An optimal transport foundation for a class of dynamically consistent risk measures

2026-05-20

AI Summary

This study addresses the construction of dynamically consistent risk measures under distributional uncertainty in Markovian reference models. Model ambiguity is characterized via a penalized worst-case expectation, and a convex monotone semigroup acting on bounded continuous payoff functions is derived from one-step convex risk evaluations. The work innovatively employs optimal transport costs as lower bounds for the penalty function, revealing first-order (drift adjustment) and second-order (volatility adjustment) structures of the risk generator under distinct scaling regimes, along with their explicit dual representations. Furthermore, stochastic control representations are established for both drift and volatility adjustments under two canonical penalty schemes, providing a rigorous theoretical foundation and computational framework for dynamic robust risk modeling.

Abstract

We study a class of dynamically consistent risk measures that robustify a time-homogeneous Markovian reference model by allowing for distributional uncertainty in its transition laws. We start from one-step convex risk evaluations in which ambiguity is captured by penalized worst-case expectations over alternative transition laws. Imposing time consistency then yields a convex monotone semigroup on bounded continuous payoff functions, and this semigroup represents the associated dynamic risk measure. The semigroup is uniquely characterized by its risk generator. Under a lower bound on the family of penalties in terms of suitable optimal transport costs relative to the reference laws, we identify the generator on smooth test functions. For optimal transport bounds with linear small-time scaling, this produces a first-order, drift-type correction given by a convex Hamiltonian acting on the gradient. Under martingale transport constraints and a different scaling, however, the leading correction is genuinely of second order and is described by a convex monotone functional acting on the Hessian. We illustrate both regimes for Wasserstein and martingale Wasserstein penalizations and derive explicit formulas via convex conjugates of the underlying transport costs. The associated dynamic risk measures admit stochastic control representations in which the control acts on the drift in the first-order case and on the volatility in the second-order case.

Problem

• optimal transport
• dynamic risk measures
• distributional uncertainty
• time consistency
• Markovian models

Innovation

• optimal transport
• dynamic risk measures
• time consistency
• convex Hamiltonian
• martingale transport