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

TL;DR

This paper develops a mathematical framework for dynamic risk measures using optimal transport theory, providing explicit formulas and stochastic control representations for models with distributional uncertainty.

Contribution

It introduces a novel connection between dynamic risk measures and optimal transport, characterizing the risk generator and deriving explicit formulas for different transport bounds.

Findings

Identifies the risk generator via optimal transport costs.

Derives explicit formulas for Wasserstein and martingale Wasserstein penalizations.

Provides stochastic control representations for the risk measures.

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.