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.

Definition

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 $\mathbb {Q}$ of the process ${\displaystyle X=(X_{t})_{t\in [0,T]}}$ if

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

Construction

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 $\mathbb {Q}$ 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 $\lor$ is the join.^[1]

Application

• If $X$ is a discounted American option payoff with Snell envelope $U$ then $U_{t}$ is the minimal capital requirement to hedge $X$ from time $t$ to the expiration date.^[1]

References

1. Föllmer, Hans; Schied, Alexander (2004). Stochastic finance: an introduction in discrete time (2 ed.). Walter de Gruyter. pp. 280–282. ISBN 9783110183467.