Burkholder–Davis–Gundy inequality

Remarks

For $p = 2$ the inequality follows from Doob's martingale inequality in its $L^2$ form and the identity $E[M_\tau^2] = E[[M]_\tau]$ for suitable $\tau$, with $c_2 = 1$ and $C_2 = 4$. For other $p$, one standard proof applies Itô's formula to $\abs{M}^p$ or uses a good-$\lambda$ inequality comparing the laws of $M^*$ and $[M]^{1/2}$. Example: for a Wiener process $W$ and a fixed time $t$, $[W]_t = t$, so $E[\sup_{s \le t} \abs{W_s}^p]$ is between $c_p t^{p/2}$ and $C_p t^{p/2}$. For càdlàg local martingales the inequality holds for $p \ge 1$, with the same form. The constants do not depend on $M$ or $\tau$, and this uniformity is what makes the inequality useful.

Going deeper

For a student

The inequality lets one bound moments of a stochastic integral $\int_0^t H_s \, dW_s$ by moments of $\int_0^t H_s^2 \, ds$. It is a key estimate in proving existence, uniqueness and moment bounds for solutions of stochastic differential equations. It also shows that a continuous local martingale with $E[[M]_\infty^{1/2}] < \infty$ is a uniformly integrable martingale.