Micro-macro approaches to age-structured heterogeneous population dynamics. Application to self-exciting processes and demography.

Summary This thesis deals with the modeling of population dynamics and its application to demography and actuarial science on the one hand, and to the study of Hawkes processes on the other hand. This thesis proposes to explore through different points of view how the structure of a population is deformed, both in terms of age distribution and in terms of its composition in terms of characteristics. Through five chapters, we present the same philosophy which, in order to understand how aggregate quantities evolve, proposes to study the dynamics of the population at a finer scale, that of the individual. After a first introductory chapter in French, detailing the motivations and the main contributions, we first propose in Chapter 2 the description of the general framework of random dynamic modeling of populations structured in characteristics and ages, based on Bensusan et al. (2010-2015), as well as several examples motivated by demographic and actuarial applications. We detail the mathematical construction of such processes as well as the link to classical deterministic equations in demography. We also discuss the impact of heterogeneity on the example of a cohort effect, as well as the role of the random environment. The next two chapters highlight the importance of the age pyramid. The general population model from Chapter 2 is declined in Chapter 3 to study Hawkes processes with general immigrants, for which we exploit the concept of the age pyramid. In this theoretical study, based on Boumezoued (2015b), we establish new results on their distribution for a class of functions that generalize the exponential case studied so far. In Chapter 4, following Arnold et al. (2015), we analyze the impact of changes in cause-of-death mortality on the dynamics of the population pyramid, and in particular on the dependency ratio which is a crucial indicator of population aging. By including the set of births in the dynamics, this simulation work, based on WHO data, complements the existing literature on causes of death which traditionally focuses on mortality indicators. The last two chapters focus on the heterogeneity of populations. Chapter 5, based on Boumezoued et al. (2015), proposes to measure mortality heterogeneity in INSEE's Permanent Demographic Sample data. As part of this contribution of adapting statistical methods and its implementation on real data, we propose a parametric maximum likelihood estimation method for multi-state models that takes into account both interval censoring, characteristic of longitudinal data from the census, and also the return in intermediate states. Finally, Chapter 6, taken from Boumezoued (2015a), repeats the general model from Chapter 2 in which individuals can give birth, change characteristics, and die. The contribution of this theoretical part is to study the behavior of the population when individual characteristics change frequently. We establish a large population limit theorem for the population pyramid process, whose behavior is then described by birth and death rates aggregated over the stable structure in terms of characteristics.