= Solution
{c}
A continuous nondecreasing function $\Phi$ with $\Phi(0)=0$ has <moderate growth> if for some fixed $a>1$ and finite $K$,
$$
\Phi(ax)\leq K\Phi(x)\qquad(x\geq0).
$$
Iteration and monotonicity then give such an inequality, with a different constant, for every fixed dilation factor. Powers $\Phi(x)=x^p$, $p>0$, are standard examples.
The <Burkholder-Davis-Gundy inequalities> say that for a real continuous <local martingale> $M$ with $M_0=0$ and any stopping time $\tau$, there are positive finite constants $c_\Phi,C_\Phi$, depending only on $\Phi$, such that
$$
\boxed{c_\Phi\,\mathbb E\Phi(\sqrt{\langle M\rangle_\tau})
\leq\mathbb E\Phi(M_\tau^*)
\leq C_\Phi\,\mathbb E\Phi(\sqrt{\langle M\rangle_\tau}).}
$$
Here $M_\tau^*=\sup_{s\leq\tau}|M_s|$. Expectations may be infinite, with the inequalities understood by localization. For $\Phi(x)=x^p$, these compare the $p$th moment of the running maximum with the $p/2$th moment of quadratic variation.
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