Dynkin formula
E262081
Dynkin formula is a fundamental result in the theory of Markov processes that expresses the expected value of a function of the process at a stopping time in terms of its generator and an integral over time.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Dynkin formula canonical | 2 |
| Dynkin formula for killed processes | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2393112 — 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: Dynkin formula Context triple: [Kolmogorov backward equation, relatedTo, Dynkin formula]
-
A.
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.
-
B.
Chapman–Kolmogorov equation
The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.
-
C.
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.
-
D.
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.
-
E.
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.
- 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: Dynkin formula Target entity description: Dynkin formula is a fundamental result in the theory of Markov processes that expresses the expected value of a function of the process at a stopping time in terms of its generator and an integral over time.
-
A.
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.
-
B.
Chapman–Kolmogorov equation
The Chapman–Kolmogorov equation is a fundamental relation in the theory of stochastic processes that expresses how transition probabilities of a Markov process over longer time intervals can be obtained by integrating over intermediate states.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in Markov process theory
ⓘ
theorem in probability theory ⓘ |
| appliesTo |
Markov process
ⓘ
stopping time ⓘ time-homogeneous Markov process ⓘ |
| assumes |
Markov property
ⓘ
integrability conditions for the stopping time ⓘ sufficient regularity of the test function ⓘ |
| component |
Markov process (X_t)
ⓘ
generator L of the Markov process ⓘ stopping time τ ⓘ test function f on the state space ⓘ |
| expresses |
expectation as initial value plus time integral of the generator applied to the function
ⓘ
expected value of a function of the process at a stopping time ⓘ |
| field |
Markov processes
ⓘ
probability theory ⓘ stochastic processes ⓘ |
| generalizationOf |
Kolmogorov backward equation for expectations
ⓘ
linked to:
Kolmogorov backward equation
|
| hasForm | E[f(X_τ)] = f(X_0) + E[∫_0^τ Lf(X_s) ds] under suitable conditions ⓘ |
| hasVersion |
Dynkin formula for killed processes
ⓘ
linked to:
Dynkin formula
local Dynkin formula ⓘ |
| holdsFor |
continuous-time Markov chains
ⓘ
diffusion processes ⓘ jump Markov processes ⓘ |
| implies | that f(X_t) − f(X_0) − ∫_0^t Lf(X_s) ds is a martingale ⓘ |
| mathematicalNature | identity for expectations of Markov processes ⓘ |
| namedAfter | Eugene Dynkin ⓘ |
| relatedTo |
Feynman–Kac formula
ⓘ
martingale problem of Stroock–Varadhan ⓘ |
| relatesConcept |
Itô formula
ⓘ
linked to:
Itô’s lemma
Kolmogorov backward equation ⓘ expected value of functionals of a process ⓘ infinitesimal generator of a Markov process ⓘ martingale methods ⓘ |
| stateSpace | general measurable state space ⓘ |
| timeDomain | continuous time ⓘ |
| typicalCondition |
bounded stopping time or suitable integrability
ⓘ
f in the domain of the generator L ⓘ |
| usedFor |
characterizing harmonic functions for Markov processes
ⓘ
deriving partial differential equations for expectations ⓘ proving uniqueness of solutions to martingale problems ⓘ solving boundary value problems in probability ⓘ verification of solutions to stochastic control problems ⓘ |
| usedIn |
applied probability
ⓘ
mathematical finance ⓘ potential theory of Markov processes ⓘ queueing theory ⓘ stochastic control theory ⓘ |
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: Dynkin formula Description of subject: Dynkin formula is a fundamental result in the theory of Markov processes that expresses the expected value of a function of the process at a stopping time in terms of its generator and an integral over time.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Dynkin formula