, F , {F n } n 0 , P be a fil- tered probability space. Suppose - - PowerPoint PPT Presentation

Martingale Theorems

• Stopping Time: Given a filtered measurable space
• Ω, F, {Fn}n≥0
• ,

a stopping time is a random variable T : Ω → Z+ with the property that {ω : T(ω) ≤ n} ∈ Fn for all n ≥ 0.

• In words: we can decide whether T ≤ n based on information

available at time n.

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• If T is a stopping time, then equivalently

θn = 1T>n, n ≥ 0, is adapted to {Fn}n≥0.

• Consequently, if {Xn}n≥0 is a martingale, then so is

Zn =

n−1

• j=0

θj

• Xj+1 − Xj
• .
• But Zn = XT∧n − X0, where T ∧ n = min(T, n).
• So {XT∧n}n≥0 is a martingale.

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• Optional Stopping Theorem. Let
• Ω, F, {Fn}n≥0, P
• be a fil-

tered probability space. Suppose that the process {Xn}n≥0 is a

• P, {Fn}n≥0
• martingale, and that T is a bounded stopping
• time. Then

E[XT|F0] = X0

and hence

E[XT] = E[X0].

• Boundedness of T gives a simple proof, but is not necessary.

– But if T is unbounded, conditions on {Xn} are needed for the theorem to hold.

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• Positive Supermartingale Convergence Theorem: If {Xn}n≥0

is a

• P, {Fn}n≥0
• supermartingale and P[Xn ≥ 0] = 1 for all

n ≥ 0, then there exists a F∞-measurable X∞ such that

P[Xn → X∞ as n → ∞] = 1.

– Note that a martingale is also a supermartingale, so this theorem also applies to positive martingales.

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• Compensation: Submartingales tend to increase over time,

and supermartingales tend to decrease.

• Suppose that {Xn}n≥0 is a
• P, {Fn}n≥0
• submartingale. Then

there is a previsible non-decreasing process {An}n≥0 such that {Xn − An}n≥0 is a

• P, {Fn}n≥0
• martingale.

– We simply set A0 = 0 and An−An−1 = E[Xn−Xn−1|Fn−1].

• Similarly, if {Xn}n≥0 is a
• P, {Fn}n≥0
• supermartingale, then

there is a previsible non-decreasing process {An}n≥0 such that {Xn + An}n≥0 is a

• P, {Fn}n≥0
• martingale.
• In each case, {An}n≥0 is the compensator.

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