HITTING DISTRIBUTIONS OF \(\alpha \)-STABLE PROCESSES VIA PATH CENSORING AND SELF-SIMILARITY

By Andreas E. Kyprianou^*,§,¶, Juan Carlos Pardo^†,‡ and Alexander R. Watson^*,§

^*University of Bath, ^†CIMAT

AMS 2000 subject classifications: 60G52, 60G18, 60G51.

Keywords and phrases: Lévy processes, stable processes, hitting distributions, hitting probabilities, killed potential, stable processes conditioned to stay positive, positive self-similar Markov processes, Lamperti transform, Lamperti-stable processes, hypergeometric Lévy processes.

We consider two first passage problems for stable processes, not necessarily symmetric, in one dimension. We make use of a novel method of path censoring in order to deduce explicit formulas for hit- ting probabilities, hitting distributions, and a killed potential meas- ure. To do this, we describe in full detail the Wiener-Hopf factor- isation of a new Lamperti-stable-type Lévy process obtained via the Lamperti transform, in the style of recent work in this area.

1. Introduction. A Lévy process is a stochastic process issued from the origin with stationary and independent increments and càdlàg paths. If \(X: = (X_t)_{t\geq 0}\) is a one-dimensional Lévy process with law \(\stP \), then the classical Lévy-Khintchine formula states that for all \(t\geq 0\) and \(\theta \in \RR \), the characteristic exponent \(\Psi (\theta ) : = -t^{-1}\log \stE (e^{\iu \theta X_t})\) satisfies

\[ \Psi (\theta ) = \iu a\theta + \frac {1}{2}\sigma ^2\theta ^2 + \int _{\RR } (1 - e^{\iu \theta x} + \iu \theta x\Indic {|x|\leq 1})\Pi (\dd x), \]

where \(a\in \mathbb {R}\), \(\sigma \geq 0\) and \(\Pi \) is a measure (the Lévy measure) concentrated on \(\RR \setminus \{0\}\) such that \(\int _{\RR }(1\wedge x^2)\Pi (\dd x)<\infty \).

\((X,\stP )\) is said to be a (strictly) \(\alpha \)-stable process if it is a Lévy process which also satisfies the scaling property: under \(\stP \), for every \(c > 0\), the process \((cX_{t c^{-\alpha }})_{t \ge 0}\) has the same law as \(X\). It is known that \(\alpha \in (0,2]\), and the case \(\alpha = 2\) corresponds to Brownian motion, which we exclude. The Lévy-Khintchine representation of such a process is as follows: \(\sigma = 0\), and \(\Pi \) is absolutely continuous with density given by

\[ c_+ x^{-(\alpha +1)} \Indic {x > 0} + c_- \abs {x}^{-(\alpha +1)} \Indic {x < 0}, \for x \in \RR , \]

where \(c_+, c_- \ge 0\), and \(c_+ = c_-\) when \(\alpha = 1\). It holds that \(a = (c_+-c_-)/(\alpha -1)\) when \(\alpha \ne 1\), and we specify that \(a = 0\) when \(\alpha = 1\); the latter condition is a restriction which ensures that \(X\) is a symmetric process when \(\alpha = 1\), so the only \(1\)-stable process we consider is the symmetric Cauchy process.

These choices mean that, up to a multiplicative constant \(c>0\), \(X\) has the characteristic exponent

\[ \Psi (\theta ) = \begin {cases} c\abs {\theta }^\alpha (1 - \iu \beta \tan \frac {\pi \alpha }{2}\sgn \theta ) & \alpha \in (0,2)\setminus \{1\}, \\ c\abs {\theta } & \alpha = 1, \end {cases} \for \theta \in \mathbb {R}, \]

where \(\beta = (c_+- c_-)/(c_+ + c_-)\). For more details, see Sato [31, §14].

For consistency with the literature we appeal to in this article, we shall always parameterise our \(\alpha \)-stable process such that

\[ c_+ = \frac {\Gamma (\alpha +1)}{\Gamma (\alpha \rho )\Gamma (1-\alpha \rho )} \quad \text {and} \quad c_- = \frac {\Gamma (\alpha +1)}{\Gamma (\alpha \rhohat )\Gamma (1-\alpha \rhohat )}, \]

where \(\rho = \stP (X_t \ge 0) = \stP (X_t > 0)\) is the positivity parameter, and \(\rhohat = 1-\rho \).

We take the point of view that the class of stable processes, with this normalisation, is parameterised by \(\alpha \) and \(\rho \); the reader will note that all the quantities above can be written in terms of these parameters. We shall restrict ourselves a little further within this class by excluding the possibility of having only one-sided jumps. Together with our assumption about the case \(\alpha = 1\), this gives us the following set of admissible parameters:

\begin{multline*} \stparamset = \bigl \{ (\alpha ,\rho ) : \alpha \in (0,1), \, \rho \in (0,1) \bigr \} \\ {} \cup \bigl \{ (\alpha ,\rho ) : \alpha \in (1,2), \, \rho \in (1-1/\alpha , 1/\alpha ) \bigr \} \cup \bigl \{ (\alpha , \rho ) = (1, 1/2) \bigr \}. \end{multline*}

After Brownian motion, \(\alpha \)-stable processes are often considered an exemplary family of processes for which many aspects of the general theory of Lévy processes can be illustrated in closed form. First passage problems, which are relatively straightforward to handle in the case of Brownian motion, become much harder in the setting of a general Lévy process on account of the inclusion of jumps. A collection of articles through the 1960s and early 1970s, appealing largely to potential analytic methods for general Markov processes, were relatively successful in handling a number of first passage problems, in particular for symmetric \(\alpha \)-stable processes in one or more dimensions. See, for example, [3, 14, 15, 26, 29] to name but a few.

However, following this cluster of activity, several decades have passed since new results on these problems have appeared. The last few years have seen a number of new, explicit first passage identities for one-dimensional \(\alpha \)-stable processes, thanks to a better understanding of the intimate relationship between the aforesaid processes and positive self-similar Markov processes. See, for example, [6, 8, 10, 20, 22].

In this paper we return to the work of Blumenthal, Getoor and Ray [3], published in 1961, which gave the law of the position of first entry of a symmetric \(\alpha \)-stable process into the unit ball. Specifically, we are interested in establishing the same law, but now for all the one-dimensional \(\alpha \)-stable processes which fall within the parameter regime \(\stparamset \); we remark that Port [26, §3.1, Remark 3] found this law for processes with one-sided jumps, which justifies our exclusion of these processes in this work. Our method is modern in the sense that we appeal to the relationship of \(\alpha \)-stable processes with certain positive self-similar Markov processes. However, there are two notable additional innovations. First, we make use of a type of path censoring. Second, we are able to describe in explicit analytical detail a non-trivial Wiener-Hopf factorisation of an auxiliary Lévy process from which the desired solution can be sourced. Moreover, as a consequence of this approach, we are able to deliver a number of additional, related identities in explicit form for \(\alpha \)-stable processes.

We now state the main results of the paper. Let \(\stP _x\) refer to the law of \(X+x\) under \(\stP \), for each \(x\in \mathbb {R}\). We introduce the first hitting time of the interval \((-1,1)\),

\[ \tau _{-1}^1 = \inf \{ t > 0 : X_t \in (-1,1) \} . \]

Note that, for \(x \notin \{-1,1\}\), \(\stP _x\bigl (X_{\tau _{-1}^1} \in (-1,1)\bigr ) = 1\) so long as \(X\) is not spectrally one-sided. However, in Proposition 1.3, we will consider a spectrally negative \(\alpha \)-stable process, for which \(X_{\tau _{-1}^1}\) may take the value \(-1\) with positive probability.

^‡Supported by CONACYT grant 128896.

^§These authors gratefully acknowledge support from the Santander Research Fund.

^¶Corresponding Author.

When \(X\) is symmetric, Theorem 1.1 reduces immediately to Theorems B and C of [3]. Moreover, the following hitting probability can be obtained.

This extends Corollary 2 of [3], as can be seen by differentiating and using the doubling formula [17, 8.335.2] for the gamma function.

The spectrally one-sided case can be found as the limit of Theorem 1.1, as we now explain. The first part of the coming proposition is due to Port [26], but we re-state it for the sake of clarity.

The following killed potential is also available.

To obtain the potential of the previous theorem for \(x < -1\) and \(y < -1\), one may easily appeal to duality. In the case that \(x<-1\) and \(y>1\), one notes that

\begin{equation} \stE _x \int _0^{\tau _{-1}^1} \Indic {X_t \in \dd y} \, \dd t = \stE _x \stE _\Delta \int _0^{\tau _{-1}^1} \Indic {X_t \in \dd y} \, \dd t , \label {DELTA} \end{equation}

where the quantity \(\Delta \) is randomised according to the distribution of
\(X_{\tau ^+_{-1}}\Indic {X_{\tau ^+_{-1}}>1}\), with

\[ \tau ^+_{-1}= \inf \{ t > 0 : X_t > -1 \}. \]

Although the distribution of \(X_{\tau ^+_{-1}}\) is available from [30], and hence the right hand side of (1) can be written down explicitly, it does not seem to be easy to find a convenient closed form expression for the corresponding potential density.

Regarding this potential, let us finally remark that our methods give an explicit expression for this potential even when \(\alpha \in (1,2)\), but again, there does not seem to be a compact expression for the density.

A further result concerns the first passage of \(X\) into the half-line \((1,\infty )\) before hitting zero. Let

\[ \tau _1^+ = \inf \{ t > 0 : X_t > 1 \} \text { and } \tau _0 = \inf \{ t > 0 : X_t = 0 \} . \]

Recall that when \(\alpha \in (0,1]\), \(\stP _x(\tau _0 = \infty ) = 1\), while when \(\alpha \in (1,2)\), \(\stP _x(\tau _0 < \infty ) = 1\), for \(x \ne 0\). In the latter case, we can obtain a hitting probability as follows.

It is not difficult to push Theorem 1.5 a little further to give the law of the position of first entry into \((1,\infty )\) on the event \(\{\tau ^+_1<\tau _0\}\). Indeed, by the Markov property, for \(x < 1\),

\begin{align} \stP _x(X_{\tau _1^+} \in \dd y, \, \tau _1^+ < \tau _0) &= \stP _x(X_{\tau _1^+} \in \dd y) - \stP _x(X_{\tau _1^+} \in \dd y, \tau _0 < \tau _1^+) \nonumber \\ \label {put-in-Rog} &= \stP _x(X_{\tau _1^+} \in \dd y) - \stP _x(\tau _0 < \tau _1^+) \stP _0(X_{\tau _1^+} \in \dd y). \end{align} Moreover, Rogozin [30] found that, for \(x < 1\) and \(y>1\),

\( \seteqnumber {3} \)

\begin{equation} \stP _x(X_{\tau _1^+} \in \dd y) = \frac {\sin (\pi \alpha \rho )}{\pi } (1-x)^{\alpha \rho } (y-1)^{-\alpha \rho } (y-x)^{-1} \, \dd y. \label {Rog-first} \end{equation}

Hence substituting (3) together with the hitting probability from Theorem 1.5 into (2) yields the following corollary.

We conclude this section by giving an overview of the rest of the paper. In Section 2, we recall the Lamperti transform and discuss its relation to \(\alpha \)-stable processes. In Section 3, we explain the operation which gives us the path-censored \(\alpha \)-stable process \(Y\), that is to say the \(\alpha \)-stable process with the negative components of its path removed. We show that \(Y\) is a positive self-similar Markov process, and can therefore be written as the exponential of a time-changed Lévy process, say \(\xi \). We show that the Lévy process \(\xi \) can be decomposed into the sum of a compound Poisson process and a so-called Lamperti-stable process. Section 4 is dedicated to finding the distribution of the jumps of this compound Poisson component, which we then use in Section 5 to compute in explicit detail the Wiener-Hopf factorisation of \(\xi \). Finally, we make use of the explicit nature of the Wiener-Hopf factorisation in Section 6 to prove Theorems 1.1 and 1.4. There we also prove Theorem 1.5 via a connection with the process conditioned to stay positive.

A positive self-similar Markov process (pssMp) with self-similarity index \(\alpha > 0\) is a standard Markov process \(Y = (Y_t)_{t\geq 0}\) with filtration \(\GGt \) and probability laws \((\stP _x)_{x > 0}\), on \([0,\infty )\), which has \(0\) as an absorbing state and which satisfies the scaling property, that for every \(x, c > 0\),

\( \seteqnumber {4} \)

\begin{equation} \label {scaling prop}\text { the law of } (cY_{t c^{-\alpha }})_{t \ge 0} \text { under } \stP _x \text { is } \stP _{cx} \text {.} \end{equation}

Here, we mean “standard” in the sense of [4], which is to say, \(\GGt \) is a complete, right-continuous filtration, and \(Y\) has càdlàg paths and is strong Markov and quasi-left-continuous.

In the seminal paper [25], Lamperti describes a one to one correspondence between pssMps and Lévy processes, which we now outline. It may be worth noting that we have presented a slightly different definition of pssMp from Lamperti; for the connection, see [34, §0].

Let \(S(t) = \int _0^t (Y_u)^{-\alpha }\, \dd u .\) This process is continuous and strictly increasing until \(Y\) reaches zero. Let \((T(s))_{s \ge 0}\) be its inverse, and define

\[ \xi _s = \log Y_{T(s)} \qquad s\geq 0. \]

Then \(\xi : = (\xi _s)_{s\geq 0}\) is a Lévy process started at \(\log x\), possibly killed at an independent exponential time; the law of the Lévy process and the rate of killing do not depend on the value of \(x\). The real-valued process \(\xi \) with probability laws \((\LevP _y)_{y \in \RR }\) is called the Lévy process associated to \(Y\), or the Lamperti transform of \(Y\).

An equivalent definition of \(S\) and \(T\), in terms of \(\xi \) instead of \(Y\), is given by taking \(T(s) = \int _0^s \exp (\alpha \xi _u)\, \dd u\) and \(S\) as its inverse. Then,

\( \seteqnumber {5} \)

\begin{equation} \label {Lamp repr} Y_t = \exp (\xi _{S(t)}) \end{equation}

for all \(t\geq 0\), and this shows that the Lamperti transform is a bijection.

Let \(T_0 = \inf \{ t > 0: Y_t = 0 \}\) be the first hitting time of the absorbing state zero. Then the large-time behaviour of \(\xi \) can be described by the behaviour of \(Y\) at \(T_0\), as follows:

• (i) If \(T_0 = \infty \) a.s., then \(\xi \) is unkilled and either oscillates or drifts to \(+ \infty \).

• (ii) If \(T_0 < \infty \) and \(Y_{T_0 -} = 0\) a.s., then \(\xi \) is unkilled and drifts to \(-\infty \).

• (iii) If \(T_0 < \infty \) and \(Y_{T_0 -} > 0\) a.s., then \(\xi \) is killed.

It is proved in [25] that the events mentioned above satisfy a zero-one law independently of \(x\), and so the three possibilites above are an exhaustive classification of pssMps.

Three concrete examples of positive self-similar Markov processes related to \(\alpha \)-stable processes are treated in Caballero and Chaumont [6]. We present here the simplest case, namely that of the \(\alpha \)-stable process absorbed at zero. To this end, let \(X\) be the \(\alpha \)-stable process as defined in the introduction, and let

\[ \tau _0^- = \inf \{ t > 0 : X_t \le 0 \} . \]

Denote by \(\LSabs \) the Lamperti transform of the pssMp \(\stproca {X_t \Indic {t < \tau _0^-}}\). Then \(\LSabs \) has Lévy density

\( \seteqnumber {6} \)

\begin{equation} \label {LSabs density} c_+ \frac {e^x}{(e^x-1)^{\alpha +1}} \Indic {x > 0} + c_- \frac {e^x}{(1-e^x)^{\alpha +1}} \Indic {x < 0} , \end{equation}

and is killed at rate \(c_-/\alpha = \frac {\Gamma (\alpha )}{\Gamma (\alpha \rhohat )\Gamma (1-\alpha \rhohat )}\).

We note here that in [6] the authors assume that \(X\) is symmetric when \(\alpha = 1\), which motivates the same assumption in this paper.

We now describe the construction of the censored \(\alpha \)-stable process that will lie at the heart of our analysis, show that it is a pssMp and discuss its Lamperti transform.

Henceforth, \(X\), with probability laws \((\stP _x)_{x \in \RR }\), will denote the \(\alpha \)-stable process defined in the introduction. Define the occupation time of \((0,\infty )\),

\[ A_t = \int _0^t \Indic {X_s > 0} \, \dd s , \]

and let \(\gamma (t) = \inf \{ s \ge 0 : A_s > t \}\) be its right-continuous inverse. Define a process \((\Ych _t)_{ t \ge 0}\) by setting \(\Ych _t = X_{\gamma (t)}\), \(t\geq 0\). This is the process formed by erasing the negative components of \(X\) and joining up the gaps.

Write \(\FFt \) for the augmented natural filtration of \(X\), and \(\GG _t = \FF _{\gamma (t)}\), \(t \ge 0\).

We now make zero into an absorbing state. Define the stopping time

\[ T_0 = \inf \{ t > 0 : \Ych _t = 0 \} \]

and the process

\[ Y_t = \Ych _t \Indic {t < T_0} , \for t \ge 0 , \]

so that \(Y :=(Y_t)_{t\geq 0}\) is \(\Ych \) absorbed at zero. We call the process \(Y\) with probability laws \((\stP _x)_{x > 0}\) the path-censored \(\alpha \)-stable process.

We now consider the pssMp \(Y\) more closely for different values of \(\alpha \in (0,2)\). Taking account of Bertoin [2, Proposition VIII.8] and the discussion immediately above it we know that for \(\alpha \in (0, 1]\), points are polar for \(X\). That is, \(T_0 = \infty \) a.s., and so in this case \(Y = \Ych \). Meanwhile, for \(\alpha \in (1,2)\), every point is recurrent, so \(T_0 < \infty \) a.s.. However, the process \(X\) makes infinitely many jumps across zero before hitting it. Therefore, in this case \(Y\) approaches zero continuously. In fact, it can be shown that, in this case, \(\Ych \) is the recurrent extension of \(Y\) in the spirit of [27] and [13].

Now, let \(\xi =(\xi _s)_{s\geq 0}\) be the Lamperti transform of \(Y\). That is,

\( \seteqnumber {7} \)

\begin{equation} \xi _s = \log Y_{T(s)} , \for s \ge 0, \label {e:LT of Y} \end{equation}

where \(T\) is a time-change. As in Section 2, we will write \(\LevP _y\) for the law of \(\xi \) started at \(y \in \RR \); note that \(\LevP _y\) corresponds to \(\stP _{\exp (y)}\). The space transformation (7), together with the above comments and, for instance, the remark on p. 34 of [2], allows us to make the following distinction based on the value of \(\alpha \).

• (i) If \(\alpha \in (0,1)\), \(T_0 = \infty \) and \(X\) (and hence \(Y\)) is transient a.s.. Therefore, \(\xi \) is unkilled and drifts to \(+ \infty \).

• (ii) If \(\alpha = 1\), \(T_0 = \infty \) and every neighbourhood of zero is an a.s. recurrent set for \(X\), and hence also for \(Y\). Therefore, \(\xi \) is unkilled and oscillates.

• (iii) If \(\alpha \in (1,2)\), \(T_0 < \infty \) and \(Y\) hits zero continuously. Therefore, \(\xi \) is unkilled and drifts to \(- \infty \).

Furthermore, we have the following result.

Before beginning the proof, let us make some preparatory remarks. Let

\[ \tau = \inf \{ t > 0 : X_t < 0 \} \quad \text {and} \quad \sigma = \inf \{ t > \tau : X_t > 0 \} \]

be hitting and return times of \((-\infty ,0)\) and \((0,\infty )\) for \(X\). Note that, due to the time-change \(\gamma \), \(Y_\tau = X_\sigma \), while \(Y_{\tau -} = X_{\tau -}\). We require the following lemma.

• Proof of Proposition 3.4. First we note that, applying the strong Markov property to the \(\GGt \)-stopping time \(\tau \), it is sufficient to study the process \((Y_t)_{t \le \tau }\).

  It is clear that the path section \((Y_t)_{ t < \tau }\) agrees with \((X_t)_{ t < \tau _0^-}\); however, rather than being killed at time \(\tau \), the process \(Y\) jumps to a positive state. Recall now that the effect of the Lamperti transform on the time \(\tau \) is to turn it into an exponential time of rate \(c_-/\alpha \) which is independent of \((\xi _s)_{s < S(\tau )}\). This immediately yields the decomposition of \(\xi \) into the sum of \(\xiLS : = (\xiLS _s)_{s\geq 0}\) and \(\xiCPP : = (\xiCPP _s)_{s\geq 0}\), where \(\xiCPP \) is a process which jumps at the times of a Poisson process with rate \(c_-/\alpha \), but whose jumps may depend on the position of \(\xi \) prior to this jump. What remains is to be shown is that the values of the jumps of \(\xiCPP \) are also independent of \(\xiLS \).

  By the remark at the beginning of the proof, it is sufficient to show that the first jump of \(\xiCPP \) is independent of the previous path of \(\xiLS \). Now, using only the independence of the jump times of \(\xiLS \) and \(\xiCPP \), we can compute

  \( \seteqnumber {8} \)

  \begin{align*} \jump Y_{\tau } := Y_{\tau } - Y_{\tau -} &= \exp (\xiLS _{S(\tau )} + \xiCPP _{S(\tau )}) - \exp (\xiLS _{S(\tau )-} + \xiCPP _{S(\tau ) -}) \\ &= \exp (\xi _{S(\tau )-}) \bigl [ \exp (\jump \xiCPP _{S(\tau )}) - 1 \bigr ] \\ &= X_{\taull } \bigl [ \exp (\jump \xiCPP _{S(\tau )}) - 1 \bigr ] , \end{align*} where \(S\) is the Lamperti time change for \(Y\), and \(\jump \xiCPP _s = \xiCPP _s - \xiCPP _{s-}\). Now,

  \[ \exp (\jump \xiCPP _{S(\tau )})= 1 + \frac {\jump Y_{\tau }}{X_{\taull }} = 1 + \frac {X_\sigma - X_{\taull }}{X_\taull } = \frac {X_\sigma }{X_\taull }. \]

  Hence, it is sufficient to show that \(\frac {X_\sigma }{X_\taull }\) is independent of \((X_t, t < \tau )\). The proof of this is essentially the same as that of part (iii) in Theorem 4 from Chaumont, Panti and Rivero [11], which we reproduce here for clarity.

  First, observe that one consequence of Lemma 3.5 is that, for \(g\) a Borel function and \(x > 0\),

  \[ \stE _x \biggl [ g\biggl ( \frac {X_\sigma }{X_\taull } \biggr )\biggr ] = \stE _1 \biggl [ g\biggl ( \frac {X_\sigma }{X_\taull } \biggr ) \biggr ] . \]

  Now, fix \(n \in \NN \), \(f\) and \(g\) Borel functions and \(s_1 < s_2 < \dotsb < s_n = t\). Then, using the Markov property and the above equality,

  \( \seteqnumber {8} \)

  \begin{align*} \stE _1 \biggl [ f(X_{s_1}, \dotsc , X_t) g\biggl ( \frac {X_\sigma }{X_\taull } \biggr ) \Indic {t < \tau } \biggr ] &= \stE _1 \biggl [ f(X_{s_1}, \dotsc , X_t) \Indic {t < \tau } \stE _{X_t} \biggl [ g\biggl ( \frac {X_\sigma }{X_\taull } \biggr ) \biggr ] \biggr ] \\ &= \stE _1 \biggl [ f(X_{s_1}, \dotsc , X_t) \Indic {t < \tau } \biggr ] \stE _{1} \biggl [ g\biggl ( \frac {X_\sigma }{X_\taull } \biggr ) \biggr ]. \end{align*} We have now shown that \(\xiLS \) and \(\xiCPP \) are independent, and this completes the proof. □

4. Jump distribution of the compound Poisson component. In this section, we express the jump distribution of \(\xiCPP \) in terms of known quantitites, and hence derive its characteristic function and density.

Before stating a necessary lemma, we establish some more notation. Let \(\hat X\) be an independent copy of the dual process \(-X\) and denote its probability laws by \((\stPhat _x)_{x \in \RR }\), and let

\[ \hat \tau = \inf \{ t > 0 : \hat X_t < 0\} . \]

Furthermore, we shall denote by \(\jump \xiCPP \) the random variable whose law is the same as the jump distribution of \(\xiCPP \).

The characteristic function of \(\jump \xiCPP \) can now be found by rewriting the expression in Proposition 4.1 in terms of overshoots and undershoots of stable Lévy processes, whose marginal and joint laws are given in Rogozin [30] and Doney and Kyprianou [12]. Following the notation of [12], let

\[ \tau _a^+ = \inf \{ t > 0 : X_t > a \} , \]

and let \(\hat \tau _a^+\) be defined similarly for \(\hat X\).

It is now possible to deduce the density of the jump distribution from its characteristic function. By substituting on the left and using the beta integral, it can be shown that

\begin{align*} \int _{-\infty }^{\infty } e^{\iu \theta x}\, \alpha e^x (1+e^x)^{-(\alpha + 1)}\, \dd x &= \frac {\Gamma (1+\iu \theta ) \Gamma (\alpha - \iu \theta )}{\Gamma (\alpha )} , \\ \int _{-\infty }^{\infty } e^{\iu \theta x}\, \frac {\sin (\pi \alpha \rho )}{\pi } e^{(1-\alpha \rho )x} (1+e^x)^{-1}\, \dd x &= \frac {\sin (\pi \alpha \rho )}{\pi } \Gamma (\alpha \rho -\iu \theta ) \Gamma (1-\alpha \rho +\iu \theta ) , \end{align*} and so the density of \(\jump \xiCPP \) can be seen as the convolution of these two functions. Moreover, it is even possible to calculate this convolution directly:

\begin{align} &\LevP _0\bigl (\jump \xiCPP \in \dd x\bigr )/\dd x \nonumber \\ &{} = \frac {\alpha }{\Gamma (\alpha \rho )\Gamma (1-\alpha \rho )} \int _{-\infty }^{\infty } e^u (1+e^u)^{-(\alpha +1)} e^{(1-\alpha \rho )(x-u)} (1+e^{x-u})^{-1} \, \dd u \nonumber \\ &{} = \frac {\alpha }{\Gamma (\alpha \rho )\Gamma (1-\alpha \rho )} e^{-\alpha \rho x} \int _0^\infty t^{\alpha \rho } (1+t)^{-(\alpha +1)} (te^{-x} + 1)^{-1}\, \dd t \nonumber \\ &{} = \frac {\alpha \Gamma (\alpha \rho +1) \Gamma (\alpha \rhohat + 1)}{\Gamma (\alpha \rho )\Gamma (1-\alpha \rho )\Gamma (\alpha +2)} e^{-\alpha \rho x} \Ghg {1}{\alpha \rho +1}{\alpha +2}{1-e^{-x}} , \label {jump density} \end{align} where the final line follows from [17, formula 3.197.5], and is to be understood in the sense of analytic continuation when \(x < 0\).

We begin with a brief sketch of the Wiener-Hopf factorisation for Lévy processes, and refer the reader to [21, Chapter 6] or [2, VI.2] for further details, including proofs.

The Wiener-Hopf factorisation describes the characteristic exponent of a Lévy process in terms of the Laplace exponents of two subordinators. For our purposes, a subordinator is defined as an increasing Lévy process, possibly killed at an independent exponentially distributed time and sent to the cemetary state \(+\infty \). If \(H\) is a subordinator with expectation operator \(\LevE \), we define its Laplace exponent \(\phi \) by the equation

\[ \LevE \bigl [ \exp (-\lambda H_1)\bigr ] = \exp (-\phi (\lambda )), \for \lambda \ge 0 . \]

Standard theory allows us to analytically extend \(\phi (\lambda )\) to \(\{\lambda \in \mathbb {C}: \Re \lambda \geq 0\}\). Similarly, let \(\xi \) be a Lévy process, again with expectation \(\LevE \), and denote its characteristic exponent by \(\CE \), so that

\[ \LevE \bigl [ \exp (\iu \theta \xi _1) \bigr ] = \exp (-\CE (\theta )) , \for \theta \in \RR . \]

The Wiener-Hopf factorisation of \(\xi \) consists of the decomposition

\( \seteqnumber {14} \)

\begin{equation} \label {the WHF}k \CE (\theta ) = \kappa (-\iu \theta ) \hat \kappa ( \iu \theta ) , \for \theta \in \RR , \end{equation}

where \(k > 0\) is a constant which may, without loss of generality, be taken equal to unity, and the functions \(\kappa \) and \(\hat \kappa \) are the Laplace exponents of certain subordinators which we denote \(H\) and \(\hat H\).

Any decomposition of the form (14) is unique, up to the constant \(k\), provided that the functions \(\kappa \) and \(\hat \kappa \) are Laplace exponents of subordinators. The exponents \(\kappa \) and \(\hat \kappa \) are termed the Wiener-Hopf factors of \(\xi \).

The subordinator \(H\) can be identified in law as an appropriate time change of the running maximum process \(\bar \xi : = (\bar \xi _t)_{t\geq 0}\), where \(\bar \xi _t = \sup \{ \xi _s, \, s \le t\}\). In particular, the range of \(H\) and \(\bar \xi \) are the same. Similarly, \(\hat H\) is equal in law to an appropriate time-change of \(-\underline {\xi }: = (-\underline {\xi })_{t\geq 0}\), with \(\underline \xi _t = \inf \{\xi _s, \, s \le t \}\), and they have the same range. Intuitively speaking, \(H\) and \(\hat H\) keep track of how \(\xi \) reaches its new maxima and minima, and they are therefore termed the ascending and descending ladder height processes associated to \(\xi \).

In Sections 5.4 and 5.5 we shall deduce in explicit form the Wiener-Hopf factors of \(\xi \) from its characteristic exponent. Analytically, we will need to distinguish the cases \(\alpha \in (0,1]\) and \(\alpha \in (1,2)\); in probabilistic terms, these correspond to the regimes where \(X\) cannot and can hit zero, respectively.

Accordingly, the outline of this section is as follows. We first introduce two classes of Lévy processes and two transformations of subordinators which will be used to identify the process \(\xi \) and the ladder processes \(H,\hat H\). We then present two subsections with the same structure: first a theorem identifying the factorisation and the ladder processes, and then a proposition collecting some further details of important characteristics of the ladder height processes, which will be used in the applications.

A process is said to be a hypergeometric Lévy process with parameters \((\beta ,\gamma ,\hat \beta ,\hat \gamma )\) if it has characteristic exponent

\[ \frac {\Gamma (1-\beta +\gamma -\iu \theta )}{\Gamma (1-\beta -\iu \theta )} \frac {\Gamma (\hat \beta +\hat \gamma +\iu \theta )}{\Gamma (\hat \beta + \iu \theta )} , \for \theta \in \RR \]

and the parameters lie in the admissible set

\[ \bigl \{ \beta \le 1, \, \gamma \in (0,1), \, \hat \beta \ge 0, \, \hat \gamma \in (0,1) \bigr \} . \]

In Kuznetsov and Pardo [20] the authors derive the Lévy measure and Wiener-Hopf factorisation of such a process, and show that the processes \(\xi ^*\), \(\xi ^\uparrow \) and \(\xi ^\downarrow \) of Caballero and Chaumont [6] belong to this class; these are, respectively, the Lévy processes appearing in the Lamperti transform of the \(\alpha \)-stable process absorbed at zero, conditioned to stay positive and conditioned to hit zero continuously.

A Lamperti-stable subordinator is characterised by parameters in the admissible set

\[ \{ (q, \mathtt {a}, \beta , c, \LSdrift ) : \mathtt {a} \in (0,1),\ \beta \leq 1+\mathtt {a}, \, q, c, \LSdrift \ge 0 \} , \]

and it is defined as the (possibly killed) increasing Lévy process with killing rate \(q\), drift \(\LSdrift \), and Lévy density

\[ c \frac {e^{\beta x}}{(e^x-1)^{\mathtt {a}+1}}, \for x > 0 . \]

It is simple to see from [7, Theorem 3.1] that the Laplace exponent of such a process is given, for \(\lambda \ge 0\), by

\( \seteqnumber {15} \)

\begin{equation} \label {LSS LE}\Phi (\lambda ) = q+ \LSdrift \lambda - c \Gamma (-\mathtt {a}) \left ( \frac {\Gamma (\lambda + 1 - \beta + \mathtt {a})}{\Gamma (\lambda + 1 - \beta )} - \frac {\Gamma (1-\beta +\mathtt {a})}{\Gamma (1-\beta )} \right ). \end{equation}

5.3. Esscher and \(\Ttrans _{\beta }\) transformations and special Bernstein functions. The Lamperti-stable subordinators just introduced will not be sufficient to identify the ladder processes associated to \(\xi \) in the case \(\alpha \in (1,2)\). We therefore introduce two transformations of subordinators in order to expand our repertoire of processes.

The first of these is the classical Esscher transformation, a generalisation of the Cameron-Girsanov-Martin transformation of Brownian motion. The second, the \(\Ttrans _\beta \) transformation, is more recent, but we will see that, in the cases we are concerned with, it is closely connected to the Esscher transform. We refer the reader to [21, §3.3] and [23, §2] respectively for details.

The following result is classical.

Before giving the next theorem, we need to introduce the notions of special Bernstein function and conjugate subordinators, first defined by Song and Vondraček [33]. Consider a function \(\phi \colon [0,\infty ) \to \RR \), and define \(\phi ^* \colon [0,\infty ) \to \RR \) by

\[ \phi ^*(\lambda ) = \lambda /\phi (\lambda ) . \]

The function \(\phi \) is called a special Bernstein function if both \(\phi \) and \(\phi ^*\) are the Laplace exponents of subordinators. In this case, \(\phi \) and \(\phi ^*\) are said to be conjugate to one another, as are their corresponding subordinators.

• Proof. The first assertion is proved in Gnedin [16, p. 124] as the result of a path transformation, and directly, for spectrally negative Lévy processes (from which the case of subordinators is easily extracted) in Kyprianou and Patie [23]. The killing rate and drift coefficient can be read off as \(\Ttrans _\beta \phi (0)\) and \(\lim _{\lambda \to \infty } \Ttrans _\beta \phi (\lambda )/\lambda \).

  The second claim can be seen immediately by rewriting (16) as

  \[ \Ttrans _\beta \phi (\lambda ) = \frac {\lambda }{\phi ^*(\lambda +\beta )} \]

  and observing that \(\phi ^*(\lambda +\beta ) = \EsscherT _\beta \phi ^*(\lambda ) + \phi ^*(\beta )\) for \(\lambda \ge 0\). □

In this section, we use the Wiener-Hopf factorisation of \(\xi \) to prove Theorems 1.1 and 1.4 and deduce Corollary 1.2. We then make use of a connection with the process conditioned to stay positive in order to prove Theorem 1.5.

Our method for proving each theorem will be to prove a corresponding result for the Lévy process \(\xi \), and to relate this to the \(\alpha \)-stable process \(X\) by means of the Lamperti transform and censoring. In this respect, the following observation is elementary but crucial. Let

\[ \tau _0^b = \inf \{ t > 0 : X_t \in (0,b) \} \]

be the first time at which \(X\) enters the interval \((0,b)\), where \(b < 1\), and

\[ S_a^- = \inf \{ s > 0 : \xi _s < a \} \]

the first passage of \(\xi \) below the negative level \(a\). Notice that, if \(e^a = b\), then

\[ S_a^- < \infty \text {, and } \xi _{S_a^-} \le x \iff \tau _0^b < \infty \text {, and } X_{\tau _0^b} \le e^x .\]

We will use this relationship several times.

Our first task is to prove Theorem 1.1. We split the proof into two parts, based on the value of \(\alpha \). In principle, the method which we use for \(\alpha \in (0,1]\) extends to the \(\alpha \in (1,2)\) regime; however, it requires the evaluation of an integral including the descending renewal measure. For \(\alpha \in (1,2)\) we have been unable to calculate this in closed form, and have instead used a method based on the Laplace transform. Conversely, the second method could be applied in the case \(\alpha \in (0,1]\); however, it is less transparent.

We now turn to the problem of first passage upward before hitting a point. To tackle this problem, we will use the stable process conditioned to stay positive. This process has been studied by a number of authors; for a general account of conditioning to stay positive, see for example Chaumont and Doney [9]. If \(X\) is the standard \(\alpha \)-stable process defined in the introduction and

\[ \tau _0^- = \inf ( t \ge 0 : X_t < 0 ) \]

is the first passage time below zero, then the process conditioned to stay positive, denoted \(\Xup \), with probability laws \((\stPup _x)_{x > 0}\), is defined as the Doob \(h\)-transform of the killed process \(\bigl ( X_t \Indic {t < \tau _0^-}, \, t \ge 0 \bigr )\) under the invariant function

\[ h(x) = x^{\alpha \rhohat } . \]

That is, if \(T\) is any a.s. finite stopping time, \(Z\) an \(\FF _T\) measurable random variable, and \(x > 0\), then

\[ \stEup _x(Z) = \stE _x \biggl [ Z \frac {h(X_T)}{h(x)} , \, T < \tau _0^- \biggr ] . \]

In fact we will make use of this construction for the dual process \(\hat X\), with invariant function \(\hat h(x) = x^{\alpha \rho }\), and accordingly we will denote the conditioned process by \(\Xhatup \) and use \((\stPhatup _x)_{x > 0}\) for its probability laws. It is known that the process \(\Xhatup \) is a strong Markov process which drifts to \(+\infty \).

Caballero and Chaumont [6] show that the process \(\Xhatup \) is a pssMp, and so we can apply the Lamperti transform to it. We will denote the Lévy process associated to \(\Xhatup \) by \(\xihatup \) with probability laws \(( \LevPhatup _y)_{y >0 }\). The crucial observation here is that \(\Xhatup \) hits the point \(1\) if and only if its Lamperti transform, \(\hat \xi ^\uparrow \), hits the point \(0\).

We now have all the apparatus in place to begin the proof.

Acknowledgements. The authors would like to thank the anonymous referees, whose comments have led to great improvements in the paper.

• [1]  Bertoin, J. (1993). Splitting at the infimum and excursions in half-lines for random walks and Lévy processes. Stochastic Process. Appl. 47 17–35. MR1232850

• [2]  Bertoin, J. (1996). Lévy processes. Cambridge Tracts in Mathematics 121. Cambridge University Press, Cambridge. MR1406564

• [3]  Blumenthal, R. M., Getoor, R. K. and Ray, D. B. (1961). On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99 540–554. MR0126885

• [4]  Blumenthal, R. M. and Getoor, R. K. (1968). Markov processes and potential theory. Pure and Applied Mathematics, Vol. 29. Academic Press, New York. MR0264757

• [5]  Bogdan, K., Burdzy, K. and Chen, Z.-Q. (2003). Censored stable processes. Probab. Theory Related Fields 127 89–152. MR2006232

• [6]  Caballero, M. E. and Chaumont, L. (2006). Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Probab. 43 967–983. MR2274630

• [7]  Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2010). On Lamperti stable processes. Probab. Math. Statist. 30 1–28. MR2792485

• [8]  Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2011). Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17 34–59. MR2797981

• [9]  Chaumont, L. and Doney, R. A. (2005). On Lévy processes conditioned to stay positive. Electron. J. Probab. 10 948–961. MR2164035

• [10]  Chaumont, L., Kyprianou, A. E. and Pardo, J. C. (2009). Some explicit identities associated with positive self-similar Markov processes. Stochastic Process. Appl. 119 980–1000. MR2499867

• [11]  Chaumont, L., Panti, H. and Rivero, V. (8 November 2011). The Lamperti representation of real-valued self-similar Markov processes. Preprint, hal-00639336, version 1. URL http://hal.archives-ouvertes.fr/hal-00639336.

• [12]  Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Probab. 16 91–106. MR2209337

• [13]  Fitzsimmons, P. J. (2006). On the existence of recurrent extensions of self-similar Markov processes. Electron. Comm. Probab. 11 230–241. MR2266714

• [14]  Getoor, R. K. (1961). First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 75–90. MR0137148

• [15]  Getoor, R. K. (1966). Continuous additive functionals of a Markov process with applications to processes with independent increments. J. Math. Anal. Appl. 13 132–153. MR0185663

• [16]  Gnedin, A. V. (2010). Regeneration in random combinatorial structures. Probab. Surv. 7 105–156. MR2684164

• [17]  Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of integrals, series, and products, Seventh ed. Elsevier/Academic Press, Amsterdam. Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. MR2360010

• [18]  Kadankova, T. V. and Veraverbeke, N. (2007). On several two-boundary problems for a particular class of Lévy processes. J. Theoret. Probab. 20 1073–1085. MR2359069

• [19]  Kuznetsov, A., Kyprianou, A. E. and Pardo, J. C. (2012). Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Probab. 22 1101–1135.

• [20]  Kuznetsov, A. and Pardo, J. C. (2010). Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. arXiv:1012.0817v1 [math.PR].

• [21]  Kyprianou, A. E. (2006). Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin. MR2250061

• [22]  Kyprianou, A. E., Pardo, J. C. and Rivero, V. (2010). Exact and asymptotic \(n\)-tuple laws at first and last passage. Ann. Appl. Probab. 20 522–564. MR2650041

• [23]  Kyprianou, A. E. and Patie, P. (2011). A Ciesielski-Taylor type identity for positive self-similar Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 47 917–928. MR2848004

• [24]  Kyprianou, A. E. and Rivero, V. (2008). Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Probab. 13 1672-1701. MR2448127

• [25]  Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 205–225. MR0307358

• [26]  Port, S. C. (1967). Hitting times and potentials for recurrent stable processes. J. Analyse Math. 20 371–395. MR0217877

• [27]  Rivero, V. (2005). Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11 471–509. MR2146891

• [28]  Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov processes, and martingales. Vol. 1. Cambridge Mathematical Library. Cambridge University Press, Cambridge. MR1796539

• [29]  Rogozin, B. A. (1971). The distribution of the first ladder moment and height and fluctuation of a random walk. Theory Probab. Appl. 16 575–595.

• [30]  Rogozin, B. A. (1972). The distribution of the first hit for stable and asymptotically stable walks on an interval. Theory Probab. Appl. 17 332–338.

• [31]  Sato, K.-i. (1999). Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge University Press, Cambridge. MR1739520

• [32]  Silverstein, M. L. (1980). Classification of coharmonic and coinvariant functions for a Lévy process. Ann. Probab. 8 539–575. MR573292

• [33]  Song, R. and Vondraček, Z. (2006). Potential theory of special subordinators and subordinate killed stable processes. J. Theoret. Probab. 19 817–847. MR2279605

• [34]  Vuolle-Apiala, J. (1994). Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 546–565. MR1288123