Concentration of dynamic risk measures in a Brownian filtration

TL;DR

This paper develops concentration inequalities for dynamic risk measures within a Brownian filtration, utilizing BSDE theory to analyze tail behavior and liquidity risk in financial losses.

Contribution

It extends concentration inequalities to time-consistent dynamic risk measures driven by Brownian motion, incorporating BSDE duality and allowing fast-growing generators.

Findings

Derived concentration inequalities for dynamic risk measures.

Characterized tail behavior of financial losses.

Linked risk measure concentration to BSDE generator properties.

Abstract

Motivated by liquidity risk in mathematical finance, D. Lacker introduced concentration inequalities for risk measures, i.e. upper bounds on the \emph{liquidity risk profile} of a financial loss. We derive these inequalities in the case of time-consistent dynamic risk measures when the filtration is assumed to carry a Brownian motion. The theory of backward stochastic differential equations (BSDEs) and their dual formulation plays a crucial role in our analysis. Natural by-products of concentration of risk measures are a description of the tail behavior of the financial loss and transport-type inequalities in terms of the generator of the BSDE, which in the present case can grow arbitrarily fast.