The duality of optimal exercise and domineering claims: a Doob–Meyer decomposition approach to the Snell envelope

Abstract

We develop a concept of a “domineering claim” and apply it to the existence, uniqueness and properties of optimal stopping times in continuous time. The notion pinpoints a key observation of pathwise optimality implicit in Davis and Karatzas. It also ties in well with several formulations of a duality in optimal stopping theory, including the minimax duality pricing formula in Rogers and Haugh and Kogan for American and Bermudan options and its multiplicative version. We give a general formulation and proof that the Snell envelope is a supermartingale. Combined with the Doob–Meyer decomposition in different numeraire measures, this gives rise to (many) domineering claims. The multiplicative decomposition, for which a formula is derived, yields a uniquely invariant domineering numeraire. A pricing formula in Kim, Jacka, Carr et al. and Jamshidian are extended and related to the additive decomposition. The iterative construction of the Snell envelope in Chen and Glasserman is partially extended to continuous time. In Bermudan case, it is complemented with construction of stopping times converging to the optimal one, reminiscent of Kolodko and Schoemakers. The perpetual American put is treated by incorporating an approach of Beibel and Lerche. Assuming smooth pasting, the jump-diffusion setting of Chiarella and Ziogas is extend based on the Itô–Meyer formula.