An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift

We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous

Describing the complexity of systems: multi-variable "set complexity" and the information basis of systems biology

Context dependence is central to the description of complexity. Keying on the pairwise definition of "set complexity" we use an information theory approach to formulate general measures of systems complexity. We examine the properties of multi-variable dependency starting with the concept of interaction information. We then present a new measure for unbiased detection of multi-variable dependency, "differential interaction information." This quantity for two variables reduces to the pairwise "set complexity" previously proposed as a context-dependent measure of information in biological systems. We generalize it here to an arbitrary number of variables. Critical limiting properties of the "differential interaction information" are key to the generalization. This measure extends previous ideas about biological information and provides a more sophisticated basis for study of complexity. The properties of "differential interaction information" also suggest new approaches to data analysis. Given a data set of system measurements differential interaction information can provide a measure of collective dependence, which can be represented in hypergraphs describing complex system interaction patterns. We investigate this kind of analysis using simulated data sets. The conjoining of a generalized set complexity measure, multi-variable dependency analysis, and hypergraphs is our central result. While our focus is on complex biological systems, our results are applicable to any complex system.Comment: 44 pages, 12 figures; made revisions after peer revie

The Problem of Respecting Higher-Order Doubt

This paper argues that higher-order doubt generates an epistemic dilemma. One has a higher-order doubt with regards to P insofar as one justifiably withholds belief as to what attitude towards P is justified. That is, one justifiably withholds belief as to whether one is justified in believing, disbelieving, or withholding belief in P. Using the resources provided by Richard Feldman’s recent discussion of how to respect one’s evidence, I argue that if one has a higher-order doubt with regards to P, then one is not justified in having any attitude towards P. Otherwise put: No attitude towards the doubted proposition respects one’s higher-order doubt. I argue that the most promising response to this problem is to hold that when one has a higher-order doubt about P, the best one can do to respect such a doubt is to simply have no attitude towards P. Higher-order doubt is thus much more rationally corrosive than non-higher-order doubt, as it undermines the possibility of justifiably having any attitude towards the doubted proposition