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
[edit]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 .[1]
Construction
[edit]Given a (discrete) 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
- for
where is the join (in this case equal to the maximum of the two random variables).[1]
Application
[edit]- If is a discounted American option payoff with Snell envelope then is the minimal capital requirement to hedge from time to the expiration date.[1]