On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

TL;DR

This paper introduces continuous-time dynamic spectral risk measures (DSR), establishing their theoretical foundation via a limit theorem and demonstrating their application in optimal portfolio allocation.

Contribution

It defines DSR using backward stochastic differential equations and proves they are limits of iterated spectral risk-measures, extending spectral risk measures to continuous time.

Findings

DSRs are strong time-consistent extensions of iterated spectral risk-measures.

A functional limit theorem connects DSRs with lattice-random walk-based iterated measures.

Application to dynamic portfolio optimization demonstrates practical relevance.

Abstract

In this paper we propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk-measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk-measures, which are obtained by iterating a given spectral risk-measure (such as Expected Shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk-measures driven by lattice-random walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.