On Dynamic Deviation Measures and Continuous-Time Portfolio Optimisation

TL;DR

This paper introduces dynamic deviation measures as a time-consistent way to quantify risk, characterizes them via backward SDEs, and applies them to formulate and solve a portfolio optimization problem with explicit equilibrium strategies.

Contribution

It develops a novel dynamic deviation measure framework, characterizes it through backward SDEs, and applies it to derive equilibrium strategies in a jump-diffusion portfolio optimization setting.

Findings

Dynamic deviation measures satisfy a generalized conditional variance formula.

They are characterized as solutions to certain backward SDEs.

A linear equilibrium strategy is derived in the jump-diffusion model.

Abstract

In this paper we propose the notion of dynamic deviation measure, as a dynamic time-consistent extension of the (static) notion of deviation measure. To achieve time-consistency we require that a dynamic deviation measures satisfies a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of backward SDEs. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively $m$-stable dual sets. Using this notion of dynamic deviation measure we formulate a dynamic mean-deviation portfolio optimisation problem in a jump-diffusion setting and identify a subgame-perfect Nash equilibrium strategy that is linear as function of wealth by deriving and solving an associated extended HJB equation.