Rigorous PDE formulation of equilibrium payoffs in asymmetric-information Dynkin games

Develop a rigorous theorem that formalizes the PDE-based characterization of the equilibrium payoff processes in the asymmetric-information Dynkin games described in Section 6.3, including (i) the partially observed dynamics model in which a diffusion X has a drift determined by an unobserved regime J∈{0,1} and the uninformed player’s posterior ψ evolves under the innovation process, and (ii) the partially observed scenarios model with a finite regime affecting payoffs. Precisely establish the variational inequality system for the informed players’ value functions u^0 and u^1 and the uninformed player’s value function v, identify the associated stopping regions, and prove equivalence of this PDE system to the martingale and saddle-point conditions defining equilibria in randomised stopping times.

Background

Section 6.3 proposes a heuristic derivation of a PDE system that connects the paper’s martingale-based sufficient conditions for equilibrium to variational inequalities for the players’ value functions. In the partially observed dynamics model, both players observe a diffusion X whose drift depends on a hidden regime J; the uninformed player’s posterior ψ evolves via the innovation process, and the authors sketch variational inequalities involving the generators of (X, ψ). In the partially observed scenarios model, the underlying diffusion X is common, while a finite regime affects only the payoffs, leading to a related system of PDE constraints.

Establishing a rigorous statement would bridge the martingale characterization developed in the paper and PDE/variational-inequality frameworks used in related literature (e.g., verification approaches), clarifying the precise conditions under which the heuristic PDE system is well-posed and equivalent to the game’s equilibrium in randomised stopping times. A formal theorem would specify the function spaces, boundary and complementarity conditions, and prove the equivalence with the equilibrium value processes and stopping regions identified by the martingale theory.