Bernstein Inequality for continous local martingale

Question

I'm looking for a simple proof of the following fact, which is somehow Bernstein inequality in continuous time.

Let $(M_t)_{t\geq 0}$ be a continuous local martingale. Then :

$$P\left(\sup_{t\in [0,T]}M_t \geq a \ \vert \ [M]_T\leq b\right) \leq \exp\left(-\frac{a^2}{2b}\right).$$

It's a special case of an inequality of Shorack and Wellner, but I think there should be an easy proof. One can use Dubins-Schwarz theorem but I don't want to, I'm looking for a direct proof, in the spirit of Bernstein Inequality.

Answer 1

You can use the Bernstein inequality along a discrete skeleton and then pass to the limit using continuity.