Lectures on random sets and their applications in economics and finance

This course introduces main concepts from the theory of random sets with emphasis on applications in economics and finance: most importantly inference for partially identified models and transaction costs modelling. The main mathematical ideas introduced in this course are that of a random closed set, its distribution and main analytical tools to handle it, selections and expectations of random sets, laws of large numbers and limit theorems, and set-valued stochastic processes. Theoretical material will be presented along with describing the following applications. • Partially identified models appearing if the available data do not suffice to uniquely identify the parameter of interest, even if the sample size grows. Possible reasons for this are interval responses in regression models or multiple equilibria in games. Using random sets, it is possible to come up with an adequate mathematical framework that makes it possible to unify a number of special cases and come up with new results. • In finance it is possible to represent the range of prices (which are always non-unique in case of transaction costs) as random sets. In the univariate case, this set is a segment with end-points being bid and ask prices. The no-arbitrage property of the dynamic model with discrete time is closely related to the existence of martingales that evolve inside the set-valued process. 1 Distributions of random sets