On the upper and lower semicontinuity of the Aumann Integral

Includes bibliographical references (p. 21-22)

Noncooperative Oligopoly in Markets with a Continuum of Traders

In this paper, we study three prototypical models of noncooperative oligopoly in markets with a continuum of traders : the model of Cournot-Walras equilibrium of Codognato and Gabszewicz (1991), the model of Cournot-Nash equilibrium of Lloyd S. Shapley, and the model of Cournot-Walras equilibrium of Busetto et al. (2008). We argue that these models are all distinct and only the Shapley's model with a continuum of traders and atoms gives an endogenous explanation of the perfectly and imperfectly competitive behavior of agents in a one-stage setting. For this model, we prove a theorem of existence of a Cournot-Nash equilibrium.

On an optimal consumption problem for p-integrable consumption plans

A generalization is presented of the existence results for an optimal consumption problem of Aumann and Perles [4] and Cox and Huang [10]. In addition, we present avery general optimality principle

Brownian motion : a graduate course in stochastic processes

"June 1985." This report constitutes the first three chapters of a book to be published by Springer-Verlag.Includes bibliography.ARO Grant DAAG-29-84-K-005by Ioannis Karatzas and Steven E. Shreve

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Stochastic Approximation and Optimization for Markov Chains

We study the convergence properties of the projected stochasticapproximation (SA) algorithm which may be used to find the root of an unknown steady state function of a parameterized family of Markov chains. The analysis is based on the ODE Method and we develop a set of application-oriented conditions which imply almost sure convergence and are verifiable in terms of typically available model data. Specific results are obtained for geometrically ergodic Markov chains satisfying a uniform Foster-Lyapunov drift inequality.Stochastic optimization is a direct application of the above root finding problem if the SA is driven by a gradient estimate of steady state performance. We study the convergence properties of an SA driven by agradient estimator which observes an increasing number of samples from the Markov chain at each step of the SA's recursion. To show almost sure convergence to the optimizer, a framework of verifiable conditions is introduced which builds on the general SA conditions proposed for the root finding problem.We also consider a difficulty sometimes encountered in applicationswhen selecting the set used in the projection operator of the SA algorithm.Suppose there exists a well-behaved positive recurrent region of the state process parameter space where the convergence conditions are satisfied; this being the ideal set to project on. Unfortunately, the boundaries of this projection set are not known a priori when implementing the SA. Therefore, we consider the convergence properties when the projection set is chosen to include regions outside the well-behaved region. Specifically, we consider an SA applied to an M/M/1 which adjusts the service rate parameter when the projection set includes parameters that cause the queue to be transient.Finally, we consider an alternative SA where the recursion is driven by a sample average of observations. We develop conditions implying convergence for this algorithm which are based on a uniform large deviation upper bound and we present specialized conditions implyingthis property for finite state Markov chains

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute

Random walks in random media

In this thesis we study three RWIRE models. A RWIRE is simply a random walk where the transition rates are themselves random variables. The first model studied is a unit jump non directed RWIRE in discrete and continuous time. Results are presented for the "speed" of the RWIRE in discrete time. It is also shown that the random environment slows down the random walk in the discrete time setting. The second model studied is a unit jump directed RWIRE in continuous time. Results for the "speed", the slow approach to infinity, and the limiting distribution are given for this model. It is also shown, as m the first model, that the random environment slows down the random walk. The final model is a multiple jump directed RWIRE in both discrete and continuous time The discrete model findings include the "speed" and the limiting distribution. For the continuous case the “speed" is derived