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Burkholder-Davis-Gundy Inequalities

The two main quantities in the Burkholder-Davis-Gundy inequalities
The Burkholder-Davis-Gundy inequalities are fundamental estimates for continuous local martingales. They compare two quantities attached to a martingale: its absolute maximum and its quadratic variation.
BURKHOLDER-DAVIS-GUNDY INEQUALITIES
For every p∈(0,∞)p \in (0,\infty), there exist constants cp>0c_p>0 and Cp<∞C_p<\infty such that, for every continuous local martingale MM vanishing at zero,
cp E ⁣[⟨M⟩∞p/2]≤E ⁣[(M∞∗)p]≤Cp E ⁣[⟨M⟩∞p/2].\begin{equation*}c_p \, \mathbb{E}\!\left[\langle M \rangle_{\infty}^{p/2}\right]\leq \mathbb{E}\!\left[(M_{\infty}^*)^p\right]\leq C_p \, \mathbb{E}\!\left[\langle M \rangle_{\infty}^{p/2}\right].\end{equation*}
Let M=(Mt)t≥0M=(M_t)_{t \geq 0} be a continuous local martingale with M0=0M_0=0. We write
Mt∗=sup⁡s≤t∣Ms∣\begin{equation*}M_t^* = \sup_{s \leq t} |M_s|\end{equation*}
for its running maximum, and ⟨M⟩t\langle M \rangle_t for its quadratic variation. The constants cpc_p and CpC_p depend only on pp. They are universal in the sense that the same constants work for all continuous local martingales on all probability spaces.
The inequality grew out of martingale maximal inequalities and square-function estimates. Burkholder, Davis, and Gundy [BurkholderDavisGundy1972] proved the original form in their 1972 paper on convex functions of martingale operators.
The process M∗M^* measures how large the martingale becomes at any time:
M∞∗=sup⁡t≥0∣Mt∣.\begin{equation*}M_{\infty}^* = \sup_{t \geq 0} |M_t|.\end{equation*}
The quadratic variation ⟨M⟩\langle M \rangle measures the accumulated random oscillation of the martingale. For Brownian motion BB, for example, ⟨B⟩t=t\langle B \rangle_t = t.
BDG inequalities are used constantly in stochastic integration. They allow estimates for stochastic integrals to be converted into estimates for quadratic variation, which is often easier to compute.
Take Mt=BtM_t=B_t, where BB is standard Brownian motion. Then
⟨B⟩t=t.\begin{equation*}\langle B \rangle_t = t.\end{equation*}
Applying BDG to the stopped martingale B⋅∧tB_{\cdot \wedge t} gives
E ⁣[sup⁡0≤s≤t∣Bs∣p]≍ptp/2.\begin{equation*}\mathbb{E}\!\left[\sup_{0 \leq s \leq t}|B_s|^p\right]\asymp_pt^{p/2}.\end{equation*}
This matches Brownian scaling: over time tt, Brownian motion typically has size t\sqrt{t}, so its pp-th moment has size tp/2t^{p/2}.
Let
Mt=∫0tHs dBswith⟨M⟩t=∫0tHs2 ds.\begin{equation*}M_t = \int_0^t H_s \, dB_s \qquad \text{with} \qquad \langle M\rangle_t = \int_0^t H_s^2 \, ds.\end{equation*}
where HH is a predictable integrand. Then BDG gives the estimate
E ⁣[(∫0tHs2 ds)p/2]≤cpE ⁣[sup⁡s≤t∣Ms∣p]≤CpE ⁣[(∫0tHs2 ds)p/2].\begin{equation*}\mathbb{E}\!\left[\left(\int_0^t H_s^2 \, ds\right)^{p/2}\right]\leq c_p\mathbb{E}\!\left[\sup_{s \leq t}|M_s|^p\right]\leq C_p\mathbb{E}\!\left[\left(\int_0^t H_s^2 \, ds\right)^{p/2}\right].\end{equation*}
For p=2p=2, this becomes especially simple:
E ⁣[sup⁡s≤t∣∫0sHu dBu∣2]≤4E∫0tHu2 du.\begin{equation*}\mathbb{E}\!\left[\sup_{s \leq t}\left|\int_0^s H_u \, dB_u\right|^2\right]\leq4\mathbb{E}\int_0^t H_u^2 \, du.\end{equation*}
The following two estimates prove the two sides of the BDG comparison only for p≥4p \geq 4. To extend the proof to 0<p≤40 < p \leq 4, see Chapter 4, Paragraph 4 of [RevuzYor2005].
Upper estimate for p≥2p\geq 2
For p≥2p\geq 2, there exists a constant CpC_p such that for every continuous local martingale MM with M0=0M_0=0,
E ⁣[(M∞∗)p]≤Cp E ⁣[⟨M,M⟩∞p/2].\begin{equation*}\mathbb{E}\!\left[(M_{\infty}^*)^p\right]\leq C_p \,\mathbb{E}\!\left[\langle M,M\rangle_{\infty}^{p/2}\right].\end{equation*}
\quad Proof. By stopping, it is enough to prove the result for bounded MM. Since the function x↦∣x∣px\mapsto |x|^p is twice differentiable, Itô's formula gives
∣M∞∣p=∫0∞p∣Ms∣p−1sgn⁡(Ms) dMs+12∫0∞p(p−1)∣Ms∣p−2 d⟨M,M⟩s.\begin{equation*}|M_{\infty}|^p=\int_0^{\infty} p |M_s|^{p-1}\operatorname{sgn}(M_s)\,dM_s+ \frac{1}{2}\int_0^{\infty} p(p-1)|M_s|^{p-2}\,d\langle M,M\rangle_s.\end{equation*}
Consequently,
E ⁣[∣M∞∣p]=p(p−1)2E ⁣[∫0∞∣Ms∣p−2 d⟨M,M⟩s]≤p(p−1)2E ⁣[(M∞∗)p−2⟨M,M⟩∞]≤p(p−1)2∥(M∞∗)p−2∥p/(p−2)∥⟨M,M⟩∞∥p/2.\begin{align*}\mathbb{E}\!\left[|M_{\infty}|^p\right]&=\frac{p(p-1)}{2}\mathbb{E}\!\left[\int_0^{\infty} |M_s|^{p-2}\,d\langle M,M\rangle_s\right] \\&\leq\frac{p(p-1)}{2}\mathbb{E}\!\left[(M_{\infty}^*)^{p-2}\langle M,M\rangle_{\infty}\right] \\&\leq\frac{p(p-1)}{2}\left\|(M_{\infty}^*)^{p-2}\right\|_{p/(p-2)}\left\|\langle M,M\rangle_{\infty}\right\|_{p/2}.\end{align*}
On the other hand, by Doob's inequality,
∥M∞∗∥p≤pp−1∥M∞∥p,⟹∥M∞∥pp≤p(p−1)2∥M∞∗∥pp−2∥⟨M,M⟩∞1/2∥p2.\begin{equation*}\left\|M_{\infty}^*\right\|_p\leq\frac{p}{p-1}\left\|M_{\infty}\right\|_p, \Longrightarrow \left\|M_{\infty}\right\|_p^p\leq\frac{p(p-1)}{2}\left\|M_{\infty}^*\right\|_p^{p-2}\left\|\langle M,M\rangle_{\infty}^{1/2}\right\|_p^2.\end{equation*}
while
∥(M∞∗)p−2∥p/(p−2)=∥M∞∗∥pp−2and∥⟨M,M⟩∞∥p/2=∥⟨M,M⟩∞1/2∥p2.\begin{equation*}\left\|(M_{\infty}^*)^{p-2}\right\|_{p/(p-2)}=\left\|M_{\infty}^*\right\|_p^{p-2}\qquad \text{and} \qquad\left\|\langle M,M\rangle_{\infty}\right\|_{p/2}=\left\|\langle M,M\rangle_{\infty}^{1/2}\right\|_p^2.\end{equation*}
Thus
∥M∞∥pp≤p(p−1)2∥M∞∗∥pp−2∥⟨M,M⟩∞1/2∥p2.\begin{equation*}\left\|M_{\infty}\right\|_p^p\leq\frac{p(p-1)}{2}\left\|M_{\infty}^*\right\|_p^{p-2}\left\|\langle M,M\rangle_{\infty}^{1/2}\right\|_p^2.\end{equation*}
Using Doob's inequality to replace ∥M∞∥p\left\|M_{\infty}\right\|_p by ∥M∞∗∥p\left\|M_{\infty}^*\right\|_p, we get
(p−1p)p∥M∞∗∥pp≤p(p−1)2∥M∞∗∥pp−2∥⟨M,M⟩∞1/2∥p2.\begin{equation*}\left(\frac{p-1}{p}\right)^p\left\|M_{\infty}^*\right\|_p^p\leq\frac{p(p-1)}{2}\left\|M_{\infty}^*\right\|_p^{p-2}\left\|\langle M,M\rangle_{\infty}^{1/2}\right\|_p^2.\end{equation*}
If ∥M∞∗∥p=0\left\|M_{\infty}^*\right\|_p=0, the estimate is trivial. Otherwise, dividing by ∥M∞∗∥pp−2\left\|M_{\infty}^*\right\|_p^{p-2} gives
∥M∞∗∥p2≤Cp∥⟨M,M⟩∞1/2∥p2.\begin{equation*}\left\|M_{\infty}^*\right\|_p^2 \leq C_p\left\|\langle M,M\rangle_{\infty}^{1/2}\right\|_p^2.\end{equation*}
Raising both sides to the power p/2p/2 proves
E ⁣[(M∞∗)p]≤CpE ⁣[⟨M,M⟩∞p/2].\begin{equation*}\mathbb{E}\!\left[(M_{\infty}^*)^p\right] \leq C_p\mathbb{E}\!\left[\langle M,M\rangle_{\infty}^{p/2}\right].\end{equation*}
□\Box
Lower estimate for p≥4p\geq 4
For p≥4p\geq 4, there exists a constant cpc_p such that for every continuous local martingale MM with M0=0M_0=0,
cp E ⁣[⟨M,M⟩∞p/2]≤E ⁣[(M∞∗)p].\begin{equation*}c_p \,\mathbb{E}\!\left[\langle M,M\rangle_{\infty}^{p/2}\right]\leq\mathbb{E}\!\left[(M_{\infty}^*)^p\right].\end{equation*}
\quad Proof. By stopping, it is enough to prove the result in the case where ⟨M,M⟩\langle M,M\rangle is bounded. In what follows, apa_p denotes a universal constant depending only on pp, but its value may vary from line to line. For instance, for two real numbers xx and yy,
∣x+y∣p≤ap(∣x∣p+∣y∣p).\begin{equation*}|x+y|^p \leq a_p\left(|x|^p+|y|^p\right).\end{equation*}
From the equality
Mt2=2∫0tMs dMs+⟨M,M⟩t,\begin{equation*}M_t^2 = 2\int_0^t M_s\,dM_s + \langle M,M\rangle_t,\end{equation*}
it follows that
E ⁣[⟨M,M⟩∞p/2]≤ap(E ⁣[(M∞∗)p]+E ⁣[∣∫0∞Ms dMs∣p/2]).\begin{equation*}\mathbb{E}\!\left[\langle M,M\rangle_{\infty}^{p/2}\right]\leq a_p\left(\mathbb{E}\!\left[(M_{\infty}^*)^p\right]+\mathbb{E}\!\left[\left|\int_0^{\infty} M_s\,dM_s\right|^{p/2}\right]\right).\end{equation*}
Applying the previous proposition to the local martingale ∫0⋅Ms dMs\int_0^\cdot M_s\,dM_s, we get
E ⁣[⟨M,M⟩∞p/2]≤ap(E ⁣[(M∞∗)p]+E ⁣[(∫0∞Ms2 d⟨M,M⟩s)p/4])≤ap(E ⁣[(M∞∗)p]+(E ⁣[(M∞∗)p]E ⁣[⟨M,M⟩∞p/2])1/2).\begin{align*}\mathbb{E}\!\left[\langle M,M\rangle_{\infty}^{p/2}\right]&\leq a_p\left(\mathbb{E}\!\left[(M_{\infty}^*)^p\right]+\mathbb{E}\!\left[\left(\int_0^{\infty} M_s^2\,d\langle M,M\rangle_s\right)^{p/4}\right]\right) \\&\leq a_p\left(\mathbb{E}\!\left[(M_{\infty}^*)^p\right]+\left(\mathbb{E}\!\left[(M_{\infty}^*)^p\right]\mathbb{E}\!\left[\langle M,M\rangle_{\infty}^{p/2}\right]\right)^{1/2}\right).\end{align*}
If we set
x=E ⁣[⟨M,M⟩∞p/2]1/2andy=E ⁣[(M∞∗)p]1/2,\begin{equation*}x =\mathbb{E}\!\left[\langle M,M\rangle_{\infty}^{p/2}\right]^{1/2}\qquad \text{and} \qquad y = \mathbb{E}\!\left[(M_{\infty}^*)^p\right]^{1/2},\end{equation*}
then the inequality above reads
x2−apxy−apy2≤0.\begin{equation*}x^2 - a_pxy - a_py^2 \leq 0.\end{equation*}
Thus xx is bounded by a constant depending only on pp times yy, which proves the proposition. □\Box

References

  • [BurkholderDavisGundy1972]D. L. Burkholder, B. J. Davis, and R. F. Gundy. Integral inequalities for convex functions of operators on martingales. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 223--240, 1972.
  • [RevuzYor2005]Daniel Revuz and Marc Yor. Continuous Martingales and Brownian Motion. Springer, 2005.