# Creeping of Lévy processes through curves

Loïc Chaumont  Loïc Chaumont, Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000
Angers, France<EMAIL_ADDRESS>and  Thomas Pellas  Thomas
Pellas, Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France
<EMAIL_ADDRESS>
###### Abstract.

A Lévy process is said to creep through a curve if, at its first passage time
across this curve, the process reaches it with positive probability. We first
study this property for bivariate subordinators. Given the graph
$\\{(t,f(t)):t\geq 0\\}$ of any continuous, non increasing function $f$ such
that $f(0)>0$, we give an expression of the probability that a bivariate
subordinator $(Y,Z)$ issued from 0 creeps through this graph in terms of its
renewal function and the drifts of the components $Y$ and $Z$. We apply this
result to the creeping probability of any real Lévy process through the graph
of any continuous, non increasing function at a time where the process also
reaches its past supremum. This probability involves the density of the
renewal function of the bivariate upward ladder process as well as its drift
coefficients. We also investigate the case of Lévy processes conditioned to
stay positive creeping at their last passage time below the graph of a
function. Then we provide some examples and we give an application to the
probability of creeping through fixed levels by stable Ornstein-Uhlenbeck
processes. We also raise a couple of open questions along the text.

###### Key words and phrases:

Lévy process; bivariate subordinator; creeping; Ornstein-Uhlenbeck process

###### 2020 Mathematics Subject Classification:

60G50

##  1\. Introduction

A real Lévy process $X$ starting from 0 under $\mathbb{P}$ is said to creep
through the level $x>0$ if

(1.1) $\mathbb{P}(X_{\tau_{x}^{+}}=x)>0\,,$

where $\tau_{x}^{+}=\inf\\{t:X_{t}>x\\}$. The first study of this property is
due to Millar [13] who, according to his own words, gives a ’reasonably
complete solution’ of the creeping problem, that is conditions bearing on the
characteristics of $X$ for (1.1) to hold. It is proved in [13] that if (1.1)
holds for some $x>0$, then it holds for all $x>0$. We will then simply say
that $X$ creeps upward. According to the same paper, when $X$ has bounded
variation, it creeps upward if and only if it has positive drift. When $X$ has
unbounded variation, it creeps upward if it has a Brownian part or if its Lévy
measure $\pi$ satisfies $\int_{0}^{1}xd\pi(x)<\infty$. It does not creep
upward if it has no Brownian part and $\int_{-1}^{0}|x|d\pi(x)<\infty$. When
$X$ has no Brownian part and
$\int_{0}^{1}xd\pi(x)=\int_{-1}^{0}|x|d\pi(x)=\infty$, both possibilities can
occur and an analytic criterion in terms of the characteristic exponent of $X$
is given. As pointed out by Vigon [19], this criterion has the disadvantage of
involving both the characteristic exponent of $X$ and its Lévy measure and we
rarely know both at the same time. This gap is filled in [19] where an
integral test only bearing on the Lévy measure is established for a Lévy
process with unbounded variation and no Brownian part to creep upward. Let us
also mention the work of Rogers [15] who deduced some results of Millar. It
seems that it was in the latter article that the term ’creeping’ first
appeared.

According to a remark first made in [13] and then used in [19], the upward
ladder height process $H$ of $X$ satisfies $H_{S_{x}^{+}}=X_{\tau_{x}^{+}}$,
$x>0$, where $S_{x}^{+}=\inf\\{t:H_{t}>x\\}$, so that creeping of general Lévy
processes can in fact be reduced to that of subordinators. More specifically,
$X$ creeps upward if and only if $H$ creeps upward, which holds if and only if
the drift $d_{H}$ of $H$ is positive according to Neveu [14] and Kesten [11].
These authors also proved that in this case, the renewal measure of the ladder
height process admits a continuous and bounded density $u(x)$ and that the
creeping probability is then given by

(1.2) $\mathbb{P}(X_{\tau_{x}^{+}}=x)=d_{H}u(x)\,.$

So the whole creeping problem for a Lévy process $X$ boils down to giving
conditions on its characteristics for $d_{H}$ to be positive. Let us also
mention a more recent result in this direction by Griffin and Maller [9] who
studied the law of the creeping time, that is the law of the first passage
time conditionally on the creeping event.

In the present work, we are interested in the probability for Lévy processes
to creep through the graph of continuous non increasing functions. We first
consider the case of a bivariate subordinator $(Y,Z)$, that is a two
dimensional Lévy process issued from 0, whose both coordinates are non
decreasing. Let $f:(0,\infty)\mapsto(0,\infty)$ be a continuous non increasing
function. Our first main result gives an expression of the probability that
the path $\\{(Y_{t},Z_{t}):t>0\\}$ crosses the graph $\\{(t,f(t)):t>0\\}$ in a
continuous way, that is

$\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}<Y_{S}<u_{1}),\;\;\;\mbox{where}\;\;\;S=\inf\\{t:Z_{t}>f(Y_{t})\\},$

for all $0\leq u_{0}<u_{1}\leq\infty$. This expression involves the drifts of
$Y$ and $Z$ and the renewal function of the process $(Y,Z)$, see Theorem 2.1.
In particular, for $(Y,Z)$ to creep through the graph of $f$ with positive
probability, it is necessary that at least one of the drifts of $Y$ and $Z$ is
positive. This result will then be applied in Theorem 2.2 to the probability
for a real Lévy process $X$ to creep at its supremum through the graph of $f$.
More formally, define the first passage time above the curve of $f$ by

$T_{f}=\inf\\{t>0:X_{t}>f(t)\\}\,.$

Then we give an explicit expression of the probability
$\mathbb{P}(X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}}),\,t_{0}<T_{f}<t_{1})$,
for all $0\leq t_{0}<t_{1}\leq\infty$, where $\overline{X}_{T_{f}}=\sup_{t\leq
T_{f}}X_{t}$. Our result shows that this probability is composed of two terms.
The first one represents the contribution of the accumulation of jumps of the
ladder height process. This shows that the function acts as a fixed
(horizontal) barrier through which the process creeps upward. Then the second
term is due to the accumulation of jumps of the ladder time process. When its
drift is positive, that is when 0 is not regular for $(-\infty,0)$, this term
represents a kind of creeping forward, as if the function acted as a vertical
barrier. Note that our result actually provides the law of $T_{f}$
conditionally on the event $\\{X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}})\\}$.
Our only assumption is that the renewal measure of the bivariate upward ladder
process has a continuous density. As proved in the present work, it is ensured
whenever $X_{t}$ has a bounded density for each $t>0$. The latter result can
then be applied to the creeping of the Lévy process $X$ conditioned to stay
positive. Thanks to a time reversal argument, we obtain in Theorem 2.3 the
probability for this process to creep at its last passage time below the
function $f$.

The problem of creeping through fixed levels by Ornstein-Uhlenbeck processes
is at the origin of our work. Indeed, the process
$Z_{t}=z+X_{t}-\gamma\int_{0}^{t}Z_{s}ds$, $t\geq 0$, $\gamma>0$, $z<0$ creeps
through $y\in(z,0)$ if and only if $X$ creeps through the adapted, continuous
decreasing functional $t\mapsto y-z+\gamma\int_{0}^{t}Z_{s}ds$. This problem
is still an open question. However, when $X$ is stable, it can be reduced to
the creeping of $X$ through a deterministic function. Then we prove, as an
application of our main result, that Ornstein-Uhlenbeck processes driven by
stable processes with index less than 1 always creep through some fixed
levels.

We present our main results in the next section. Theorems 2.1, 2.2 and 2.3
will be proved in Sections 5, 6 and 7, respectively. We give some examples for
which the creeping probabilities can be computed explicitly in Section 3, and
in Section 4 we present the above-mentioned application of our results to the
creeping property of stable Ornstein-Uhlenbeck processes.

##  2\. Main results

###  2.1. Creeping of bivariate subordinators through curves

A bivariate subordinator $(Y,Z)$ is a possibly killed two dimensional Lévy
process starting from 0, whose coordinates $Y$ and $Z$ are non decreasing. We
set $Y_{t}=Z_{t}=\infty$, for $t\in[\zeta,\infty]$, where $\zeta$ is the
lifetime of $(Y,Z)$. In what follows, the drift of a (univariate) subordinator
$Y$ will be denoted by $d_{Y}$. Moreover, a measure on some subset of
$\mathbb{R}^{d}$ is said to be absolutely continuous if it is absolutely
continuous with respect to the Lebesgue measure on this subset.

Our first theorem concerns the creeping of a bivariate subordinator $(Y,Z)$
through the curve defined by the graph of a continuous non increasing function
$f$. More specifically, we are interested in the probability that the path
$\\{(Y_{t},Z_{t}):t\geq 0\\}$ crosses continuously the graph
$\\{(u,f(u)):u\geq 0\\}$ of $f$. This event can be written as
$\\{Z_{S}=f(Y_{S})\\}$, where $S=\inf\\{t:Z_{t}>f(Y_{t})\\}$, see Figure 1.
When it holds, we will say that $(Y,Z)$ creeps through the graph of $f$.

######  Theorem 2.1.

Let $(Y,Z)$ be a bivariate subordinator with semigroup $\mathbb{P}(Y_{t}\in
dy,\,Z_{t}\in dz)=p_{t}(dy,dz)$, $t>0$, $y,z\in[0,\infty)$. Assume that the
renewal measure $v(dy,dz):=\int_{0}^{+\infty}p_{t}(dy,dz)\,dt$ is absolutely
continuous on $(0,\infty)^{2}$ and that its density $(y,z)\mapsto v(y,z)$ is
continuous on $(0,\infty)^{2}$. Let $f:(0,\infty)\rightarrow(0,\infty)$ be a
continuous, non increasing function. Set $f(0)=\lim_{t\rightarrow 0+}f(t)$,
$f(\infty)=\lim_{t\rightarrow\infty}f(t)$ and define,

$S=\inf\\{t:Z_{t}>f(Y_{t})\\}\,.$

  * $1.$

Then almost surely, $0<S<\infty$, $Y_{S-}<\infty$ and $Z_{S-}<\infty$.
Moreover, the process $(Y,Z)$ can neither jump on the graph
$\\{(u,f(u)):u>0\\}$ nor jump from this graph, that is

$\mathbb{P}(Z_{S-}=f(Y_{S-}),\Delta(Y,Z)_{S}\neq
0)=\mathbb{P}(Z_{S}=f(Y_{S}),\Delta(Y,Z)_{S}\neq 0)=0\,,$

where $\Delta(Y,Z)_{S}=(Y,Z)_{S}-(Y,Z)_{S-}$.

  * $2.$

Let $u_{0},u_{1}$ be such that $0\leq u_{0}<u_{1}\leq\infty$. Then the
creeping probability of $(Y,Z)$ through the graph of $f$ is given by,

(2.3)
$\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}<Y_{S}<u_{1})=d_{Z}\int_{u_{0}}^{u_{1}}v(u,f(u))\,du-
d_{Y}\int_{u_{0}}^{u_{1}}v(u,f(u))\,df(u)\,.$

  * $3.$

Assume that the measure $p_{t}(dy,dz)$ is absolutely continuous on
$(0,\infty)^{2}$ for all $t>0$ and that the densities $(t,y,z)\mapsto
p_{t}(y,z)$ are continuous on $(0,\infty)^{3}$. Then for all
$t_{0},t_{1},u_{0},u_{1}$ such that $0\leq t_{0}<t_{1}\leq\infty$ and $0\leq
u_{0}<u_{1}\leq\infty$,

$\displaystyle\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}<Y_{S}<u_{1},\,t_{0}<S<t_{1})=$
(2.4) $\displaystyle\qquad\qquad
d_{Z}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,du-
d_{Y}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,df(u)\,.$

Let us mention that an intermediary result between those of part 2. and part
3. of Theorem 2.1 can be stated as follows:

  * $2^{\prime}.$

Assume that for some $0\leq t_{0}<t_{1}\leq\infty$, the measure
$\int_{t_{0}}^{t_{1}}p_{t}(dy,dz)\,dt$ is absolutely continuous on
$(0,\infty)^{2}$, with density $\bar{v}(y,z)$ and that $\bar{v}$ is continuous
on $(0,\infty)^{2}$. Then for all $u_{0},u_{1}$ such that $0\leq
u_{0}<u_{1}\leq\infty$,

$\displaystyle\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}<Y_{S}<u_{1},\,t_{0}<S<t_{1})=$
$\displaystyle\qquad\qquad d_{Z}\int_{u_{0}}^{u_{1}}\bar{v}(u,f(u))\,du-
d_{Y}\int_{u_{0}}^{u_{1}}\bar{v}(u,f(u))\,df(u)\,.$

The proof of $2^{\prime}.$ is quite similar to this of part 3. of Theorem 2.1,
see subsections 5.3 and 5.4, so it is omitted.

$0$$Y_{S}$$Y_{t}$$Z_{t}$$Z_{S}=f(Y_{S})$

Figure 1. A sample path of $(Y,Z)$ creeping through the graph of $f$

Let us now focus on two direct applications of Theorem 2.1. The first one
gives the probability for the Euclidian norm $\|(Y,Z)\|=\sqrt{Y^{2}+Z^{2}}$ of
a bivariate subordinator to creep through a fixed level, that is
$\mathbb{P}(\|(Y,Z)_{U_{a}}\|=a)$, for $a>0$ and
$U_{a}=\inf\\{t:\|(Y,Z)_{t}\|>a\\}$. This is simply obtained from Theorem 2.1
by choosing $f(y)=\sqrt{a^{2}-y^{2}}$, so that $S=U_{a}$.

######  Corollary 2.1.

Let $(Y,Z)$ be a bivariate subordinator. Then with the same notation and
assumptions than in Theorem 2.1,

(2.5)
$\mathbb{P}(\|(Y,Z)\|_{U_{a}}=a)=d_{Z}\int_{0}^{a}v(u,\sqrt{a^{2}-u^{2}})\,du+d_{Y}\int_{0}^{a}\frac{uv(u,\sqrt{a^{2}-u^{2}})}{\sqrt{a^{2}-u^{2}}}\,du\,.$

The second direct application concerns the probability for a (one dimensional)
subordinator to creep through the graph of a continuous non increasing
function. Let $X$ be a subordinator whose semigroup has continuous densities
$(t,x)\mapsto p_{t}(x)$ on $(0,\infty)^{2}$ and let us apply identity (2.3) to
the bivariate subordinator $(Y_{t},Z_{t})=(t,X_{t})$. The renewal measure of
$(Y,Z)$ is then $v(t,x)\,dt\,dx=p_{t}(x)\,dt\,dx$. Hence it satisfies the
assumption of part 2. of Theorem 2.1. Moreover, since $Y_{t}=t$, one has
$Y_{S}=\inf\\{t:X_{t}>f(t)\\}$. This leads to the following corollary.

######  Corollary 2.2.

Assume that $X$ is a subordinator whose semigroup $\mathbb{P}(X_{t}\in dx)$ is
absolutely continuous on $(0,\infty)$ for each $t>0$, with continuous
densities $(t,x)\mapsto p_{t}(x)$ on $(0,\infty)^{2}$. Set
$T_{f}=\inf\\{t>0:X_{t}>f(t)\\}$, where $f$ is as in the statement of Theorem
2.1. Then for all $t_{0}$ and $t_{1}$ such that $0\leq t_{0}<t_{1}\leq\infty$,

(2.6)
$\displaystyle\mathbb{P}(X_{T_{f}}=f({T_{f}}),t_{0}<{T_{f}}<t_{1})=d_{X}\int_{t_{0}}^{t_{1}}p_{u}(f(u))\,du-\int_{t_{0}}^{t_{1}}p_{u}(f(u))\,df(u)\,.$

Let us mention that the creeping probability for subordinators with no drift
has been determined in [6] when $f$ is differentiable. We will actually extend
(2.6) to the creeping probability of any Lévy process at its supremum in the
next section.

Note that if $f$ is decreasing, then (2.3) can also be written as

$\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}<Y_{S}<u_{1})=d_{Y}\int_{u_{0}}^{u_{1}}v(f^{-1}(u),u)\,du-
d_{Z}\int_{u_{0}}^{u_{1}}v(f^{-1}(u),u)\,df^{-1}(u)\,,$

and this applies to (2.5) and (2.6).

The question of finding necessary and sufficient conditions for a bivariate
subordinator $(Y,Z)$ to creep through a given continuous, non increasing
function is still open. In particular, nothing is known when the renewal
measure $v(dy,dz)$ is not absolutely continuous. Looking at (2.3) in Theorem
2.1, one is tempted to think that when $d_{Y}>0$ and $d_{Z}>0$, the
probability $\mathbb{P}(Z_{S}=f(Y_{S}))$ is always positive and when $d_{Y}=0$
and $d_{Z}=0$, this probability is always equal to 0. Let us also mention that
the absolute continuity condition of Theorem 2.1 is discussed in Subsection
5.6.

It seems hardly possible to obtain an expression of the creeping probability
when $f$ is a general continuous function. When $f$ is continuous and non
decreasing we can still obtain an upper bound for the creeping probability as
the following proposition shows.

######  Proposition 2.1.

Let $(Y,Z)$ be a bivariate subordinator which satisfies the same assumptions
as in part $2$. of Theorem 2.1. Keep the same notation as in this theorem. Let
$f:(0,\infty)\rightarrow(0,\infty)$ be a continuous non decreasing function
such that $\lim_{t\rightarrow 0+}f(t)>0$. Set $f(0)=\lim_{t\rightarrow
0+}f(t)$, $f(\infty)=\lim_{t\rightarrow\infty}f(t)$ and define
$S=\inf\\{t:Z_{t}>f(Y_{t})\\}$.

  * $1.$

Then $\mathbb{P}(S>0)=1$ and the process $(Y,Z)$ can neither jump on the graph
$\\{(u,f(u)):u>0\\}$ nor jump from this graph, that is

$\mathbb{P}(Z_{S-}=f(Y_{S-}),\Delta(Y,Z)_{S}\neq
0,Y_{S}<\infty)=\mathbb{P}(Z_{S}=f(Y_{S}),\Delta(Y,Z)_{S}\neq
0,Y_{S}<\infty)=0\,.$

  * $2.$

Let $u_{0},u_{1}$ be such that $0\leq u_{0}<u_{1}\leq\infty$. Then the
creeping probability of $(Y,Z)$ through the graph of $f$ is bounded from above
as follows,

(2.7) $\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}<Y_{S}<u_{1})\leq
d_{Z}\int_{u_{0}}^{u_{1}}v(u,f(u))\,du\,.$

Proposition 2.1 implies that when $d_{Z}=0$, the process $(Y,Z)$ never creeps
through the graph of any continuous non decreasing function. This result and
its consequence in Proposition 2.2 will be useful for our application to
Ornstein-Uhlenbeck processes in Section 4.

###  2.2. Creeping of real Lévy processes at their supremum

Let us now consider a non killed real Lévy process $X$. We will always assume
in the sequel that $X_{0}=0$, a.s. Since our interest lies in creeping upward
of $X$, it is natural to assume throughout this paper that $-X$ is not a
subordinator. Moreover the case of subordinators has already been dealt with
in Corollary 2.2. Therefore in this subsection, we will assume that $|X|$ is
not a subordinator. We write $\overline{X}$ for the supremum process, that is,

$\overline{X}_{t}=\sup_{s\leq t}X_{s}\,,\;\;\;t\geq 0\,.$

Then let us first recall a few basic definitions. It is well known that the
reflected process $\overline{X}-X$ is strongly Markovian under $\mathbb{P}$.
Let $L$ be the local time at 0 of $\overline{X}-X$ (a proper definition is
given in Subsection 6.1) and denote by $(\tau,H)$ the upward ladder process of
$X$, that is the bivariate Lévy process whose coordinates are the following
(possibly killed) subordinators:

$\tau_{t}=\inf\\{s:L_{s}>t\\}\;\;\;\mbox{and}\;\;\;H_{t}=X_{\tau_{t}}\,,\;\;t\geq
0\,,$

where $\tau_{t}=H_{t}=\infty$, for $t\geq L_{\infty}$. The process $\tau$
(resp. $H$) is called the upward ladder time (resp. height) process of $X$. We
will denote by $d_{\tau}$ and $d_{H}$ their respective drift coefficients. Let
us introduce the renewal measure $U$ on $[0,\infty)^{2}$ of the ladder process
$(\tau,H)$, that is,

$U(dt,dh)=\int_{0}^{\infty}\mathbb{P}(\tau_{u}\in dt,\,H_{u}\in dh)\,du\,.$

We specify that when the measure $U(dt,dh)$ is absolutely continuous on
$(0,\infty)^{2}$, its density will be denoted by $q_{t}^{*}(h)$. This notation
may seem unnatural but it comes from the fact that this density corresponds to
the entrance law of the reflected excursions, see Subsection 6.1, which is
thus noted in older references, see [2] and [4] for instance.

Throughout this paper, for any function $f:(0,\infty)\rightarrow\mathbb{R}$,
we set

$T_{f}:=\inf\\{t>0:X_{t}>f(t)\\}\,.$

When a Lévy process $X$ satisfies
$\mathbb{P}(X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}}),\,t_{0}<{T_{f}}<t_{1})>0$,
we will say that $X$ creeps at its supremum through the function $f$ over the
interval $(t_{0},t_{1})$. Omission of $(t_{0},t_{1})$ will simply mean that X
creeps over the whole half line $(0,\infty)$. The following result gives an
expression of the probability of this event when $f$ is a continuous, non
increasing function.

######  Theorem 2.2.

Let $X$ be a real Lévy process and $f:(0,\infty)\rightarrow(0,\infty)$ be a
continuous, non increasing function.

  * $1.$

Then $\mathbb{P}(T_{f}>0)=1$ and the process $X$ can neither jump on the graph
$\\{(u,f(u)):u>0\\}$ nor jump from this graph, that is

$\mathbb{P}(X_{T_{f}-}=f(T_{f})<X_{T_{f}},\,T_{f}<\infty)=\mathbb{P}(X_{T_{f}-}<X_{T_{f}}=f(T_{f}),\,T_{f}<\infty)=0\,.$

  * $2.$

Assume that the renewal measure $U(dt,dx)$ of the ladder process $(\tau,H)$
has a continuous density $(t,x)\mapsto q_{t}^{*}(x)$ on $(0,\infty)^{2}$. Then
$X$ creeps at its supremum through $f$ if and only if its ladder process
$(\tau,H)$ creeps through the graph of $f$. Moreover, for all $t_{0}$ and
$t_{1}$ such that $0\leq t_{0}<t_{1}\leq\infty$,

(2.8)
$\mathbb{P}(X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}}),\,t_{0}<{T_{f}}<t_{1})=d_{H}\int_{t_{0}}^{t_{1}}q^{*}_{u}(f(u))\,du-
d_{\tau}\int_{t_{0}}^{t_{1}}q^{*}_{u}(f(u))\,df(u)\,.$

The creeping probability of part 2. of Theorem 2.2 reveals two types of
creeping at the supremum, each of which corresponds to one of the two terms
involved. When $d_{H}>0$, the process creeps upward either continuously (in
absence of positive jumps) or through an accumulation of jumps of $H$ as in
the case of a fixed barrier. In the case where $d_{\tau}>0$ (which is
equivalent to the fact that 0 is not regular for $(-\infty,0)$), another type
of creeping occurs. Then the process creeps forward through an accumulation of
jumps of $\tau$ provided $f$ decreases at the creeping time. Recall that the
later jumps correspond to the lengths of the excursions of $\overline{X}-X$.
This new kind of creeping happens as if $f$ were acting as a vertical barrier.

$0$$T_{f}$$X_{T_{f}}=\overline{X}_{T_{f}}=f\left({T_{f}}\right)$

Figure 2. A sample path of $X$ creeping at its supremum through $f$

We point out that the assumption of absolute continuity in part 2. of the
above result is not as strong as it may appear. Indeed, we will see in
Proposition 6.1, that it is satisfied whenever the transition semigroup of $X$
admits densities, $x\mapsto p_{t}(x)$, $x\in\mathbb{R}$, which are bounded for
all $t>0$, and for all $c\geq 0$, the process $(|X_{t}-ct|,\,t\geq 0)$ is not
a subordinator. Moreover, from the same proposition, $q_{t}^{*}$ is positive
on $(0,\infty)$, for all $t>0$, which ensures the positivity of the creeping
probability (2.8) whenever $d_{H}>0$ or $d_{\tau}>0$ and $f$ is decreasing. On
the other hand, when the assumption of Theorem 2.2 is not satisfied, we can
still expect that the process creeps at its supremum through $f$ with positive
probability, but finding necessary and sufficient conditions for this to hold
is an open question which is closely related to the question raised by Theorem
2.1, see the discussion after Corollary 2.2.

Recall from $(3.3)$ in [4] that the function $x\mapsto
u(x)=\int_{0}^{\infty}q_{t}^{*}(x)\,dt$ is the potential density of $H$
$($this is true by definition when $X$ is a subordinator since
$q_{t}^{*}(x)=p_{t}(x))$. Hence by taking $f\equiv x$ in the statement of the
above theorem, as well as in Corollary 2.2, we recover the classical creeping
result for Lévy processes recalled in the introduction.

When $\mathbb{P}(X_{T_{f}}=f({T_{f}}),\,t_{0}<{T_{f}}<t_{1})>0$ we will say
that $X$ creeps through the function $f$ (over the interval $(t_{0},t_{1})$).
Finding an expression for the probability
$\mathbb{P}(X_{T_{f}}=f({T_{f}}),\,t_{0}<{T_{f}}<t_{1})$ seems more
complicated than for
$\mathbb{P}(X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}}),\,t_{0}<{T_{f}}<t_{1})$.
Indeed, in the second case, we only need to ensure that $X$ stays below its
past supremum before time $T_{f}$, whereas in the first case the condition is
that the whole path of $X$ stays below the curve of $f$ before time $T_{f}$.
Therefore, the expression of this probability must strongly depend on the
behaviour of whole paths of $X$ with respect to the curve of $f$ before time
$T_{f}$ and it seems hardly possible to perform this computation for a general
function. However the example below and Corollary 2.3 give conditions for $X$
to creep through a function $f$ in some particular cases.

Let $X$ be the Lévy process $X_{t}=S_{t}-at$, $t\geq 0$, where $S$ is a
subordinator without drift and $a>0$. Then 0 is not regular for $(0,\infty)$
and it is regular for $(-\infty,0)$, so that $d_{H}=d_{\tau}=0$ (see
Subsection 6.1) and from Theorem 2.2, the process does not creep at its
supremum through any continuous non increasing function. However, it does
creep through the function $f(t)=1-t$, since for any $a<1$, the subordinator
with positive drift $S_{t}+(1-a)t$ creeps through the level 1. This simply
means that the creeping time of $X$ through $f$ is a.s. not a time at which
$X$ reaches its past supremum. From this example, we can build a more general
result regarding the probability to creep through a function (not necessarily
at the supremum) of Lévy processes with bounded variation.

######  Corollary 2.3.

Let $X$ be a Lévy process with bounded variation and nonnegative drift. Assume
that for all $c\geq 0$, the process $(|X_{t}-ct|,\,t\geq 0)$ is not a
subordinator and that for all $t>0$, the distribution $\mathbb{P}(X_{t}\in
dx)$ is absolutely continuous on $\mathbb{R}$ with a bounded density. Let
$f:(0,\infty)\rightarrow\mathbb{R}$ be a function such that for some $0\leq
t_{0}<t_{1}\leq\infty$ and $a>0$, $f(t)+at$ is positive, continuous and non
increasing over the interval $(t_{0},t_{1})$. Then $X$ creeps through $f$ over
the interval $(t_{0},t_{1})$, that is

$\mathbb{P}(X_{T_{f}}=f({T_{f}}),\,t_{0}<{T_{f}}<t_{1})>0\,.$

The above corollary will be useful for our application to the creeping of
Ornstein-Uhlenbeck processes in Section 4.

From Theorem 2.2, for $X$ to creep at its supremum through $f$, it is
necessary that $d_{H}>0$ or $d_{\tau}>0$. In particular, if $0$ is not regular
for $[0,\infty)$, then $d_{\tau}=d_{H}=0$ (see Subsection 6.1) and $X$ never
creeps at its supremum through a continuous non increasing function. On the
other hand, when $X$ has unbounded variation, $d_{\tau}=0$ and hence the
process can creep at its supremum through $f$ only if $d_{H}>0$. We conjecture
that when $X$ has unbounded variation and $d_{H}=0$, then $X$ never creeps
through any continuous, non increasing function. Indeed, from the results of
this section it is reasonable to think that the only chance for a Lévy process
$X$ to creep through such a function is that either $d_{H}>0$ or $X$ has
bounded variation.

We end this subsection with the case of a continuous non decreasing function.

######  Proposition 2.2.

Let $X$ be a real Lévy process and $f:(0,\infty)\rightarrow(0,\infty)$ be a
continuous, non decreasing function such that $\lim_{t\rightarrow 0+}f(t)>0$.

  * $1.$

Then $\mathbb{P}(T_{f}>0)=1$ and the process $X$ can neither jump on the graph
$\\{(u,f(u)):u>0\\}$ nor jump from this graph, that is

$\mathbb{P}(X_{T_{f}-}=f(T_{f})<X_{T_{f}},\,T_{f}<\infty)=\mathbb{P}(X_{T_{f}-}<X_{T_{f}}=f(T_{f}),\,T_{f}<\infty)=0\,.$

  * $2.$

Assume that the renewal measure $U(dt,dx)$ of the ladder process $(\tau,H)$
has a continuous density $(t,x)\mapsto q_{t}^{*}(x)$ on $(0,\infty)^{2}$. Then
for all $t_{0}$ and $t_{1}$ such that $0\leq t_{0}<t_{1}\leq\infty$,

$\displaystyle\mathbb{P}(X_{T_{f}}=f({T_{f}}),\,t_{0}<{T_{f}}<t_{1})\leq
d_{H}\int_{t_{0}}^{t_{1}}q^{*}_{u}(f(u))\,du\,.$

Proposition 2.2 implies that a Lévy process such that $d_{H}=0$ never creeps
through a continuous non decreasing function. Note that since $f$ is non
decreasing, when the process $X$ creeps through $f$, it necessarily creeps at
its supremum, that is
$\\{X_{T_{f}}=f({T_{f}})\\}=\\{X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}})\\}$.

###  2.3. Creeping of Lévy processes conditioned to stay positive

Recall that $X$ is a non killed real Lévy process such that $X_{0}=0$, a.s.
Moreover, we assume again that $|X|$ is not a Lévy process. The process $X$
conditioned to stay positive is a Doob $h$-transform of the process killed at
its first hitting time of the negative half-line. Let us briefly recall the
definition of this process of which one will find a more complete description
in [3]. Let us set,

$\tau_{-x}^{-}=\inf\\{t>0:X_{t}<-x\\}\,,$

for $x\geq 0$ and let
$h(x)=\int_{0}^{\infty}\mathbb{P}(H_{t}^{*}\in[0,x])\,dt$ be the renewal
function of the downward ladder height process $H^{*}$ of $X$ (see Subsection
6.1 for a full definition). Then the process $X$ conditioned to stay positive
is a (possibly non conservative) strong Markov process with state space
$(0,\infty)$, which we denote by $X^{x,\uparrow}$ when it is issued from $x$
and whose semigroup is given by

$\mathbb{P}(X^{x,\uparrow}_{t}\in dy)=\frac{1}{h(x)}\mathbb{P}(x+X_{t}\in
dy,t<\tau_{-x}^{-})\,,\;\;\;x,y>0\,.$

It is proved in [3] that when 0 is regular for $(0,\infty)$ and $X$, the
process $X^{x,\uparrow}$ converges weakly in the Skohorod’s space, as $x$
tends to 0 toward a non degenerate process which will be denoted here by
$X^{\uparrow}$. This process satisfies $X^{\uparrow}_{0}=0$, a.s. and if
moreover $\limsup_{t\rightarrow\infty}X_{t}=\infty$, a.s., then
$(X^{x,\uparrow},x>0)$ is conservative and
$\lim_{t\rightarrow\infty}X^{\uparrow}_{t}=\infty$, a.s. Let us now define the
future infimum process of $X^{\uparrow}$ and its last passage time below the
graph of a function $f$ respectively by,

$\underline{\underline{X}}_{t}^{\uparrow}=\inf_{s\geq
t}X_{s}^{\uparrow}\,,\;\;t\geq
0\;\;\;\mbox{and}\;\;\;\sigma_{f}=\sup\\{t:X^{\uparrow}_{t}\leq f(t)\\}\,.$

######  Theorem 2.3.

Let $X$ be a real Lévy process such that
$\limsup_{t\rightarrow\infty}X_{t}=\infty$, a.s. and assume that $0$ is
regular for $(0,\infty)$. Let $X^{\uparrow}$ be the process $X$ conditioned to
stay positive and let $f:(0,\infty)\rightarrow(0,\infty)$ be a continuous, non
increasing function.

  * $1.$

Then $0<\sigma_{f}<\infty$, a.s. and at time $\sigma_{f}$, the process
$X^{\uparrow}$ can neither jump on the graph $\\{(u,f(u)):u>0\\}$ nor jump
from this graph, that is

$\mathbb{P}(X_{\sigma_{f}-}^{\uparrow}=f(\sigma_{f})<X_{\sigma_{f}}^{\uparrow})=\mathbb{P}(X_{\sigma_{f}-}^{\uparrow}<X_{\sigma_{f}}^{\uparrow}=f(\sigma_{f}))=0\,.$

  * $2.$

Assume that the renewal measure $U(dt,dx)$ of the ladder process $(\tau,H)$
has a continuous density $(t,x)\mapsto q_{t}^{*}(x)$ on $(0,\infty)^{2}$. Then
for all $t_{0}$ and $t_{1}$ such that $0\leq t_{0}<t_{1}\leq\infty$,

$\mathbb{P}(X_{\sigma_{f}}^{\uparrow}=\underline{\underline{X}}_{\sigma_{f}}^{\uparrow}=f({\sigma_{f}}),\,t_{0}<{\sigma_{f}}<t_{1})=d_{H}\int_{t_{0}}^{t_{1}}q^{*}_{u}(f(u))\,du-
d_{\tau}\int_{t_{0}}^{t_{1}}q^{*}_{u}(f(u))\,df(u)\,.$

It appears that the creeping probability of $X^{\uparrow}$ at its future
infimum through the graph of $f$ has the same expression as the creeping
probability of $X$ at its past supremum stated in Theorem 2.2. This can be
explained by the following time reversal property

$[(X_{T_{f}}-X_{(T_{f}-t)-}\,,0\leq t\leq
T_{f})\,|\,X_{T_{f}}=\overline{X}_{T_{f}}]\,\stackrel{{\scriptstyle\mbox{\tiny$(d)$}}}{{=}}\,[(X_{t}^{\uparrow}\,,0\leq
t\leq\sigma_{f})\,|\,X_{\sigma_{f}}^{\uparrow}=\underline{\underline{X}}_{\sigma_{f}}^{\uparrow}]\,,$

where $X_{0-}=X_{0}=0$. The later identity in law is itself a consequence of
the proof of Theorem 2.3, see Section 7.

##  3\. Examples

Illustrating Theorem 2.1 with a bivariate subordinator $(Y,Z)$ such that none
of the coordinates $Y$ and $Z$ is a pure drift and whose renewal density
measure is explicit seems to be quite challenging. Our first example regards
Corollary 2.2, that is the case when one of the subordinators is a pure drift.

a) Let $X$ be a stable subordinator with index $1/2$. Then for $t>0$, $X_{t}$
has density

$p_{t}(x)=\frac{t}{\sqrt{2\pi
x^{3}}}e^{\frac{-t^{2}}{2x}}\mathbbm{1}_{(0,\infty)}(x)\,.$

From Corollary 2.2, the distribution of the creeping time is then

$\displaystyle\mathbb{P}(X_{T_{f}}=f({T_{f}}),\,T_{f}\in
dt)=-\frac{t}{\sqrt{2\pi f(t)^{3}}}e^{-t^{2}/(2f(t))}\,df(t)\,,\;\;\;t>0\,.$

Taking $f(t)=1/t^{2}$ gives the creeping probability through $f$ over the
half-line $(0,\infty)$ and the distribution of the creeping time conditionally
on this event:

$\displaystyle\mathbb{P}(X_{T_{f}}=f({T_{f}}))$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}te^{-t^{4}/2}\,dt=\frac{1}{2}$
$\displaystyle\mathbb{P}(T_{f}\in dt\,|\,X_{T_{f}}=f({T_{f}}))$
$\displaystyle=$ $\displaystyle
2\sqrt{\frac{2}{\pi}}te^{-t^{4}/2}\,dt\,,\;\;\;t>0\,.$

Our second example illustrates Theorem 2.2.

b) Standard Brownian motion creeping at its supremum. When $X$ is standard
Brownian motion, $0$ is regular for both half-lines $(-\infty,0)$ and
$(0,\infty)$ and hence $d_{\tau}=0$. On the other hand, our normalisation of
the local time $L$ (see Subsection 6.1) gives $d_{H}=1/\sqrt{2}$ and for
$t>0$,

$q_{t}^{*}(x)=\frac{x}{\sqrt{\pi
t^{3}}}e^{-x^{2}/2t}\mathbbm{1}_{(0,\infty)}(x)\,.$

From part 2. of Theorem 2.2, the distribution of the creeping time at the
supremum with respect to any continuous non increasing function
$f:(0,\infty)\rightarrow(0,\infty)$ is given by

$\displaystyle\mathbb{P}(X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}}),\,T_{f}\in
dt)=\frac{f(t)}{\sqrt{2\pi t^{3}}}e^{-f(t)^{2}/2t}\,dt\,,\;\;\;t>0\,.$

For instance, the probability for the standard Brownian motion to creep at its
supremum through the function $f(t)=1/t$ on $(0,\infty)$ and the distribution
of the creeping time $T_{f}$ conditionally on this event are respectively,

$\displaystyle\mathbb{P}(X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}}))$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}\frac{1}{\sqrt{2\pi
t^{5}}}e^{-1/2t^{3}}\,dt=\frac{1}{3}$ $\displaystyle\mathbb{P}(T_{f}\in
dt\,|\,X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}}))$ $\displaystyle=$
$\displaystyle\frac{3}{\sqrt{2\pi}t^{5/2}}e^{-1/2t^{3}}\,dt\,,\;\;\;t>0\,.$

A slight generalisation of the above computation can be obtained by
considering the case of Brownian motion with drift $\mu\in\mathbb{R}$. The
expression of $q_{t}^{*}(x)$ is then $\frac{x}{\sqrt{\pi
t^{3}}}e^{-(x+\mu)^{2}/2t}$. The density $(t,x)\mapsto q_{t}^{*}(x)$ of the
renewal measure can actually be made explicit in very few cases and the few
known expressions are too complicated to allow the calculation of the creeping
probability.

Actually both examples a) and b) are closely related to each other. Indeed,
for standard Brownian motion, the ladder process $(\tau,H)$ is such that
$\tau$ is a stable subordinator with index $1/2$ and $H$ is a pure drift
process, that is $H_{t}=d_{H}\cdot t$, $t\geq 0$. So, example b) is also a
consequence of Corollary 2.2.

##  4\. Application : creeping of $\alpha$-stable Ornstein-Uhlenbeck processes

Let $X$ be a non killed real Lévy process such that $|X|$ is not a
subordinator and $X_{0}=0$, a.s. The Ornstein-Uhlenbeck process starting from
$z\in\mathbb{R}$ and driven by $X$ is the unique strong solution of the
following stochastic differential equation:

(4.9) $Z_{t}=z+X_{t}-\gamma\int_{0}^{t}Z_{s}ds\,,\;\;t\geq
0\,,\;\;\gamma>0\,.$

It is explicitly given by $Z_{t}=e^{-\gamma t}(z+\int_{0}^{t}e^{\gamma
s}dX_{s})$. For $x\in\mathbb{R}$, we set $T^{+}_{x}=\inf\\{t:Z_{t}>x\\}$ and
$T^{-}_{x}=\inf\\{t:Z_{t}<x\\}$ and we say that $Z$ creeps through the state
$x\neq z$ if either $x>z$ and
$\mathbb{P}(Z_{T_{x}^{+}}=x,\,T_{x}^{+}<\infty)>0$ or $x<z$ and
$\mathbb{P}(Z_{T_{x}^{-}}=x,\,T_{x}^{-}<\infty)>0$. We are interested here in
the creeping property of $Z$, when $X$ is a stable Lévy process with index
$\alpha\in(0,1)$. Actually stable Lévy processes never creep across any level.
Creeping of an Ornstein-Uhlenbeck process $Z$ driven by a stable Lévy process
with index $\alpha\in(0,1)$ is actually due to the smoothing effect of the
functional $t\mapsto\int_{0}^{t}Z_{s}ds$, as the proof of the next proposition
shows. Recall that when $X$ is stable with any index $\alpha\in(0,2]$, $Z$
fulfils the following representation:

(4.10) $(Z_{t},\,t\geq
0)\stackrel{{\scriptstyle\mbox{\tiny$(d)$}}}{{=}}\left(e^{-\gamma
t}(z+X_{\frac{e^{\alpha\gamma t}-1}{\alpha\gamma}}),\,t\geq 0\right)\,,$

see for instance E 18.17, p.116 in [16]. When moreover $X$ is symmetric and
$\alpha\in(0,1)$, it is proved in [10] (see Proposition 12 on p.618) that $Z$
creeps through $x\in(z,0)$ when $z<0$ and through $x\in(0,z)$, when $z>0$.
Proposition 4.1 below provides an extension of this result.

######  Proposition 4.1.

Let $Z$ be the Ornstein-Uhlenbeck process solution of $(\ref{2227})$ with
$\gamma>0$, starting point $z\in\mathbb{R}$ and driven by a stable Lévy
process $X$ with index $\alpha\in(0,1)$ such that $|X|$ is not a subordinator.

  * $1.$

If $z\in(-\infty,0)$, then $Z$ creeps through $x$ if and only if $x\in(z,0)$.

  * $2.$

If $z\in(0,\infty)$, then $Z$ creeps through $x$ if and only if $x\in(0,z)$.

  * $3.$

If $z=0$, then $Z$ does not creep through any level.

###### Proof.

From the representation $(\ref{2562})$, we can write by setting
$s=(e^{\alpha\gamma t}-1)/(\alpha\gamma)$,

$\displaystyle T^{+}_{x}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha\gamma}\log(\alpha\gamma\inf\\{s:X_{s}>x(\alpha\gamma
s+1)^{1/\alpha}-z\\}+1)$ $\displaystyle T^{-}_{x}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha\gamma}\log(\alpha\gamma\inf\\{s:X_{s}<x(\alpha\gamma
s+1)^{1/\alpha}-z\\}+1)\,,$

so that $Z$ creeps through $x$ if and only if $X$ creeps through
$f(s)=x(\alpha\gamma s+1)^{1/\alpha}-z$.

Let $z\in(-\infty,0)$ and $x\in(z,0)$. Then recall that $X$ has bounded
variation and zero drift and that for all $t>0$, $\mathbb{P}(X_{t}\in dx)$ is
absolutely continuous on $\mathbb{R}$ with a bounded density, see Theorem
53.1, p.404 in [16]. Therefore, $X$ satisfies the conditions of Corollary 2.3.
On the other hand, note that $f(0)=x-z>0$ and that $f$ is continuous.
Moreover, $f(s)+as$ is non increasing on $(0,\infty)$ whenever $0<a<-x\gamma$.
Let us take such a value $a$ and note that there is $t_{1}>0$ such that
$f(s)+as$ is positive on $(0,t_{1})$. Then the result follows from Corollary
2.3.

If $z<0\leq x$, then $f(0)>0$ and $f$ is non decreasing. Since $X$ satisfies
$d_{H}=0$ ($H$ is a stable subordinator), the result follows from Proposition
2.2.

If $x<z<0$ then $f(0)<0$ and $f$ is non increasing and the result follows
again from Proposition 2.2 by considering $-X$ and $-f$.

The proof of $2.$ is deduced from this of $1.$ by considering the Ornstein-
Uhlenbeck process $-Z$ which is solution of $(\ref{2227})$ with $\gamma>0$,
starting point $-z\in\mathbb{R}\setminus\\{0\\}$ and driven by the stable
process $-X$.

If $z=0$ and $x>0$, then $f$ is non decreasing and the result follows again
from Proposition 2.2. For $x<0$, we derive the result by considering $-Z$ from
the same argument as above. ∎

Looking at the previous proof and in accordance with the remarks made in
Subsection 2.2, we conjecture that when $\alpha>1$, since $X$ has unbounded
variation, the process $Z$ never creeps across any level. However, as proved
in [5], points are not polar for $Z$ when the leading Lévy process is stable
with any index $\alpha\in(0,2]$.

Finally let us mention that dealing with the creeping through $x\in(z,0)$ of
an Ornstein-Uhlenbeck process $Z$ starting from $z<0$ and driven by any Lévy
process $X$ would require to study the creeping property of $X$ through the
continuous non increasing adapted functional $t\mapsto
x-z+\gamma\int_{0}^{t}Z_{s}ds$, as equation (4.9) shows. This raises the more
general question of the creeping of a Lévy process through any adapted,
continuous and non increasing functional.

##  5\. Proof of Theorem 2.1 and Proposition 2.1

Let us note that from monotone convergence, we do not lose any generality by
proving Theorem 2.1 in the case where $t_{0},u_{0}>0$ and
$t_{1},u_{1}<\infty$. Therefore in all this section, $t_{0}$, $u_{0}$, $t_{1}$
and $u_{1}$ will be chosen so that

$0<u_{0}<u_{1}<\infty\;\;\;\mbox{and}\;\;\;0<t_{0}<t_{1}<\infty\,.$

###  5.1. Proof of the first assertion in Theorem 2.1

The following lemma actually shows a much stronger result than the first
assertion of Theorem 2.1. It will also be used in further results.

######  Lemma 5.1.

Let ${\mathcal{L}}$ be any $\mathbb{R}^{d}$-valued Lévy process with infinite
lifetime and set
$\Delta{\mathcal{L}}_{t}={\mathcal{L}}_{t}-{\mathcal{L}}_{t-}$, $t>0$. Let
$G:\mathbb{R}^{d}\rightarrow\mathbb{R}$ be any Borel function such that for
all $x\in\mathbb{R}^{d}$, $\mathbb{P}(G({\mathcal{L}}_{t}+x)=0)=0$, for
$\lambda$-a.e. $t\geq 0$ $($$\lambda$ being the Lebesgue measure on
$[0,\infty)$$)$. Then

(5.11) $\mathbb{P}\left(\exists\;t\geq
0,\;G({\mathcal{L}}_{t-})=0,\,\Delta{\mathcal{L}}_{t}\neq
0\right)=\mathbb{P}(\exists\;t\geq
0,\;G({\mathcal{L}}_{t})=0,\,\Delta{\mathcal{L}}_{t}\neq 0)=0\,.$

###### Proof.

Recall that $t\mapsto\Delta{\mathcal{L}}_{t}$ is a Poisson point process with
characteristic measure $\nu$, where $\nu$ is the Lévy measure of
${\mathcal{L}}$. Then from the compensation formula for Poisson point
processes, for every $\varepsilon>0$,

$\mathbb{E}\left(\sum_{t\geq 0}\mbox{\rm
1I}_{\\{G({\mathcal{L}}_{t-})=0\\}}\mbox{\rm
1I}_{\\{\|\Delta{\mathcal{L}}_{t}\|>\varepsilon\\}}\right)=\nu(\\{x:\|x\|>\varepsilon\\})\mathbb{E}\left(\int_{0}^{\infty}\mbox{\rm
1I}_{\\{G({\mathcal{L}}_{t})=0\\}}\,dt\right).$

But from our assumption, the last term is 0. Then the equality

$\mathbb{P}\left(\exists\;t\geq
0,\;G({\mathcal{L}}_{t-})=0,\,\Delta{\mathcal{L}}_{t}\neq 0\right)=0$

in (5.11) is obtained by taking $\varepsilon$ to 0 and using monotone
convergence.

We prove the second equality through the same argument by writing

$\displaystyle\mathbb{E}\left(\sum_{t\geq 0}\mbox{\rm
1I}_{\\{G({\mathcal{L}}_{t-}+\Delta{\mathcal{L}}_{t})=0\\}}\mbox{\rm
1I}_{\\{\|\Delta{\mathcal{L}}_{t}\|>\varepsilon\\}}\right)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dt\mathbb{E}\left(\int_{\\{\|x\|>\varepsilon\\}}\nu(dx)\mbox{\rm
1I}_{\\{G({\mathcal{L}}_{t-}+x)=0\\}}\right)$ $\displaystyle=$
$\displaystyle\int_{\\{\|x\|>\varepsilon\\}}\nu(dx)\int_{0}^{\infty}\mathbb{P}(G({\mathcal{L}}_{t}+x)=0)\,dt=0$

and the conclusion follows as for the first equality. ∎

Let us now prove the first assertion of Theorem 2.1. First of all, since $f$
is positive and non increasing, it is clear that $0<S<\infty$, a.s., so that
$Y_{S-}$, $Z_{S-}$ are well defined and $Y_{S-}<\infty$ and $Z_{S-}<\infty$,
a.s. Then let us apply Lemma 5.1 to the Lévy process $\mathcal{L}=(Y,Z)$ and
the Borel function $G(y,z)=f(y)-z$ under the assumption that $\mathcal{L}$ has
infinite lifetime. The condition: for all $x=(y,z)\in\mathbb{R}^{2}$,
$\mathbb{P}(G(\mathcal{L}_{t}+x)=0)=\mathbb{P}(f(Y_{t}+y)=Z_{t}+z)=0$, for
$\lambda$-a.e. $t\geq 0$ is satisfied since, by assumption, the renewal
measure $v(dy,dz)$ of $(Y,Z)$ is absolutely continuous. Then part 1. of
Theorem 2.1 follows from Lemma 5.1 and the equalities

$\displaystyle\mathbb{P}(Z_{S-}=f(Y_{S-}),\Delta(Y,Z)_{S}\neq 0)$
$\displaystyle=$ $\displaystyle\mathbb{P}(\exists t\geq
0,\,Z_{t-}=f(Y_{t-}),\Delta(Y,Z)_{t}\neq 0)$
$\displaystyle\mathbb{P}(Z_{S}=f(Y_{S}),\Delta(Y,Z)_{S}\neq 0)$
$\displaystyle=$ $\displaystyle\mathbb{P}(\exists t\geq
0,\,Z_{t}=f(Y_{t}),\Delta(Y,Z)_{t}\neq 0)\,.$

Now let $\tilde{\mathcal{L}}=(\tilde{Y},\tilde{Z})$ be a Lévy process with
infinite lifetime such that $\mathcal{L}$ is obtained by killing
$\tilde{\mathcal{L}}$ at an independent exponential time ${\bf e}$. Denote by
$\zeta$ the lifetime of ${\mathcal{L}}$. Then,

$\displaystyle\mathbb{P}(Z_{S-}=f(Y_{S-}),\Delta(Y,Z)_{S}\neq 0)$
$\displaystyle=$ $\displaystyle\mathbb{P}(\exists
t\leq\zeta,\,Z_{t-}=f(Y_{t-}),\Delta\mathcal{L}_{t}\neq 0)$ $\displaystyle=$
$\displaystyle\mathbb{P}(\exists t\leq{\bf
e},\,\tilde{Z}_{t-}=f(\tilde{Y}_{t-}),\Delta\tilde{\mathcal{L}}_{t}\neq 0)$
$\displaystyle\leq$
$\displaystyle\mathbb{P}(\tilde{Z}_{S-}=f(\tilde{Y}_{S-}),\Delta(\tilde{Y},\tilde{Z})_{S}\neq
0)=0\,.$

The case of $\mathbb{P}(Z_{S}=f(Y_{S}),\Delta(Y,Z)_{S}\neq 0)=0$ is handled in
the same way.

###  5.2. Preliminary lemmas

Recall that $(Y,Z)$ is a bivariate subordinator as defined in Subsection 2.1.
Its semigroup and its renewal measure are respectively denoted by
$p_{t}(dy,dz)$ and $v(dy,dz)$. In all the remainder of this paper we set for
all $y,z\geq 0$,

$T_{y}^{Y}=\inf\\{t:Y_{t}>y\\}\;\;\;\mbox{and}\;\;\;T_{z}^{Z}=\inf\\{t:Z_{t}>z\\}\,.$

######  Lemma 5.2.

Assume that the renewal measure $v(dy,dz)$ is absolutely continuous on
$(0,\infty)^{2}$ and let us denote by $v(y,z)$ its density, then for
$y,z\in(0,\infty)$,

(5.12) $\displaystyle\mathbb{P}(Y_{T_{y}^{Y}}=y,\,Z_{T_{y}^{Y}}\in
dz)\,dy=d_{Y}v(y,z)\,dy\,dz$ (5.13)
$\displaystyle\mathbb{P}(Z_{T_{z}^{Z}}=z,\,Y_{T_{z}^{Z}}\in
dy)\,dz=d_{Z}v(y,z)\,dy\,dz\,.$

Assume that for all $t>0$, the semigroup $p_{t}(dy,dz)$ is absolutely
continuous on $(0,\infty)^{2}$ and let us denote by $p_{t}(y,z)$ its
densities, then for $t,y,z\in(0,\infty)$,

(5.14) $\displaystyle\mathbb{P}(Y_{T_{y}^{Y}}=y,\,Z_{T_{y}^{Y}}\in
dz,\,T_{y}^{Y}\in dt)\,dy=d_{Y}p_{t}(y,z)\,dt\,dy\,dz$ (5.15)
$\displaystyle\mathbb{P}(Z_{T_{z}^{Z}}=z,\,Y_{T_{z}^{Z}}\in dy,\,T_{z}^{Z}\in
dt)\,dz=d_{Z}p_{t}(y,z)\,dt\,dy\,dz\,.$

###### Proof.

Let us prove (5.12) and (5.14). Note that the time $T_{y}^{Y}$ satisfies
$d_{Y}T_{y}^{Y}=\int_{0}^{y}\mbox{\rm 1I}_{\\{Y_{T_{s}^{Y}}=s\\}}\,ds$, for
all $y\geq 0$. Hence for $\alpha,\beta,\gamma>0$,

$\displaystyle\int_{y=0}^{\infty}\int_{t,z\in[0,\infty)^{2}}e^{-\alpha y-\beta
z-\gamma t}\mathbb{P}(Y_{T_{y}^{Y}}=y,\,Z_{T_{y}^{Y}}\in dz,\,T_{y}^{Y}\in
dt)\,dy$ $\displaystyle=$
$\displaystyle\mathbb{E}\left(\int_{0}^{\infty}e^{-\alpha Y_{T_{y}^{Y}}-\beta
Z_{T^{Y}_{y}}-\gamma
T_{y}^{Y}}d_{Y}dT_{y}^{Y}\right)=\mathbb{E}\left(\int_{0}^{\infty}e^{-\alpha
Y_{t}-\beta Z_{t}-\gamma t}d_{Y}dt\right)$ $\displaystyle=$
$\displaystyle\int_{y=0}^{\infty}\int_{t,z\in[0,\infty)^{2}}e^{-\alpha y-\beta
z-\gamma t}d_{Y}p_{t}(dy,dz)\,dt\,,$

which proves both identities. The proof of (5.13) and (5.15) is the same. ∎

Let us mention that expressions similar to (5.12) and (5.13) have been
obtained in Section 4 of [9], see Theorem 4.2 therein. We also use the latter
theorem for the proof of the next Lemma.

######  Lemma 5.3.

Assume that the renewal measure $v(dy,dz)$ is absolutely continuous on
$(0,\infty)^{2}$ and that its density $(y,z)\mapsto v(y,z)$ is locally
bounded. Then there are functions $\varepsilon_{1}$ and $\varepsilon_{2}$
satisfying $\varepsilon_{1}(h),\varepsilon_{2}(h)\rightarrow 0$, as
$h\rightarrow 0$ and such that for all $h\in(0,u_{1}-u_{0})$ and
$u\in[u_{0},u_{1}-h]$,

$\displaystyle\mathbb{P}(Y_{T_{u+h}^{Y}-}\in(u,u+h),\,Z_{T_{u+h}^{Y}-}\in(f(u+h),f(u)))\leq
h\varepsilon_{1}(h)\,,$
$\displaystyle\mathbb{P}(Z_{T_{f(u)}^{Z}-}\in(f(u+h),f(u)),\,Y_{T_{f(u)}^{Z}-}\in(u,u+h))\leq
h\varepsilon_{2}(h)\,.$

###### Proof.

Let us denote by $\pi_{Y}$ and $\pi_{Z}$ the Lévy measures of $Y$ and $Z$ and
set $\bar{\pi}_{Y}(y)=\pi_{Y}(y,\infty)$ and $\bar{\pi}_{Z}(z)=\pi(z,\infty)$.
Recall that $\zeta$ is the lifetime of $(Y,Z)$ and let us denote by
$q\in[0,\infty)$ its rate. Applying Theorem 4.2 in [9] we obtain,

$\displaystyle\mathbb{P}(Y_{T_{u+h}^{Y}-}\in(u,u+h),\,Z_{T_{u+h}^{Y}-}\in(f(u+h),f(u)),\,T_{u+h}^{Y}<\zeta)$
(5.16)
$\displaystyle=\int_{z=f(u+h)}^{f(u)}\int_{y=u}^{u+h}v(y,z)\bar{\pi}_{Y}(y)\,dydz\leq\bar{\pi}_{Y}(u_{0})Ch\sup_{u\in[u_{0},u_{1}-h]}(f(u)-f(u+h))\,,$

where $C$ is a bound for $v$ on $[u_{0},u_{1}]\times[f(u_{1}),f(u_{0})]$
(recall that $u_{0}>0$). On the other hand,

$\displaystyle\mathbb{P}(Y_{T_{u+h}^{Y}-}\in(u,u+h),\,Z_{T_{u+h}^{Y}-}\in(f(u+h),f(u)),\,T_{u+h}^{Y}=\zeta)$
$\displaystyle\quad\leq\mathbb{P}(Y_{\zeta-}\in(u,u+h),\,Z_{\zeta-}\in(f(u+h),f(u)))$
(5.17)
$\displaystyle\quad=q\int_{z=f(u+h)}^{f(u)}\int_{y=u}^{u+h}v(y,z)\,dydz\leq
qCh\sup_{u\in[u_{0},u_{1}-h]}(f(u)-f(u+h))\,.$

Then the first inequality of the statement follows from (5.16), (5.17) and the
uniform continuity of $f$ on $[u_{0},u_{1}]$.

Similarly, we have from Theorem 4.2 in [9],

$\displaystyle\mathbb{P}(Z_{T_{f(u)}^{Z}-}\in(f(u+h),f(u)),\,Y_{T_{f(u)}^{Z}-}\in(u,u+h),\,T_{f(u)}^{Z}<\zeta)$
$\displaystyle=\int_{z=f(u+h)}^{f(u)}\int_{y=u}^{u+h}v(y,z)\bar{\pi}_{Z}(z)\,dydz\leq\bar{\pi}_{Z}(f(u_{1}))Ch\sup_{u\in[u_{0},u_{1}-h]}(f(u)-f(u+h))\,,$

and the same conclusion follows from the same argument. ∎

###  5.3. Proof of the lower bound in identities (2.3) and (2.4)

Let us first prove the lower bound in equation (2.4), that is the inequality,

$\displaystyle\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}<Y_{S}<u_{1},\,t_{0}<S<t_{1})\geqslant$
(5.18) $\displaystyle\qquad\qquad
d_{Z}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,du-
d_{Y}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,df(u)\,.$

We define the sequence $(\sigma_{n})_{n\geq 1}$ of subdivisions of
$[u_{0},u_{1}]$,

(5.19) $\sigma_{n}=\left\\{u_{k}^{n}=u_{0}+\frac{k}{n}(u_{1}-u_{0}),\
0\leqslant k\leqslant n\right\\},\;\;\;n\geq 1\,,$

and we set, for all $n\geq 1$, $\displaystyle
B_{n}=\bigcup_{k=0}^{n-1}B_{n,k}^{(1)}\cup B_{n,k}^{(2)}$, where

(5.20) $\displaystyle B_{n,k}^{(1)}$ $\displaystyle=$
$\displaystyle\left\\{Z_{T_{f(u_{k}^{n})}^{Z}}=f(u_{k}^{n}),u_{k}^{n}\leqslant
Y_{T_{f(u_{k}^{n})}^{Z}}<u_{k+1}^{n},\,t_{0}<T_{f(u_{k}^{n})}^{Z}<t_{1}\right\\}$
(5.21) $\displaystyle B_{n,k}^{(2)}$ $\displaystyle=$
$\displaystyle\left\\{Y_{T_{u_{k+1}^{n}}^{Y}}=u_{k+1}^{n},f(u_{k+1}^{n})\leqslant
Z_{T_{u_{k+1}^{n}}^{Y}}<f(u_{k}^{n}),\,t_{0}<T_{u_{k+1}^{n}}^{Y}<t_{1}\right\\}\,.$

Then let us check that

(5.22) $\limsup_{n}B_{n}\subset\\{Z_{S}=f(Y_{S}),\,u_{0}\leq Y_{S}\leq
u_{1},\,t_{0}\leq S\leq t_{1}\\}\,,\;\;\;\mbox{almost surely}\,.$

Let us first note that

$\displaystyle\omega\in
B_{n,k}^{(1)}\Rightarrow\|(Y_{T_{f(u_{k}^{n})}^{Z}},Z_{T_{f(u_{k}^{n})}^{Z}})-(u_{k}^{n},f(u_{k}^{n}))\|\leq
u_{k+1}^{n}-u_{k}^{n}=\frac{u_{1}-u_{0}}{n}\;\;\;\mbox{and}$
$\displaystyle\omega\in
B_{n,k}^{(2)}\Rightarrow\|(Y_{T_{u_{k+1}^{n}}^{Y}},Z_{T_{u_{k+1}^{n}}^{Y}})-(u_{k+1}^{n},f(u_{k+1}^{n}))\|\leq
f(u_{k}^{n})-f(u_{k+1}^{n})\leq\varepsilon(n)\,,$

where in the last inequality, the mapping $n\mapsto\varepsilon(n)$ does not
depend on $k$ and satisfies $\lim_{n\rightarrow\infty}\varepsilon(n)=0$. This
follows from the uniform continuity of $f$ on $[u_{0},u_{1}]$. Therefore if
$\omega\in\limsup_{n}B_{n}$, then there is a subsequence
$u_{j_{n}}^{i_{n}}(\omega):=u_{n}$ of $(u_{k}^{n})_{n\geq 1,\,0\leq k\leq n}$
which converges to $u\in[u_{0},u_{1}]$ and such that
$(Y_{T_{f(u_{n})}^{Z}},Z_{T_{f(u_{n})}^{Z}})$ converges to $(u,f(u))$ or
$(Y_{T_{u_{n}}^{Y}},Z_{T_{u_{n}}^{Y}})$ converges to $(u,f(u))$. Assume for
instance that $(Y_{T_{f(u_{n})}^{Z}},Z_{T_{f(u_{n})}^{Z}})$ converges to
$(u,f(u))$. Since $Y$ and $Z$ are non decreasing, this means that
$T_{f(u_{n})}^{Z}$ tends to some value $S$ such that
$(Y_{S-},Z_{S-})=(u,f(u))$. Moreover $t_{0}\leq S\leq t_{1}$ and $u_{0}\leq
Y_{S-}\leq u_{1}$. In particular, $Z_{S-}=f(Y_{S-})$, so that
$S=\inf\\{t:Z_{t}\geq f(Y_{t})\\}$, since $Y$ and $Z$ are non decreasing. If
$(Y_{T_{u_{n}}^{Y}},Z_{T_{u_{n}}^{Y}})$ converges to $(u,f(u))$, then the same
argument leads to the same conclusion. Then (5.22) follows from part 1. of
Theorem 2.1.

Now, let us prove that

(5.23)
$\lim_{n\rightarrow+\infty}\mathbb{P}(B_{n})=d_{Z}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,du-
d_{Y}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,df(u)\,.$

We first notice that, for all $n\in\mathbb{N}$ and all integer $0\leqslant
k\leqslant n-1$, $B_{n,k}^{(1)}\cap B_{n,k}^{(2)}=\emptyset$. Moreover,
$\sigma_{n}$ being a subdivision of $[u_{0},u_{1}]$, for all integers $n\geq
2$, $k^{\prime}$, $k^{\prime\prime}$ such that $k^{\prime}\neq
k^{\prime\prime}$ and $0\leqslant k^{\prime},k^{\prime\prime}\leqslant n-1$,
$(B_{n,k^{\prime}}^{(1)}\cup
B_{n,k^{\prime}}^{(2)})\cap(B_{n,k^{\prime\prime}}^{(1)}\cup
B_{n,k^{\prime\prime}}^{(2)})=\emptyset$. On the other hand, from
$(\ref{2622})$ and $(\ref{8946})$ in Lemma 5.2, we obtain

$\mathbb{P}\left(B_{n,k}^{(1)}\right)=d_{Z}\int_{u_{k}^{n}}^{u_{k+1}^{n}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u_{k}^{n}))\,dt\,du\;\;\;\mbox{and}\;\;\;\mathbb{P}\left(B_{n,k}^{(2)}\right)=d_{Y}\int_{f(u_{k+1}^{n})}^{f(u_{k}^{n})}\int_{t_{0}}^{t_{1}}p_{t}(u_{k+1}^{n},s)\,dt\,ds\,,$

so that

$\displaystyle\mathbb{P}(B_{n})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}\left[\mathbb{P}\left(B_{n,k}^{(1)}\right)+\mathbb{P}\left(B_{n,k}^{(2)}\right)\right]$
$\displaystyle=$ $\displaystyle
d_{Z}\sum_{k=0}^{n-1}\int_{u_{k}^{n}}^{u_{k+1}^{n}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u_{k}^{n}))\,dt\,du+d_{Y}\sum_{k=0}^{n-1}\int_{f(u_{k+1}^{n})}^{f(u_{k}^{n})}\int_{t_{0}}^{t_{1}}p_{t}(u_{k+1}^{n},s)\,dt\,ds\,.$

Recall that $0<t_{0}<t_{1}<\infty$ and $0<u_{0}<u_{1}<\infty$ and assume first
that $f(u_{1})<f(u_{0})$, in particular $0<f(u_{1})<f(u_{0})<\infty$. Then we
derive from the continuity of $(t,y,z)\mapsto p_{t}(y,z)$ on
$[t_{0},t_{1}]\times[u_{0},u_{1}]\times[f(u_{1}),f(u_{0})]$ that the mapping
$(y,z)\mapsto\int_{t_{0}}^{t_{1}}p_{t}(y,z)\,dt$ is uniformly continuous on
$[u_{0},u_{1}]\times[f(u_{1}),f(u_{0})]$. Set
$\bar{v}(y,z):=\int_{t_{0}}^{t_{1}}p_{t}(y,z)\,dt$. We derive from the uniform
continuity of $(y,z)\mapsto\bar{v}(y,z)$ on
$[u_{0},u_{1}]\times[f(u_{1}),f(u_{0})]$ and this of the mapping $u\mapsto
f(u)$ on $[u_{0},u_{1}]$ that there are functions
$\alpha(n),\,\beta(n)\rightarrow 0$, as $n\rightarrow\infty$, such that for
all $0\leq k\leq n-1$,
$|\bar{v}(u,f(u_{k}^{n}))-\bar{v}(u_{k}^{n},f(u_{k}^{n}))|\leq\alpha(n)$, for
all $u\in[u_{k}^{n},u_{k+1}^{n}]$ and
$|\bar{v}(u_{k+1}^{n},s)-\bar{v}(u_{k+1}^{n},f(u_{k+1}^{n}))|\leq\beta(n)$,
for all $s\in[f(u_{k+1}^{n}),f(u_{k}^{n})]$. These arguments and the Riemann-
Stieltjes integrability of the function $u\mapsto\bar{v}(u,f(u))$ with respect
to the measures $du$ and $df(u)$ imply that

$\lim_{n\rightarrow\infty}\sum_{k=0}^{n-1}\int_{u_{k}^{n}}^{u_{k+1}^{n}}\bar{v}(u,f(u_{k}^{n}))\,du=\lim_{n\rightarrow\infty}\sum_{k=0}^{n-1}\bar{v}(u_{k}^{n},f(u_{k}^{n}))(u_{k+1}^{n}-u_{k}^{n})=\int_{u_{0}}^{u_{1}}\bar{v}(u,f(u))\,du\,,$

and that

(5.24)
$\displaystyle\lim_{n\rightarrow\infty}\sum_{k=0}^{n-1}\int_{f(u_{k+1}^{n})}^{f(u_{k}^{n})}\bar{v}(u_{k+1}^{n},s)\,ds$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\sum_{k=0}^{n-1}\bar{v}(u_{k+1}^{n},f(u_{k+1}^{n}))(f(u_{k}^{n})-f(u_{k+1}^{n}))$
$\displaystyle=$ $\displaystyle-\int_{u_{0}}^{u_{1}}\bar{v}(u,f(u))\,df(u)\,,$

which proves (5.23) in this case. If $f(u_{0})=f(u_{1}):=w$, then the result
follows from the same arguments. We only need to replace the mapping
$(y,z)\mapsto\int_{t_{0}}^{t_{1}}p_{t}(y,z)\,dt$ by the mapping
$y\mapsto\int_{t_{0}}^{t_{1}}p_{t}(y,w)\,dt$ and the function
$u\mapsto\bar{v}(u,f(u))$ by the function $u\mapsto\bar{v}(u,w)$. (The term
(5.24) is then clearly equal to 0.) Finally, recall that
$\displaystyle\limsup_{n\longrightarrow\infty}B_{n}\subset\\{Z_{S}=f(Y_{S}),\,u_{0}\leq
Y_{S}\leq u_{1},\,t_{0}\leq S\leq t_{1}\\}$, a.s. Then by Fatou’s lemma,

$\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}\leq Y_{S}\leq u_{1},\,t_{0}\leq S\leq
t_{1})\geqslant\mathbb{P}(\limsup_{n\longrightarrow\infty}B_{n})\geqslant\lim_{n\longrightarrow\infty}\mathbb{P}(B_{n})\,,$

and (5.23) implies that

$\displaystyle\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}\leq Y_{S}\leq
u_{1},\,t_{0}\leq S\leq t_{1})\geqslant$ $\displaystyle\qquad\qquad
d_{Z}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,du-
d_{Y}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,df(u)\,.$

Since the latter inequality is valid for all $u_{0}$, $u_{1}$, $t_{0}$ and
$t_{1}$, large inequalities in the left member can be replaced by strict
inequalities, so that (5.18) holds.

The proof of the corresponding inequality in (2.3) uses $(\ref{2621})$ and
$(\ref{8945})$ instead of $(\ref{2622})$ and $(\ref{8946})$ in the above
arguments. It is actually simpler so it is omitted.

###  5.4. Proof of the upper bound in identities (2.3) and (2.4)

Let us first prove the upper bound in equation (2.4), that is the inequality

$\displaystyle\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}<Y_{S}<u_{1},\,t_{0}<S<t_{1})\leqslant$
(5.25) $\displaystyle\qquad\qquad
d_{Z}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,du-
d_{Y}\int_{u_{0}}^{u_{1}}\int_{t_{0}}^{t_{1}}p_{t}(u,f(u))\,dt\,df(u)\,.$

Recall the definition (5.19) of the sequence $(\sigma_{n})_{n\geq 1}$ of
subdivisions of $[u_{0},u_{1}]$. We define the two following sequences of
events which are slightly different from those introduced in (5.20) and
(5.21),

$\displaystyle\overline{B}_{n,k}^{(1)}$ $\displaystyle=$
$\displaystyle\left\\{Z_{T_{f(u_{k}^{n})}^{Z}}=f(u_{k}^{n}),u_{k}^{n}\leqslant
Y_{T_{f(u_{k}^{n})}^{Z}}\leqslant u_{k+1}^{n},\,t_{0}\leqslant
T_{f(u_{k}^{n})}^{Z}\leqslant t_{1}+o_{1}(n)\right\\}$
$\displaystyle\overline{B}_{n,k}^{(2)}$ $\displaystyle=$
$\displaystyle\left\\{Y_{T_{u_{k+1}^{n}}^{Y}}=u_{k+1}^{n},f(u_{k+1}^{n})\leqslant
Z_{T_{u_{k+1}^{n}}^{Y}}\leqslant f(u_{k}^{n}),\,t_{0}\leqslant
T_{u_{k+1}^{n}}^{Y}\leqslant t_{1}+o_{2}(n)\right\\}\,,$

where $o_{1}(n)=d_{Z}^{-1}\max_{0\leq k\leq
n-1}(f(u_{k}^{n})-f(u_{k+1}^{n}))$, if $d_{Z}>0$ and $o_{2}(n)=(nd_{Y})^{-1}$,
if $d_{Y}>0$. If $d_{Y}=0$ (resp. $d_{Z}=0$), we set $o_{2}(n)=\infty$ (resp.
$o_{1}(n)=\infty$), for all $n\geq 1$. Note that from uniform continuity of
$f$ on $[u_{0},u_{1}]$, if $d_{Z}>0$, then $o_{1}(n)\rightarrow 0$, as
$n\rightarrow\infty$ (which is also clearly verified for $o_{2}$, when
$d_{Y}>0$). Then let us also define the sequences,

$\displaystyle B_{n,k}^{(3)}$ $\displaystyle=$
$\displaystyle\left\\{f(u_{k+1}^{n})\leqslant
Z_{T_{f(u_{k}^{n})}^{Z}-}\leqslant f(u_{k}^{n}),\,u_{k}^{n}\leqslant
Y_{T_{f(u_{k}^{n})}^{Z}-}\leqslant u_{k+1}^{n}\right\\}$ $\displaystyle
B_{n,k}^{(4)}$ $\displaystyle=$ $\displaystyle\left\\{u_{k}^{n}\leqslant
Y_{T_{u_{k+1}^{n}}^{Y}-}\leqslant u_{k+1}^{n},\,f(u_{k+1}^{n})\leqslant
Z_{T_{u_{k+1}^{n}}^{Y}-}\leqslant f(u_{k}^{n})\right\\}\,,$

for $n\geq 1$ and $k=0,\dots,n-1$ and let us check the inclusion,

(5.26) $A:=\\{Z_{S}=f(Y_{S}),\,u_{k}^{n}\leqslant
Y_{S}<u_{k+1}^{n},\,t_{0}<S<t_{1}\\}\subset\overline{B}_{n,k}^{(1)}\cup\overline{B}_{n,k}^{(2)}\cup
B_{n,k}^{(3)}\cup B_{n,k}^{(4)}\,.$

Assume that $A$ holds and note that $T_{f(u_{k}^{n})}^{Z}\geq S$ and
$T_{u_{k+1}^{n}}^{Y}\geq S$. Then, only the two events described below in
$(a)$ and $(b)$ can occur:

$(a)$ In the first case, either $Z$ creeps through the level $f(u_{k}^{n})$
before time $T_{u_{k+1}^{n}}^{Y}$ or $Y$ creeps through the level
$u_{k+1}^{n}$ before time $T_{f(u_{k}^{n})}^{Z}$.

Let us consider the subcase where $Z$ creeps through the level $f(u_{k}^{n})$
before time $T_{u_{k+1}^{n}}^{Y}$. Then we have
$Z_{T_{f(u_{k}^{n})}^{Z}}=f(u_{k}^{n})$ and since $u_{k}^{n}\leqslant
Y_{S}<u_{k+1}^{n}$ and $S\leq T_{f(u_{k}^{n})}^{Z}\leq T_{u_{k+1}^{n}}^{Y}$,
we also have $u_{k}^{n}\leqslant Y_{T_{f(u_{k}^{n})}^{Z}}\leqslant
u_{k+1}^{n}$. Moreover, recall that $S\geq t_{0}$, therefore
$T_{f(u_{k}^{n})}^{Z}\geq t_{0}$. On the other hand since $Z_{S}=f(Y_{S})$,
then from time $S$, the process $Z$ will reach the level $f(u_{k}^{n})$ in a
time which is at most equal to $d_{Z}^{-1}(f(u_{k}^{n})-f(Y_{S}))$. Hence
$T_{f(u_{k}^{n})}^{Z}\leq S+d_{Z}^{-1}(f(u_{k}^{n})-f(Y_{S}))\leq
t_{1}+o_{1}(n)$. We conclude that $\overline{B}_{n,k}^{(1)}$ holds in this
subcase.

In the subcase where $Y$ creeps through the level $u_{k+1}^{n}$ before time
$T_{f(u_{k}^{n})}^{Z}$, we prove in the same way that
$\overline{B}_{n,k}^{(2)}$ holds.

$(b)$ In the second case, either $Z$ jumps through the level $f(u_{k}^{n})$
and the event $B_{n,k}^{(3)}$ holds or $Y$ jumps through the level
$u_{k+1}^{n}$ and the event $B_{n,k}^{(4)}$ holds. Note the both events can
occur at the same time.

Then the inclusion (5.26) is proved.

$u_{k}^{n}$$u_{k+1}^{n}$$f\left(u_{k+1}^{n}\right)$$f\left(u_{k}^{n}\right)$$Y_{T_{f\left(u_{k}^{n}\right)}^{Z}-}$$Z_{T_{f(u_{k}^{n})}^{Z}-}$

$u_{k}^{n}$$u_{k+1}^{n}$$f\left(u_{k+1}^{n}\right)$$f\left(u_{k}^{n}\right)$$Y_{T_{u_{k+1}^{n}}^{Y}-}$$Z_{T_{u_{k+1}^{n}}^{Y}-}$

Figure 3. An occurrence of the event $B_{n,k}^{(3)}$ on the left and
$B_{n,k}^{(4)}$ on the right

Let us set $v_{1}^{(n)}(y,z):=\int_{t_{0}}^{t_{1}+o_{1}(n)}p_{t}(y,z)\,dt$ and
$v_{2}^{(n)}(y,z):=\int_{t_{0}}^{t_{1}+o_{2}(n)}p_{t}(y,z)\,dt$. Then the
inclusion (5.26) implies that

$\displaystyle\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{0}<Y_{S}<u_{1},\,t_{0}<S<t_{1})$
$\displaystyle\leqslant$
$\displaystyle\sum_{k=0}^{n-1}\mathbb{P}(Z_{S}=f(Y_{S}),\,u_{k}^{n}\leq
Y_{S}\leq u_{k+1}^{n},\,t_{0}<S<t_{1})$ $\displaystyle\leqslant$
$\displaystyle\sum_{k=0}^{n-1}\mathbb{P}(\overline{B}_{n,k}^{(1)})+\sum_{k=0}^{n-1}\mathbb{P}(\overline{B}_{n,k}^{(2)})+\sum_{k=0}^{n-1}\mathbb{P}(B_{n,k}^{(3)})+\sum_{k=0}^{n-1}\mathbb{P}(B_{n,k}^{(4)})\,,$

and Lemmas 5.2 and 5.3 imply that

$\displaystyle\sum_{k=0}^{n-1}\mathbb{P}(\overline{B}_{n,k}^{(1)})+\sum_{k=0}^{n-1}\mathbb{P}(\overline{B}_{n,k}^{(2)})+\sum_{k=0}^{n-1}\mathbb{P}(B_{n,k}^{(3)})+\sum_{k=0}^{n-1}\mathbb{P}(B_{n,k}^{(4)})$
$\displaystyle\leqslant$
$\displaystyle\sum_{k=0}^{n-1}d_{Z}\int_{u_{k}^{n}}^{u_{k+1}^{n}}v_{1}^{(n)}(u,f(u_{k}^{n}))\,du+\sum_{k=0}^{n-1}d_{Y}\int_{f(u_{k+1}^{n})}^{f(u_{k}^{n})}v_{2}^{(n)}(u_{k+1}^{n},s)\,ds$
$\displaystyle+\sum_{k=0}^{n-1}\frac{1}{n}(\varepsilon_{1}(n)+\varepsilon_{2}(n))\,.$

We prove in a similar way as in the previous subsection that the sum of the
two first sums in the last expression tends to
$d_{Z}\int_{u_{0}}^{u_{1}}v(u,f(u))\,du-
d_{Y}\int_{u_{0}}^{u_{1}}v(u,f(u))\,df(u)$, when $n$ tend to $\infty$. The
third sum tends clearly to 0, as $n$ tends to $\infty$, so that inequality
(5.25) is proved.

The upper bound in equality (2.3) is proved in the same way. This achieves the
proof of Theorem 2.1.

###  5.5. Proof of Proposition 2.1

Part 1. is proved as part 1. of Theorem 2.1, see Subsection 5.1. (Note that
$\mathbb{P}(S>0)=1$ since $\lim_{t\rightarrow 0+}f(t)>0$. In particular,
$Y_{S-}$ and $Z_{S-}$ are well defined.)

The proof of part 2. follows very similar arguments as those used for the
proof of the upper bound in identities (2.3) and (2.4) which is given in
Subsection 5.4. We only describe it here in broad outline. Recall the sequence
$(\sigma_{n})_{n\geq 1}$ of subdivisions of $[u_{0},u_{1}]$ defined in (5.19).
Then define the two following sequences of events

$\displaystyle C_{n,k}^{(1)}$ $\displaystyle=$
$\displaystyle\left\\{Z_{T_{f(u_{k+1}^{n})}^{Z}}=f(u_{k+1}^{n}),u_{k}^{n}\leqslant
Y_{T_{f(u_{k+1}^{n})}^{Z}}\leqslant u_{k+1}^{n}\right\\}$ $\displaystyle
C_{n,k}^{(2)}$ $\displaystyle=$ $\displaystyle\left\\{f(u_{k}^{n})\leqslant
Z_{T_{f(u_{k+1}^{n})}^{Z}-}<f(u_{k+1}^{n}),\,u_{k}^{n}\leqslant
Y_{T_{f(u_{k+1}^{n})}^{Z}-}<u_{k+1}^{n}\right\\}\,.$

Since $f$ is non decreasing, the inclusion,

(5.27) $C:=\\{Z_{S}=f(Y_{S}),\,u_{k}^{n}\leqslant Y_{S}<u_{k+1}^{n}\\}\subset
C_{n,k}^{(1)}\cup C_{n,k}^{(2)}$

holds for all $n$ sufficiently large and $k=0,\dots,n-1$. Indeed, by
definition of $S$, there is $\varepsilon>0$ such that the path of $(Y,Z)$
stays above the graph of $f$ on the interval $[S,S+\varepsilon]$. This implies
that for all $n$ sufficiently large, if $t\geq 0$ is such that
$Y_{S+t}\in[u_{k}^{n},u_{k+1}^{n}]$ for some $k=0,1,\dots,n-1$, then
$Z_{S+t}\geq f(S+t)$ so that either the process $Z$ creeps above the level
$f(u_{k+1}^{n})$, that is $C_{n,k}^{(1)}$ holds, or $Z$ jumps above this
level, that is $C_{n,k}^{(2)}$ holds. There is no other possibility.

The remainder of the proof is then very similar to the arguments of the end of
Subsections 5.1 and 5.4, the main difference being that the term due to the
creeping of $Y$ is now absent.

###  5.6. On assumptions of Theorem 2.1

The aim of this subsection is to show that Theorem 2.1 holds whenever the
transition semigroup of $(Y,Z)$ is absolutely continuous with bounded
densities. Let $d\geq 1$ and let ${\mathcal{L}}$ be any possibly killed
$\mathbb{R}^{d}$ valued Lévy process.

######  Proposition 5.1.

Let $\Psi$ be the characteristic exponent of ${\mathcal{L}}$, that is
$\mathbb{E}(e^{i\langle\xi,{\mathcal{L}}_{1}\rangle},1<\zeta)=e^{-\Psi(\xi)}$,
$\xi\in\mathbb{R}^{d}$, where $\zeta$ is the lifetime of ${\mathcal{L}}$. Then
the following assertions are equivalent:

  * $(i)$

The transition semigroup of ${\mathcal{L}}$ is absolutely continuous and there
is a version of its densities, denoted by $x\longmapsto l_{t}(x)$,
$x\in\mathbb{R}^{d}$, which are bounded for all $t>0$.

  * $(ii)$

For all $t>0$,

$e^{-t\Psi(\xi)}\in\mathbf{L}^{1}(\mathbb{R}^{d})\,.$

If these conditions are satisfied, then the function $(t,x)\mapsto l_{t}(x)$
is continuous on $(0,\infty)\times\mathbb{R}^{d}$. In particular, if all
coordinates of ${\mathcal{L}}$ are subordinators, then $l_{t}(0)=0$, for all
$t>0$.

###### Proof.

The proof follows similar arguments as those used in the beginning of Section
2 in [4]. Boundness of $l_{t}$ implies that
$l_{t}\in\mathbf{L}^{2}(\mathbb{R}^{d})$ and consequently
$e^{-t\Psi(\xi)}\in\mathbf{L}^{2}(\mathbb{R}^{d})$, for all $t>0$, which
implies that $e^{-t\Psi(\xi)}\in\mathbf{L}^{1}(\mathbb{R}^{d})$, for all
$t>0$. Conversely, if $e^{-t\Psi(\xi)}\in\mathbf{L}^{1}(\mathbb{R}^{d})$, for
all $t>0$, then Fourier inversion theorem implies $(i)$. Indeed, by the
Riemann-Lebesgue lemma, $l_{t}\in\mathcal{C}_{0}(\mathbb{R}^{d})$.
Furthermore, from Fourier inversion theorem we have, for all
$(t,x)\in(0,\infty)\times\mathbb{R}^{d}$,

(5.28) $l_{t}(x)=\frac{1}{2\pi}\int_{\mathbb{R}^{d}}e^{-i\langle
x,\xi\rangle}e^{-t\Psi(\xi)}d\xi\,.$

Let $(t_{0},x_{0})\in(0,\infty)\times\mathbb{R}^{d}$ and $K$ be a compact
neighborhood of $(t_{0},x_{0})$ such that
$K\subset(0,\infty)\times\mathbb{R}^{d}$. Then, there exists $t_{1}>0$ such
that for all $(t,x,\xi)\in K\times\mathbb{R}^{d}$, $|e^{-i\langle
x,\xi\rangle}e^{-t\Psi(\xi)}|=e^{-t\mbox{\tiny
Re}(\Psi(\xi))}\leqslant|e^{-t_{1}\Psi(\xi)}|=e^{-t_{1}\mbox{\tiny
Re}(\Psi(\xi))}$, (recall that $\mbox{Re}(\Psi(\xi))\geq 0$). Since
$e^{-t_{1}\Psi(\xi)}\in\mathbf{L}^{1}(\mathbb{R}^{d})$, we obtain from (5.28)
and Lebesgue’s dominated convergence theorem, that $(t,x)\mapsto l_{t}(x)$ is
continuous in $(t_{0},x_{0})$ and finally on $(0,\infty)\times\mathbb{R}^{d}$.
∎

Recall from Subsection 5.1 that the absolute continuity of $v(dy,dz)$ is
actually enough for part 1. of Theorem 2.1 to hold. Therefore, condition $(i)$
(or $(ii)$) of Proposition 5.1 implies part 1. of Theorem 2.1. Moreover,
condition $(i)$ (or $(ii)$) of Proposition 5.1 applied to the bivariate
subordinator $(Y,Z)$ immediately implies the assumption of part 3. of Theorem
2.1. The mapping $(t,y,z)\mapsto p_{t}(y,z)$ is then locally bounded on
$(0,\infty)^{3}$. Hence the assumption of assertion 2’. (stated after Theorem
2.1) is satisfied for all $t_{0}$, $t_{1}$ such that $0<t_{0}<t_{1}<\infty$
and this implies part 2. of Theorem 2.1 by monotone convergence.

Let us finally notice that assumptions of Theorem 2.1 could be weakened.
Indeed, it appears from the limits in (5.24) that the continuity assumption of
$v$ could be replaced by an assumption closed to the Riemann integrability of
this function. (Recall that continuity of $v$ is not required in lemmas 5.3
and 5.2.)

##  6\. Proof of Theorem 2.2, Corollary 2.3 and Proposition 2.2

Recall that $X$ is a real non killed Lévy process and that we have assumed
that $|X|$ is not a subordinator.

###  6.1. Some reminders in fluctuation theory

Recall that $\overline{X}_{t}=\sup\\{X_{s}:0\leq s\leq t\\}$ and let
$\underline{X}_{t}=\inf\\{X_{s}:0\leq s\leq t\\}$. It is well known that both
processes $\overline{X}-X$ and $\underline{X}-X$ are strongly Markovian. The
state 0 is regular for itself, for the process $\overline{X}-X$ (resp.
$\underline{X}-X$) if and only if 0 is regular for $[0,\infty)$ (resp.
$(-\infty,0]$) for the process $X$. When it is the case, we will simply write
that $[0,\infty)$ (resp. $(-\infty,0]$) is regular. Similarly, we will write
that $(0,\infty)$ (resp. $(-\infty,0)$) is regular when 0 is regular for
$(0,\infty)$ (resp. $(-\infty,0)$) for the process $X$. Recall also that when
$X$ is not a compound Poisson process, at least one of the half lines
$(-\infty,0)$ or $(0,\infty)$ is regular. Moreover compound Poisson processes
are the only Lévy processes for which $[0,\infty)$ and $(-\infty,0]$ are
regular but not $(0,\infty)$ and $(-\infty,0)$.

Let us briefly recall the definition of the local time at 0 of the process
$\overline{X}-X$ which we denote by $L$. If $[0,\infty)$ is regular, $L$ is a
continuous, increasing, additive functional such that $L_{0}=0$, a.s., and the
support of the measure $dL_{t}$ is the set
$\overline{\\{t:X_{t}=\overline{X}_{t}\\}}$. Moreover $L$ is the unique
process, up to a multiplicative constant, satisfying these properties. We can
normalize it for instance in the following way,

$\mathbb{E}\left(\int_{0}^{\infty}e^{-t}\,dL_{t}\right)=1\,.$

When $[0,\infty)$ is not regular, the set $\\{t:(X-\overline{X})_{t}=0\\}$ is
discrete and the local time at 0 of the process $\overline{X}-X$ is defined as
follows

$L_{t}=\sum_{k=0}^{R_{t}}{\rm\bf e}^{(k)}\,,$

where $R_{0}=0$, for $t>0$,
$R_{t}=\mbox{Card}\\{s\in(0,t]:X_{s}=\overline{X}_{s}\\}$ and ${\rm\bf
e}^{(k)}$, $k=0,1,\dots$ is a sequence of independent and exponentially
distributed random variables with parameter

$\gamma=\left(1-\mathbb{E}(e^{-\tau^{+}_{0}})\right)^{-1}\,,$

and $\tau_{0}^{+}=\inf\\{t>0:X_{t}>0\\}$. The choice of the parameter $\gamma$
is consistent with the normalisation
$\mathbb{E}\left(\int_{0}^{\infty}e^{-t}\,dL_{t}\right)=1$ of the regular
case. The process $L$ is called the local time at the supremum of $X$. The
local time at the infimum is defined in the same way with respect to the
process $\underline{X}-X$ and is denoted by $L^{*}$. It corresponds to the
local time at the supremum of the dual process $X^{*}:=-X$.

Let us now recall the definition of the ladder processes. The ladder time
processes $\tau$ and $\tau^{*}$, and the ladder height processes $H$ and
$H^{*}$ are the following (possibly killed) subordinators:

$\tau_{t}=\inf\\{s:L_{s}>t\\}\,,\;\;\tau^{*}_{t}=\inf\\{s:L_{s}^{*}>t\\}\,,\;\;H_{t}=X_{\tau_{t}}\,,\;\;H^{*}_{t}=-X_{\tau_{t}^{*}}\,,\;\;t\geq
0\,,$

where $\tau_{t}=H_{t}=+\infty$, for $t\geq L_{\infty}$ and
$\tau_{t}^{*}=H_{t}^{*}=+\infty$, for $t\geq L^{*}_{\infty}$. Recall that the
processes $(\tau,H)$ and $(\tau^{*},H^{*})$ are bivariate subordinators. The
renewal measures of the ladder processes $(\tau,H)$ and $(\tau^{*},H^{*})$ are
the measures on $[0,\infty)^{2}$ defined by,

$U(dt,dh)=\int_{0}^{\infty}\mathbb{P}(\tau_{u}\in dt,\,H_{u}\in
dh)\,du\;\;\;\mbox{and}\;\;\;U^{*}(dt,dh)=\int_{0}^{\infty}\mathbb{P}(\tau^{*}_{u}\in
dt,\,H^{*}_{u}\in dh)\,du\,.$

The drifts $d_{\tau}$ and $d_{\tau^{*}}$ of the subordinators $\tau$ and
$\tau^{*}$ satisfy

(6.29) $\int_{0}^{t}\mbox{\rm
1I}_{\\{X_{s}=\overline{X}_{s}\\}}\,ds=d_{\tau}L_{t}\,,\;\;\;\int_{0}^{t}\mbox{\rm
1I}_{\\{X_{s}=\underline{X}_{s}\\}}\,ds=d_{\tau^{*}}L_{t}^{*}.$

Recall that $d_{\tau}>0$ if and only if $(-\infty,0)$ is not regular.
Similarly, $d_{\tau^{*}}>0$ if and only if 0 is not regular for $(0,\infty)$.
Hence if $X$ is not a compound Poisson process, then $d_{\tau}d_{\tau^{*}}=0$
holds since 0 is necessarily regular for at least one of the half lines. If
$X$ is a compound Poisson process, then $d_{\tau}>0$ and $d_{\tau^{*}}>0$.

Given our normalisation of the local time $L$, the drifts $d_{H}$ and
$d_{H^{*}}$ of $H$ and $H^{*}$ satisfy $2d_{H}d_{H^{*}}=\sigma^{2}$, where
$\sigma$ represents the Brownian part of $X$ in its Lévy-Khintchine
decomposition (this follows from the Wiener-Hopf factorisation of the
characteristic exponent of $X$, see p.166 in [1]). When $X$ has bounded
variation, we define its drift as the almost sure limit $d:=\lim_{t\rightarrow
0}X_{t}/t$. It was proved in Section 2. of [13] that in this case, $d>0$ if
and only if $d_{H}>0$. When $X$ has unbounded variation and no Brownian part,
Theorem Kaa of [19] asserts that $d_{H}>0$ if and only if

$\int_{0}^{1}\frac{x}{\int_{-x}^{0}\bar{\bar{\pi}}_{1}(u)du}\hat{\pi}(x)\,dx<\infty\,,$

where $\pi$ is the Lévy measure of $X$, for $x\in\mathbb{R}$,
$\hat{\pi}(x)=\int_{\mathbb{R}}\left(\mbox{\rm 1I}_{\\{0<x\leq u\\}}+\mbox{\rm
1I}_{\\{u\leq x<0\\}}\right)d\pi(u)$ and for $x\in[-1,0)$,
$\bar{\bar{\pi}}_{1}(x)=\int_{-1}^{x}\hat{\pi}(u)du$. Note also that if
$d_{\tau}>0$ and $d_{H}>0$, then $d_{H}=dd_{\tau}$, since $\lim_{t\rightarrow
0}\frac{H_{t}}{t}=d_{H}=\lim_{t\rightarrow
0}\frac{X_{\tau_{t}}}{\tau_{t}}\frac{\tau_{t}}{t}=dd_{\tau}$, a.s.

Let us assume for the remainder of this section that $X$ is not a compound
Poisson process. The Itô measure $n^{*}$ of the excursions away from 0 of the
process $X-\underline{X}$ is the characteristic measure of the Poisson point
process

$t\mapsto\left\\{\begin{array}[]{ll}\\{(X-\underline{X})_{\tau_{t-}^{*}+s},\,0\leq
s<\tau_{t}^{*}-\tau_{t-}^{*}\\}\;\;\;\mbox{if}\;\;\;\tau_{t-}^{*}<\tau_{t}^{*}\\\
\delta\;\;\;\mbox{if}\;\;\;\tau_{t-}^{*}=\tau_{t}^{*}\,,\end{array}\right.$

where $\delta$ is some isolated point added to the space of excursions. We
refer to [1], Chap. IV, [12], Chap. 6 and [7] for a more detailed definition
$n^{*}$. The measure $n^{*}$ is a Markovian measure whose semigroup is this of
the killed Lévy process when it enters in the negative half line. More
specifically, for $x>0$, let us denote by $\mathbb{Q}_{x}^{*}$ the law of the
process $(X_{t}\mbox{\rm 1I}_{\\{t<\tau_{0}^{-}\\}}+\infty\cdot\mbox{\rm
1I}_{\\{t\geq\tau_{0}^{-}\\}},\,t\geq 0)$ under $\mathbb{P}_{x}$, where
$\tau_{0}^{-}=\inf\\{t>0:X_{t}<0\\}$. That is for $\Lambda\in\mathcal{F}_{t}$,

(6.30)
$\mathbb{Q}_{x}^{*}(\Lambda,t<\zeta)=\mathbb{P}_{x}(\Lambda,\,t<\tau_{0}^{-})\,.$

Then for all Borel positive functions $f$ and $g$ and for all $s,t>0$,

(6.31)
$n^{*}(f(X_{t})g(X_{s+t}),s+t<\zeta)=n^{*}(f(X_{t})\mathbb{E}^{\mathbb{Q}^{*}}_{X_{t}}(g(X_{s})),s<\zeta)\,,$

where $\mathbb{E}_{x}^{\mathbb{Q}^{*}}$ means the expectation under
$\mathbb{Q}_{x}^{*}$ and $\zeta$ is the lifetime of the excursions. Moreover,
we define the entrance law $(q_{t}^{*}(dx),t>0)$ of the excursion measure
$n^{*}$ as

(6.32) $n^{*}(f(X_{t}),t<\zeta)=\int_{0}^{\infty}f(x)q_{t}^{*}(dx)\,.$

The Itô measure of the excursions away from 0 of the reflected process at its
supremum $\overline{X}-X=X^{*}-\underline{X}^{*}$ is defined in the same way
as for $X-\underline{X}$. It will be denoted by $n$ and its entrance law by
$q_{t}(dx)$. We define the probability measures $\mathbb{Q}_{x}$ in the same
way as in (6.30) with respect to the dual process $X^{*}=-X$. Let us finally
recall that the entrance laws $q_{t}(dh)$ and $q_{t}^{*}(dh)$ are related to
the renewal measures $U(dt,dh)$ and $U^{*}(dt,dh)$ of $(\tau,H)$ and
$(\tau^{*},H^{*})$ as follows,

(6.33) $\displaystyle
d_{\tau^{*}}\delta_{\\{(0,0)\\}}(dt,dh)+q_{t}^{*}(dh)\,dt$ $\displaystyle=$
$\displaystyle U(dt,dh)$ (6.34) $\displaystyle
d_{\tau}\delta_{\\{(0,0)\\}}(dt,dh)+q_{t}(dh)\,dt$ $\displaystyle=$
$\displaystyle U^{*}(dt,dh).$

These identities between measures on $[0,\infty)^{2}$ can be found in Lemma 1
of [2]. In particular, under the assumption of Theorem 2.2, that is when
$U(dt,dh)$ has a density on $(0,\infty)^{2}$, this density is the same as this
of $q_{t}^{*}(dh)\,dt$, which explains the notation $q_{t}^{*}(h)$,
$t,h\in(0,\infty)$ in this theorem.

###  6.2. Proof of Theorem 2.2

Part 1. of Theorem 2.2 is a direct application of Lemma 5.1. It suffices to
consider $\mathcal{L}_{t}=(X_{t},t)$ and $G(u,v)=u-f(v)$ in this lemma. (Note
that $\mathbb{P}(T_{f}>0)=1$ follows from the fact that $f$ is positive. In
particular, $X_{T_{f}-}$ is well defined on $\\{T_{f}<\infty\\}$.)

Then part 2. is a consequence of Lemma 5.1 and Theorem 2.1. Recall first from
Subsection 6.1 that $(\tau,H)$ is a bivariate subordinator. If the time
$T_{f}=\inf\\{t:X_{t}>f(t)\\}$ satisfies
$X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}})$, then it is a zero of
$\overline{X}-X$ and hence there is $S^{\prime}$ such that
$T_{f}=\tau_{S^{\prime}}$ or $T_{f}=\tau_{S^{\prime}-}$. Therefore
$H_{S^{\prime}}=f(\tau_{S^{\prime}})$ or
${X}_{\tau_{S^{\prime}-}}=\overline{X}_{\tau_{S^{\prime}-}}=f(\tau_{S^{\prime}-})$
but we know, applying again Lemma 5.1 to the function $G$ as above, that in
the latter case, $X$ is necessarily a.s. continuous at time
$\tau_{S^{\prime}-}$. In particular,
$H_{S^{\prime}-}={X}_{\tau_{S^{\prime}-}-}=f(\tau_{S^{\prime}-})$. On the
other hand, applying again Lemma 5.1, it follows that if
$H_{S^{\prime}-}=f(\tau_{S^{\prime}-})$, then $(\tau,H)$ is necessarily a.s.
continuous at time $S^{\prime}$, so that
$\tau_{S^{\prime}}=\tau_{S^{\prime}-}$. This proves that
$H_{S^{\prime}}=f(\tau_{S^{\prime}})$, a.s. Moreover, since the subordinator
$H$ is not lattice, the time $S^{\prime}$ is necessarily unique, so that
$S^{\prime}=S=\inf\\{t\geq 0:H_{t}>f(\tau_{t})\\}$. This proves in particular
that $\tau_{S}=T_{f}$.

Conversely assume that $S$ satisfies
$H_{S}=X_{\tau_{S}}=\overline{X}_{\tau_{S}}=f(\tau_{S})$. Since $X$ is not
lattice, and $f$ is non increasing, a time $T$ satisfying
$X_{T}=\overline{X}_{T}=f(T)$ is necessarily unique. Moreover, when $X$ is not
a compound Poisson process, $\tau_{S}=\inf\\{t:X_{t}>X_{\tau_{S}}\\}$. Hence,
since $f$ is non increasing,
$\tau_{S}=\inf\\{t:\overline{X}_{t}>f(t)\\}=T_{f}$, so that
$X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}})$ and we have proved the identity

$\\{H_{S}=f(\tau_{S}),\,u_{0}<\tau_{S}<u_{1}\\}=\\{X_{T_{f}}=\overline{X}_{T_{f}}=f({T_{f}}),\,u_{0}<T_{f}<u_{1}\\}$

which allows us to conclude thanks to part 2. of Theorem 2.1.

###  6.3. On assumptions of Theorem 2.2

It is assumed in part 2. of Theorem 2.2 that the renewal measure of the ladder
process $(\tau,H)$ has continuous densities $(t,x)\mapsto q_{t}^{*}(x)$ on
$(0,\infty)^{2}$. The next proposition provides conditions for this to hold.
Moreover, when the later are satisfied, $q_{t}^{*}$ is positive on
$(0,\infty)$, for all $t>0$. This ensures the positivity of the creeping
probability (2.8) whenever $d_{H}>0$ or $d_{\tau}>0$ and $f(t_{1})<f(t_{0})$.

######  Proposition 6.1.

Assume that $X$ is a non killed real Lévy process which satisfies the
equivalent conditions $(i)$ and $(ii)$ of Proposition 5.1, then

  * $1.$

the semigroup $q_{t}^{*}(x,dy)$ of the killed process $(X,\mathbb{Q}^{*}_{x})$
defined in $(\ref{4524})$ is absolutely continuous and its densities
$(t,x,y)\mapsto q_{t}^{*}(x,y)$ are continuous in
$(0,\infty)^{2}\times[0,\infty)$.

  * $2.$

Assume moreover that for all $c\geq 0$, the process $(|X_{t}-ct|,\,t\geq 0)$
is not a subordinator. Then the entrance law, $q_{t}^{*}(dx)$ is absolutely
continuous on $[0,\infty)$ for all $t>0$ and there is a version
$q_{t}^{*}(x)$, $x\geq 0$, $t>0$ of its densities such that the function
$(t,x)\mapsto q^{*}_{t}(x)$ is continuous on $(0,\infty)\times[0,\infty)$.
Moreover, for all $t>0$, $q^{*}_{t}$ is positive on $(0,\infty)$.

Denote by $h$ the renewal function of $H$, that is
$h(x)=\int_{0}^{\infty}\mathbb{P}(H_{t}\leq x)\,dt$, then $\lim_{y\rightarrow
0+}q_{t}^{*}(x,y)=h(0)q_{t}(x)$. Moreover for all $t>0$, $\lim_{x\rightarrow
0+}q_{t}^{*}(x)=h(0)p_{t}(0)/t$. We then set $q_{t}^{*}(x,0)=h(0)q_{t}(x)$ and
$q_{t}^{*}(0)=h(0)p_{t}(0)/t$, for all $t,x>0$.

###### Proof.

The first assertion is a slight reinforcement of Lemma 2 in [18] and our proof
follows the same arguments. Recall first the definition (6.30) of the killed
process $(X,\mathbb{Q}^{*}_{x})$ and let us denote by $q_{t}^{*}(x,dy)$ its
semigroup. Then from Lemma 2 in [18], under our assumption and thanks to
Proposition 5.1, this semigroup admits densities $q_{t}^{*}(x,y)$ which
satisfy for all $t>0$, $x>0$, $y\geq 0$,

$q_{t}^{*}(x,y)=\mathbb{P}_{x,y}^{t}(\Lambda_{t})p_{t}(y-x)\,,$

where $\mathbb{P}_{x,y}^{t}$ is the law of the bridge of $(X,\mathbb{P})$
starting from $x$ and ending in $y$ at time $t$ and
$\Lambda_{t}=\\{\underline{X}_{t}>0\\}\cup\\{X_{s}>0,\,0\leq
s<t,\,X_{t}=0\\}$. Here, in order to define $q_{t}^{*}(x,0)$ for all $t>0$ and
$x>0$, we have included $y=0$ since the right member is then well defined.
From Corollary 1 of [5], the measures $\mathbb{P}_{x,y}^{t}$ are weakly
continuous in $(t,x,y)\in(0,\infty)\times\mathbb{R}^{2}$. Then from the same
arguments as in Lemma 2 of [18], these measures satisfy
$\mathbb{P}_{x,y}^{t}(\partial\Lambda_{t})=0$ for all $t,x>0$ and $y\geq 0$.
Hence from portemanteau theorem and Proposition 5.1, the mapping
$(t,x,y)\mapsto\mathbb{P}_{x,y}^{t}(\Lambda_{t})p_{t}(y-x)$ is continuous in
$(0,\infty)^{2}\times[0,\infty)$.

In order to prove $2.$, recall from part 3. of Lemma 1 in [2] that
$q_{t}^{*}(dx)$ is absolutely continuous under our assumptions. Let us denote
by $q_{t}^{*}(x)$ the corresponding densities and recall the Chapman-
Kolmogorov equation,

(6.35)
$q_{t+s}^{*}(y)=\int_{0}^{\infty}q_{s}^{*}(x)q_{t}^{*}(x,y)\,dx\,,\;\;\;s,t,y>0\,,$

which can be derived for instance from (6.31). From Proposition 1 in [4], for
all $x>0$ and $t>0$, $\lim_{y\rightarrow 0+}q_{t}^{*}(x,y)=h(0)q_{t}(x)$. Let
us set $q_{t}^{*}(x,0)=h(0)q_{t}(x)$. Let $\Psi$ be the characteristic
exponent of $X$, that is $\mathbb{E}(e^{i\xi X_{1}})=e^{-\Psi(\xi)}$. From
Lemma 2 in [18], $q_{t}^{*}(x,y)\leq p_{t}(y-x)$, for all $t>0$ and
$x,y\in(0,\infty)^{2}$, so that,

(6.36) $q_{t}^{*}(x,y)\leq
p_{t}(y-x)=\frac{1}{2\pi}\int_{\mathbb{R}}e^{-i(y-x)\xi}e^{-t\Psi(\xi)}d\xi\leq\frac{1}{2\pi}\int_{\mathbb{R}}|e^{-t\Psi(\xi)}|d\xi\,,$

and this bound can be extended to $y=0$. From the same arguments as in the
proof of Proposition 5.1, the mapping
$t\mapsto\int_{\mathbb{R}}|e^{-t\Psi(\xi)}|d\xi$ is continuous on
$(0,\infty)$. Let $(t_{0},y_{0})\in(0,\infty)\times[0,\infty)$, then from
(6.36) and the continuity of $t\mapsto\int_{\mathbb{R}}|e^{-t\Psi(\xi)}|d\xi$,
there is $\varepsilon>0$ and $C_{\varepsilon}>0$ such that $q_{t}^{*}(x,y)\leq
C_{\varepsilon}$, for all $x>0$, $y\geq 0$ and
$t\in(t_{0}-\varepsilon,t_{0}+\varepsilon)$. Moreover, from (6.31)
$\int_{0}^{\infty}q_{s}^{*}(x)\,dx=n^{*}(s<\zeta)<\infty$, hence, (6.35) can
be extended to $y=0$ and we can set $q_{t}^{*}(0)=\lim_{y\rightarrow
0+}q_{t}^{*}(y)$, $t>0$. Moreover, continuity of $(t,y)\mapsto q_{t}^{*}(y)$
at $(t_{0},y_{0})$ follows from (6.35), continuity of $(t,x,y)\mapsto
q_{t}^{*}(x,y)$ in $(0,\infty)^{2}\times[0,\infty)$ proved in assertion 1. and
Lebesgue theorem of dominated convergence.

The positivity of $q_{t}^{*}$ for all $t>0$ follows from Lemma 2 in [4] and
the two last assertions in $2.$ follow from Proposition 1 in [4]. ∎

###  6.4. Proof of Corollary 2.3

As recalled in Subsection 6.1, since the process $Y_{t}:=X_{t}+at$, $t\geq 0$
has bounded variation and positive drift, its upward ladder height process
also has positive drift. Moreover, by assumption, for all $t>0$ the
distribution $\mathbb{P}(X_{t}\in dx)$ is absolutely continuous with a bounded
density, and for all $c\geq 0$, the process $(|X_{t}-ct|,\,t\geq 0)$ is not a
subordinator. Then these conditions are clearly also satisfied by the process
$Y$. Let us denote by $q_{t}^{*,a}(x)$ the density of the renewal measure of
the ladder process associated to $Y$. Then from part 2. of Proposition 6.1,
$(t,x)\mapsto q_{t}^{*,a}(x)$ is continuous on $(0,\infty)^{2}$. Moreover,
$q_{t}^{*,a}$ is positive on $(0,\infty)$ for all $t>0$. These properties
imply, from part 2. of Theorem 2.2, that the process $Y$ creeps at its
supremum through the function $f(t)+at$ over the interval $(t_{0},t_{1})$, so
that $X$ creeps through the function $f(t)$ over the interval $(t_{0},t_{1})$.

###  6.5. Proof of Proposition 2.2

Proposition 2.2 is a consequence of Proposition 2.1 and its proof follows the
same arguments as those used for the proof of Theorem 2.2 in Subsection 6.2.
The details are omitted.

##  7\. Proof of Theorem 2.3

The proof is a direct application of the extension of Tanaka’s construction of
random walks conditioned to stay positive obtained in [8]. Let us first state
this transformation. Recall that $\overline{X}_{t}=\sup_{s\leq t}X_{s}$ and
let us set for $t>0$,

$g_{t}=\sup\\{s<t:X_{s}=\overline{X}_{s}\\}\;\;\;\mbox{and}\;\;\;d_{t}=\inf\\{s>t:X_{s}=\overline{X}_{s}\\}\,.$

Then the process defined by

$\displaystyle W_{0}$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle W_{t}$
$\displaystyle=$ $\displaystyle\overline{X}_{d_{t}}+\mbox{\rm
1I}_{\\{d_{t}>g_{t}\\}}(\overline{X}-X)_{(d_{t}+g_{t}-t)-}\,,\;\;\;t>0\,,$

has the law of $X$ conditioned to stay positive, $X^{\uparrow}$. According to
this construction, the process $W$ is obtained by time reversing of each
excursion of the reflected process $\overline{X}-X$ and then by gluing them
back together.

The assertion $0<\sigma_{f}<\infty$, a.s. is straightforward since
$\lim_{t\rightarrow 0+}f(t)>0$ and
$\lim_{t\rightarrow\infty}X_{t}^{\uparrow}=\infty$, almost surely. Then we
derive from the above construction that almost surely,

$\overline{X}_{t}=\underline{\underline{W}}_{t}\,,\;\;\;\mbox{for all $t\geq
0$.}$

Moreover, the sets $\\{t\geq 0:X_{t}=\overline{X}_{t}\\}$ and $\\{t\geq
0:W_{t}=\underline{\underline{W}}_{t}\\}$ are identical. It follows that
$\overline{X}_{T_{f}}=\underline{\underline{W}}_{T_{f}}$, almost surely and
that $X_{T_{f}}=\overline{X}_{T_{f}}$ if and only if
$W_{T_{f}}=\underline{\underline{W}}_{T_{f}}$. Then it remains to observe that
on the set
$\\{X_{T_{f}}=\overline{X}_{T_{f}}\\}=\\{W_{T_{f}}=\underline{\underline{W}}_{T_{f}}\\}$,
$T_{f}=\sup\\{t:W_{t}\leq f(t)\\}=\sigma_{f}$ and this proves parts 1. and 2.
of Theorem 2.3.

## References

  * [1] Bertoin, J. Lévy processes. Cambridge University Press, 1996.
  * [2] Chaumont, L. (2013) On the law of the supremum of Lévy processes. Ann. Probab., Vol. 41, No. 3A, 1191-1217.
  * [3] Chaumont, L. and Doney, R.A. (2005) On Lévy processes conditioned to stay positive. Electron. J. Probab. 10, no. 28, 948–961.
  * [4] Chaumont, L. and Małecki, J. On the asymptotic behavior of the density of the supremum of Lévy processes. Annales de l’Institut Henri Poincaré, Vol. 52, No. 3, 1178-1195, (2016).
  * [5] Chaumont, L., and Uribe Bravo, G. Markovian Bridges: Weak continuity and pathwise constructions. Ann. Probab., Vol. 39, No. 2, 609–647, (2011).
  * [6] Chi, Z. On exact sampling of the first passage event of a Lévy process with infinite Lévy measure and bounded variation. Stochastic Process. Appl., 126, no. 4, 1124–1144, (2016).
  * [7] Doney, R.A. Fluctuation theory for Lévy processes. Lectures from the 35th Summer School on Probability Theory held in Saint-Flour, July 6–23, 2005. Lecture Notes in Mathematics, 1897. Springer, Berlin, 2007.
  * [8] Doney, R.A. Tanaka’s construction for random walks and Lévy processes. Séminaire de Probabilités XXXVIII, 1–4, Lecture Notes in Math., 1857, Springer, Berlin, 2005.
  * [9] Griffin, P.S., and Maller, R.A. The time at which a Lévy process creeps. Electronic Journal of Probability, Vol 16, (2011).
  * [10] Jakubowski, T. On Harnack inequality for $\alpha$-stable Ornstein-Uhlenbeck processes. Math. Z., 258(3) : 609-628, 2008.
  * [11] Kesten, H. Hitting probabilities of single points for processes with stationary independent increments. Memoirs of the American Mathematical Society, No. 93 American Mathematical Society, Providence, R.I. 1969.
  * [12] Kyprianou, A.E. Fluctations of Lévy processes with applications. Springer, 2014.
  * [13] Millar, P. W. Exit properties of stochastic processes with stationary independent increments. Trans. Amer. Math. Soc. 178, 459–479, (1973).
  * [14] Neveu, J. Une généralisation des processus à accroissements positifs indépendants. Abh. Math. Sem. Univ. Hamburg 25, 36–61, (1961).
  * [15] Rogers, L. C. G. A new identity for real Lévy processes. Ann. Inst. H. Poincaré Probab. Statist. 20, no. 1, 21–34, (1984).
  * [16] Sato, K. Lévy processes and infinitely divisible distribution. Cambridge University Press, 1999.
  * [17] Savov, M, and Winkel, M Right inverse of Lévy processes : the excursion measure in the general case. Electron. comm. Probab. 15, 572–584, (2010).
  * [18] Uribe, G. G. Bridges of Lévy processes conditioned to stay positive. Bernoulli (1) 190–206, (2014).
  * [19] Vigon, V. Votre Lévy rampe-t-il ? J. London Math. Soc. (2) 65, 243–256, (2002).