TY - JOUR
T1 - On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration
AU - Backhoff-Veraguas, Julio
AU - Tangpi, Ludovic
N1 - Funding Information:
We thank Beatrice Acciaio, Joaqu?n Fontbona, Asgar Jamneshan and Michael Kupper for their feedback on this article.
Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/6/1
Y1 - 2020/6/1
N2 - 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.
AB - 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.
KW - Dynamic programming principle
KW - HJB equation
KW - Optimized certainty equivalent
KW - Risk measures
KW - Singular Hamiltonian
KW - Time-inconsistency
KW - Unbounded stochastic control
KW - Viscosity solution
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U2 - 10.1007/s11579-020-00261-2
DO - 10.1007/s11579-020-00261-2
M3 - Article
AN - SCOPUS:85084693892
SN - 1862-9679
VL - 14
SP - 433
EP - 460
JO - Mathematics and Financial Economics
JF - Mathematics and Financial Economics
IS - 3
ER -