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## Bidding Games with Several Risky Assets

We investigate a model of one-stage bidding between two differently informed stockmarket agents for a risky asset (share). The random liquidation price of a share may take two values: the integer positive m with probability p and 0 with probability 1−p. Player 1 (insider) is informed about the price, Player 2 is not. Both players know the probability p. Player 2 knows that Player 1 is an insider. Both players propose simultaneously their bids. The player who posts the larger bid buys one share from his opponent for this price. Any integer bids are admissible. The model is reduced to a zero-sum game with lack of information on one side. We construct the solution of this game for any p and m: we find the optimal strategies of both players and describe recurrent mechanism for calculating the game value. The results are illustrated by means of computer simulation.

This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation principle for random walks was first established by the author in 1967, under the restrictive condition that the distribution tails decay faster than exponentially. (A close assertion was proved by S.R.S. Varadhan in 1966, but only in a rather special case.) Since then, the principle has always been treated in the literature only under this condition. Recently, the author jointly with A.A. Mogul'skii removed this restriction, finding a natural metric for which the large deviation principle for random walks holds without any conditions. This new version is presented in the book, as well as a new approach to studying large deviations in boundary crossing problems. Many results presented in the book, obtained by the author himself or jointly with co-authors, are appearing in a monograph for the first time.

We study Bertrand competition models with incomplete information about rivals' costs, where uncertainty is given by independent identically distributed random variables. It turns out that Bayesian Nash equilibria of the simplest of these games are described as Cournot prices. Then we discuss general conditions when Cournot prices give Bayesian Nash equilibria for Bertrand games with incomplete information about rivals' costs.

In the first part of the paper we consider a "random flight" process in \(R^d\) and obtain the weak limits under different transformations of the Poissonian switching times. In the second part we construct diffusion approximations for this process and investigate their accuracy. To prove the weak convergence result we use the approach of Stroock and Varadhan (1979). We consider more general model which may be called "random walk over ellipsoids in \(R^d\)". For this model we establish the Edgeworth type expansion. The main tool in this part is the parametrix method (Konakov (2012), Konakov and Mammen (2009)).

The paper investigates the optimization methods used by an investor working on the Russian stork market. The efficient sets, corresponding for the two different states of the market (with «moderate» and «rapid» growth rates), are build. The paper denies the necessity of the «deep» diversification of the portfolio on the Russian stork market. Some recommendations concerning the investment portfolio management are formulated.

We consider the problem of selecting a predetermined number of objects from a given finite set. It is assumed that the preferences of the decisionmaker on this set are only partially known. Our solution approach is based on the notions of optimal and non-dominated subsets. The properties of such subsets and the objects they contain are investigated. The implementation of the developed approach is discussed and illustrated by various examples.