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 1 Let X be a nonnegative cadlag submartingale. Then,
- for all .
- for all .
We also saw the following much stronger (sub)martingale inequality in the post on the maximum maximum of martingales with known terminal distribution.
Theorem 2 Let X be a cadlag submartingale. Then, for any real K and nonnegative real t,
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
for certain choices of functions . The aim of this post is to show how they have a more general `pathwise’ form,
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,
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 3 Let X be a cadlag process and t be a nonnegative time.
- For real ,
- If X is nonnegative and p,q are positive reals with then,
- If X is nonnegative then,
The pathwise inequalities can be stated quite succinctly in discrete time, in which case they become simple inequalities for finite sequences of real numbers. Given any such sequence , set . By considering the cadlag process over , the inequalities stated in theorem 3 are of the form,
The more general continuous-time inequalities can also be derived as a limit of the discrete-time case, so this is still quite general. Furthermore, if is a discrete-time submartingale, then taking expectations of (8) gives the discrete version of Doob’s inequalities, from which the continuous-time versions can be obtained by taking limits. Alternatively, we can show directly that Doob’s inequalities follow from the pathwise versions by taking expectations in continuous time.
Lemma 4 Let X be a cadlag submartingale taking values in an interval and satisfying (3) for some , where G is convex and increasing, and for some nonnegative and locally bounded predictable process . Then,
for all bounded stopping times .
Proof: As is locally bounded and nonnegative, is locally a submartingale and, hence, there exists a sequence of stopping times increasing to infinity such that are submartingales. Therefore, . Next, as G is convex and increasing, is a submartingale. So, taking expectations of (3) at time ,
Letting n increase to infinity and applying monotone convergence on the left hand side gives the result. ⬜
See Gushchin, On pathwise counterparts of Doob’s maximal inequalities, and the references contained therein, for more information on the pathwise approach.
To prove the pathwise inequalities, I start with the following simple, but very neat, identity. This actually extends to all measurable and locally bounded functions h in the case where X is a semimartingale (applying the monotone class theorem). However, I state it only for the required situation where is right-continuous and increasing, in which case the integral makes sense for all cadlag processes X.
Lemma 5 Let X be a cadlag process and be increasing and right-continuous. Then,
Proof: Applying (4),
So, to prove (9) it only needs to be shown that
and, as is an increasing process, this integral is understood in the pathwise Lebesgue-Stieltjes sense.
For any let be the last time before t at which ,
Then, is constant on and, hence,
As , we see that whenever and, hence, the above equality is identically zero. Next, if s is any time at which then, by right-continuity, for all sufficiently close to s, giving . Hence, using the fact that the positive rationals are dense in , by countable additivity,
as required. ⬜
We obtain the following consequence of lemma 9. I am using to denote the right-hand derivative which, as is convex, is right-continuous and increasing.
Corollary 6 Let X be a cadlag process and be convex with . Then,
Proof: By (9) it is sufficient to show that . See, for example, lemma 1 of the post on local times. Alternatively, by approximating by continuous functions, we can suppose that it is continuous (and increasing). So, by convexity,
Hence, for any sequence of times ,
Letting the mesh go to zero, bounded convergence shows that the right hand side of the above inequality tends to , giving the result. ⬜
I did not include terms involving the starting value in the pathwise inequalities, just in order to have very simple statements. However, they can be included if desired, in which case we should subtract from the right hand side of (10) and, the proof given still holds. In fact, if we do this, then (10) is very sharp, and is an equality whenever is continuous. I finally give proofs of the pathwise inequalities, as an application of corollary 6.
Dividing through by gives (5).
Next, apply (10) with and ,
Finally, apply (10) with and ,