Month: June 2020

Pathwise Burkholder-Davis-Gundy Inequalities

As covered earlier in my notes, the Burkholder-David-Gundy inequality relates the moments of the maximum of a local martingale M with its quadratic variation,

\displaystyle c_p^{-1}{\mathbb E}[[M]^{p/2}_\tau]\le{\mathbb E}[\bar M_\tau^p]\le C_p{\mathbb E}[[M]^{p/2}_\tau]. (1)

Here, {\bar M_t\equiv\sup_{s\le t}\lvert M_s\rvert} is the running maximum, {[M]} is the quadratic variation, {\tau} is a stopping time, and the exponent {p} is a real number greater than or equal to 1. Then, {c_p} and {C_p} 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 {p > 0}.

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 {[M]_0=M_0^2}. 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 {M_0^2} everywhere to the quadratic variation {[M]}.

In keeping with the theme of the previous post on Doob’s inequalities, such martingale inequalities should have pathwise versions of the form

\displaystyle c_p^{-1}[M]^{p/2}+\int\alpha dM\le\bar M^p\le C_p[M]^{p/2}+\int\beta dM (2)

for predictable processes {\alpha,\beta}. 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 {X\le Y-N} for a local (sub)martingale N starting from zero. Then, {{\mathbb E}[X_\tau]\le{\mathbb E}[Y_\tau]} for all stopping times {\tau}.

Proof: Let {\tau_n} be an increasing sequence of bounded stopping times increasing to infinity such that the stopped processes {N^{\tau_n}} are submartingales. Then,

\displaystyle {\mathbb E}[1_{\{\tau_n\ge\tau\}}X_\tau]\le{\mathbb E}[X_{\tau_n\wedge\tau}]={\mathbb E}[Y_{\tau_n\wedge\tau}]-{\mathbb E}[N_{\tau_n\wedge\tau}]\le{\mathbb E}[Y_{\tau_n\wedge\tau}]\le{\mathbb E}[Y_\tau].

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 {r=1/2}. As usual, I am using {x\vee y} to represent the maximum of two numbers.

Theorem 2 Let X and Y be nonnegative continuous processes with {X_0=Y_0}. For any {0 < r\le1} we have,

\displaystyle (1-r)\bar X^r\le (3-2r)\bar Y^r+r\int(\bar X\vee\bar Y)^{r-1}d(X-Y) (3)

and, if X is increasing, this can be improved to,

\displaystyle \bar X^r\le (2-r)\bar Y^r+r\int(\bar X\vee\bar Y)^{r-1}d(X-Y). (4)

If {r\ge1} and X is increasing then,

\displaystyle \bar X^r\le r^{r\vee 2}\,\bar Y^r+r^2\int(\bar X\vee\bar Y)^{r-1}d(X-Y). (5)

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Pathwise Martingale Inequalities

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 {\bar X_t\equiv\sup_{s\le t}X_s} to denote the running maximum of a process.

Theorem 1 Let X be a nonnegative cadlag submartingale. Then,

  • {{\mathbb P}\left(\bar X_t \ge K\right)\le K^{-1}{\mathbb E}[X_t]} for all {K > 0}.
  • {\lVert\bar X_t\rVert_p\le (p/(p-1))\lVert X_t\rVert_p} for all {p > 1}.
  • {{\mathbb E}[\bar X_t]\le(e/(e-1)){\mathbb E}[X_t\log X_t+1]}.

In particular, for a cadlag martingale X, then {\lvert X\rvert} is a submartingale, so theorem 1 applies with {\lvert X\rvert} 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 2 Let X be a cadlag submartingale. Then, for any real K and nonnegative real t,

\displaystyle {\mathbb P}(\bar X_t\ge K)\le\inf_{x < K}\frac{{\mathbb E}[(X_t-x)_+]}{K-x}. (1)

This is particularly sharp, in the sense that for any distribution for {X_t}, 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 {x=0} 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

\displaystyle {\mathbb E}[F(\bar X_t)]\le{\mathbb E}[G(X_t)] (2)

for certain choices of functions {F,G\colon{\mathbb R}\rightarrow{\mathbb R}^+}. The aim of this post is to show how they have a more general `pathwise’ form,

\displaystyle F(\bar X_t)\le G(X_t) - \int_0^t\xi\,dX (3)

for some nonnegative predictable process {\xi}. 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, {\xi_t} will be of the form {h(\bar X_{t-})} for an increasing right-continuous function {h\colon{\mathbb R}\rightarrow{\mathbb R}}, so integration by parts can be used,

\displaystyle \int h(\bar X_-)\,dX = h(\bar X)X-h(\bar X_0)X_0 - \int X\,dh(\bar X). (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 {h(\bar X)}, so can be used as the definition of {\int h(\bar X_-)dX}. In the case where X is a semimartingale, integration by parts ensures that this agrees with the stochastic integral {\int\xi\,dX}. 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 {t\mapsto X_t} 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 {x=0}, we do not need to explicitly include this here.

Theorem 3 Let X be a cadlag process and t be a nonnegative time.

  1. For real {K > x},
    \displaystyle 1_{\{\bar X_t\ge K\}}\le\frac{(X_t-x)_+}{K-x}-\int_0^t\xi\,dX (5)

    where {\xi=(K-x)^{-1}1_{\{\bar X_-\ge K\}}}.

  2. If X is nonnegative and p,q are positive reals with {p^{-1}+q^{-1}=1} then,
    \displaystyle \bar X_t^p\le q^p X^p_t-\int_0^t\xi dX (6)

    where {\xi=pq\bar X_-^{p-1}}.

  3. If X is nonnegative then,
    \displaystyle \bar X_t\le\frac{e}{e-1}\left( X_t \log X_t +1\right)-\int_0^t\xi\,dX (7)

    where {\xi=\frac{e}{e-1}\log(\bar X_-\vee1)}.

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