Here, is the running maximum, is the quadratic variation, is a stopping time, and the exponent is a real number greater than or equal to 1. Then, and are positive constants depending on p, but independent of the choice of local martingale and stopping time. Furthermore, for continuous local martingales, which are the focus of this post, the inequality holds for all .
Since the quadratic variation used in my notes, by definition, starts at zero, the BDG inequality also required the local martingale to start at zero. This is not an important restriction, but it can be removed by requiring the quadratic variation to start at . Henceforth, I will assume that this is the case, which means that if we are working with the definition in my notes then we should add everywhere to the quadratic variation .
In keeping with the theme of the previous post on Doob’s inequalities, such martingale inequalities should have pathwise versions of the form
for predictable processes . Inequalities in this form are considerably stronger than (1), since they apply on all sample paths, not just on average. Also, we do not require M to be a local martingale — it is sufficient to be a (continuous) semimartingale. However, in the case where M is a local martingale, the pathwise version (2) does imply the BDG inequality (1), using the fact that stochastic integration preserves the local martingale property.
Lemma 1 Let X and Y be nonnegative increasing measurable processes satisfying for a local (sub)martingale N starting from zero. Then, for all stopping times .
Proof: Let be an increasing sequence of bounded stopping times increasing to infinity such that the stopped processes are submartingales. Then,
Letting n increase to infinity and using monotone convergence on the left hand side gives the result. ⬜
Moving on to the main statements of this post, I will mention that there are actually many different pathwise versions of the BDG inequalities. I opt for the especially simple statements given in Theorem 2 below. See the papers Pathwise Versions of the Burkholder-Davis Gundy Inequality by Bieglböck and Siorpaes, and Applications of Pathwise Burkholder-Davis-Gundy inequalities by Soirpaes, for slightly different approaches, although these papers do also effectively contain proofs of (3,4) for the special case of . As usual, I am using to represent the maximum of two numbers.
Theorem 2 Let X and Y be nonnegative continuous processes with . For any we have,
and, if X is increasing, this can be improved to,
If and X is increasing then,