On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration

Julio Backhoff-Veraguas, Ludovic Tangpi

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. 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 an expression of the risk measure as the value of a stochastic control problem on an extended 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.

Original languageEnglish (US)
Pages (from-to)433-460
Number of pages28
JournalMathematics and Financial Economics
Issue number3
StatePublished - Jun 1 2020

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Finance
  • Statistics, Probability and Uncertainty


  • Dynamic programming principle
  • HJB equation
  • Optimized certainty equivalent
  • Risk measures
  • Singular Hamiltonian
  • Time-inconsistency
  • Unbounded stochastic control
  • Viscosity solution


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