Abstract
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 language | English (US) |
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Pages (from-to) | 433-460 |
Number of pages | 28 |
Journal | Mathematics and Financial Economics |
Volume | 14 |
Issue number | 3 |
DOIs | |
State | Published - Jun 1 2020 |
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Finance
- Statistics, Probability and Uncertainty
Keywords
- Dynamic programming principle
- HJB equation
- Optimized certainty equivalent
- Risk measures
- Singular Hamiltonian
- Time-inconsistency
- Unbounded stochastic control
- Viscosity solution