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Created by Knut M. Synstadfrom the Noun Project

Martingale Convergence Theorem

Theorem (1st Martingale Convergence Theorem)

Let (Xn)n∈N(X_{n})_{n\in\mathbb{N}} be a (Fn)n∈N(\mathcal{F}_{n})_{n\in\mathbb{N}}-martingale on (Ω,F,P)(\Omega,\mathcal{F},P)

  1. If (Xn)n∈N(X_{n})_{n\in\mathbb{N}} is uniformly integrable, then l=lim⁡n→∞Xn\mathscr{l}=\lim_{ n \to \infty }X_{n} a.s. exists and is integrable and furthermore Xn→l in L1X_{n}\to \mathscr{l}\text{ in }L^{1}and closes (Xn)n∈N(X_{n})_{n\in\mathbb{N}} from the right i.e. Xn=E[l∣Fn] a.s. , ∀n∈NX_{n}=E[\mathscr{l}|\mathcal{F}_{n}]\text{ a.s. }, \ \forall n\in\mathbb{N}
  2. Conversely, if ∃X∞∈L1(Ω,F,P)\exists X_{\infty}\in\mathscr{L}^1(\Omega,\mathcal{F},P) which closes (Xn)n∈N(X_{n})_{n\in\mathbb{N}} to the right (i.e. Xn=E[X∞∣Fn]X_{n}=E[X_{\infty}|\mathcal{F}_{n}] a.s. ∀n∈N\forall n\in\mathbb{N}) then (Xn)n∈N(X_{n})_{n\in\mathbb{N}} is uniformly integrable and for l∈L1(Ω,F,P)\mathscr{l}\in\mathscr{L}^{1}(\Omega,\mathcal{F},P) given by l=lim⁡n→∞Xn\mathscr{l}=\lim_{ n \to \infty }X_{n} a.s. and in L1 satisfies l=E[X∞∣F∞−] a.s.\mathscr{l}=E[X_{\infty}|\mathcal{F}_{\infty^{-}}]\text{ a.s.}where F∞−=σ(⋃n∈NFn)\mathcal{F_{\infty^{-}}}=\sigma\left( \bigcup_{n\in\mathbb{N}}\mathcal{F}_{n} \right)

Intuition

By X∞X_{\infty} “closing XnX_{n} to the right” we mean that ∀n∈N\forall n\in\mathbb{N} (for every time step) Xn=E[X∞∣Fn]X_{n}=E[X_{\infty}|\mathcal{F}_{n}]holds. So that means for our martingale, X∞X_{\infty} is the limiting value.

For the first one what we’re saying is: “If X=(Xn)n∈NX=(X_{n})_{n\in\mathbb{N}} is u.i. then XnX_{n} converges to some limit, l\mathscr{l}, and this value closes XnX_{n} to the right ∀n∈N\forall n\in\mathbb{N}” i.e. XnX_{n} u.i.   ⟹  \implies Xn→lX_{n}\to \mathscr{l} & l\mathscr{l} closes XnX_{n} to the right For the second one what we’re saying is: “If we have some function X∞X_{\infty} closing X=(Xn)n∈NX=(X_{n})_{n\in\mathbb{N}} to the right then (Xn)n∈N(X_{n})_{n\in\mathbb{N}} is u.i. and X=(Xn)n∈NX=(X_{n})_{n\in\mathbb{N}} converges to some l\mathscr{l} and also satisfies martingale property”. i.e. X∞X_{\infty} closes XnX_{n} to the right   ⟹  \implies XnX_{n} u.i. & Xn→lX_{n}\to \mathscr{l} & l=E[X∞∣F∞−]\mathscr{l}=E[X_{\infty}|\mathcal{F}_{\infty^{-}}]

Theorem (2nd Martingale Convergence Theorem)

Let (Xn)n∈Z−(X_{n})_{n\in\mathbb{Z}^{-}} be a (Fn)n∈Z−(\mathcal{F}_{n})_{n\in\mathbb{Z}^{-}}-martingale (i.e. (Fn)n∈Z−(\mathcal{F}_{n})_{n\in\mathbb{Z}^{-}} is a filtration on (Ω,F,P)(\Omega,\mathcal{F},P)) or Fn+m⊂Fn ∀m,n∈Z−\mathcal{F}_{n+m}\subset \mathcal{F}_{n} \ \forall m,n\in\mathbb{Z}^{-} e.g. F0⊃F−1⊃F−2⊃…\mathcal{F}_{0}\supset\mathcal{F}_{-1}\supset\mathcal{F}_{-2}\supset\dots and (Xn)n∈Z−⊂L1(Ω,F,P)(X_{n})_{n\in\mathbb{Z}^{-}}\subset \mathscr{L}^{1}(\Omega,\mathcal{F},P) and XnX_{n} be Fn\mathcal{F}_{n}-measurable ∀n∈Z−\forall n\in\mathbb{Z}^{-} (i.e. XnX_{n} is (Fn)n∈Z−(\mathcal{F}_{n})_{n\in\mathbb{Z}^-}-adapted or XnX_{n} is a RV on Fn, ∀n∈Z−\mathcal{F_{n}}, \ \forall n\in\mathbb{Z}^{-}) and E[Xn∣Fn+m]=XmE[X_{n}|\mathcal{F}_{n+m}]=X_{m} a.s. ∀n,m∈Z−,m≤n\forall n,m\in\mathbb{Z}^{-}, m\le n. Then (Xn)n∈Z−(X_{n})_{n\in\mathbb{Z}^{-}} is uniformly integrable and ∃l∈L1(Ω,F,P)\exists \mathscr{l}\in\mathscr{L}^1(\Omega,\mathcal{F},P) such that Xn→lX_{n}\to \mathscr{l} a.s. and in L1. Furthermore l\mathscr{l} is F−∞\mathcal{F}_{-\infty}-measurable and l=E[X0∣F−∞]\mathscr{l}=E[X_{0}|\mathcal{F}_{-\infty}]