# Snell envelope

Canonical: https://web4.xpay.bi/en/wiki/wp-en-35652136/

Language: en

Published: 2026-10-25T21:06:29.251Z

Updated: 2026-10-25T21:06:29.251Z

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.

## Key facts

## Overview

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

(         Ω         ,

F

,         (

F

t

)

t             ∈             \[             0             ,             T             \]

,

P

)

\{\\displaystyle (\\Omega ,\{\\mathcal \{F\}\},(\{\\mathcal \{F\}\}\_\{t\})\_\{t\\in \[0,T\]\},\\mathbb \{P\} )\}

and an absolutely continuous probability measure

Q

≪

P

\{\\displaystyle \\mathbb \{Q\} \\ll \\mathbb \{P\} \}

then an adapted process

U         =         (

U

## Construction

Given a (discrete) filtered probability space

(         Ω         ,

F

,         (

F

n

)

n             =             0

N

,

P

)

\{\\displaystyle (\\Omega ,\{\\mathcal \{F\}\},(\{\\mathcal \{F\}\}\_\{n\})\_\{n=0\}^\{N\},\\mathbb \{P\} )\}

and an absolutely continuous probability measure

Q

≪

P

\{\\displaystyle \\mathbb \{Q\} \\ll \\mathbb \{P\} \}

then the Snell envelope

(

## Application

If

X

\{\\displaystyle X\}

is a discounted American option payoff with Snell envelope

U

\{\\displaystyle U\}

then

U

t

\{\\displaystyle U\_\{t\}\}

is the minimal capital requirement to hedge

X

\{\\displaystyle X\}

from time

t

\{\\displaystyle t\}

to the expiration date.

## Sources

- Snell envelope · Wikipedia: https://en.wikipedia.org/?curid=35652136

Source: Wikipedia contributors — https://en.wikipedia.org/?curid=35652136&oldid=1342433681

Licence: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/

Adaptation notice: Reformatted for XPay; the source may be abridged, restructured or translated. Consult the cited revision for the original.