic of
"Stochastic modelling and statistical inference for stochastic and point processes"
. The organziers of the symposium,

Univ.-Prof. Dr. Gerhard Larcher

A.Univ.-Prof. Dr. Jürgen Maaß, and

die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Stochastic modelling and statistical inference for stochastic and point processes

Over the last decades, stochastic differential equations (SDEs) have become an established and powerful tool for modelling time-dependent, real-world phenomena with underlying random effects. In several applications, it is of primary interest to determine the epochs when a diffusion process (solution of an underlying SDE) hits a time-varying threshold for the first time (the so-called first passage time problem, closely related to stopping or exit times). Depending on the underlying assumptions on the process and on the threshold, the collection of hitting times yield renewal or nonrenewal point processes. My interest is in the study of stochastic processes and point processes from a modelling, numerical, probabilistic and statistical point of view. The results are mainly discussed in the framework of neuroscience, but the same scenarios can be found in many fields, ranging from biology over finance to physics, psychology and others.

First, we will focus on the stochastic modelling side with two examples, seeing how the Jacobi process (a univariate diffusion with multiplicative noise which keeps the process confined between two boundaries) and multivariate Gaussian models with additive noise can be used for the single neuron modelling and for the visual identification of briefly presented, mutually confusable single stimuli in pure accuracy tasks, respectively.

Second, we will focus on the statistical side, performing statistical inference of relevant underlying model parameters from fully/partially observed stochastic processes (e.g. discrete observations of one or more coordinates) and (perturbed) point processes. We will briefly discuss a couple of examples (with application on lung cancer data and visual data) where the underlying likelihood function can be derived, leading to maximum likelihood estimation. Since the underlying likelihoods are often unknown or intractable, we will then focus on likelihood-free methods, and in particular on Approximate Bayesian Computation (ABC) method for Hamiltonian SDEs. We will show how only the combination of statistics based on the properties of the model that are less sensitive to its intrinsic stochasticity and a proper (measure-preserving) numerical (splitting) scheme preserving them yields a successful inference, while the commonly used Euler-Maruyama drastically fails.

Keywords: diffusion processes; hitting/stopping times; mathematical neuroscience; likelihood-free inference; Approximate Bayesian Computation; stochastic numerics.