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 $(\Omega,\mathcal,(\mathcal_t)_,\mathbb)$ and an absolutely continuous probability measure $\mathbb \ll \mathbb$ then an adapted process $U = (U_t)_$ is the Snell envelope with respect to $\mathbb$ of the process $X = (X_t)_$ if
# $U$ is a $\mathbb$-supermartingale
# $U$ dominates $X$, i.e. $U_t \geq X_t$ $\mathbb$-almost surely for all times t \in ,T/math>
# If $V = (V_t)_$ is a $\mathbb$-supermartingale which dominates $X$, then $V$ dominates $U$.

Construction

Given a (discrete) filtered probability space $(\Omega,\mathcal,(\mathcal_n)_^N,\mathbb)$ and an absolutely continuous probability measure $\mathbb \ll \mathbb$ then the Snell envelope $(U_n)_^N$ with respect to $\mathbb$ of the process $(X_n)_^N$ is given by the recursive scheme
:$U_N := X_N,$
:U_n := X_n \lor \mathbb^ _ \mid \mathcal_n/math> 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.

References

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Mathematical finance