Snell envelope
E284683
The Snell envelope is a stochastic process that represents the smallest supermartingale dominating a given process and is fundamental in optimal stopping theory and the valuation of American-style options.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Snell envelope canonical | 1 |
| Snell envelope method for American option pricing | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2631372 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Snell envelope Context triple: [Doob–Meyer decomposition, relatedTo, Snell envelope]
-
A.
Doob–Meyer decomposition
The Doob–Meyer decomposition is a fundamental result in stochastic process theory that uniquely expresses a submartingale as the sum of a martingale and a predictable, increasing process.
-
B.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
C.
Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
-
D.
martingale representation theorem
The martingale representation theorem is a fundamental result in stochastic calculus stating that, under suitable conditions, every martingale can be expressed as a stochastic integral with respect to a Brownian motion (or more generally, a fundamental martingale).
-
E.
Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Snell envelope Target entity description: The Snell envelope is a stochastic process that represents the smallest supermartingale dominating a given process and is fundamental in optimal stopping theory and the valuation of American-style options.
-
A.
Doob–Meyer decomposition
The Doob–Meyer decomposition is a fundamental result in stochastic process theory that uniquely expresses a submartingale as the sum of a martingale and a predictable, increasing process.
-
B.
Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
-
C.
Itô’s lemma
Itô’s lemma is a fundamental result in stochastic calculus that generalizes the chain rule to functions of stochastic processes, especially Brownian motion.
-
D.
martingale representation theorem
The martingale representation theorem is a fundamental result in stochastic calculus stating that, under suitable conditions, every martingale can be expressed as a stochastic integral with respect to a Brownian motion (or more generally, a fundamental martingale).
-
E.
Kolmogorov backward equation
The Kolmogorov backward equation is a fundamental partial differential equation in stochastic processes that characterizes the time evolution of expected values of functionals of Markov processes, complementary to the Fokker–Planck (forward) equation.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
object in optimal stopping theory ⓘ stochastic process ⓘ supermartingale ⓘ |
| appearsIn |
risk-neutral valuation of contingent claims
ⓘ
theory of stopping times ⓘ |
| appliesTo |
continuous-time stochastic processes
ⓘ
discrete-time stochastic processes ⓘ |
| associatedWith |
Doob–Meyer decomposition
ⓘ
martingale theory ⓘ supermartingale theory ⓘ |
| characterizes | value process of an optimal stopping problem ⓘ |
| comparisonProperty | any supermartingale dominating the process dominates the Snell envelope ⓘ |
| constructionMethod |
Snell envelope as essential supremum over conditional expectations of stopped process
ⓘ
backward recursion in discrete time ⓘ |
| dominates | given adapted process ⓘ |
| ensures | existence of optimal stopping times under suitable conditions ⓘ |
| field |
mathematical finance
ⓘ
optimal stopping theory ⓘ probability theory ⓘ stochastic processes ⓘ |
| guarantees | supermartingale property of the value process ⓘ |
| isDefinedAs | smallest supermartingale dominating a given process ⓘ |
| mathematicalNature | defined up to almost sure equality ⓘ |
| minimalityProperty | smallest supermartingale greater than or equal to the process almost surely at all times ⓘ |
| namedAfter | J. L. Snell ⓘ |
| optimalStoppingRule | optimal stopping time is first time Snell envelope equals reward process under regularity conditions ⓘ |
| property |
adapted to the underlying filtration
ⓘ
right-continuous with left limits under standard assumptions ⓘ supermartingale dominating the reward process ⓘ |
| relatedTo |
American option pricing
ⓘ
backward induction in discrete time ⓘ dynamic programming principle ⓘ |
| requires |
filtered probability space
ⓘ
integrable or bounded reward process under standard formulations ⓘ |
| roleInControl | tool in stochastic control problems with stopping ⓘ |
| roleInFinance | represents value process of an American-style derivative under no-arbitrage ⓘ |
| timeIndex | can be indexed by discrete or continuous time ⓘ |
| usedFor |
characterizing value processes of stopping problems
ⓘ
deriving optimal stopping rules ⓘ optimal stopping problems ⓘ valuation of American-style options ⓘ |
| usedIn |
Snell envelope method for American option pricing
ⓘ
linked to:
Snell envelope
proofs of existence of optimal stopping times ⓘ |
| yields | optimal stopping time via first hitting time of the reward process ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Snell envelope Description of subject: The Snell envelope is a stochastic process that represents the smallest supermartingale dominating a given process and is fundamental in optimal stopping theory and the valuation of American-style options.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Snell envelope