# 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

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 .
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## 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.^{[1]}

## 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.
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## References

- ↑
^{1.0}^{1.1}^{1.2}Föllmer, Hans; Schied, Alexander (2004).*Stochastic finance: an introduction in discrete time*(2 ed.). Walter de Gruyter. pp. 280–282. ISBN 9783110183467.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>