The best constant in a martingale moment inequality

Suppose we have a stochastic integral of the form $M_{t}=\int_{0}^{t}{H_{u}dW_{u}}$ and we know $M$ is a (true) martingale. It is known that for all $p\geq1$, $$\mathbb{E}[|M_{t}|^{p}]^{1/p} \leq C(p)\mathbb{E}[\langle M,M\rangle_{t}^{p/2}]^{1/p},$$ where $\langle M,M\rangle$ is the quadratic variation of the process and $C(p)$ is a constant that depends only on $p$. I am interested in finding the best possible/sharpest constant $C(p)$, both for small $p$ and (asymptotically) for large $p$.

I have read the literature and found some interesting results due to Novikov (1973), Burkholder (1984), Carlen and Kree (1991) etc. For instance, Burkholder found $C(p)=p-1$, for $p\geq2$ (the inequality is sharp for discrete time/cadlag martingales, but not in the continuous time case), whereas Carlen and Kree found $C(p)=2\sqrt{p}$ (the constant $2$ in front of $\sqrt{p}$ is the best possible), for $p\geq1$. Also, supposedly Pardoux ("Integrales Stochastiques Hilbertiennes", 1976) found $C(p)=\sqrt{p(p-1)/2}$, however this result was mentioned briefly in a different work and I could not find the original paper.

Is this last bound reliable, and for what values of $p$ does it hold (e.g. $p\geq2$, or $p$ integer)? Also, do you know of a sharp constant $C(p)$, especially for $2<p\leq4$?

• You might get an answer here, but this question is probably better suited to quant.stackexchange.com – GeoMatt22 Sep 11 '16 at 1:54