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 $$(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,T]},\mathbb{P})$$ 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
 * 1) $$U$$ is a $$\mathbb{Q}$$-supermartingale
 * 2) $$U$$ dominates $$X$$, i.e. $$U_t \geq X_t$$ $$\mathbb{Q}$$-almost surely for all times $$t \in [0,T]$$
 * 3) If $$V = (V_t)_{t \in [0,T]}$$ is a $$\mathbb{Q}$$-supermartingale which dominates $$X$$, then $$V$$ dominates $$U$$.

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).

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.