the Burkholder-Davis-Gundy inequalities
/ Burkholder = BURK-holder; Gundy = GUN-dee /
The Burkholder-Davis-Gundy (BDG) inequalities say that, for a martingale, the size of the running maximum and the size of the square root of the quadratic variation are comparable in every L^p norm: controlling one controls the other, up to universal constants depending only on p. They are the master moment inequalities of martingale theory — the tool that converts a bound on the quadratic variation [M] (often computable, since it is a pathwise sum of squared increments) into a bound on the maximal function sup |M|, and vice versa. They underpin the L^p theory of stochastic integrals and the moment estimates for SDE solutions.
Statement: for every p in (0, infinity) there are universal constants c_p, C_p > 0, depending only on p, such that for every cadlag local martingale M with M_0 = 0 and every stopping time T, c_p E[ [M]_T^{p/2} ] <= E[ (sup_{t <= T} |M_t|)^p ] <= C_p E[ [M]_T^{p/2} ]. In words, the p-th moment of the maximal function is sandwiched between constant multiples of the (p/2)-th moment of the quadratic variation. The continuous-martingale case can use [M] = <M> (they agree), so the inequalities read in terms of the angle bracket. The remarkable strength is the range of p: unlike Doob's maximal inequality, which needs p > 1, BDG holds for ALL p > 0, including the hard small-p (and p = 1) regimes where Doob's inequality fails — this is precisely the Davis half, the p = 1 case being Davis's theorem.
Why they matter: in stochastic integration, [integral H dM]_t = integral H^2 d[M]_t, so BDG immediately bounds the L^p norm of a stochastic integral by an L^{p/2} norm of integral H^2 d[M] — the workhorse estimate for proving continuity of integrals, completeness of the H^p martingale spaces, and existence/uniqueness and moment bounds for SDEs (via Gronwall after a BDG step). The honest points: the constants c_p, C_p are universal but not 1 (the inequality is a two-sided comparison, not an identity — only at p = 2 does it collapse to the exact identity E[M_T^2] = E([M]_T) when M^2 - [M] is a martingale); the optimal constants are known in some regimes (Burkholder's work) and grow like sqrt(p) for large p; and BDG requires the quadratic variation [M], so it is genuinely a statement about martingales (or local martingales), not arbitrary processes.
For a stochastic integral X_t = integral_0^t H_s dB_s against Brownian motion, [X]_t = integral_0^t H_s^2 ds, so BDG gives E[ sup_{s <= t} |X_s|^p ] <= C_p E[ (integral_0^t H_s^2 ds)^{p/2} ]. This single inequality is what feeds the Gronwall argument in the existence-uniqueness proof for SDEs and the moment estimates for their solutions.
BDG turns a bound on the integrated quadratic variation into a bound on the whole running maximum, for every p > 0.
BDG is a two-sided comparison with universal constants, not an identity; only at p = 2 does it become the exact equality E[M_T^2] = E([M]_T), and unlike Doob's inequality it crucially holds for all p > 0 including p = 1 (the Davis case).