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

Farshid Jamshidian
2007 Stochastics: An International Journal of Probability and Stochastic Processes  
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 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
more » ... ulation 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, 5, 19, 16] 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 Itô-Meyer formula. 1 We refer to [14] and references therein for other Monte-Carlo methods such as the stochastic mesh method, optimal exercise policy approximation, and regression on basis functions. 2 I thank F. Delbaen for clarifying this point at the Symposium. 3 I thank A. Irle for informing me that this result conjectured at my talk has long been known for decades. I thank R. Dalang for pointing out to me the fallacy in an incorrect converse that I also conjectured.
doi:10.1080/17442500601051914 fatcat:5ms5xu4gibevvdth4ib2cjyblu