martingale representation theorem
E59640
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).
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Martingale representation theorem | 0 |
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf | theorem in stochastic calculus ⓘ |
| appliesTo |
continuous martingales
ⓘ
martingales adapted to the Brownian filtration ⓘ square-integrable martingales ⓘ |
| assumes |
completeness of probability space
ⓘ
probability space with filtration ⓘ right-continuous filtration ⓘ usual conditions on filtration ⓘ |
| conclusion |
Brownian motion
ⓘ
surface form:
Brownian motion is a fundamental martingale for its natural filtration
every square-integrable martingale can be represented as a stochastic integral ⓘ martingales are generated by a fundamental martingale ⓘ |
| dealsWith |
Brownian motion
ⓘ
adapted processes ⓘ filtrations ⓘ martingales ⓘ stochastic integrals ⓘ |
| field |
probability theory
ⓘ
stochastic analysis ⓘ stochastic calculus ⓘ |
| generalizationOf | representation of martingales in Brownian filtration ⓘ |
| hasVariant |
martingale representation for Lévy processes
ⓘ
martingale representation for Poisson random measures ⓘ martingale representation in general semimartingale setting ⓘ |
| implies |
any L2-martingale is an Itô integral with respect to Brownian motion
ⓘ
uniqueness of integrand up to indistinguishability ⓘ |
| importance |
central in theory of stochastic integration
ⓘ
fundamental structural result for martingales ⓘ key tool in continuous-time finance ⓘ |
| relatedTo |
Brownian filtration
ⓘ
Clark–Ocone formula ⓘ Doob–Meyer decomposition ⓘ
surface form:
Doob–Meyer decomposition theorem
Itô calculus ⓘ
surface form:
Itô integral
Itô’s lemma ⓘ
surface form:
Itô's lemma
|
| representationWithRespectTo |
Brownian motion
ⓘ
fundamental martingale ⓘ |
| requires |
existence of stochastic integral with respect to Brownian motion
ⓘ
square-integrability of the martingale ⓘ |
| typicalFormulation | every L2-martingale adapted to the Brownian filtration is an Itô integral of a predictable process ⓘ |
| usedIn |
Girsanov theorem applications
ⓘ
backward stochastic differential equations ⓘ completeness of financial markets ⓘ derivation of Black–Scholes formula ⓘ hedging theory ⓘ mathematical finance ⓘ stochastic control ⓘ |
Referenced by (1)
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