Recall Doob’s inequalities, covered earlier in these notes, which bound expectations of functions of the maximum of a martingale in terms of its terminal distribution. Although these are often applied to martingales, they hold true more generally for cadlag submartingales. Here, I use to denote the running maximum of a process.

Theorem 1LetXbe a nonnegative cadlag submartingale. Then,

- for all .
- for all .
- .

In particular, for a cadlag martingale *X*, then is a submartingale, so theorem 1 applies with in place of *X*.

We also saw the following much stronger (sub)martingale inequality in the post on the maximum maximum of martingales with known terminal distribution.

Theorem 2LetXbe a cadlag submartingale. Then, for any realKand nonnegative realt,

(1)

This is particularly sharp, in the sense that for *any* distribution for , there exists a martingale with this terminal distribution for which (1) becomes an equality simultaneously for all values of *K*. Furthermore, all of the inequalities stated in theorem 1 follow from (1). For example, the first one is obtained by taking in (1). The remaining two can also be proved from (1) by integrating over *K*.

Note that all of the submartingale inequalities above are of the form

(2) |

for certain choices of functions . The aim of this post is to show how they have a more general `pathwise’ form,

(3) |

for some nonnegative predictable process . It is relatively straightforward to show that (2) follows from (3) by noting that the integral is a submartingale and, hence, has nonnegative expectation. To be rigorous, there are some integrability considerations to deal with, so a proof will be included later in this post.

Inequality (3) is required to hold almost everywhere, and not just in expectation, so is a considerably stronger statement than the standard martingale inequalities. Furthermore, it is not necessary for *X* to be a submartingale for (3) to make sense, as it holds for all semimartingales. We can go further, and even drop the requirement that *X* is a semimartingale. As we will see, in the examples covered in this post, will be of the form for an increasing right-continuous function , so integration by parts can be used,

(4) |

The right hand side of (4) is well-defined for *any* cadlag real-valued process, by using the pathwise Lebesgue–Stieltjes integral with respect to the increasing process , so can be used as the definition of . In the case where *X* is a semimartingale, integration by parts ensures that this agrees with the stochastic integral . Since we now have an interpretation of (3) in a pathwise sense for all cadlag processes *X*, it is no longer required to suppose that *X* is a submartingale, a semimartingale, or even require the existence of an underlying probability space. All that is necessary is for to be a cadlag real-valued function. Hence, we reduce the martingale inequalities to straightforward results of real-analysis not requiring any probability theory and, consequently, are much more general. I state the precise pathwise generalizations of Doob’s inequalities now, leaving the proof until later in the post. As the first of inequality of theorem 1 is just the special case of (1) with , we do not need to explicitly include this here.

Theorem 3LetXbe a cadlag process andtbe a nonnegative time.

- For real ,

(5) where .

- If
Xis nonnegative andp,qare positive reals with then,

(6) where .

- If
Xis nonnegative then,

(7) where .