Weighted L\'epingle inequality

TL;DR

This paper establishes a new estimate for weighted p-th moments of martingale pathwise r-variation, relating it to the A_p characteristic of the weight, using a novel proof technique that avoids real interpolation.

Contribution

It introduces a new proof method for weighted inequalities in martingale variation without relying on real interpolation techniques.

Findings

Derived an estimate linking weighted p-th moments to A_p characteristic.

Provided a proof avoiding traditional real interpolation methods.

Enhanced understanding of weighted martingale inequalities.

Abstract

We prove an estimate for weighted $p$-th moments of the pathwise $r$-variation of a martingale in terms of the $A_{p}$ characteristic of the weight. The novelty of the proof is that we avoid real interpolation techniques.

Equations

$$Q_p(w):=\underset{\tau }{\sup }\Vert \mathbb{E}(w\text{nonscript}|\text{nonscript}{\mathcal{F}}_{\tau })\mathbb{E}(w^{-1/(p-1)}\text{nonscript}|\text{nonscript}{\mathcal{F}}_{\tau }{)}^{p-1}{\Vert }_{L^{\infty }(w)},$$

$$V^{r}X(\omega):=V^{r}_{n}X_{n}(\omega):=\sup_{u_{1}<u_{2}<\dotsb}\Bigl{(}\sum_{j}\lvert X_{u_{j-1}}(\omega)-X_{u_{j}}(\omega)\rvert^{r}\Bigr{)}^{1/r},$$

$$\Vert V^{r}X{\Vert }_{L^p(\Omega ,w)}\le C_p\sqrt{\frac{r}{r-2}}{Q}_p(w{)}^{\max (1,1/(p-1))}\Vert X{\Vert }_{L^p(\Omega ,w)}.$$

$$\sqrt{\log \frac{r}{r-2}}\text{when}r\to 2.$$

$$V^{2}X\le N^{1/2-1/r}{V}^{r}X\le C{V}^{r}X.$$

$${\tau }_0^{(m)}(\omega ):=0,{\tau }_{j+1}^{(m)}(\omega ):=\inf \{t\ge {\tau }_j^{(m)}(\omega )\text{nonscript}|\text{nonscript}d(X_t(\omega ),X_{{\tau }_j(\omega )}(\omega ))\ge 2^{-m}{M}_t(\omega )\}.$$

$$2<d(X_{t^{\prime }}(\omega ),X_t(\omega ))/(2^{-m}{M}_t(\omega ))\le 4.$$

$$d(X_{t^{\prime }}(\omega ),X_t(\omega ))\le 8d(X_{{\tau }_{j-1}^{(m)}(\omega )}(\omega ),X_{{\tau }_j^{(m)}(\omega )}(\omega )).$$

$$2\cdot 2^{-m}{M}_t<d(X_{t^{\prime }},X_t)\le d(X_{{\tau }^{\prime }},X_{t^{\prime }})+d(X_{{\tau }^{\prime }},X_t)<2^{-m}{M}_{t^{\prime }}+2^{-m}{M}_t\le 2\cdot 2^{-m}{M}_t,$$

$$d(X_{{\tau }_{j-1}^{(m)}},X_{{\tau }_j^{(m)}})\ge 2^{-m}{M}_{{\tau }^{\prime }}\ge 2^{-m}{M}_t/2\ge d(X_{t^{\prime }},X_t)/8.\text{qed}$$

$$V^{r}_{t}(X_{t}(\omega))^{r}\leq 8^{\rho}\sum_{m=2}^{\infty}\bigl{(}2^{-(m-2)}M_{\infty}(\omega)\bigr{)}^{r-\rho}\sum_{j=1}^{\infty}d(X_{\tau^{(m)}_{j-1}(\omega)}(\omega),X_{\tau^{(m)}_{j}(\omega)}(\omega))^{\rho}.$$

$$2<d(X_{u_l},X_{u_{l+1}})/(2^{-m}{M}_{u_{l+1}})\le 4.$$

$$d(X_{u_l},X_{u_{l+1}}{)}^r\le 8^{\rho }d(X_{{\tau }_{j-1}^{(m)}},X_{{\tau }_j^{(m)}}{)}^{\rho }\cdot (4\cdot 2^{-m}{M}_{u_{l+1}}{)}^{r-\rho }.$$

$$\sum _{l}d(X_{u_l},X_{u_{l+1}}{)}^r\le 8^{\rho }\sum _{m,j}d(X_{{\tau }_{j-1}^{(m)}},X_{{\tau }_j^{(m)}}{)}^{\rho }\cdot (2^{-(m-2)}{M}_{\infty }{)}^{r-\rho }.$$

$$\boxed{V_t^r(X_t(\omega ){)}^{\rho }\le 8^{\rho }\sum _{m=2}^{\infty }{2}^{-(m-2)(r-\rho )}\sum _{j=1}^{\infty }d(X_{{\tau }_{j-1}^{(m)}(\omega )}(\omega ),X_{{\tau }_j^{(m)}(\omega )}(\omega ){)}^{\rho }.}$$

$$M_{\infty }(\omega )\le V_t^r(X_t(\omega ))<\infty .$$

$$\Big{\lVert}\bigl{(}\sum_{j=1}^{\infty}\lvert X_{j}-X_{j-1}\rvert^{2}\bigr{)}^{1/2}\Big{\rVert}_{L^{p}(\Omega,w)}\leq C_{p}Q_{p}(w)^{\max(1,1/(p-1))}\lVert X\rVert_{L^{p}(\Omega,w)},$$

$$S_{(m)}(\omega):=\Bigl{(}\sum_{j=1}^{\infty}\lvert X_{\tau^{(m)}_{j-1}(\omega)}(\omega)-X_{\tau^{(m)}_{j}(\omega)}(\omega)\rvert^{2}\Bigr{)}^{1/2}$$

$$V^{r}X\leq 8\Bigl{(}\sum_{m=2}^{\infty}2^{-(m-2)(r-2)}S_{(m)}^{2}\Bigr{)}^{1/2}.$$

$$\lVert V^{r}X\rVert_{L^{p}(\Omega,w)}\leq 8\Bigl{(}\sum_{m=2}^{\infty}2^{-(m-2)(r-2)}\lVert S_{(m)}\rVert_{L^{p}(\Omega,w)}^{2}\Bigr{)}^{1/2}.$$

$$\Vert V^{r}X{\Vert }_{L^p(\Omega ,w)}$$

$$\displaystyle=8C_{p}\bigl{(}1-2^{-(r-2)}\bigr{)}^{-1/2}Q_{p}(w)\lVert X\rVert_{L^{p}(\Omega,w)}.$$

$$\Vert V^{r}X{\Vert }_{L^p(\Omega )}\le C_{\mathcal{X},p}\frac{r}{r-\rho }\Vert X{\Vert }_{L^p(\Omega )}.$$

Full text

PZ was partially supported by the Hausdorff Center for Mathematics (DFG EXC 2047)

1. Introduction

Lépingle’s inequality [Lép76] is a moment estimate for the pathwise $r$-variation of martingales. Finite $r$-variation is a parametrization-invariant version of Hölder continuity of order $1/r$ and plays a central role in Lyons’s theory of rough paths [Lyo98].

Lépingle’s inequality also found applications in ergodic theory [Bou89] and harmonic analysis [NOT10], see [MSZ18] and [DOP17, DDU18] and references therein, respectively, for recent developments in these directions. Weighted inequalities in harmonic analysis go back to [Muc72], and weighted variational inequalities have been studied since [Cre+09]. A major motivation of the weighted theory is the Rubio de Francia extrapolation theorem that allows to obtain vector-valued $L^{p}$ inequalities for all $1<p<\infty$ from scalar-valued weighted $L^{p}$ inequalities for a single $p$, see [Duo11, Section 3] for the most basic version of that result and [DPW17, Theorem 8.1] for a version applicable to martingales.

In this article, we prove a weighted version of Lépingle’s inequality for martingales with asymptotically sharp dependence on the $A_{p}$ characteristic of the weight. For dyadic martingales, weighted variational inequalities were first obtained in [DL12, Lemma 6.1] using the real interpolation approach as in [PX88, Bou89, JSW08, MSZ18]. The argument in the dyadic case relied on the so-called open property of $A_{p}$ classes, see e.g. [HPR12, Theorem 1.2], that is in general false for martingale $A_{p}$ classes, see the example in [BL79, §3] and [BB78]. Therefore, we use a new stopping time argument that is also simpler than the previous proofs of Lépingle’s inequality even in the classical, unweighted, case.

1.1. Notation

Let $(\Omega,(\mathcal{F}_{n})_{n=0}^{\infty},\mu)$ be a filtered probability space and $\mathcal{F}_{\infty}:=\vee_{n=0}^{\infty}\mathcal{F}_{n}$. A weight is a positive $\mathcal{F}_{\infty}$-measurable function $w:\Omega\to(0,\infty)$. The corresponding weighted $L^{p}$ norm is given by \lVert X\rVert_{L^{p}(\Omega,w)}:=\bigl{(}\int_{\Omega}\lvert X\rvert^{p}w\mathop{}\!\mathrm{d}\mu\bigr{)}^{1/p}. For $1<p<\infty$, the martingale $A_{p}$ characteristic of the weight $w$ is defined by

[TABLE]

where the supremum is taken over all adapted stopping times $\tau$. For comparison of our main result with the unweighted case, note that for $w\equiv 1$ we have $Q_{p}(w)=1$ for all $1<p<\infty$.

For $0<r<\infty$, a sequence of random variables $X=(X_{n})_{n}$, and $\omega\in\Omega$, the $r$-variation of $X$ at $\omega$ is defined by

[TABLE]

where the supremum is taken over arbitrary increasing sequences.

1.2. Main result

For an integrable $\mathcal{F}_{\infty}$-measurable function $X:\Omega\to\mathbb{R}$, the associated martingale is defined by $X_{n}:=\mathbb{E}\lparen X\nonscript\>|\allowbreak\nonscript\>\mathopen{}\mathcal{F}_{n}\rparen$. We have the following weighted moment estimate for the pathwise $r$-variation of this martingale.

Theorem 1.1**.**

For every $1<p<\infty$, there exists a constant $C_{p}<\infty$ such that, for every $r>2$, every filtered probability space $\Omega$, every weight $w$ on $\Omega$, and every integrable function $X:\Omega\to\mathbb{R}$, we have

[TABLE]

Remark 1.2*.*

By the monotone convergence theorem, Theorem 1.1 extends to càdlàg martingales.

Remark 1.3*.*

The example in [Qia98, Theorem 2.1] shows that, for $p=2$, the constant in (1.2) must diverge at least as

[TABLE]

Indeed, it is proved there that, if $(X_{n})_{n=0}^{N}$ is a martingale with i.i.d. increments that are Gaussian random variables with zero expectation and unit variance, then $(V^{2}X)^{2}\geq cN\log\log N$ with probability converging to $1$ as $N\to\infty$ for every $c<1/12$. In this case, choosing $r$ such that $r-2=1/\log N$, by Hölder’s inequality, we obtain

[TABLE]

This would lead to a contradiction if the constant in (1.2) diverges slower than stated in (1.3). The growth rate of the constant in (1.2) as $r\to 2$ is important e.g. in Bourgain’s multi-frequency lemma, as explained in [Zor15, §3.2].

Remark 1.4*.*

The growth rate of the constant in (1.2) as $r\to 2$ is also related to endpoint estimates, in which the $\ell^{r}$ norm in (1.1) is replaced by an Orlicz space norm. The results of [Tay72] for the Brownian motion suggest that it might be possible to use a Young function that decays as $x^{2}/\log\log x^{-1}$ when $x\to 0$. Such an estimate would imply an estimate of the form (1.3) for the constant in (1.2), and it would have useful consequences for rough differential equations, see [Dav08, Remark 5]. Our method allows to use Young functions that decay as $x^{2}/(\log x^{-1})^{1+\epsilon}$ when $x\to 0$.

Remark 1.5*.*

A Fefferman–Stein type weighted estimate that substitutes (1.2) in the case $p=1$ can be deduced from Corollary 2.4 and [Osȩ17, Theorem 1.1].

2. Stopping times and a pathwise $r$-variation bound

In this section, we estimate the $r$-variation of an arbitrary adapted process pathwise by a linear combination of square functions. We consider an adapted process $(X_{n})_{n}$ with values in an arbitrary metric space $(\mathcal{X},d)$ and extend the definition of $r$-variation (1.1) by replacing the absolute value of the difference by the distance. We have the following metric spaces $\mathcal{X}$ in mind.

1. (1)

In Theorem 1.1, we will use $\mathcal{X}=\mathbb{R}$ (and $\rho=2$ below). 2. (2)

In applications to the theory of rough paths, one takes $\mathcal{X}$ to be a free nilpotent group, see [FV10, §9]. 3. (3)

When $\mathcal{X}$ is a Banach space with martingale cotype $\rho\in[2,\infty)$, Corollary 2.4 can be used to recover [PX88, Theorem 4.2].

Definition 2.1**.**

Let $M_{t}:=\sup_{t^{\prime\prime}\leq t^{\prime}\leq t}d(X_{t^{\prime}},X_{t^{\prime\prime}})$. For each $m\in\mathbb{N}$, define an increasing sequence of stopping times by

[TABLE]

Lemma 2.2**.**

Let $0\leq t^{\prime}<t<\infty$ and $m\geq 2$. Suppose that

[TABLE]

Then there exists $j$ with $t^{\prime}<\tau^{(m)}_{j}(\omega)\leq t$ and

[TABLE]

Proof.

We fix $\omega$ and omit it from the notation. Let $j$ be the largest integer with $\tau^{\prime}:=\tau^{(m)}_{j}\leq t$. We claim that $\tau^{\prime}>t^{\prime}$. Suppose for a contradiction that $\tau^{\prime}<t^{\prime}$ (the case $\tau^{\prime}=t^{\prime}$ is similar but easier). By the hypothesis (2.2) and the assumption that $t,t^{\prime}$ are not stopping times, we obtain

[TABLE]

a contradiction. This shows $\tau^{\prime}>t^{\prime}$.

It remains to verify (2.3). Assume that $M_{\tau^{\prime}}<M_{t}/2$. Then, for some $\tau^{\prime}<\tau^{\prime\prime}\leq t$, we have $d(X_{\tau^{\prime}},X_{\tau^{\prime\prime}})\geq M_{t}/2\geq 2^{-m}M_{\tau^{\prime\prime}}$, contradicting maximality of $\tau^{\prime}$. It follows that

[TABLE]

Lemma 2.3**.**

For every $0<\rho<r<\infty$, we have the pathwise inequality

[TABLE]

Proof.

We fix $\omega$ and omit it from the notation. Let $(u_{l})$ be any increasing sequence. For each $l$ with $d(X_{u_{l}},X_{u_{l+1}})\neq 0$, let $m=m(l)\geq 2$ be such that

[TABLE]

Such $m$ exists because the distance is bounded by $M_{u_{l+1}}$.

Let $j$ be given by Lemma 2.2 with $t^{\prime}=u_{l}$ and $t=u_{l+1}$. Then

[TABLE]

Since each pair $(m,j)$ occurs for at most one $l$, this implies

[TABLE]

Taking the supremum over all increasing sequences $(u_{l})$, we obtain (2.4). ∎

Corollary 2.4**.**

For every $0<\rho<r<\infty$, we have the pathwise inequality

[TABLE]

Proof.

By the monotone convergence theorem, we may assume that $X_{n}$ becomes independent of $n$ for sufficiently large $n$. In this case,

[TABLE]

Substituting this inequality in (2.4) and canceling $V^{r}_{t}(X_{t}(\omega))^{r-2}$ on both sides, the claim follows. ∎

3. Proof of the weighted Lépingle inequality

Estimates in weighted spaces $L^{p}(\Omega,w)$ for differentially subordinate martingales with sharp dependence on the characteristic $Q_{p}(w)$ were obtained in [TTV15] in the discrete case (a simpler alternative proof is in [Lac17]) and [DP19] in the continuous case (a simpler alternative proof is in [DP16]). By Khintchine’s inequality, these results imply the following weighted estimate for the martingale square function.

Theorem 3.1** (cf. [DP16]).**

Let $(X_{j})_{j=0}^{\infty}$ be a martingale on a probability space $\Omega$. Then, for every $1<p<\infty$, we have

[TABLE]

where the constant $C_{p}<\infty$ depends only on $p$, but not on the martingale $X$ or the weight $w$.

An alternative proof that deals directly with the square function (3.1) appears in [BO18], but it is carried out only for continuous time martingales with continuous paths.

Proof of Theorem 1.1.

By extrapolation, see [DPW17, Theorem 8.1], it suffices to consider $p=2$. We will in fact give a direct proof for $2\leq p<\infty$. A similar argument also works for $1<p<2$, but gives a poorer dependence on $r$ than claimed in (1.2).

Let $\tau^{(m)}_{j}$ be the stopping times constructed in (2.1), and let

[TABLE]

denote the square function of the sampled martingale $(X_{\tau^{(m)}_{j}})_{j}$. Then Corollary 2.4 with $\mathcal{X}=\mathbb{R}$ and $\rho=2$ gives

[TABLE]

Since $2\leq p<\infty$, by Minkowski’s inequality, this implies

[TABLE]

Inserting the square function estimates (3.1) for the sampled martingales $(X_{\tau^{(m)}_{j}})_{j}$ on the right-hand side above, we obtain

[TABLE]

This implies (1.2). ∎

Remark 3.2*.*

One can also directly apply Theorem 3.1 for $1<p<2$, without passing through the extrapolation theorem. But this seems to lead to a faster growth rate of the constant in (1.2) as $r\to 2$.

Remark 3.3*.*

The unweighted Lépingle inequality (Theorem 1.1 with $w\equiv 1$) follows from Corollary 2.4 and the usual Burkholder–Davis–Gundy (BDG) inequality.

Remark 3.4*.*

Corollary 2.4 can be used to recover the $p$-variation rough path BDG inequality [CF19, Theorem 4.7]. For convex moderate functions $F(x)=x^{p}$ with $1<p<\infty$, the required estimate for the square function appearing in (2.5) can be deduced from the usual BDG inequality and [KZ19, Proposition 3.1]. The latter result can be extended to arbitrary convex moderate functions $F$ using the Davis martingale decomposition.

Remark 3.5*.*

Let $\rho\in[2,\infty)$, and let $\mathcal{X}$ be a Banach space with martingale cotype $\rho$. Using Corollary 2.4 and the $\rho$-function bounds for $\mathcal{X}$-valued martingales in [Pis16, Theorem 10.59], we see that, for every $1<p<\infty$, $r>\rho$, every filtered probability space $\Omega$, and every integrable function $X:\Omega\to\mathcal{X}$, we have

[TABLE]

In fact, it is possible to obtain a slightly better dependence on $r$, which we omit for simplicity. There is also an endpoint version of (3.2) at $p=1$, in which $X$ is replaced by the martingale maximal function on the right-hand side.

The vector-valued estimate (3.2) was first proved in [PX88, Theorem 4.2], with an unspecified dependence on $r$. The dependence on $r$ stated in (3.2) can also be obtained using Theorem 1.3 and Lemma 2.17 in [MSZ18], as well as real interpolation, but this method does not work at the endpoint $p=1$.

Acknowledgment.

This work was partially supported by the Hausdorff Center for Mathematics (DFG EXC 2047).