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.[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.