On Misapplications of Diffusion Approximations in Birth and Death Processes of Noisy Evolution

Birth and death processes with a finite number of states are used in modeling different kinds of noisy learning processes in economics. To analyze the long run properties one looks a the corresponding stationary distribution. When the number of states is large, the stationary distribution becomes bulky and difficult to analyze. To simplify the analysis in such a situation and hence to make the long run properties of the learning process more transparent, a diffusion approximation has been suggested. Unfortunately, quite often such such approximation is not correctly done. Why this happens and how the situation can be fixed is discussed in this note

Strong Convergence of Stochastic Approximation Without Lyapunov Functions

We prove convergence with probability one of a multivariate Markov stochastic approximation procedure of the Robbins-Monro type with several roots. The argument exploits convergence of the corresponding system of ordinary differential equations to its stationary points. If the points are either linearly stable or linearly unstable, we prove convergence with probability 1 of the procedure to a random vector whose distribution concentrates on the set of stable stationary points. This generalizes for procedures with several roots the approach suggested by L. Ljung for processes with a single root. Along with stochastic approximation processes as such, the result can be applied to generalized urn schemes and stochastic models of technological and economic dynamics based on them, in particular, evolutionary games with incomplete information

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

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

Non-standard Limit Theorems for Stochastic Approximation Procedures and Their Applications for Urn Schemes

A limit theorem for the Robbins-Monro stochastic approximation procedure is proved in the case of a non-smooth regression function. Using this result a conditional limit theorem is given for the case when the regression function has several stable roots. The first result shows that the rate of convergence for the stochastic approximation-type procedures (including Monte-Carlo optimization algorithms and adaptive processes of growth being modelled by the generalized urn scheme) decreases as the smoothness increases. The second result demonstrates that in the case of several stable roots, there is no convergence rate for the procedure as whole, but for each of stable roots there exists its specific rate of convergence. The latter allows to derive several conceptual results for applied problems in biology, physical chemistry and economics which can be described by the generalized urn scheme

Price Expectations, Cobwebs and Stability

There is given a market for several perishable goods, supplied under technological randomness and price uncertainty. We study whether and how producers eventually may learn rational price expectations. The model is of cobweb type. Its dynamics fit standard forms of stochastic approximation. Relying upon quite weak and natural assumptions we prove new convergence results

The Method of Generalized Urn Scheme in the Analysis of Technological and Economic Systems

Adaptive (path dependent) processes of growth modeled by urn schemes are important for several fields of applications: biology, physics, chemistry, economics. In this paper we present a review of studies that have been done in the technological dynamics by means of the urn schemes. Also several new macroeconomic models of technological dynamics are analysed by the same machinery and its new modification allowing to tackle non-homogeneity of the face space. We demonstrate the phenomena of multiple equilibria, different convergence rates for different limit patterns, locally positive and locally negative feedbacks, limit behavior associated with non-homogeneity of economic environment where producers (firms) are operating