|
3 | Clarified notation | ||
|
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. |
||||
|
|
2 | edited body; edited body | ||
|
The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |M|^{p})^{1/p} X|^{p})^{1/p} \le O(p) \cdot (\mathsf{E} \langle X \rangle ^{p/2})^{1/p}$ 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 $|M|$, |X|$, which is clearly suboptimal, since they should be Gaussian. |
||||
|
|
1 |
|
||
What is the optimal growth of the constant in BDG?The "standard" proof of Burkholder-Davis-Gundy inequalities found in books yields $(\mathsf{E} |M|^{p})^{1/p} \le O(p) \cdot (\mathsf{E} \langle X \rangle ^{p/2})^{1/p}$ 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 $|M|$, which is clearly suboptimal, since they should be Gaussian.
|
||||
