I am considering a local martingale $M_t$ with continuous sample paths (i.e. $M_t\in\mathcal{M}_{C,\,\text{loc}}$) and with quadratic variation $\langle M\rangle_t$ (and also $M_0=\langle M\rangle_0$ a.s.). I am required to find a sharp estimate for $\mathbb{P}(M_t\geq \alpha)$ for $\alpha\in\mathbb{R}$. I am told to consider two points (with $\beta>0$):
i) Prove that if $\langle M\rangle_t\leq t$, then $\mathbb{E}\left(\mathrm{e}^{\beta M_t-\frac12\beta^2t}\right)\leq1$; and
ii) Prove that $\mathbb{P}\left(\mathrm{e}^{\beta M_t}>\mathrm{e}^{\alpha\beta}\right)\leq \mathrm{e}^{\frac12\beta^2t-\alpha\beta}$. (Typo corrected, thanks @UBM !)
Point i) suggests that somewhere in the proof, I must claim that non-negative local martingales are supermartingales, but I am not sure how to go about using this claim. Point ii) suggests to me that I should be doing some form of Markov inequality with i) (although I can't really see it) and minimising the right hand side with respect to $\beta$, is this correct? Any guidance is greatly appreciated.
The Doleans-Dade exponential of the process $\beta M^t, 0 \leq t \leq T$ is
$$\mathscr E (M)_t:=\exp \left(\beta M_t -\frac{1}{2} \beta^2\langle M \rangle_t \right), 0 \leq t \leq T$$ which solves the SDE $$\mathscr E (M)_t = 1 + \int_0^t \mathscr E (M)_s dM_s.$$ Thus, since $M$ is a continuous local martingale, so is $\mathscr E (M).$ Also note that is an exponential so it is a non-negative process and therefore a supermartingale. Since $E\mathscr E (M)_0 = 1,$ we must have $$E\mathscr E (M)_t \leq 1 \quad \text{ for all } t \in [0,T]. \tag{1}$$ So part $(i)$ follows from (1) and the hypothesis $\langle M \rangle_t \leq t.$
For part $(ii),$ first note that part $(i)$ is equivalent to $$E[e^{\beta M_t}] \leq e^{\frac{1}{2}\beta^2 t} \tag{2}.$$
Thus, using the Markov inequality and (2) we have $$P(e^{\beta M_t} > e^{\alpha \beta}) \leq \frac{E[e^{\beta M_t}]}{e^{\alpha \beta}} \leq e^{\frac{1}{2} \beta^2 t} e^{-\alpha \beta}.$$