Excursions of Markov Processes: A Large Deviation Approach

This paper is dedicated to the memory of Nikita Dmitrievna Vve-denskaya. These authors had a privilege to collaborate with Vvedenskaya during the last period of her life. The paper considers a buyer-seller interaction model based on a two-dimensional process. The first coordinate is the number of buyers and the second one is the number of sellers. The rates of the income and outcome of sellers and buyers depend on their numbers polynomially at a given time. To make the process ergodic we require that the intensity of outcome has a higher degree. The problem associated with positive excursions of the trajectories of such processes is being solved. A local principle of large deviations for these excursions is obtained.