Ergodic BSDEs with jumps and time dependence

In this paper we look at ergodic BSDEs in the case where the forward dynamics are given by the solution to a non-autonomous (time-periodic coefficients) Ornstein-Uhlenbeck SDE with L\'evy noise, taking values in a separable Hilbert space. We establish the existence of a unique bounded solution to an infinite horizon discounted BSDE. We then use the vanishing discount approach, together with coupling techniques, to obtain a Markovian solution to the EBSDE. We also prove uniqueness under certain growth conditions. Applications are then given, in particular to risk-averse ergodic optimal control and power plant evaluation under uncertainty

Nash equilibria for non zero-sum ergodic stochastic differential games

In this paper we consider non zero-sum games where multiple players control the drift of a process, and their payoffs depend on its ergodic behaviour. We establish their connection with systems of Ergodic BSDEs, and prove the existence of a Nash equilibrium under the generalised Isaac's conditions. We also study the case of interacting players of different type

Implementing competing risks in discrete event simulation:the event-specific probabilities and distributions approach

Background: Although several strategies for modelling competing events in discrete event simulation (DES) exist, a methodological gap for the event-specific probabilities and distributions (ESPD) approach when dealing with censored data remains. This study defines and illustrates the ESPD strategy for censored data. Methods: The ESPD approach assumes that events are generated through a two-step process. First, the type of event is selected according to some (unknown) mixture proportions. Next, the times of occurrence of the events are sampled from a corresponding survival distribution. Both of these steps can be modelled based on covariates. Performance was evaluated through a simulation study, considering sample size and levels of censoring. Additionally, an oncology-related case study was conducted to assess the ability to produce realistic results, and to demonstrate its implementation using both frequentist and Bayesian frameworks in R.Results: The simulation study showed good performance of the ESPD approach, with accuracy decreasing as sample sizes decreased and censoring levels increased. The average relative absolute error of the event probability (95%-confidence interval) ranged from 0.04 (0.00; 0.10) to 0.23 (0.01; 0.66) for 60% censoring and sample size 50, showing that increased censoring and decreased sample size resulted in lower accuracy. The approach yielded realistic results in the case study. Discussion: The ESPD approach can be used to model competing events in DES based on censored data. Further research is warranted to compare the approach to other modelling approaches for DES, and to evaluate its usefulness in estimating cumulative event incidences in a broader context. </p

Topics in ergodic control and backward stochastic differential equations

The core of this thesis focuses on a number of different aspects of ergodic stochastic control in connection with backward stochastic differential equations (BSDEs for short). Chapter 1 serves as an introduction to the problem formulation in various contexts and states a number of results we will be using in the sequel. Chapter 2 deals with the so called weak formulation, where the control is represented as a change of measure. The optimal value and feedback control are obtained using a relatively recent object called ergodic BSDEs. In order to achieve this we establish the existence and uniqueness of solutions to these equations along the way. Chapter 3 is concerned with non zero-sum games where multiple players control the drift of a process, and their payoffs depend on its ergodic behaviour. We establish their connection with systems of Ergodic BSDEs, and prove the existence of a Nash equilibrium under general conditions. In Chapter 4 we show a novel duality between the existence of a solution to an infinite horizon adjoint BSDE and strong dissipativity of the forward process. Thus the link between ergodicity of the controlled process and the infinite horizon stochastic maximum principle is established. Finally, in Chapter 5 we provide conclusions, conjectures and directions for future research.</p

Disease progression modelling of Alzheimer’s disease using probabilistic principal components analysis

The recent biological redefinition of Alzheimer’s Disease (AD) has spurred the development of statistical models that relate changes in biomarkers with neurodegeneration and worsening condition linked to AD. The ability to measure such changes may facilitate earlier diagnoses for affected individuals and help in monitoring the evolution of their condition. Amongst such statistical tools, disease progression models (DPMs) are quantitative, data-driven methods that specifically attempt to describe the temporal dynamics of biomarkers relevant to AD. Due to the heterogeneous nature of this disease, with patients of similar age experiencing different AD-related changes, a challenge facing longitudinal mixed-effects-based DPMs is the estimation of patient-realigning time-shifts. These time-shifts are indispensable for meaningful biomarker modelling, but may impact fitting time or vary with missing data in jointly estimated models. In this work, we estimate an individual’s progression through Alzheimer’s disease by combining multiple biomarkers into a single value using a probabilistic formulation of principal components analysis. Our results show that this variable, which summarises AD through observable biomarkers, is remarkably similar to jointly estimated time-shifts when we compute our scores for the baseline visit, on cross-sectional data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI). Reproducing the expected properties of clinical datasets, we confirm that estimated scores are robust to missing data or unavailable biomarkers. In addition to cross-sectional insights, we can model the latent variable as an individual progression score by repeating estimations at follow-up examinations and refining long-term estimates as more data is gathered, which would be ideal in a clinical setting. Finally, we verify that our score can be used as a pseudo-temporal scale instead of age to ignore some patient heterogeneity in cohort data and highlight the general trend in expected biomarker evolution in affected individuals