Stochastic resonance in noisy nondynamical systems

We have analyzed the effects of the addition of external noise to nondynamical systems displaying intrinsic noise, and established general conditions under which stochastic resonance appears. The criterion we have found may be applied to a wide class of nondynamical systems, covering situations of different nature. Some particular examples are discussed in detail.

Divergent signal-to-noise ratio and stochastic resonance in monostable systems

We present a class of systems for which the signal-to-noise ratio always increases when increasing the noise and diverges at infinite noise level. This new phenomenon is a direct consequence of the existence of a scaling law for the signal-to-noise ratio and implies the appearance of stochastic resonance in some monostable systems. We outline applications of our results to a wide variety of systems pertaining to different scientific areas. Two particular examples are discussed in detail.

Stochastic multiresonance

We present a class of systems for which the signal-to-noise ratio as a function of the noise level may display a multiplicity of maxima. This phenomenon, referred to as stochastic multiresonance, indicates the possibility that periodic signals may be enhanced at multiple values of the noise level, instead of at a single value which has occurred in systems considered up to now in the framework of stochastic resonance.

Spatiotemporal stochastic resonance in the Swift-Hohenberg equation

We show the appearance of spatiotemporal stochastic resonance in the Swift-Hohenberg equation. This phenomenon emerges when a control parameter varies periodically in time around the bifurcation point. By using general scaling arguments and by taking into account the common features occurring in a bifurcation, we outline possible manifestations of the phenomenon in other pattern-forming systems.

Cross-cultural adaptation, reliability, internal consistency and validation of the Spinal Function Sort (SFS) for French- and German-speaking patients with back complaints.

Introduction Functional subjective evaluation through questionnaire is fundamental, but not often realized in patients with back complaints, lacking validated tools. The Spinal Function Sort (SFS) was only validated in English. We aimed to translate, adapt and validate the French (SFS-F) and German (SFS-G) versions of the SFS. Methods Three hundred and forty-four patients, experiencing various back complaints, were recruited in a French (n = 87) and a German-speaking (n = 257) center. Construct validity was estimated via correlations with SF-36 physical and mental scales, pain intensity and hospital anxiety and depression scales (HADS). Scale homogeneities were assessed by Cronbach's α. Test-retest reliability was assessed on 65 additional patients using intraclass correlation (IC). Results For the French and German translations, respectively, α were 0.98 and 0.98; IC 0.98 (95% CI: [0.97; 1.00]) and 0.94 (0.90; 0.98). Correlations with physical functioning were 0.63 (0.48; 0.74) and 0.67 (0.59; 0.73); with physical summary 0.60 (0.44; 0.72) and 0.52 (0.43; 0.61); with pain -0.33 (-0.51; -0.13) and -0.51 (-0.60; -0.42); with mental health -0.08 (-0.29; 0.14) and 0.25 (0.13; 0.36); with mental summary 0.01 (-0.21; 0.23) and 0.28 (0.16; 0.39); with depression -0.26 (-0.45; -0.05) and -0.42 (-0.52; -0.32); with anxiety -0.17 (-0.37; -0.04) and -0.45 (-0.54; -0.35). Conclusions Reliability was excellent for both languages. Convergent validity was good with SF-36 physical scales, moderate with VAS pain. Divergent validity was low with SF-36 mental scales in both translated versions and with HADS for the SFS-F (moderate in SFS-G). Both versions seem to be valid and reliable for evaluating perceived functional capacity in patients with back complaints.

Regression analysis of spatial data.

Many of the most interesting questions ecologists ask lead to analyses of spatial data. Yet, perhaps confused by the large number of statistical models and fitting methods available, many ecologists seem to believe this is best left to specialists. Here, we describe the issues that need consideration when analysing spatial data and illustrate these using simulation studies. Our comparative analysis involves using methods including generalized least squares, spatial filters, wavelet revised models, conditional autoregressive models and generalized additive mixed models to estimate regression coefficients from synthetic but realistic data sets, including some which violate standard regression assumptions. We assess the performance of each method using two measures and using statistical error rates for model selection. Methods that performed well included generalized least squares family of models and a Bayesian implementation of the conditional auto-regressive model. Ordinary least squares also performed adequately in the absence of model selection, but had poorly controlled Type I error rates and so did not show the improvements in performance under model selection when using the above methods. Removing large-scale spatial trends in the response led to poor performance. These are empirical results; hence extrapolation of these findings to other situations should be performed cautiously. Nevertheless, our simulation-based approach provides much stronger evidence for comparative analysis than assessments based on single or small numbers of data sets, and should be considered a necessary foundation for statements of this type in future.

On the problematic of reserving and stochastic loss models

Abstract Traditionally, the common reserving methods used by the non-life actuaries are based on the assumption that future claims are going to behave in the same way as they did in the past. There are two main sources of variability in the processus of development of the claims: the variability of the speed with which the claims are settled and the variability between the severity of the claims from different accident years. High changes in these processes will generate distortions in the estimation of the claims reserves. The main objective of this thesis is to provide an indicator which firstly identifies and quantifies these two influences and secondly to determine which model is adequate for a specific situation. Two stochastic models were analysed and the predictive distributions of the future claims were obtained. The main advantage of the stochastic models is that they provide measures of variability of the reserves estimates. The first model (PDM) combines one conjugate family Dirichlet - Multinomial with the Poisson distribution. The second model (NBDM) improves the first one by combining two conjugate families Poisson -Gamma (for distribution of the ultimate amounts) and Dirichlet Multinomial (for distribution of the incremental claims payments). It was found that the second model allows to find the speed variability in the reporting process and development of the claims severity as function of two above mentioned distributions' parameters. These are the shape parameter of the Gamma distribution and the Dirichlet parameter. Depending on the relation between them we can decide on the adequacy of the claims reserve estimation method. The parameters have been estimated by the Methods of Moments and Maximum Likelihood. The results were tested using chosen simulation data and then using real data originating from the three lines of business: Property/Casualty, General Liability, and Accident Insurance. These data include different developments and specificities. The outcome of the thesis shows that when the Dirichlet parameter is greater than the shape parameter of the Gamma, resulting in a model with positive correlation between the past and future claims payments, suggests the Chain-Ladder method as appropriate for the claims reserve estimation. In terms of claims reserves, if the cumulated payments are high the positive correlation will imply high expectations for the future payments resulting in high claims reserves estimates. The negative correlation appears when the Dirichlet parameter is lower than the shape parameter of the Gamma, meaning low expected future payments for the same high observed cumulated payments. This corresponds to the situation when claims are reported rapidly and fewer claims remain expected subsequently. The extreme case appears in the situation when all claims are reported at the same time leading to expectations for the future payments of zero or equal to the aggregated amount of the ultimate paid claims. For this latter case, the Chain-Ladder is not recommended.

Coherent stochastic resonance

We consider Brownian motion on a line terminated by two trapping points. A bias term in the form of a telegraph signal is applied to this system. It is shown that the first two moments of survival time exhibit a minimum at the same resonant frequency.

Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality

A stochastic nonlinear partial differential equation is constructed for two different models exhibiting self-organized criticality: the Bak-Tang-Wiesenfeld (BTW) sandpile model [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)] and the Zhang model [Phys. Rev. Lett. 63, 470 (1989)]. The dynamic renormalization group (DRG) enables one to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.

Essays on the treatment of cash flows under stochastic interest rates

Abstract This paper shows how to calculate recursively the moments of the accumulated and discounted value of cash flows when the instantaneous rates of return follow a conditional ARMA process with normally distributed innovations. We investigate various moment based approaches to approximate the distribution of the accumulated value of cash flows and we assess their performance through stochastic Monte-Carlo simulations. We discuss the potential use in insurance and especially in the context of Asset-Liability Management of pension funds.

Stochastic resonance in a suspension of magnetic dipoles under shear flow

We show that a magnetic dipole in a shear flow under the action of an oscillating magnetic field displays stochastic resonance in the linear response regime. To this end, we compute the classical quantifiers of stochastic resonance, i.e., the signal to noise ratio, the escape time distribution, and the mean first passage time. We also discuss the limitations and role of the linear response theory in its applications to the theory of stochastic resonance.

Test of the fluctuation theorem for stochastic entropy production in a nonequilibrium steady state

We derive a simple closed analytical expression for the total entropy production along a single stochastic trajectory of a Brownian particle diffusing on a periodic potential under an external constant force. By numerical simulations we compute the probability distribution functions of the entropy and satisfactorily test many of the predictions based on Seiferts integral fluctuation theorem. The results presented for this simple model clearly illustrate the practical features and implications derived from such a result of nonequilibrium statistical mechanics.

Stochastic resonance in nonpotential systems

We propose a method to analytically show the possibility for the appearance of a maximum in the signal-to-noise ratio in nonpotential systems. We apply our results to the FitzHugh-Nagumo model under a periodic external forcing, showing that the model exhibits stochastic resonance. The procedure that we follow is based on the reduction to a one-dimensional dynamics in the adiabatic limit and in the topology of the phase space of the systems under study. Its application to other nonpotential systems is also discussed.

Stochastic resonance in a system of magnetic particles

We show that a dispersion of monodomain ferromagnetic particles in a solid phase exhibits stochastic resonance when a driven linearly polarized magnetic field is applied. By using an adiabatic approach, we calculate the power spectrum, the distribution of residence times, and the mean first passage time. The behavior of these quantities is similar to the behavior of corresponding quantities in other systems where stochastic resonance has also been observed.