Multivariate risk measures in the non-convex setting

Abstract

The family of admissible positions in a transaction costs model is a random closed set, which is convex in case of proportional transaction costs. However, the convexity fails, e.g. in case of fixed transaction costs or when only a finite number of transfers are possible. The paper presents an approach to measure risks of such positions based on the idea of considering all selections of the portfolio and checking if one of them is acceptable. Properties and basic examples of risk measures of non-convex portfolios are presented.

Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations

“…The notion of a random set gives meaning to random objects X whose realizations X(ω), ω ∈ Ω, take values as subsets of some space X ; see [2]. These objects have an important role in mathematical finance and stochastic optimization; see e.g., [6][7][8][9][10]. By considering the Boolean valued model associated with the underlying probability space, we show that a random set corresponds to a Borel set in this model.…”