Renewal theory began development in the early 1940s, as the need for it in the industrial engineering sub-discipline operations research had risen. In time, the theory found applications in many stochastic processes. In this thesis I investigated the effect of seasonal effects on Poisson and non-Poisson renewal processes in the form of perturbations. It was determined that the statistical analysis methods developed at UNT Center for Nonlinear Science can be used to detect the effects of seasonality on the data obtained from Poisson/non-Poisson renewal systems. It is proved that a perturbed Poisson process can serve as a paradigmatic model for a case where seasonality is correlated to the noise and that diffusion entropy method can be utilized in revealing this relation. A renewal model making a connection with the stochastic resonance phenomena is used to analyze a previous neurological experiment, and it was shown that under the effect of a nonlinear perturbation, a non-Poisson system statistics may make a transition and end up in the of Poisson basin of statistics. I determine that nonlinear perturbation of the power index for a complex system will lead to a change in the complexity characteristics of the system, i.e., the system will reach a new form of complexity.

In dieser These untersuche ich die Wirkungsmechanismen stochastischer Volatilität in einem neoklassischem Wachstumsmodel mit Arbeitsmarktfriktionen, Anpassungskosten, variabler Kapitalintensität und kurzfristigen Einkommenseffekt. Nominale Rigiditäten werden in diesem Modell nicht betrachtet. Im gegebenen allgemeinen Gleichgewicht generiert stochastische Volatilität Konjunkturzyklen in den wesentlichen makroökonomischen Aggregaten. Dies ist das Resultat eines vorbeugenden Sparmotives der risiko-aversen Haushalte, dennoch sind die quantitativen Effekten auf die unbedingten Momente der makroökonomischen Aggregate vernachlässigbar. / In this thesis I examine the propagation mechanism of stochastic volatility in a neoclassical growth model that incorporates labor market search, adjustment cost to investment, variable capital utilization and a weak short-run wealth effect, but no nominal frictions such as sticky wage and price. In this general equilibrium environment, stochastic volatility generates business cycle fluctuations in major macroeconomic aggregates due to the precautionary motive of risk-averse agents, yet it has no significant effects on these major aggregates as suggested by the numerical analysis of the model.

In this thesis we investigate a model of nonlinearly perturbed continuous-time renewal equation. Some characteristics of the renewal equation are assumed to have non-polynomial perturbations, more specifically they can be expanded with respect to a non-polynomial asymptotic scale. The main result of the present study is exponential asymptotic expansions for the solution of the perturbed renewal equation. These asymptotic results are also applied to various applied probability models like perturbed risk processes, perturbed M/G/1 queues and perturbed dam/storage processes. The thesis is based on five papers where the model described above is successively studied.