proof of Martingale criterion (continuous time)

Proof.

1. Let $X$ be a martingale. By the optional sampling theorem we have $E(X_c|{ℱ}_{\tau })=X_{c\wedge \tau }=X_{\tau }\forall \tau \le c$. Since conditional expectations are uniformly integrable the first direction follows.

2. Let ${({\tau }_k)}_{k\ge 1}$ be a local sequence of stopping times (i.e. ${\tau }_k\uparrow \mathrm{\infty }$ a.s. and $X^{{\tau }_k}$ martingale $\forall k\in \mathbb{N}$). For each $t\in {ℝ}_{+}$ we have $X_{{\tau }_k\wedge t}\to X_t,k\to \mathrm{\infty }$ almost surely. The set

$\mathrm{\{}{X}_{{\tau }_k\wedge t}:k\in \mathbb{N}\}$ $\subset \{X_{\tau }:\tau \text{stopping time},\tau \le c\}$

is uniformly integrable (take $c=t$). It follows that $X_t^{{\tau }_k}\stackrel{\begin{array}{c}{ℒ}^1\end{array}}{⟶}{X}_t,k\to \mathrm{\infty }$. Since the martingale property is stable under ${ℒ}^1$ convergence, $X$ is a martingale. ∎

Title	proof of Martingale criterion (continuous time)
Canonical name	ProofOfMartingaleCriterioncontinuousTime
Date of creation	2013-03-22 18:54:28
Last modified on	2013-03-22 18:54:28
Owner	karstenb (16623)
Last modified by	karstenb (16623)
Numerical id	4
Author	karstenb (16623)
Entry type	Proof
Classification	msc 60G07
Classification	msc 60G48