Clark–Ocone formula
E284688
The Clark–Ocone formula is a key result in stochastic calculus and Malliavin calculus that provides an explicit integral representation of square-integrable random variables with respect to Brownian motion.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Clark–Ocone formula canonical | 2 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
result in Malliavin calculus ⓘ result in stochastic calculus ⓘ |
| appliesTo |
functionals of Brownian motion
ⓘ
square-integrable random variables ⓘ |
| assumes |
adaptedness to Brownian filtration
ⓘ
square-integrability ⓘ |
| characterizes | square-integrable functionals as stochastic integrals plus constants ⓘ |
| context |
Brownian filtration
ⓘ
Wiener space ⓘ |
| field |
Malliavin calculus
ⓘ
probability theory ⓘ stochastic analysis ⓘ |
| formalSetting | L^2 space of the underlying probability space ⓘ |
| generalizationOf | martingale representation for Brownian motion ⓘ |
| gives |
integral representation of random variables
ⓘ
martingale representation ⓘ |
| hasComponent |
constant term equal to the expectation of the variable
ⓘ
stochastic integral term with predictable integrand ⓘ |
| hasVersion |
formula for Brownian motion in \/R
ⓘ
formula for Poisson random measures ⓘ formula for multidimensional Brownian motion ⓘ formula under change of measure ⓘ |
| integrandGivenBy | conditional expectation of the Malliavin derivative given the filtration ⓘ |
| involves | conditional expectation of the Malliavin derivative ⓘ |
| namedAfter |
Daniel Ocone
ⓘ
John Michael Clark ⓘ |
| relatedTo |
Girsanov theorem
ⓘ
Itô integral ⓘ Itô’s lemma ⓘ Malliavin calculus ⓘ
surface form:
Malliavin integration by parts
martingale representation theorem ⓘ |
| requires | Malliavin differentiability of the functional ⓘ |
| timePeriod | late 20th century ⓘ |
| type | representation theorem ⓘ |
| typicalAssumption | complete probability space with Brownian filtration ⓘ |
| usedFor |
computing Greeks via Malliavin calculus
ⓘ
explicit computation of hedging strategies ⓘ representation of payoffs in terms of Brownian motion ⓘ |
| usedIn |
derivative pricing
ⓘ
filtering theory ⓘ hedging theory ⓘ mathematical finance ⓘ sensitivity analysis of financial derivatives ⓘ stochastic control ⓘ |
| uses |
Brownian motion
ⓘ
Malliavin calculus ⓘ
surface form:
Malliavin derivative
stochastic integral ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Martingale representation theorem