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 (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. References 

Mathematical finance