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 ![{\displaystyle t\in [0,T]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7ea7b28971838e52f450c48053939e81daa26f)
- If
is a
-supermartingale which dominates
, then
dominates
.[1]
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
![{\displaystyle U_{N}:=X_{N},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1e1198557ebc6fc75908b7c11b2d6b3a2e1f2c)
for ![{\displaystyle n=N-1,...,0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/593c3611c16e20d22c02ad0672e394556b733866)
where
is the join (in this case equal to the maximum of the two random variables).[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.[1]
References
- ^ 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.