Topological Sorting of Finitely Near Sets

This paper introduces a natural approach in the evaluation of the nearness of sets in topological spaces. The objective is to classify levels of nearness of sets relative to each given set. The main result is a proximity measure of nearness for disjoint sets in an extremally disconnected topological space. Another result is that if A, B are nonempty semi-open sets such that \(A\ \delta \ B\), then \((\text{int }A)\ \delta \ (\text{int }B)\).     

On some properties of bitopological QHC spaces

Some important questions connected with bitopological QHC spaces are investigated. New conditions are found, under which such spaces are compact with respect to one component of the topology. It is shown that a pairwise extremal disconnected bitopological QHC space is S-closed in the sense of [4]. Theorems on the second category of a base set and on the almost Baire property of bitopological QHC spaces are proved. Also, several properties of QHC bitopological spaces are found under some known bitopological mappings. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 186–192, April–June, 2006.     

On optimal stopping of inhomogeneous standard Markov processes

The connection between the optimal stopping problems for inhomogeneous standard Markov process and the corresponding homogeneous Markov process constructed in the extended state space is established. An excessive characterization of the value-function and the limit procedure for its construction in the problem of optimal stopping of an inhomogeneous standard Markov process is given. The form of -optimal (optimal) stopping times is also found.     

Some comments on regular and normal bitopological spaces

Some properties of regular and normal bitopological spaces are established. The classes of sets inheriting the bitopological properties of regularity and normality are found. A theorem on a finite covering of pairwise normal spaces is proved. We also study the behavior of individual multivalued mappings, taking the axioms of bitopological regularity and normality into account. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1720–1724, December, 2006.     

Stochastic variational inequalities and optimal stopping: applications to the robustness of the portfolio/consumption processes

We give a new characterization of the Snell envelope of a given process as the unique solution of the stochastic variational inequality (SVI) in this article. This approach leads to several a priori estimates for the Snell envelopes and their components. The valuation for American Contingent Claims (ACC) in general financial market model is considered as an application. The robustness of the optimal portfolio/consumption processes with respect to the payoff function is established.