The Duality of Optimal Exercise and Domineering Claims: a Doob-Meyer Decomposition Approach to the Snell Envelope

Abstract

We develop a concept of a 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 in Davis and Karatzas [8]. It also ties in well with several formulations of a duality in optimal stopping theory, including the minimax duality pricing formula in Rogers [30] and Haugh and Kogan [15] 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 (24,16,5,19] is extended and related to the additive decomposition. The iterative construction of the Snell envelope in Chen and Glasserman [6] is partially extended to continuous time. In Bermudan case, it is complemented with construction of stopping times converging to the optimal one, reminiscent of [26]. The perpetual American put is treated by incorporating an approach of [2]. Assuming smooth pasting, the jump-diffusion setting of [7] is extend based on the Ito-Meyer formula.