# What is the optimal growth of the constant in BDG?

Let $X$ be a continuous local martingale, and $\langle X \rangle$ be its quadratic variation process. The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |X|^{p})^{1/p} \le O(p) \cdot (\mathsf{E} \langle X \rangle ^{p/2})^{1/p}$ for large $p$.

Can the growth rate be improved to, say, $O(p^{1/2})$? For example, if $\langle X \rangle$ is bounded, this estimate gives exponential tails for $|X|$, which is clearly suboptimal, since they should be Gaussian.

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What is $\langle X \rangle$? –  Bill Johnson May 31 '12 at 20:49
Quadratic variation. Updated the post to clarify this. –  Alexander Shamov May 31 '12 at 20:52
The best constants are known, and you can't do better than p-1 for p > 2. This was proven by Davis I think, but I'm not sure if that applies specifically to continuous martingales. –  George Lowther May 31 '12 at 21:56

I know a version which exactly gives the constant $O(p^{1/2})$ for $p\ge 2$. It is contained in a lecture note by D. Khoshnevisan on SPDE.

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Do you mean Theorem 5.27 in math.utah.edu/~davar/ps-pdf-files/SPDE.pdf? On the bottom of p.17 he says, in his notation, something equivalent to $\left(\phi\left(t\right)\right)^{1/p}\le a_{p}^{1/2}\left(\mathsf{E}\left\langle N\right\rangle _{t}^{p/2}\right)^{1/p}$, where $\phi\left(t\right)=\mathsf{E}\sup_{\left[0,t\right]}\left|N\right|^{p}$ and $a_{p}=\frac{p\left(p-1\right)}{2}\left(\frac{p}{p-1}\right)^{p}$. This seems to give the $O(p)$ bound that I was talking about, not $O(p^{1/2})$. –  Alexander Shamov Jan 23 '14 at 23:19
Please see p. 196 of the file: stt.msu.edu/CBMS2013/D_Khoshnevisan_Lecture.pdf –  epsilon Jan 24 '14 at 14:40