Why are driftless diffusion processes local martingales?

Question

According to Wikipedia, all driftless diffusion processes are local martingales. How can you prove this? A driftless diffusion process $X$ is given by $$dX_t = \sigma(X_t)dB_t,$$ for a Lipschitz continuous function $\sigma$. I suspect that you should construct a localizing sequence of stopping times $\tau_n$ such that each $X^{\tau_n}$ is a martingale, but I don't know how to do that in this general setting.

Answer 1

Here's a proof.

First, we define a localizing sequence of stopping times $\tau^n$ by: $$\tau_n= \min(\{|X_t|\geq n\}),$$ if such a $t$ exists, and $$\tau_n=n,$$ elsewise. Now since $X$ is finite and continuous, we have $\tau_n\rightarrow\infty$ a.s.

Now we have to prove that $X^{\tau_n}$ is a martingale for all $n$. Observe that $X^{\tau_n}$ equals $\int_0^{t}\sigma^n(X_t^{\tau_n})d W_t$, where $\sigma^n$ is set to zero when $|X_t|\geq n$, and equals $\sigma$ elsewhere. The Lipschitz-continuity of $\sigma$ now implies that $\sigma^n(X_t^{\tau_n})$ is a bounded process. It is also progressive, because $X_t$ and $\sigma$ are continuous. Since it's bounded and progressive, by the dominated convergence theorem, we can approximate $\sigma^n(X_t^{\tau_n})$ by elementary processes $Y^{n,m}$, which converge to $\sigma^n(X_t^{\tau_n})$ in the L2 sense. These elementary processes are constant on a finite set of time intervals. For these processes we can easily verify that $\int_0^t Y^{n,m}dW_t$ is a martingale. By definition of stochastic integral, we have $$X^{\tau^n}=\int_0^t \sigma^n(X_t^{\tau_n})dW_t=\lim_{m\rightarrow\infty} \int_0^t Y^{n,m}dW_t $$ in the L2 sense, so this limit $X^{\tau_n}$ is also a martingale, as we needed to prove.