Neofunctionalization

2014-01-20T22:01:11Z

69.91.186.162: /* Neosubfunctionlization */

{{refimprove|date=April 2012}}

The '''Snell envelope''', used in [[stochastics]] and [[mathematical finance]], is the smallest [[Martingale (probability theory)|supermartingale]] dominating a [[stochastic process]]. The Snell envelope is named after [[James Laurie Snell]].

== Definition ==
Given a [[filtered probability space]] <math>(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in [0,T]},\mathbb{P})</math> and an [[absolutely continuous]] [[probability measure]] <math>\mathbb{Q} \ll \mathbb{P}</math> then an [[adapted process]] <math>U = (U_t)_{t \in [0,T]}</math> is the Snell envelope with respect to <math>\mathbb{Q}</math> of the process <math>X = (X_t)_{t \in [0,T]}</math> if
# <math>U</math> is a <math>\mathbb{Q}</math>-supermartingale
# <math>U</math> dominates <math>X</math>, i.e. <math>U_t \geq X_t</math> <math>\mathbb{Q}</math>-[[almost surely]] for all times <math>t \in [0,T]</math>
# If <math>V = (V_t)_{t \in [0,T]}</math> is a <math>\mathbb{Q}</math>-supermartingale which dominates <math>X</math>, then <math>V</math> dominates <math>U</math>.<ref name="FollmerSchied">{{cite book|first1=Hans|last1=Föllmer|first2=Alexander|last2=Schied|title=Stochastic finance: an introduction in discrete time|publisher=Walter de Gruyter|year=2004|edition=2|isbn=9783110183467|pages=280-282}}</ref>

== Construction ==
Given a (discrete) [[filtered probability space]] <math>(\Omega,\mathcal{F},(\mathcal{F}_n)_{n = 0}^N,\mathbb{P})</math> and an [[Absolutely continuous#Absolute continuity of measures|absolutely continuous]] [[probability measure]] <math>\mathbb{Q} \ll \mathbb{P}</math> then the Snell envelope <math>(U_n)_{n = 0}^N</math> with respect to <math>\mathbb{Q}</math> of the process <math>(X_n)_{n = 0}^N</math> is given by the recursive scheme
:<math>U_N := X_N,</math>
:<math>U_n := X_n \lor \mathbb{E}^{\mathbb{Q}}[U_{n+1} \mid \mathcal{F}_n]</math> for <math>n = N-1,...,0</math>
where <math>\lor</math> is the [[join and meet|join]].<ref name="FollmerSchied" />

== Application ==
* If <math>X</math> is a discounted [[American option]] payoff with Snell envelope <math>U</math> then <math>U_t</math> is the minimal capital requirement to hedge <math>X</math> from time <math>t</math> to the expiration date.<ref name="FollmerSchied" />

== References ==
{{Reflist}}

[[Category:Mathematical finance]]

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