Snell envelope

The Snell envelope, used in stochastics and mathematical finance, is the smallest supermartingale dominating a stochastic process. The Snell envelope is named after James Laurie Snell.

Definition

Given a filtered probability space and an absolutely continuous probability measure then an adapted process is the Snell envelope with respect to of the process if

is a -supermartingale
dominates, i.e. -almost surely for all times
If is a -supermartingale which dominates, then dominates.

Construction

Given a filtered probability space and an absolutely continuous probability measure then the Snell envelope with respect to of the process is given by the recursive scheme
where is the join.

Application

If is a discounted American option payoff with Snell envelope then is the minimal capital requirement to hedge from time to the expiration date.