# proof of Martingale criterion (continuous time)

###### Proof.

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

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

 $\displaystyle\{X_{\tau_{k}\wedge t}:k\in\mathbb{N}\}$ $\displaystyle\subset\{X_{\tau}:\tau\ \text{stopping time},\tau\leq c\}$

is uniformly integrable (take $c=t$). It follows that $X_{t}^{\tau_{k}}\lx@stackrel{{\scriptstyle\begin{subarray}{c}\mathscr{L}^{1}% \end{subarray}}}{{\longrightarrow}}X_{t},k\to\infty$. Since the martingale property is stable under $\mathscr{L}^{1}$ convergence, $X$ is a martingale. ∎

Title proof of Martingale criterion (continuous time) ProofOfMartingaleCriterioncontinuousTime 2013-03-22 18:54:28 2013-03-22 18:54:28 karstenb (16623) karstenb (16623) 4 karstenb (16623) Proof msc 60G07 msc 60G48