Martingale energy inequality

Question

I am reading a book on BMO martingales which uses a so-called energy inequality. I have not been able to find a solid reference for this. Can someone please give a reference to these inequalities. Hopefully someone has heard of them, apparently they are "well known."

Answer 1

The only reference I know of is P. A. Meyer's Probability and Potentials. It is not an easy read. Chapter VII Section 6 "A few Results on Energy" can resolve your questions. Let $(X_t)$ be a right-continuous potential, i.e. there is an integrable increasing process $(A_t)$ ($A_0 = 0$, increasing paths, right continuous, $E A_\infty<\infty$), and $(M_t)$ is a right continuous modification of $E[A_\infty | \mathscr{F}_t]$, and $X_t = M_t-A_t$. Note: $(X_t)$ is a positive supermartingale with $\lim_{t \to \infty} X_t = 0$ (the "usual" definition of potential). For $p>1$ an integer we define the $p$-energy $$ e_p[(X_t)] := \frac{1}{p!}E[(A_\infty)^p]. $$

The $p$-energy inequality says that if $(X_t)$ is dominated by a constant $c$, then $e_p[(X_t)] \leq c^p$, i.e. $$ E[(A_\infty)^p] \leq p!\,c^p. $$

If you are reading Kazamaki's Continuous Exponential Martingales and BMO, he says for a BMO martingale $M$ "the energy inequalities" give $$ E[\langle M\rangle_\infty^n] \leq n! ||M||^{2n}_{BMO_2}. $$ Note that this is a straightforward application of what P.A. Meyer spends a chapter proving in his book (well, he proves for $p=2$ and suggests how to do it for $p>1$, but he does not assume $A$ is continuous, where I suspect there may a much shorter proof). Take $A = \langle M \rangle$, $p=n$ and $X_t$ the right continuous version of $E[\langle M\rangle_\infty- \langle M\rangle_t| \mathscr{F}_t]$. The assumption that the BMO norm is finite tells us exactly that $|X_t| \leq ||M||_{BMO_2}^2$. Another method of proving this fact is suggested in Revuz and Yor and uses optional projections and the fact that $A_\infty= p! \int_{0}^\infty \int_{u_1}^\infty\cdots \int_{u_{p-1}}^\infty d A_{u_{p-1}} \cdots dA_{u_2} d A_{u_1}$.