The Brownian motion on Aff(R) and quasi-local theorems
This paper is concerned with Random walk approximations of the
Brownian motion on the Affine group Aff(R). We are in particular interested
in the case where the innovations are discrete. In this framework, the return
probabilities of the walk have fractional exponential decay in large time, as
opposed to the polynomial one of the continuous object. We prove that in
tegrating those return probabilities on a suitable neighborhood of the origin,
the expected polynomial decay is restored. This is what we call a Quasi-local
This note states several results on the exponential functionals of the Brownian motion and their approximations by Markov chains. Starting from M.Yor, such functionals were studied in mathematical finance. At the same time, they play a significant role in different settings: the analysis of diffusions on the class of solvable Lie groups, in particular on the group of (2 X 2) upper triangular matrices, with positive diagonal elements. The discrete random walks cannot properly describe the local structure of diffusion. However, instead of the usual local limit theorem (which is not applicable) its weaker form, namely quasi-local form is given.
We find the exact optimal decision rule in the problem of testing two hypotheses about the drift of a Brownian motion in the setting of Kiefer and Weiss.
This paper contains detailed exposition of the results presented in the short communication [M. V. Zhitlukhin and A. A. Muravlev, Russian Math. Surveys, 66 (2011), pp. 1012–1013]. We consider Chernoff’s problem of sequential testing of two hypotheses about the sign of the drift of a Brownian motion under the assumption that it is normally distributed. We obtain an integral equation which characterizes the optimal decision rule and find its solution numerically.
We formulate a general Bayesian disorder detection problem, which generalizes models considered in the literature. We study properties of basic statistics, which allow us to reduce problems of quickest detection of disorder moments to optimal stopping problems. Using general results, we consider in detail a disorder problem for Brownian motion on a finite time segment.
The content of this volume is mainly based on selected talks that were given at the “International Meeting on Game Theory (ISDG12-GTM2019),” as joint meeting of “12th International ISDG Workshop” and “13th International Conference on Game Theory and Management,” held in St. Petersburg, Russia on July 03–05, 2019. The meeting was organized by St. Petersburg State University and International Society of Dynamic Games (ISDG). Every year starting from 2007, an international conference “Game Theory and Management” (GTM) has taken place at the Saint Petersburg State University. Among the plenary speakers of this conference series were the Nobel Prize winners Robert Aumann, John Nash, Reinhard Selten, Roger Myerson, Finn Kidland, Eric Maskin, and many other famous game theorists. The underlying theme of the conferences is the promotion of advanced methods for modeling the behavior that each agent (also called player) has to adopt in order to maximize his or her reward once the reward does not only depend on the individual choices of a player (or a group of players), but also on the decisions of all agents that are involved in the conflict (game).
In this paper we introduce stochastic parameters into the network game model with production and knowledge externalities. This model was proposed by V. Matveenko and A. Korolev as a generalization of the two-period Romer model. Agents differ in their productivities which have deterministic and stochastic (Wiener) components. We study the dynamics of a single agent and the dynamics of a dyad where two agents are aggregated. We derive explicit expressions for the dynamics of a single agent and dyad dynamics in the form of Brownian random processes, and qualitatively analyze the solutions of stochastic equations and systems of stochastic equations.