Snell envelope

Overview

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

( Ω ,

F

, (

F

t

)

t ∈ [ 0 , T ]

,

P

)

{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in [0,T]},\mathbb {P} )}

and an absolutely continuous probability measure

Q

≪

P

{\displaystyle \mathbb {Q} \ll \mathbb {P} }

then an adapted process

U = (

U

Construction

Given a (discrete) filtered probability space

( Ω ,

F

, (

F

n

)

n = 0

N

,

P

)

{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n=0}^{N},\mathbb {P} )}

and an absolutely continuous probability measure

Q

≪

P

{\displaystyle \mathbb {Q} \ll \mathbb {P} }

then the Snell envelope

(

Application

If

X

{\displaystyle X}

is a discounted American option payoff with Snell envelope

U

{\displaystyle U}

then

U

t

{\displaystyle U_{t}}

is the minimal capital requirement to hedge

X

{\displaystyle X}

from time

t

{\displaystyle t}

to the expiration date.