Limit Theorems for Proportions of Balls in a Generalized Urn Scheme

In this paper the authors continue to study the process of growth modeled by urn schemes containing balls of different colors. The rate of convergence for proportions of balls to the limit state is investigated. It is shown that Gaussian as well as non-Gaussian Markov random processes may describe the asymptotic behavior

Generalized Urn Schemes and Technological Dynamics

Adaptive (path dependent) processes of growth modeled by urn schemes are important for several fields of applications: biology, physics, chemistry, economics. In this paper several macroeconomic models of technological dynamics are studied by the means of adaptive processes of growth. One of the models tackles the case when there is a separation within the pool of adopters which can be interpreted as the outcome of adaptive learning on the features of the new technologies by imperfectly informed agents. Others deal with dependence of final market shares of two technologies on the pricing policies of the firms which produce them. The stochasticity of the processes is caused by some mixed strategies used by the adopters or/and imperfectness of the information which they posses. To study these conceptual problems some modifications of the basic results concerning the generalized urn scheme are given

Non-Linear Urn Processes: Asymptotic Behavior and Applications

Adaptive (path dependent) processes of growth modeled by urn schemes are important for several fields of applications: biology, physics, chemistry, economics. In this paper the authors continue their previous investigations of generalized urn schemes with balls of different colors and path-dependent increments. They consider processes with random additions of balls at each time. These processes are similar to the branching processes, however the classical theory of branching processes does not consider the case when the composition of added balls depends on the state of the process. These state dependent processes are very important for the applications and are considered in this paper. The asymptotic behavior of the proportions of balls of each color in the total population is studied. It appears that these proportions can be expressed through the so-called urn functions which define the probabilities of adding the new ball depending on the current composition of urn population. The dynamics of the urn scheme is written in terms of a stochastic finite differential equation. The trajectories defined by this equation are attracted to the urn functions fixed points. The techniques used to obtain these results have much in common with convergence analysis of the stochastic approximation type procedures for solving systems of nonlinear equations with discontinuous functions and stochastic quasi gradient procedures of stochastic optimization. In the general case the convergence results are formulated through nondifferentiable Lyapunov functions

Spatial Modeling of Resource Allocation and Agriculture Production under Environmental Risk and Uncertainty

Within the limits of the physical production potential, effective land performance is largely determined by anthropogenic factors, e.g., the availability of infra-structure, market access and a complex interaction of behavioral, socio-economic, cultural and technological conditions. Any discussion of interactions with the environment must come to grips with uncertainties and resulting risks. The paper outlines a possible approach to modeling agricultural production in a spatial setting under environmental risk and uncertainty. The methodology proposes the distinction of "compartments", i.e., geo-referenced collections of homogenous land units which are coherent in terms of natural resources and economic production conditions. The compartments interact through commodity markets and "trade" of mobile resources, compete for allocation of limited public resources, and are jointly affected by government policies, regulations and other regional constraints. The proposed model incorporates two types of decisions: strategic ex-ante decisions and flexible adaptive ex-post decisions. The notion of strategic decisions is especially relevant when dealing with the risks of virtually irreversible impacts, such as placing a dam, clearing rainforests for agricultural purposes, or diverting agriculture land to urban and industrial uses. Adaptive ex-post decisions allow to correct the strategic decisions relying on the ability to learn from experience and to adapt to observed situations. From a formal point of view, this process can be interpreted as a stochastic decomposition procedure, integrating individual agents through market clearing conditions and allocation of public resources in order to find strategic decisions that are robust and most effective in an uncertain setting

Nonlinear Adaptive Processes of Growth with General Increments: Attainable and Unattainable Components of Terminal Set

A local asymptotic theory of adaptive processes of growth with general increments is developed for the case when a terminal set consists of more than one connected component. The notions of an attainable and unattainable component are introduced. Sufficient conditions for attainability and unattainability are derived. The limit theorems are applied in the investigation of the rate of convergence to singleton stable components. The relation between the obtained results and the study of asymptotic properties of stochastic quasi-gradient algorithms in non-convex multiextremum problems is discussed. Specifically, the developed approach is used to explore the limit behavior of iterations in the Fabian modification of the Kiefer-Wolfowitz algorithm

Convex Optimization by Radial Search

A convex nonsmooth optimization problem is replaced by a sequence of line search problems along recursively updated rays. Convergence of the method is proved and applications to linear inequalities, constraint aggregation and saddle point seeking indicated

Constrained Optimization of Discontinuous Systems

In this paper we extend the results of Ermoliev, Norkin and Wets [8] and Ermoliev and Norkin [7] to the case of constrained discontinuous optimization problems. In contrast to [7] the attention is concentrated on the proof of general optimality conditions for problems with nonconvex feasible sets. Easily implementable random search technique is proposed

An Approach to the Simulation of International Oil Trade

This paper describes the Simulation Model for International Oil Trade (SMIOT) developed at IIASA in 1979. The model is designed to calculate balanced states for the oil market taking into account the conflicts of interests between exporters and between importers. One of the main objectives of this report is to discuss the philosophy behind this approach; particular attention is also paid to the gaming algorithm used to simulate the process of trade