Soft bounds on diffusion produce skewed distributions and Gompertz growth

Constraints can affect dramatically the behavior of diffusion processes. Recently, we analyzed a natural and a technological system and reported that they perform diffusion-like discrete steps displaying a peculiar constraint, whereby the increments of the diffusing variable are subject to configuration-dependent bounds. This work explores theoretically some of the revealing landmarks of such phenomenology, termed "soft bound". At long times, the system reaches a steady state irreversibly (i.e., violating detailed balance), characterized by a skewed "shoulder" in the density distribution, and by a net local probability flux, which has entropic origin. The largest point in the support of the distribution follows a saturating dynamics, expressed by the Gompertz law, in line with empirical observations. Finally, we propose a generic allometric scaling for the origin of soft bounds. These findings shed light on the impact on a system of such "scaling" constraint and on its possible generating mechanisms.Comment: 9 pages, 6 color figure

Valuation and hedging of the ruin-contingent life annuity (RCLA)

This paper analyzes a novel type of mortality contingent-claim called a ruin-contingent life annuity (RCLA). This product fuses together a path-dependent equity put option with a "personal longevity" call option. The annuitant's (i.e. long position) payoff from a generic RCLA is \$1 of income per year for life, akin to a defined benefit pension, but deferred until a pre-specified financial diffusion process hits zero. We derive the PDE and relevant boundary conditions satisfied by the RCLA value (i.e. the hedging cost) assuming a complete market where No Arbitrage is possible. We then describe some efficient numerical techniques and provide estimates of a typical RCLA under a variety of realistic parameters. The motivation for studying the RCLA on a stand-alone basis is two-fold. First, it is implicitly embedded in approximately \$1 trillion worth of U.S. variable annuity (VA) policies; which have recently attracted scrutiny from financial analysts and regulators. Second, the U.S. administration - both Treasury and Department of Labor - have been encouraging Defined Contribution (401k) plans to offer stand-alone longevity insurance to participants, and we believe the RCLA would be an ideal and cost effective candidate for that job

An exact analytical solution for generalized growth models driven by a Markovian dichotomic noise

Logistic growth models are recurrent in biology, epidemiology, market models, and neural and social networks. They find important applications in many other fields including laser modelling. In numerous realistic cases the growth rate undergoes stochastic fluctuations and we consider a growth model with a stochastic growth rate modelled via an asymmetric Markovian dichotomic noise. We find an exact analytical solution for the probability distribution providing a powerful tool with applications ranging from biology to astrophysics and laser physics

A generalized model of mutation-selection balance with applications to aging

A probability model is presented for the dynamics of mutation-selection balance in a haploid infinite-population infinite-sites setting sufficiently general to cover mutation-driven changes in full age-specific demographic schedules. The model accommodates epistatic as well as additive selective costs. Closed form characterizations are obtained for solutions in finite time, along with proofs of convergence to stationary distributions and a proof of the uniqueness of solutions in a restricted case. Examples are given of applications to the biodemography of aging, including instabilities in current formulations of mutation accumulation.Comment: 20 pages Updated to include more historical comment and references to the literature, as well as to make clear how our non-linear, non-Markovian model differs from previous linear, Markovian particle system and measure-valued diffusion models. Further updated to take into account referee's comment

Some Notes about Inference for the Lognormal Diffusion Process with Exogenous Factors

Different versions of the lognormal diffusion process with exogenous factors have been used in recent years to model and study the behavior of phenomena following a given growth curve. In each case considered, the estimation of the model has been addressed, generally by maximum likelihood (ML), as has been the study of several characteristics associated with the type of curve considered. For this process, a unified version of the ML estimation problem is presented, including how to obtain estimation errors and asymptotic confidence intervals for parametric functions when no explicit expression is available for the estimators of the parameters of the model. The Gompertz-type diffusion process is used here to illustrate the application of the methodology.This work was supported in part by the Ministerio de Economía, Industria y Competitividad, Spain, under Grants MTM2014-58061-P and MTM2017-85568-P