Snell envelope

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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

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

References[edit]

  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.