Approximation of stochastic differential equations driven by alpha-stable Levy motion

In this paper we present a result on convergence of approximate solutions of stochastic differential equations involving integrals with respect to alpha-stable Levy motion. We prove an appropriate weak limit theorem, which does not follow from known results on stability properties of stochastic differential equations driven by semimartingales. It assures convergence in law in the Skorokhod topology of sequences of approximate solutions and justifies discrete time schemes applied in computer simulations. An example is included in order to demonstrate that stochastic differential equations with jumps are of interest in constructions of models for various problems arising in science and engineering, often providing better description of real life phenomena than their Gaussian counterparts. In order to demonstrate the usefulness of our approach, we present computer simulations of a continuous time alpha-stable model of cumulative gain in the Duffie–Harrison option pricing framework.Stable distribution, Simulation, Stochastic differential equation (SDE), Option pricing

The impact of stochastic lead times on the bullwhip effect – An empirical insight

In this article, we review the research state of the bullwhip effect in supply chains with stochastic lead times. We analyze problems arising in a supply chain when lead times are not deterministic. Using real data from a supply chain, we confirm that lead times are stochastic and can be modeled by a sequence of independent identically distributed random variables. This underlines the need to further study supply chains with stochastic lead times and model the behavior of such chains

The impact of stochastic lead times on the bullwhip effect–a theoretical insight

In this article, we analyze the models quantifying the bullwhip effect in supply chains with stochastic lead times and find advantages and disadvantages of their approaches to the bullwhip problem. Moreover, using computer simulation, we find interesting insights into the bullwhip behavior for a particular instance of a multi-echelon supply chain with constant customer demands and random lead times. We confirm the recent finding of Michna and Nielsen that under certain circumstances lead time signal processing is by itself a fundamental cause of bullwhip effect just like demand-signal processing is. The simulation also shows that in this supply chain the delay parameter of demand forecasting smooths the bullwhip effect at the manufacturer level much faster than the delay parameter of lead time forecasting. Additionally, in the supply chain with random demands, the reverse behavior is observed, that is, the delay parameter of lead time forecasting smooths bullwhip effect at the retailer stage much faster than the delay parameter of demand forecasting. At the manufacturer level, the delay parameter of demand forecasting and the delay parameter of lead time forecasting dampen the effect with a similar strength