Let $M$ be a continuous real valued local martingale with respect to a filtration $\Bbb{F}$. Then I want to prove that for any $t\geq 0$ and positive $C,K$ we have $$\Bbb{P}(M_t^*>C)\leq \frac{4K}{C^2} +\Bbb{P}([M]_t>K)$$.
I first thought about the Doob inequality, but this only works for expectations.
I know that since $M$ is a local martingale there exists a localizing sequence i.e. a sequence of stopping times $\tau_n$ s.t. $\tau_n\nearrow \infty$ and for all $n$ $M^{\tau_n}$ is a càdlàg martingale. But I also don't see how this could be useful to check the above inequality.
Can someone help me further?
By the law of total probability, we can write $$\Bbb{P}(M_t^*>C)= \Bbb{P}(M_t^*>C,[M]_t\le K) + \Bbb{P}(M_t^*>C,[M]_t>K) $$
Clearly the second summand is bounded by $\Bbb{P}([M]_t>K)$, so what we need to show is that $\Bbb{P}(M_t^*>C,[M]_t\le K) \le \frac{4K}{C^2} $.
Following the elegant argument given in saz's answer, we first assume that $M$ is a continuous martingale, and define the stopping time $\tau$ as $$\tau := \inf \{t>0: [M_t]> K\}$$ Such that we now have $$\begin{align*}\Bbb{P}(M_t^*>C,[M]_t\le K) = \Bbb{P}(M_t^*>C,\tau>t) &= \Bbb{P}(M_{t\wedge\tau}^*>C,\tau>t)\\ &\le \Bbb{P}(M_{t\wedge\tau}^*>C) = \Bbb{P}\big((M_{t\wedge\tau}^*)^2>C^2\big)\tag1\end{align*} $$ By Jensen's inequality, we have for all $0\le s\le t$ $$\mathbb E\big[(M_t)^2\mid\mathscr F_s\big] \geq \mathbb E\big[M_t\mid\mathscr F_s\big]^2 = M_s^2 $$ Hence $M^2$ is a submartingale and so is the stopped process $(M^2)^\tau $. By Doob's submartingale inequality applied to $(1)$ we find $$\Bbb{P}(M_t^*>C,[M]_t\le K)\le \frac{\mathbb E\left[(M_{t\wedge \tau}^2)^+\right]}{C^2} = \frac{\mathbb E\left[M_{t\wedge \tau}^2\right]}{C^2} \le \frac{\mathbb E\left[(M_{t\wedge \tau}^*)^2\right]}{C^2} \le \frac{4\mathbb E\big[[M_{t\wedge \tau}]\big]}{C^2}$$ Where the rightmost inequality follows from Doob's maximal inequality. Now, because by definition of $\tau$, $[M^\tau]$ is almost surely bounded by $K$, the desired result follows.
If we now only require $M$ to be continuous local martingale, we can apply the above result to $M^{\tau_n}$ where $(\tau_n)$ is a localization sequence, and the same conclusion holds by letting $n\to\infty$.