CRC/TRR 388/1: "Statistics for population models with stochastic (partial) delay differential equations" (SP B07)

Stochastic (partial) delay differential equations (henceforth S(P)delayDEs) have ap- peared recently in population models arising in population genetics and ecology. Here the delay term can be intrinsically motivated to describe the evolutionary force known as dormancy, a mechanism particularly prevalent in microbial populations and believed to be crucial for their evolution and ecology. This project focuses on estimating and testing the presence or form of the delay and other key parameters for models described by Wright-Fisher type ordinary and partial stochastic differential equations with time delay.
To this end, novel statistical methodology will be developed to construct estimators and tests for relevant biological quantities. This will require a profound understand- ing of the solution processes, e.g. their regularity or ergodicity properties. The main mathematical challenge is to analyse the laws of estimators and test statistics built on the complex data-generating processes provided by S(P)delayDEs. One major infer- ence problem is to test whether a delay term is present versus the null hypothesis of Markovianity. Often, a parametric form of the delay (like simple exponential weighting) is assumed, which allows to consider Markovian dynamics on a larger state space and which calls for parametric statistical inference. For general delay weight functions the solution theory and the long-time behaviour are more delicate and will be studied in de- tail. The construction and analysis of nonparametric estimators for the weight function require further extensions of the existing statistical and probabilistic theory. Special emphasis in this project will be given on provable statistical efficiency of inference pro- cedures relevant for S(P)delayDE models in science with an emphasis on population models possibly involving dormancy.