[Paper Review] On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration

Julio Backhoff‐Veraguas, Ludovic Tangpi|arXiv (Cornell University)|Aug 26, 2016

Risk and Portfolio Optimization36 references3 citations

TL;DR

This paper establishes a dynamic representation for time-inconsistent risk measures—specifically Optimized Certainty Equivalent (OCE) measures—using PDE techniques in a Brownian filtration. By extending the state space and applying stochastic control with viscosity solutions, it derives a dynamic programming principle and a non-linear PDE (HJB-type) that enables dynamic computation of OCE risk measures, even when they lack time consistency, covering cases like conditional value-at-risk and monotone mean-variance.

ABSTRACT

It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures do not admit a time-consistent representation. In this work we show that in a Brownian filtration the "Optimized Certainty Equivalent" risk measures of Ben-Tal and Teboulle can be computed through PDE techniques, i.e. dynamically. This can be seen as a substitute of sorts whenever they lack time consistency, and covers the cases of conditional value-at-risk and monotone mean-variance. Our method consists of focusing on the convex dual representation, which suggests extending the state space. With this we can obtain a dynamic programming principle and use stochastic control techniques, along with the theory of viscosity solutions, which we must adapt to cover the present singular situation.

Motivation & Objective

• To address the lack of time consistency in widely used risk measures such as conditional value-at-risk and monotone mean-variance.
• To develop a dynamic representation for time-inconsistent risk measures using PDE techniques in a Brownian filtration.
• To extend the state space to recover a dynamic programming principle for OCE risk measures.
• To adapt viscosity solution theory to handle singular Hamiltonians arising in unbounded stochastic control problems.
• To provide a substitute for time consistency by enabling dynamic computation of otherwise non-time-consistent risk measures.

Proposed method

• Focus on the convex dual representation of OCE risk measures to suggest an extended state space for dynamic programming.
• Derive a dynamic programming principle in the enlarged state space to model the risk measure dynamically.
• Use stochastic control techniques and adapt viscosity solution theory to handle singular Hamiltonians in unbounded control problems.
• Establish a connection between the dynamic risk measure and a non-linear PDE (HJB equation) via the dynamic programming principle.
• Apply Itô’s formula and martingale representation to construct the control process β in terms of the value function’s gradient.
• Use mollification and approximation arguments to reduce the general case to smooth, bounded processes, enabling rigorous PDE analysis.

Experimental results

Research questions

• RQ1 Can time-inconsistent risk measures such as conditional value-at-risk and monotone mean-variance be represented dynamically in a Brownian filtration?
• RQ2 Is it possible to derive a dynamic programming principle for OCE risk measures despite their lack of time consistency?
• RQ3 Can viscosity solution techniques be adapted to handle singular Hamiltonians arising in unbounded stochastic control problems related to risk measures?
• RQ4 How can the state space be extended to recover dynamic representation for law-invariant, time-inconsistent risk measures?
• RQ5 To what extent does the proposed method extend to utility-based expected shortfall, a class outside the OCE framework?

Key findings

• A dynamic representation for OCE risk measures is established via a non-linear PDE (HJB equation) in a Brownian filtration, even when time consistency fails.
• The dynamic programming principle is derived in an enlarged state space, enabling dynamic computation of risk measures such as conditional value-at-risk and monotone mean-variance.
• Viscosity solution techniques are successfully adapted to handle singular Hamiltonians arising from unbounded stochastic control problems in this context.
• The control process β is constructed explicitly as the ratio of the gradient of the value function to the value function itself, ensuring boundedness through approximation.
• The method applies to OCE risk measures with power-type penalty functions, including Rényi divergence-based measures, and covers key risk measures used in finance and regulation.
• The approach does not fully extend to utility-based expected shortfall, indicating limitations in broader applicability beyond the OCE class.