The Burkholder-Davis-Gundy inequality is a remarkable result relating the maximum of a local martingale with its quadratic variation. Recall that [X] denotes the quadratic variation of a process X, and is its maximum process.
Theorem 1 (Burkholder-Davis-Gundy) For any there exist positive constants such that, for all local martingales X with and stopping times , the following inequality holds.
Furthermore, for continuous local martingales, this statement holds for all .
A proof of this result is given below. For , the theorem can also be stated as follows. The set of all cadlag martingales X starting from zero for which is finite is a vector space, and the BDG inequality states that the norms and are equivalent.
The special case p=2 is the easiest to handle, and we have previously seen that the BDG inequality does indeed hold in this case with constants , . The significance of Theorem 1, then, is that this extends to all .
One reason why the BDG inequality is useful in the theory of stochastic integration is as follows. Whereas the behaviour of the maximum of a stochastic integral is difficult to describe, the quadratic variation satisfies the simple identity . Recall, also, that stochastic integration preserves the local martingale property. Stochastic integration does not preserve the martingale property. In general, integration with respect to a martingale only results in a local martingale, even for bounded integrands. In many cases, however, stochastic integrals are indeed proper martingales. The Ito isometry shows that this is true for square integrable martingales, and the BDG inequality allows us to extend the result to all -integrable martingales, for .
Theorem 2 Let X be a cadlag -integrable martingale for some , so that for each t. Then, for any bounded predictable process , is also an -integrable martingale.