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. (April 2012)
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.
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 .
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
where is the join.
- If is a discounted American option payoff with Snell envelope then is the minimal capital requirement to hedge from time to the expiration date.
- ^ 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.