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

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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

  1.   is a  -supermartingale
  2.   dominates  , i.e.    -almost surely for all times  
  3. If   is a  -supermartingale which dominates  , then   dominates  .[1]

Construction

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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

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  • 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]

References

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  1. ^ a b c Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. pp. 280–282. ISBN 9783110183467.