Section: New Results

Estimation and control for Markov Processes

Participants: R. Azais, F. Bouguet, T. Bastogne

We have developed statistical inference techniques for estimating the jump rate of PDMPs (piecewise-deterministic Markov processes) [2] which is an essential step to build relevant application models. In [2], we state a new characterization of the jump rate when the transition kernel only charges a discrete subset of the state space and deduce from it a competitive nonparametric technique for estimating this feature of interest. Our methodologies have been illustrated on numerical examples and real data. We also investigated the probabilistic properties of the PDMPs [5] or more general Markov processes [31] that could be useful to study properties of estimators.

A bit more generally, we have made contributions to a variety of specific estimation problems. We considered the problem of estimation of integrals under Markov design, which has a large variety of applications, in particular in biology and climatology. In [24], we have developed and analyzed a technique for estimating the average value over space when sensors describe a Markovian trajectory; this method leads to rates that are better than the traditional “root $n$”-rate, where $n$ is the sample size, and was applied to the evaluation of the average temperature of oceans.

Control of stochastic processes is also a way to optimise administration (dose, frequency) of therapy. In [8], we have presented the design and validation of a real time controller able to track a preset photobleaching trajectory by modulating the width of light impulses during the treatment sessions, which is useful in a Photodynamic therapy context. This innovative solution was validated by in vivo experiments that have shown a significantly improvement of reproducibility of the inter-individual photobleaching kinetics. This innovative controller is the first personalized solution able to adapt in realtime the dose of light to be applied in photodynamic therapy.