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If L1 Convergent, Then Martingale is Closable 📂Probability Theory

If L1 Convergent, Then Martingale is Closable

Theorem

Given a probability space (Ω,F,P)( \Omega , \mathcal{F} , P) and a martingale {(Xn,Fn)}\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}, if a stochastic process {(Xn,Fn)}\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\} converges to a random variable YY through L1\mathcal{L}_{1}, then {(Xn,Fn):n=1,⋯ ,∞}\left\{ ( X_{n} , \mathcal{F}_{n} ): n = 1 , \cdots , \infty \right\} is a closable martingale.

Description

Even if XnX_{n} converges to YY through L1\mathcal{L}_{1} and almost surely converges to X∞X_{\infty}, it cannot be guaranteed that YY and X∞X_{\infty} have some relation. Xn→L1Y∧Xn→a.s.X∞⇏Y=X∞ X_{n} \overset{\mathcal{L}_{1}}{\to} Y \land X_{n} \overset{\text{a.s.}}{\to} X_{\infty} \nRightarrow Y = X_{\infty} In formula terms, it can be restated as above, and during the proof, it is verified that in the case of martingales, Y=a.s.X∞Y \overset{\text{a.s.}}{=} X_{\infty} holds.

Proof

Part 1. E∣Xn∣→E∣Y∣E |X_{n}| \to E |Y|

It is said that {Xn}\left\{ X_{n} \right\} converges to some random variable YY through L1\mathcal{L}_{1}, as follows. lim⁡n→∞∫Ω∣Xn−Y∣dP=lim⁡n→∞E∣Xn−Y∣=0 \lim_{n \to \infty} \int_{\Omega} | X_{n} - Y | dP = \lim_{n \to \infty} E | X_{n} - Y | = 0 Without loss of generality, since ∣∣a∣−∣b∣∣≤∣a−b∣\left| |a| - |b| \right| \le | a - b|, ∣E∣Xn∣−E∣Y∣∣≤∣E(∣Xn∣−∣Y∣∣)≤E∣∣Xn∣−∣Y∣∣≤E∣Xn−Y∣ \begin{align*} \left| E | X_{n} | - E | Y | \right| \le & \left| E \left( | X_{n} | - | Y | \right| \right) \\ \le & E \left| |X_{n}| - |Y| \right| \\ \le & E \left| X_{n} - Y \right| \end{align*} That is, when n→∞n \to \infty, E∣Xn∣→E∣Y∣E |X_{n}| \to E |Y| holds.

Part 2. Y=a.s.X∞Y \overset{\text{a.s.}} = X_{\infty}

Submartingale Convergence Theorem: Given a probability space (Ω,F,P)( \Omega , \mathcal{F} , P) and a submartingale {(Xn,Fn)}\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}.

If it is assumed that sup⁡n∈NEXn+<∞\displaystyle \sup_{n \in \mathbb{N}} E X_{n}^{+} < \infty, then XnX_{n} almost surely converges to some random variable X∞:Ω→RX_{\infty}: \Omega \to \mathbb{R}, and EX∞<EX∞+<∞E X_{\infty} < E X_{\infty}^{+} < \infty

Since it is E∣Xn∣→E∣Y∣E |X_{n}| \to E |Y|, it becomes sup⁡n∈NE∣Xn∣<∞\displaystyle \sup_{n \in \mathbb{N}} E | X_{n} | < \infty. Of course, EXn+≤E∣Xn∣<∞E X_{n}^{+} \le E | X_{n} | < \infty and since a martingale is a submartingale, according to the submartingale convergence theorem, the stochastic process {Xn}\left\{ X_{n} \right\} almost surely converges to some random variable X∞X_{\infty}. Hence, Y=a.s.X∞Y \overset{\text{a.s.}}{=} X_{\infty} holds.

Part 3. E(X∞∣Fn)=XnE \left( X_{\infty} | \mathcal{F}_{n} \right)= X_{n}

Without loss of generality, since Z≥0∧EZ=0  ⟹  Z≤0Z \ge 0 \land E Z = 0 \implies Z \le 0, it is sufficient to show that E∣E(X∞∣Fn)−Xn∣=0E \left| E \left( X_{\infty} | \mathcal{F}_{n} \right) - X_{n} \right| = 0 holds. Considering m>nm > n, E∣E(X∞∣Fn)−Xn∣=E∣E(X∞∣Fn)−E(Xn∣Fn)∣=E∣E(X∞∣Fn)−E(Xm∣Fn)∣=E∣E(X∞−Xm∣Fn)∣≤E[E(∣X∞−Xm∣∣Fn)]=E∣X∞−Xm∣ \begin{align*} E \left| E \left( X_{\infty} | \mathcal{F}_{n} \right) - X_{n} \right| =& E \left| E \left( X_{\infty} | \mathcal{F}_{n} \right) - E \left( X_{n} | \mathcal{F}_{n} \right) \right| \\ =& E \left| E \left( X_{\infty} | \mathcal{F}_{n} \right) - E \left( X_{m} | \mathcal{F}_{n} \right) \right| \\ =& E \left| E \left( X_{\infty} - X_{m} | \mathcal{F}_{n} \right) \right| \\ \le & E \left[ E \left( \left| X_{\infty} - X_{m} \right| | \mathcal{F}_{n} \right) \right] \\ =& E | X_{\infty} - X_{m} | \end{align*} Applying lim⁡m→∞\displaystyle \lim_{m \to \infty} to both sides and according to Parts 1 and 2, E∣E(X∞∣Fn)−Xn∣=lim⁡m→∞E∣E(X∞∣Fn)−Xn∣≤lim⁡m→∞E∣X∞−Xm∣≤0 \begin{align*} E \left| E \left( X_{\infty} | \mathcal{F}_{n} \right) - X_{n} \right| &= \lim_{m \to \infty} E \left| E \left( X_{\infty} | \mathcal{F}_{n} \right) - X_{n} \right| \\ \le & \lim_{m \to \infty} E | X_{\infty} - X_{m} | \\ \le & 0 \end{align*} Therefore, it is confirmed that {(Xn,Fn):n=1,⋯ ,∞}\left\{ ( X_{n} , \mathcal{F}_{n} ): n = 1 , \cdots , \infty \right\} is a closable martingale.

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