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

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

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

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

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

where ${\displaystyle \lor }${\displaystyle \lor } is the join (in this case equal to the maximum of the two random variables).^[1]

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

• Mathematical finance

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