Snell envelope

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 ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\in [0,T]},\mathbb {P} )}$and an absolutely continuous probability measure ${\displaystyle \mathbb {Q} \ll \mathbb {P} }$then an adapted process ${\displaystyle U=(U_{t})_{t\in [0,T]}}$is the Snell envelope with respect to ${\displaystyle \mathbb {Q} }$of the process ${\displaystyle X=(X_{t})_{t\in [0,T]}}$if

1. ${\displaystyle U}$is a ${\displaystyle \mathbb {Q} }$-supermartingale
2. ${\displaystyle U}$dominates ${\displaystyle X}$, i.e. ${\displaystyle U_{t}\geq X_{t}}$${\displaystyle \mathbb {Q} }$-almost surely for all times ${\displaystyle t\in [0,T]}$
3. If ${\displaystyle V=(V_{t})_{t\in [0,T]}}$is a ${\displaystyle \mathbb {Q} }$-supermartingale which dominates ${\displaystyle X}$, then ${\displaystyle V}$dominates ${\displaystyle U}$.

Given a (discrete) filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{n})_{n=0}^{N},\mathbb {P} )}$and an absolutely continuous probability measure ${\displaystyle \mathbb {Q} \ll \mathbb {P} }$then the Snell envelope ${\displaystyle (U_{n})_{n=0}^{N}}$with respect to ${\displaystyle \mathbb {Q} }$of the process ${\displaystyle (X_{n})_{n=0}^{N}}$is given by the recursive scheme

${\displaystyle U_{N}:=X_{N},}$

${\displaystyle U_{n}:=X_{n}\lor \mathbb {E} ^{\mathbb {Q} }[U_{n+1}\mid {\mathcal {F}}_{n}]}$for ${\displaystyle n=N-1,...,0}$

where ${\displaystyle \lor }$is the join (in this case equal to the maximum of the two random variables).

• If ${\displaystyle X}$is a discounted American option payoff with Snell envelope ${\displaystyle U}$then ${\displaystyle U_{t}}$is the minimal capital requirement to hedge ${\displaystyle X}$from time ${\displaystyle t}$to the expiration date.