Section: Research Program

Stochastic modeling

Our aim is to propose relevant stochastic frameworks for the modeling and the understanding of biological systems. The stochastic processes are particularly suitable for this purpose. Among them, Markov chains give a first framework for the modeling of population of cells [105], [69]. Piecewise deterministic processes are non diffusion processes also frequently used in the biological context [53], [68], [61], [56]. Among Markov model, we also developed strong expertise about processes derived from Brownian motion and Stochastic Differential Equations [94], [67], [96]. For instance, knowledge about Brownian or random walk excursions [104], [93] helps to analyse genetic sequences and to develop inference about it. However, nature provides us with many examples of systems such that the observed signal has a given Hölder regularity, which does not correspond to the one we might expect from a system driven by ordinary Brownian motion. This situation is commonly handled by noisy equations driven by Gaussian processes such as fractional Brownian motion or (in higher dimensions of the parameter) fractional fields. The basic aspects of these differential equations are now well understood, mainly thanks to the so-called rough paths tools [80], but also invoking the Russo-Vallois integration techniques [95]. The specific issue of Volterra equations driven by fBm, which is central for the subdiffusion within proteins problem, is addressed in [54]. Many generalizations (Gaussian or not) of this model have been recently proposed, see for instance [44] for some Gaussian locally self-similar fields, [73] for some non-Gaussian models, [47] for anisotropic models. Our team has thus contributed [52], [74], [73], [75], [87] and still contributes [46], [48], [47], [76], [64] to this theoretical study: Hölder continuity, fractal dimensions, existence and uniqueness results for differential equations, study of the laws to quote a few examples. On the other hand, because of the observation of longitudinal data for each subject in medicine, we have to care about the random effect due to the subject and to choose adapted models like mixed effect models [77], [42], [43]. In the context of health-care and cost-effectiveness analysis, we are also interested in model of aggregation of different criteria. For this purpose, we develop research about fuzzy binary measures and Choquet integral [63], [81].