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

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

Construction

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

U_{N}:=X_{N},

U_{n}:=X_{n}\lor \mathbb {E} ^{\mathbb {Q} }[U_{n+1}\mid {\mathcal {F}}_{n}]

for

n=N-1,...,0

where $\lor$ is the join (in this case equal to the maximum of the two random variables).^[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]

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