Some stochastic inventory models with deterministic variable lead time

In this paper we determine optimal reduction in the procurement lead time duration for some stochastic inventory models, jointly with the optimal ordering decisions. The models are developed with complete and partial information about the lead time demand distribution. The stochastic models analyzed in this paper are the classical continuous and periodic review models with a mixture of backorders and lost sales and the base stock model. For each of these models, we provide sufficient conditions for the uniqueness of the optimal operating policy. We also develop algorithms for solving these models and provide illustrative numerical examples.     

Random and stochastic disturbances on the input flow in chemostat models with wall growth

In this paper, we analyze a chemostat model with wall growth where the input flow is perturbed by two different stochastic processes: the well-known standard Wiener process, which leads into several drawbacks from the biological point of view, and a suitable Ornstein-Uhlenbeck process depending on some parameters which allow us to control the noise to be bounded inside some interval that can be fixed previously by practitioners. Thanks to this last approach, which has already proved to be very realistic when modeling other simplest chemostat models, it will be possible to prove the persistence and coexistence of the species in the model without needing the theory of random dynamical systems and pullback attractors needed when dealing with the Wiener process. This is an advantage since the theoretical framework in this paper is much less complicated and provides us much more information than the other.     

Efficient deterministic numerical simulation of stochastic asset-liability management models in life insurance

New regulations, stronger competitions and more volatile capital markets have increased the demand for stochastic asset-liability management (ALM) models for insurance companies in recent years. The numerical simulation of such models is usually performed by Monte Carlo methods which suffer from a slow and erratic convergence, though. As alternatives to Monte Carlo simulation, we propose and investigate in this article the use of deterministic integration schemes, such as quasi-Monte Carlo and sparse grid quadrature methods. Numerical experiments with different ALM models for portfolios of participating life insurance products demonstrate that these deterministic methods often converge faster, are less erratic and produce more accurate results than Monte Carlo simulation even for small sample sizes and complex models if the methods are combined with adaptivity and dimension reduction techniques. In addition, we show by an analysis of variance (ANOVA) that ALM problems are often of very low effective dimension which provides a theoretical explanation for the success of the deterministic quadrature methods.     

Contribution of calcium fluxes to astrocyte spontaneous calcium oscillations in deterministic and stochastic models

Spontaneous Ca²⁺ oscillation is a special kind of astrocyte signal that is crucial in several brain functions. However, the underlying mechanism of its initiation is still unclear. Experimental investigations reveal quite different results of the contribution of Ca²⁺ influxes from the extracellular space. We developed a new biophysical model, which includes two kinds of extracellular Ca²⁺ influx: through voltage gated Ca²⁺ channels and through capacitative Ca²⁺ entry. Besides, two stochastic models are constructed to discuss the influence of channel noise in voltage gated influx and stochastic IP₃ receptor dynamics on the contribution of Ca²⁺ fluxes. Simulation results reveal that the contribution of a given flux can be considerably strengthened or weakened by special cellular states, such as the change of membrane potential, inter-flux feedback; the special cellular condition (e.g. endoplasmic reticulum depletion, other flux inhibition). In stochastic environment, the contribution of fluxes may be strongly altered and the respond of different fluxes varies qualitatively. Our results may help to understand the different results obtained in experiments.     

Adaptation in a stochastic multi-resources chemostat model

We are interested in modeling the Darwinian evolution resulting from the interplay of phenotypic variation and natural selection through ecological interactions, in the specific scales of the biological framework of adaptive dynamics. Adaptive dynamics so far has been put on a rigorous footing only for direct competition models (Lotka–Volterra models) involving a competition kernel which describes the competition pressure from one individual to another one. We extend this to a multi-resources chemostat model, where the competition between individuals results from the sharing of several resources which have their own dynamics. Starting from a stochastic birth and death process model, we prove that, when advantageous mutations are rare, the population behaves on the mutational time scale as a jump process moving between equilibrium states (the polymorphic evolution sequence of the adaptive dynamics literature). An essential technical ingredient is the study of the long time behavior of a chemostat multi-resources dynamical system. In the small mutational steps limit this process in turn gives rise to a differential equation in phenotype space called canonical equation of adaptive dynamics. From this canonical equation and still assuming small mutation steps, we prove a rigorous characterization of the evolutionary branching points.     

Evolution program for deterministic and stochastic optimizations

This paper presents an evolution program for deterministic and stochastic optimizations. To overcome premature convergence and stalling of the solution, we suggest an exponential-fitness scaling scheme. To avoid the chromosomes jamming into a corner, we introduce mutation-1 which mutates the chromosomes in a free direction. To improve the chromosomes, we introduce mutation-2 which mutates the chromosomes in the gradient direction or its negative, according to the kind of problem. Monte Carlo simulation will be employed to solve the multiple integral which is the most difficult task in the stochastic optimization. Finally, some numerical examples are discussed.     

From deterministic to stochastic surrender risk models: Impact of correlation crises on economic capital

In this paper we raise the matter of considering a stochastic model of the surrender rate instead of the classical S-shaped deterministic curve (in function of the spread), still used in almost all insurance companies. For extreme scenarios, due to the lack of data, it could be tempting to assume that surrenders are conditionally independent with respect to a S-curve disturbance. However, we explain why this conditional independence between policyholders decisions, which has the advantage to be the simplest assumption, looks particularly maladaptive when the spread increases. Indeed the correlation between policyholders decisions is most likely to increase in this situation. We suggest and develop a simple model which integrates those phenomena. With stochastic orders it is possible to compare it to the conditional independence approach qualitatively. In a partially internal Solvency II model, we quantify the impact of the correlation phenomenon on a real life portfolio for a global risk management strategy.     

Distributed modelling of deterministic and stochastic population dynamics

In many problems of practical purpose, one is interested not only in the study of the total population of the overall system, but also in the distribution of the population depending upon some characteristic parameters. This necessity is patent in biological systems defined on a wide range of distributed features, but it is also in order when one tries to analyze social systems by means of population models. As a matter of fact, with this objective in mind, we deal with infinite species population models.This study discusses this question, proposes a distributed version for the logistic equation, and examines the existence of stationary solutions. The problem of the influence of environmental noise on the dynamics of the distributed population is considered, and a model is proposed. A detailed analysis is carried out around the stationary solution via a linearization technique, and the covariance equation is derived.     

Recurrence and strong stochastic persistence of a stochastic single-species model

A stochastic generalized logistic model is considered in this paper. The condition of the existence of its stationary distribution is generalized. Recurrence and strong stochastic persistence are obtained. Finally some numerical simulations are carried out to support our results.     

Multigroup deterministic and stochastic SEIRI epidemic models with nonlinear incidence rates and distributed delays: A stability analysis

In this paper, we investigate the dynamics of a multigroup disease propagation model with distributed delays and nonlinear incidence rates, which accounts for the relapse of recovered individuals. The main concern is the stability of the equilibria, sufficient conditions for global stability being obtained by applying Lyapunov‐LaSalle invariance principle and using Lyapunov functionals, which are constructed using their single‐group counterparts. The situation in which the deterministic model is subject to perturbations of white noise type is also investigated from a stability viewpoint.     

Nonnormal deterministic equivalents and a transformation in stochastic mathematical programming

This paper considers random variables of the continuous type in a stochastic programming problem and presents (1) a general approach to the development of deterministic equivalents of constraints to be satisfied within certain probability limits, and (2) a deterministic transformation of a stochastic programming problem with random variables in the objective function. Deterministic equivalents are developed for constraints containing uniform random variables, but the approach used can be applied to other types of continuous random variables, as well. When the random variables appear in the objective function, a deterministic transformation of the stochastic programming problem is obtained to yield a closed-form solution without resort to a Monte Carlo computer simulation. Extension of this approach to stochastic problems with discrete random variables and integer decision variables is discussed briefly. A numerical example is presented.     

Cooperation of deterministic and stochastic mechanisms resulting in the intermittent behavior

Intermittent behavior near the boundary of chaotic phase synchronization in the presence of noise (when deterministic and stochastic mechanisms resulting in intermittency take place simultaneously) is studied. The noise of small intensity is shown to do not affect on the characteristics of intermittency whereas the noise of large amplitude induces new effects near the boundary of the synchronous regime. In the first case the eyelet intermittency takes place near the boundary of the synchronous regime, in the second one the ring intermittency or coexistence of both types of intermittency is realized. Main results are illustrated using the example of two unidirectionally coupled Rössler systems. Similar effects are shown to be observed in coupled spatially distributed Pierce beam–plasma systems.     

Invariance of stochastic control systems with deterministic arguments

We prove that a closed set K of a finite-dimensional space is invariant under the stochastic control system

dX=b(X,v(t))dt+σ(X,v(t))dW(t),v(t)∈U,     

Confidence domain in the stochastic competition chemostat model with feedback control

This paper studies a stochastically forced chemostat model with feedback control in which two organisms compete for a single growth-limiting substrate. In the deterministic counterpart, previous researches show that the coexistence of two competing organisms may be achieved as a stable positive equilibrium or a stable positive periodic solution by different feedback schedules. In the stochastic case, based on the stochastic sensitivity function technique,we construct the confidence domains for different feedback schedules which allow us to find the configurational arrangements of the stochastic attractors and analyze the dispersion of the random states of the stochastic model.     

Permanent coexistence in chemostat models with delayed feedback control

Feedback control for chemostat model first appears in [P. De Leenher, H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48–70] where the authors consider a dilution rate as a feedback control variable, keeping the input nutrient concentration to be constant. They showed that the coexistence of two organisms can be achieved under single nutrient source. In this paper, we improve the model in consideration of a time delay arising from controlling a dilution rate. It is shown that coexistence of two organisms can also be achieved in the improved model. The coexistence is mathematically insisted by solving permanence problem. We prove the problem by using the theory of “average Liapunov function”.     

Combining stochastic and deterministic approaches within high efficiency molecular simulations

Generalized Shadow Hybrid Monte Carlo (GSHMC) is a method for molecular simulations that rigorously alternates Monte Carlo sampling from a canonical ensemble with integration of trajectories using Molecular Dynamics (MD). While conventional hybrid Monte Carlo methods completely re-sample particle’s velocities between MD trajectories, our method suggests a partial velocity update procedure which keeps a part of the dynamic information throughout the simulation. We use shadow (modified) Hamiltonians, the asymptotic expansions in powers of the discretization parameter corresponding to timestep, which are conserved by symplectic integrators to higher accuracy than true Hamiltonians. We present the implementation of this method into the highly efficient MD code GROMACS and demonstrate its performance and accuracy on computationally expensive systems like proteins in comparison with the molecular dynamics techniques already available in GROMACS. We take advantage of the state-of-the-art algorithms adopted in the code, leading to an optimal implementation of the method. Our implementation introduces virtually no overhead and can accurately recreate complex biological processes, including rare event dynamics, saving much computational time compared with the conventional simulation methods.