Burkholder-Davis-Gundy inequalities

For every $p>0$ there are constants $c_p,C_p>0$ such that a continuous local martingale $M$ with $M_0=0$ satisfiesfor every stopping time $T$ for which the quantities are finite.

Real and imaginary parts of an analytic Brownian image, after subtracting their starting values, have equal quadratic variation. The Burkholder-Davis-Gundy inequalities compare their centered maximal moments with constants depending only on $p$. A boundary-point normalization $P(0)=0$ does not remove the starting imaginary value. For $P(z)=iz$ on a wedge of opening $\alpha \pi <\pi /2$, $\sup |\mathrm{Im}z|\le \tan (\alpha \pi )\sup |\mathrm{Re}z|$. If $\alpha p<1$ both relevant moments are finite, so an uncentered comparison requires a constant at least ${\cot }^p(\alpha \pi )$; no bound depending only on $p$ can hold over all such wedges. For fixed $\alpha >0$, polynomials with finite real-part maximal moment have degree $d<1/(\alpha p)$ by the wedge radial-maximum tail. They form a finite-dimensional vector space, and equivalence of norms in finite dimensions gives a valid constant depending on $p,\alpha$.

For Brownian motion started at zero, condition at time ${\lambda }^2$. If the stopping time exceeds ${\beta }^2{\lambda }^2$ while the running absolute maximum stays below $\delta \lambda$, the path must remain in a fixed interval for another $({\beta }^2-1){\lambda }^2$ time units. Its expected exit time from this interval is at most ${\delta }^2{\lambda }^2$. Markov's inequality yields the displayed bound. Integrating it against $d\Phi$ for a function of moderate growth, then absorbing the small tail term, proves $\mathbb{E}\Phi (\sqrt{\tau })\le C_{\Phi }\mathbb{E}\Phi (B_{\tau }^{\ast })$.

A continuous nondecreasing function $\Phi :[0,\infty )\to [0,\infty )$ with $\Phi (0)=0$ has moderate growth if the displayed inequality holds for some $a>1$ and finite $K$, uniformly in $x\ge 0$. Iteration gives a bound for any fixed positive dilation, and at most polynomial growth at infinity. The powers $x^p$, $p>0$, satisfy the condition. It allows a good-lambda tail comparison to be integrated into a moment comparison.

For $p\ge 2$ and a continuous local martingale starting at zero, the Itô formula for $|X{|}^p$, the Doob Lp maximal inequality and Hölder's inequality yield $\mathbb{E}{\sup }_{s\le t}|X_s{|}^p\le C_p\mathbb{E}⟨X{⟩}_t^{p/2}$. One possible constant is $C_p=[p(p-1)(p/(p-1){)}^p/2{]}^{p/2}$. Stop at increasing absolute-value levels to remove an initial boundedness assumption. This proves the upper half of the Burkholder-Davis-Gundy inequalities in this range.