Time-Consistent Dynamic Risk Measures on State-Dependent Musielak–Orlicz Hearts

Dynamic risk evaluation in insurance and finance is typically formulated on fixed integrability classes, such as L^p spaces or Orlicz hearts generated by a single Young function. A uniform tail regime is thereby imposed across scenarios, which is difficult to reconcile with actuarial portfolios in which loss severity and admissible model perturbations depend on observable risk factors. Regional catastrophe exposures and lapse or surrender behaviour under stress are concrete instances in which tail control should vary locally with the realised state.
Admissible terminal payoffs at time t are modelled on a conditional Musielak–Orlicz heart $M^{\Phi_{S_{t}}}(\mathcal{F}_{T}\mid\mathcal{F}_{t})$ generated by an $\mathcal{F}_{t}$-measurable random Young function driven by an adapted state process S. Dynamic evaluation is developed in the stopping-time two-parameter sense on adapted càdlàg processes, using the $L^{0}$-module viewpoint and the conditional expectation pairing to handle scalarisation cleanly in conditional duality.
A robust dual representation is obtained on state-dependent hearts as an essential supremum of penalised conditional expectations over a state-indexed dual family. Time consistency is characterised by pasting the stability of the dual classes and, in the convex case, by a cocycle property of the minimal penalty. The $\sigma$-order continuous conditional Köthe dual is identified and supports a stable weak topology under which attainment of a worst-case measure is obtained. A discrete-time Snell envelope on varying hearts follows from a backward-admissibility mechanism controlled by a conditional modular bound; it yields a minimal dominating $\mathcal{E}$-supermartingale and a first-hitting optimal exercise rule.
The construction integrates state-dependent tail control with time-consistent dynamic programming for unbounded rewards, providing a rigorous foundation for actuarial and financial evaluation under heterogeneous tail regimes. It covers situations where classical fixed-integrability frameworks are restrictive, and it supports explicit dynamic-programming implementations on the admissible state-dependent domain.