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 (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).
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.
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