Excursions away from the Lipschitz minorant of a Lévy process

Recall that a function $g:\mathbb{R}\to\mathbb{R}$ is $\alpha$-Lipschitz for some $\alpha>0$ if $|g(s)-g(t)|\leq\alpha|s-t|$ for all $s,t\in\mathbb{R}$. Given a function $f:\mathbb{R}\to\mathbb{R}$, we say that $f$ dominates the $\alpha$-Lipschitz function $g$ if $g(t)\leq f(t)$ for all $t\in\mathbb{R}$. A necessary and sufficient condition that $f$ dominates some $\alpha$-Lipschitz function is that $f$ is bounded below on compact intervals and satisfies $\liminf_{t\to-\infty}f(t)-\alpha t>-\infty$ and $\liminf_{t\to+\infty}f(t)+\alpha t>-\infty$. When the function $f$ dominates some $\alpha$-Lipschitz function there is an $\alpha$-Lipschitz function $m$ dominated by $f$ such that $g(t)\leq m(t)$ for all $t\in\mathbb{R}$ for any $\alpha$-Lipschitz function $g$ dominated by $f$; we call $m$ the $\alpha$-Lipschitz minorant of $f$. The $\alpha$-Lipschitz minorant is given concretely by

The purpose of the present paper is to continue the study of the $\alpha$-Lipschitz minorants of the sample paths of a two-sided Lévy process begun in [AE14].

A two-sided Lévy process is a real-valued stochastic process indexed by the real numbers that has càdlàg paths, stationary independent increments, and takes the value $0$ at time $0$. The distribution of a two-sided Lévy process $X$ is characterized by the Lévy-Khintchine formula $\mathbb{E}[e^{i\theta(X_{t}-X_{s})}]=e^{-(t-s)\Psi(\theta)}$ for $\theta\in\mathbb{R}$ and $-\infty<s\leq t<\infty$, where

with $a\in\mathbb{R}$, $\sigma\in\mathbb{R}_{+}$, and $\Pi$ a $\sigma$-finite measure concentrated on $\mathbb{R}\setminus\{0\}$ satisfying $\int_{\mathbb{R}}(1\wedge x^{2})\,\Pi(dx)<\infty$ (see [Ber96, Sat99] for information about (one-sided) Lévy processes — the two-sided case involves only trivial modifications). In order to avoid having to consider annoying, but trivial, special cases in what follows, we henceforth assume that $X$ is not just deterministic linear drift $X_{t}=at$, $t\in\mathbb{R}$, for some $a\in\mathbb{R}$; that is, we assume that there is a non-trivial Brownian component ($\sigma>0$) or a non-trivial jump component ($\Pi\neq 0$).

The sample paths of $X$ have bounded variation almost surely if and only if $\sigma=0$ and $\int_{\mathbb{R}}(1\wedge|x|)\,\Pi(dx)<\infty$. In this case $\Psi$ can be rewritten as

We call $d\in\mathbb{R}$ the drift coefficient.

We now recall a few facts about the $\alpha$-Lipschitz minorants of the sample paths of $X$ from [AE14].

Either the $\alpha$-Lipschitz minorant exists for almost all sample paths of $X$ or it fails to exist for almost sample paths of $X$. A necessary and sufficient condition for the $\alpha$-Lipschitz minorant to exist for almost all sample paths is that $\mathbb{E}[|X_{1}|]<\infty$ and $|\mathbb{E}[X_{1}]|<\alpha$. We assume from now on that this condition holds and denote the corresponding minorant process by $(M_{t})_{t\in\mathbb{R}}$. Figure 1.1 shows an example of a typical Brownian motion sample path and its associated $\alpha$-Lipschitz minorant.

Set $\mathcal{Z}:=\{t\in\mathbb{R}:M_{t}=X_{t}\wedge X_{t-}\}$. We call $\mathcal{Z}$ the contact set. The random closed set $\mathcal{Z}$ is non-empty, stationary, and regenerative in the sense of [FT88] (see Definition 2.1 below for a re-statement of the definition). Such a random closed set either has infinite Lebesgue measure almost surely or zero Lebesgue measure almost surely.

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  If the sample paths of $X$ have unbounded variation almost surely, then $\mathcal{Z}$ has zero Lebesgue measure almost surely.

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  If $X$ has sample paths of bounded variation and $|d|>\alpha$, then $\mathcal{Z}$ has zero Lebesgue measure almost surely.

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  If $X$ has sample paths of bounded variation and $|d|<\alpha$, then $\mathcal{Z}$ has infinite Lebesgue measure almost surely.

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  If $X$ has sample paths of bounded variation and $|d|=\alpha$, then whether the Lebesgue measure of $\mathcal{Z}$ is infinite or zero is determined by an integral condition involving the Lévy measure $\Pi$ that we omit. In particular, if $\sigma=0$, $\Pi(\mathbb{R})<\infty$, and $|d|=\alpha$, then the Lebesgue measure of $\mathcal{Z}$ is almost surely infinite.

If $\mathcal{Z}$ has zero Lebesgue measure, then $\mathcal{Z}$ is either almost surely a discrete set or almost surely a perfect set with empty interior.

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  If $\sigma>0$, then $\mathcal{Z}$ is almost surely discrete.

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  If $\sigma=0$ and $\Pi(\mathbb{R})=\infty$, then $\mathcal{Z}$ is almost surely discrete if and only if

  $$\int_{0}^{1}t^{-1}\mathbb{P}\{X_{t}\in[-\alpha t,\alpha t]\}\,dt<\infty.$$

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  If $\sigma=0$ and $\Pi(\mathbb{R})<\infty$, then $\mathcal{Z}$ is almost surely discrete if and only if $|d|>\alpha$.

The outline of the remainder of the paper is as follows.

In Section 2 we show that the pair $((X_{t})_{t\in\mathbb{R}},\mathcal{Z})$ is a space-time regenerative system in the sense that if $D_{t}:=\inf\{s\geq t:s\in\mathcal{Z}\}$ for any $t\in\mathbb{R}$, then $((X_{D_{t}+u}-X_{D_{t}})_{u\geq 0},\mathcal{Z}\cap[D_{t},\infty)-D_{t})$ is independent of $((X_{u})_{u\leq D_{t}},\mathcal{Z}\cap(-\infty,D_{t}])$ with a distribution that does not depend on $t\in\mathbb{R}$. It follows that if $\mathcal{Z}$ is discrete, we write $0<T_{1}<T_{2}<\ldots$ for the successive positive elements of $\mathcal{Z}$, and we set $Y^{n}=(X_{T_{n}+t}-X_{T_{n}},\,0\leq t\leq T_{n+1}-T_{n})$, $n\in\mathbb{N}$, for the corresponding sequence of excursions away from the contact set, then these excursions are independent and identically distributed. When $\mathcal{Z}$ is not discrete there is a “local time” on $\mathcal{Z}\cap[0,\infty)$ and we give a description of the corresponding excursions away from the contact set as the points of a Poisson point process that is analogous to Itô’s description of the excursions of a Markov process away from a regular point.

Because $((X_{t})_{t\in\mathbb{R}},\mathcal{Z})$ is stationary, the key to establishing the space-time regenerative property is to show that if $D:=D_{0}$ is the first positive point in $\mathcal{Z}$, then $((X_{D+t}-X_{D})_{t\geq 0},\mathcal{Z}\cap[D,\infty)-D)$ is independent of $((X_{t})_{t\leq D},\mathcal{Z}\cap(-\infty,D])$. This is nontrivial because $D$ is most definitely not a stopping time for the canonical filtration of $X$ and so we can’t just apply the strong Markov property. We derive the claimed fact in Section 3 using a result from [Mil78] on the path decomposition of a real-valued Markov process at the time it achieves its global minimum. This result in turn is based on general last-exit decompositions from [PS72, GS74].

When the contact set is discrete we obtain some information about the excursion away from the $\alpha$-Lipschitz minorant that contains the time zero in Section 4 using ideas from [Tho00]. If $G$ is the last contact time before zero and $D$, as above, is the first contact time after zero, we show that $\frac{D}{D-G}$ is independent of $(X_{t}-X_{G},\,G\leq t<D)$ and uniformly distributed on $[0,1]$. This observation allows us to describe the finite-dimensional distributions of $(X_{t},\,G\leq t<D)$ in terms of those of $(X_{t},\,0\leq t<D)$, and we are able to determine the latter explicitly. The argument here is based on a generalization of the fact that if $V$ is a nonnegative random variable, $U$ is uniformly distributed on $[0,1]$, and $U$ and $V$ are independent, then it is possible to express the distribution of $V$ in terms of that of $UV$.

As before, write $Y_{n}$, $n\in\mathbb{N}$, for the independent, identically distributed sequence of excursions away from the contact set that occur at positive times in the case where the contact set is discrete. When $X$ is Brownian motion with drift $\beta$, where $|\beta|<\alpha$ in order for the $\alpha$-Lipschitz minorant to exist, we establish a path decomposition description for the common distribution of the $Y_{n}$ in Section 5. Using this path decomposition we can determine the distributions of quantities such as the length $T_{n+1}-T_{n}$ and the distribution of the final value $X_{T_{n+1}}-X_{T_{n}}$. Moreover, if we write $Y_{0}$ for the excursion straddling time zero, then we have the “size-biasing” relationship $\mathbb{E}[f(Y_{0})]=\mathbb{E}[f(Y_{n})(T_{n+1}-T_{n})]/\mathbb{E}[T_{n+1}-T_{n}]$, $n\in\mathbb{N}$, for nonnegative measurable functions $f$, and this allows us to recover information about the distribution of $Y_{0}$ from a knowledge of the common distribution of the “generic” excursions $Y_{n}$, $n\in\mathbb{N}$.

As we noted above, the random time $D$ is not a stopping time for the canonical filtration of $X$. In Section 6 we investigate the filtration obtained by enlarging the Brownian filtration in such way that $D$ becomes a stopping time. Martingales for the Brownian filtration become semimartingales in the enlarged filtration and we are able to describe their canonical semimartingale decompositions quite explicitly.

The paper finishes with two auxiliary sections. Section 7 contains some (deterministic) results about the $\alpha$-Lipschitz minorant construction that are used throughout the paper. Section 8 details two general lemmas about random times for Lévy processes that are used in Section 3 and Section 6.

Let $\Omega^{\leftrightarrow}$ (resp. $\Omega^{\rightarrow}$) denote the space of càdlàg $\mathbb{R}$-valued paths indexed by $\mathbb{R}$ (resp. $\mathbb{R}_{+}$). For $t\in\mathbb{R}$, define $\tau_{t}:\Omega^{\leftrightarrow}\to\Omega^{\rightarrow}$ by

For $t\in\mathbb{R}$ define $x_{t}:\Omega^{\leftrightarrow}\to\mathbb{R}$ by

For $t\in\mathbb{R}$, define $k_{t}:\Omega^{\leftrightarrow}\to\Omega^{\leftrightarrow}$ by

Let $\tilde{\Omega}^{\leftrightarrow}$ (resp. $\tilde{\Omega}^{\rightarrow}$) denote the class of closed subsets of $\mathbb{R}$ (resp. $\mathbb{R}_{+}$). For $t\in\mathbb{R}$ define $\tilde{\tau}_{t}:\tilde{\Omega}^{\leftrightarrow}\to\tilde{\Omega}^{\rightarrow}$ by

For $t\in\mathbb{R}$ define $d_{t}:\tilde{\Omega}^{\leftrightarrow}\to\mathbb{R}\cup\{+\infty\}$ by

and $r_{t}:\tilde{\Omega}^{\leftrightarrow}\to\mathbb{R}_{+}\cup\{+\infty\}$ by

With a slight abuse of notation, also use $d_{t}$ and $r_{t}$, $t\in\mathbb{R}_{+}$, to denote the analogously defined maps from $\tilde{\Omega}^{\rightarrow}$ to $\mathbb{R}_{+}\cup\{+\infty\}$.

Put $\bar{\Omega}^{\leftrightarrow}:=\Omega^{\leftrightarrow}\times\tilde{\Omega}^{\leftrightarrow}$ and $\bar{\Omega}^{\rightarrow}:=\Omega^{\rightarrow}\times\tilde{\Omega}^{\rightarrow}$. Define $\bar{\tau}_{t}:\bar{\Omega}^{\leftrightarrow}\to\bar{\Omega}^{\rightarrow}$ by

Define $\bar{d}_{t}:\bar{\Omega}^{\leftrightarrow}\to\mathbb{R}\cup\{+\infty\}$ by

Finally, for $t\in\mathbb{R}$ define the following $\sigma$-fields on $\bar{\Omega}^{\leftrightarrow}$:

and

Define $\bar{\mathcal{G}_{t}}^{\rightarrow}$ and $\bar{\mathcal{G}}^{\rightarrow}$ analogously.

Write $\tilde{\pi}^{\leftrightarrow}:\bar{\Omega}^{\leftrightarrow}\to\tilde{\Omega}^{\leftrightarrow}$ for the projection $\bar{\omega}^{\leftrightarrow}=(\omega^{\leftrightarrow},\tilde{\omega}^{\leftrightarrow})\mapsto\tilde{\omega}^{\leftrightarrow}$. Define $\tilde{\pi}^{\rightarrow}:\bar{\Omega}^{\rightarrow}\to\tilde{\Omega}^{\rightarrow}$ similarly. If $\bar{\mathbb{Q}}^{\leftrightarrow}$ is space-time regenerative with regeneration law $\bar{\mathbb{Q}}^{\rightarrow}$, then, in the sense of [FT88], the push-forward of $\bar{\mathbb{Q}}^{\leftrightarrow}$ by $\tilde{\pi}^{\leftrightarrow}$ is regenerative with regeneration law the push-forward of $\bar{\mathbb{Q}}^{\rightarrow}$ by $\tilde{\pi}^{\rightarrow}$. It follows that $\bar{\mathbb{Q}}^{\rightarrow}\{(\omega^{\rightarrow},\tilde{\omega}^{\rightarrow}):\tilde{\omega}^{\rightarrow}\;\text{is discrete}\}$ is either $1$ or $0$.

Suppose that the probability in question is $1$. Define $(\bar{\mathcal{G}}^{\leftrightarrow}_{t})_{t\in\mathbb{R}}$-stopping times $T_{1},T_{2},\ldots$ with $0<T_{1}<T_{2}<\ldots$ almost surely by

and

where $\bar{\theta}_{t}:\bar{\Omega}^{\leftrightarrow}\to\bar{\Omega}^{\leftrightarrow}$, $t\in\mathbb{R}$, are the shift maps given by $\bar{\theta}(\omega^{\leftrightarrow},\tilde{\omega}^{\leftrightarrow})=((\omega^{\leftrightarrow}_{t+u})_{u\in\mathbb{R}},\tilde{\omega}^{\leftrightarrow}-t)$. Let $\partial$ be an isolated cemetery state adjoined to $\mathbb{R}$. Define càdlàg $\mathbb{R}\cup\{\partial\}$-valued processes $Y^{n}=(Y^{n}_{t})_{t\in\mathbb{R}_{+}}$, $n\in\mathbb{N}$, by

where $\pi^{\rightarrow}:\bar{\Omega}^{\rightarrow}\to\Omega^{\rightarrow}$ is the projection $(\omega^{\rightarrow},\tilde{\omega}^{\rightarrow})\mapsto\omega^{\rightarrow}$. Then, under $\bar{\mathbb{Q}}^{\leftrightarrow}$, the sequence $Y^{n}$, $n\in\mathbb{N}$, is independent and identically distributed.

The path of of each $Y_{n}$ lies in the set $\Omega^{0,\partial}$ consisting of càdlàg functions $f:\mathbb{R}_{+}\to\mathbb{R}\cup\{\partial\}$ such that $f(0)=0$, $0<\inf\{s\geq 0:f(s)=\partial\}<\infty$, and $f(t)=\partial$ for all $t\geq\inf\{s\geq 0:f(s)=\partial\}$.

When the probability in question is $0$ there is a local time on our regenerative set and we can construct a Poisson random measure on the set $\mathbb{R}\times\Omega^{0,\partial}$ that records the excursions away from the contact set and the order in which they occur. We use the following theorem which is a restatement of [GP80, Corollary 3.1].

We can apply this theorem if we consider $O$ to be the space $\Omega^{0,\partial}$ of càdlàg paths that vanish at the origin and have finite lifetimes, and take $O_{k}$ to be the subspace of paths with lifetime at least $\frac{1}{k}$. We define $\mathbf{V}$ to be the point process of the excursions such that, for every $t\geq 0$, $\mathbf{V}_{t}$ is equal to the excursion whose right end point is $t$, with the convention that $\mathbf{V}_{t}={\dagger}$ if $t$ is not the right end point of an excursion. In the case where $\mathcal{Z}$ is not discrete, all the conditions of Theorem 2.6 can readily be checked and we obtain a time-changed Poisson point process.