Doob–Meyer decomposition

E59636

result in probability theory result in stochastic process theory theorem

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.

Aliases (3)

Statements (48)

Predicate	Object
instanceOf	result in probability theory → result in stochastic process theory → theorem →
appliesTo	submartingales →
characterizes	submartingales →
concludesExistenceOf	unique martingale part → unique predictable increasing compensator →
ensures	martingale part starts at the initial value of the submartingale → predictable increasing part is null at time zero →
field	martingale theory → probability theory → stochastic processes →
generalizes	Lebesgue decomposition of measures in a stochastic setting →
hasProperty	linearity with respect to submartingale addition and scalar multiplication → uniqueness up to indistinguishability →
hasVersion	discrete-time Doob decomposition →
involvesConcept	adapted process → càdlàg process → filtration → increasing process → martingale → predictable process → predictable sigma-algebra → submartingale →
namedAfter	Joseph L. Doob → Paul-André Meyer →
relatedTo	Doob decomposition for discrete-time submartingales → Girsanov theorem → Snell envelope → semimartingale decomposition →
requiresCondition	integrable submartingale → right-continuous filtration with complete probability space → submartingale of class D for the classical version →
statesThat	every suitable submartingale can be written as the sum of a martingale and a predictable increasing process →
timeSetting	continuous time →
typicalAssumptionOnProcess	adapted to a right-continuous filtration → càdlàg submartingale →
usedIn	compensated Poisson processes → credit risk modeling → martingale representation theorems → mathematical finance → optional stopping and optimal stopping problems → point process theory → semimartingale theory → stochastic calculus → stochastic integration → theory of compensators →
yieldsDecomposition	submartingale = martingale + predictable increasing process →

Referenced by (5)

Subject (surface form when different)	Predicate
Doob–Meyer decomposition ("Doob decomposition for discrete-time submartingales") → Girsanov theorem → Martingale representation theorem ("Doob–Meyer decomposition theorem") →	relatedTo
Doob–Meyer decomposition ("discrete-time Doob decomposition") →	hasVersion
Itô calculus →	relatedConcept