Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems

We study linear-quadratic stochastic optimal control problems with bilinear state dependence for which the underlying stochastic differential equation (SDE) consists of slow and fast degrees of freedom. We show that, in the same way in which the underlying dynamics can be well approximated by a reduced order effective dynamics in the time scale limit (using classical homogenziation results), the associated optimal expected cost converges in the time scale limit to an effective optimal cost. This entails that we can well approximate the stochastic optimal control for the whole system by the reduced order stochastic optimal control, which is clearly easier to solve because of lower dimensionality. The approach uses an equivalent formulation of the Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs (FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares Monte Carlo algorithm and show its applicability by a suitable numerical example

Coarse-graining of non-reversible stochastic differential equations: quantitative results and connections to averaging

This work is concerned with model reduction of stochastic differential equations and builds on the idea of replacing drift and noise coefficients of preselected relevant, e.g. slow variables by their conditional expectations. We extend recent results by Legoll & Lelièvre [Nonlinearity 23, 2131, 2010] and Duong et al. [Nonlinearity 31, 4517, 2018] on effective reversible dynamics by conditional expectations to the setting of general non-reversible processes with non-constant diffusion coefficient. We prove relative entropy and Wasserstein error estimates for the difference between the time marginals of the effective and original dynamics as well as an entropy error bound for the corresponding path space measures. A comparison with the averaging principle for systems with time-scale separation reveals that, unlike in the reversible setting, the effective dynamics for a non-reversible system need not agree with the averaged equations. We present a thorough comparison for the Ornstein-Uhlenbeck process and make a conjecture about necessary and sufficient conditions for when averaged and effective dynamics agree for nonlinear non-reversible processes. The theoretical results are illustrated with suitable numerical examples

Variational approach to rare event simulation using least-squares regression

We propose an adaptive importance sampling scheme for the simulation of rare events when the underlying dynamics is given by a diffusion. The scheme is based on a Gibbs variational principle that is used to determine the optimal (i.e. zero-variance) change of measure and exploits the fact that the latter can be rephrased as a stochastic optimal control problem. The control problem can be solved by a stochastic approximation algorithm, using the Feynman-Kac representation of the associated dynamic programming equations, and we discuss numerical aspects for high-dimensional problems along with simple toy examples.Comment: 28 pages, 7 figure

Structured physical activity interventions as a complementary therapy for patients with inflammatory bowel disease – a scoping review and practical implications

Background: Patients with Inflammatory Bowel Disease (IBD) also suffer from a wide range of additional disorders, which may be caused by the disease, the side effect of the medication, or a lack of physical activity (PA). This results in reduced physical and psychological wellbeing. However, as known from other chronic diseases exercise could be utilized as supportive therapy for IBD patients. Main goals of this article are (a) collecting data of the effects structured physical activity interventions have on validated clinical parameters of IBD and health related symptoms, (b) developing activity recommendations for this clientele. Methods: A scoping review was conducted, searching for relevant articles published until May 2018, which investigated the effects of structured exercise interventions in IBD patients. The heterogeneity of the outcomes and the interventions did not support a quantitative synthesis thus, a qualitative discussion of the studies is provided. Results: After reviewing 353 records, 13 eligible articles were identified. Five studies investigated aerobic exercise, three studies resistance exercise, three studies mind-body therapies and two studies yoga. The quality of the studies is mixed, and the duration is rather short for exercise interventions. Only few studies assessed validated IBD activity markers or inflammatory biomarkers. Nevertheless, the patients showed an increase in fitness, bone mineral density (BMD), quality of life and a decrease of IBD induced stress and anxiety. No severe adversial events were reported. Conclusion: Even though the evidence is limited the application of exercise interventions in IBD patients can be assumed to be safe and beneficial for the patients‘ overall-health, and IBD specific physical and psychosocial symptoms. But there is still a high demand for more thoroughly conducted studies, focussing on important clinical outcome parameters

Model-independent view on the low-mass proton-antiproton enhancement

We present a simple interpretation of the recently observed near-threshold proton-antiproton enhancement. It is described by a set of low-energy parameters deduced from the analysis of NantiN experiments at LEAR. We predict a related effect in photoproduction reaction under study by CLAS collaboration.Comment: 10 pages, 2 figure