Probability and its applications in physics and finance

Abstract

(I) We shall study properties of universal portfolios, a concept introduced by T.M. Cover for processes of independent returns and later studied in the framework of the Information Theory. More precisely, we shall consider a similar problem when the stochastic market is represented by a Markov chain or by its generalization -- a process of infinite memory, albeit with a controlled decay. The class B of considered portfolios includes functions of a return sample over a past history. The main result is that a sample-wise optimization, over portfolios within this class, leads, with probability one, to a certain wealth rate W. The latter can be obtained by maximizing, again over class B, the largest eigen-value of a Perron-Frobenius operator naturally related to a given portfolio. (II) We shall construct a model for the trading via a Limit Order Book. This model represents a queuing system and will be studied by asymptotical methods of the theory of Markov chains focusing on properties of recurrence, and transience and convergence to the stationary distribution. These properties will be interpreted in terms of the real world market giving insights into the market stability when trading is controlled by the Limit Order Book. (III) We shall initiate the study of the aspects related to the High Frequency Data Trading. The study will include the following stages: creation of a reliable data base of the asks and bids that occurred in a real market for a sufficiently long period of time, statistical analysis of this data-base, construction of trading algorithms based on stochastic prediction models and testing these algorithms on the time series from the data-base. (IV) There will be composed and submitted -- to journals form the field of Stochastic Processes and Applications -- two manuscripts presenting our achievements in studying universal portfolios and the Limit Order Book dynamics. Two monographs will be submitted for publication: “Multi-Scale Analysis for Random Quantum Systems with Interaction” and “Information Theory and Coding by Example”. (V) A work will begin on a text-book containing problems and solutions that help students to prepare for admission exams to the Master Program in Statistics. (VI) There will be created one graduate and one post-graduate course and the existing courses will be completed with exercises and solutions similar to those form the series of books “Probability and Statistics by Example”. There will be offered a series of seminars presenting recent results about Random Quantum Systems with Interaction. (VII) Students and researches will be encouraged to join research in these directions. (AU)