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.

Keywords:

• Duality
• Optimal stopping time
• Snell envelope
• Supermartingale
• Doob–Meyer decomposition
• Multiplicative decomposition

MS Classification::

• 60G40,
• 60H30.

Acknowledgements

I am grateful to the anonymous referees for valuable comments. I would also like to thank many of the participants at the Jan-06 Symposium on Optimal Stopping, especially A. Shiryaev and R. Lerche, for interesting discussions, helpful comments and constructive criticism.

Notes

Version 4-October-2006. For possible future updates visit www.home.math.utwente.nl/ ˜jamshidianf.

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.

I thank F. Delbaen for clarifying this point at the Symposium.

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.

Of course in the finite horizon case , any martingale is automatically uniformly integrable.

A process is of class D if the family of random variables is uniformly integrable.

An exception is when Z is a submartingale; then there is only one domineering claim, namely .

Replacing Z with − Z, one also concludes that .

Indeed, . Hence, as .

Recall this means Z has left limits and does not jump at any predictable time T, i.e. a.s. It implies that a.s., whenever with . (See, e.g. Proposition I.2.26 in Ref. [17]).

An alternative argument uses the fact any numeraire B is pathwise bounded above and bounded below above zero. This easily implies that if and only if . But, the latter condition is equivalent to B domineering Z by Proposition 2.5.

To show , it suffices to show , which follows because both sides equal . To show , we must show . The jump process of the LHS is and that of the RHS is , which are the same. The continuous local martingale part of LHS is and that of the RHS is , which are equal, as presence of jump in integrand does not contribute to stochastic integral of a continuous semimartingale. Therefore, the LHS and RHS processes are equal.

Let be a localizing sequence for M. Applying the optional sampling theorem to the claim gives, . But, a.s. Thus by Fatou's lemma, .

Also, because, using and the estimate for on ,