1 Introduction
First passage theory for random walks is a classic topic, excellently treated in, for example, [Reference Borovkov10], [Reference Feller15], [Reference Spitzer31], and [Reference Takàcs33], and this includes the right-continuous random walk, or what is the upwards skip-free compound binomial model of the actuarial literature. However, in light of recent developments in the parallel continuous-time theory of spectrally negative/upwards skip-free Lévy and Markov additive processes (see, for example, [Reference Albrecher, Ivanovs and Zhou2], [Reference Avram, Grahovac and Vardar-Acar4], [Reference Avram, Kyprianou and Pistorius5], [Reference Ivanovs19], [Reference Ivanovs and Palmowski20], and [Reference Vidmar35]) it seems worthwhile to revisit this topic.
Recall that in the Lévy case the scale functions
$$\smash{W^{(q)}}$$
and
$$\smash{Z^{(q)}}$$
have been known since [Reference Avram, Kyprianou and Pistorius5] and [Reference Suprun32], and that these functions intervene in important optimization problems. For example,
$$\smash{W^{(q)}}$$
provides the value function of the classic de Finetti problem of optimizing expected dividends until ruin with discount factor q [Reference Avram, Palmowski and Pistorius6], while
$$\smash{Z^{(q)}(\cdot,\theta)}$$
appears for instance in the moment generating function (with
$$\theta$$
as argument) of the capital injections [Reference Ivanovs and Palmowski20] and in the combined dividend payout-capital injections problem for a doubly reflected process [Reference Albrecher and Ivanovs1], [Reference Avram, Palmowski and Pistorius6]. These are just two examples from an ever increasing list of problems [Reference Avram, Grahovac and Vardar-Acar4], [Reference Kyprianou23], [Reference Pistorius28], which can now be tackled by simple lookup in the list and using off-shelf packages computing the functions W and Z [Reference Ivanovs19].
It was expected that the first passage theory developed in the world of spectrally negative Lévy processes, which we call the scale functions—or the
$$\Phi,W,Z$$
—paradigm, should have parallels for other classes of spectrally negative/skip-free Markov processes. In particular, the three cases listed below, being precisely the processes with stationary independent increments that exhibit nonrandom overshoots [Reference Vidmar34] (modulo processes with monotone paths), were expected to be very similar:
(i) (discrete-time, discrete-space) right-continuous (i.e. skip-free to the right) random walks, also known in insurance as the compound binomial model;
(ii) (continuous-time, discrete-space) compound Poisson processes that live on a lattice
$$h\mathbb{Z}$$
,
$$h\in (0,\infty)$$
, jumping up only by h (what were called upwards skip-free Lévy chains in [Reference Vidmar35]);(iii) (continuous-time, continuous-space) spectrally negative Lévy processes.
However, important steps were missing for the fully discrete setup. Notably, the second scale function
$$Z_v(\cdot,w)$$
was absent from the previous literature and we provide below for the first time its generating function (z-transform) (14). Another contribution of our paper is spelling out the connections between the three types of first passage problems listed above. In particular, in Appendix B we provide a concise table featuring side-by-side some of the salient features of the
$$\Phi,W,Z$$
theory for the three types of process (i)-(ii)-(iii) delineated above. It may serve as an inexhaustive summary and a quick reference; for the complete exposition, the main body of the text must be consulted.
Now, the doubly discrete (in time and space) random walk risk model [Reference Gerber16], [Reference Shiu30] is defined by
$$
X_n =X_0 + c n- \sum_{i=1}^n {C}_i, \qquad n\in {\mathbb N}_0=\{0,1,2,\ldots\},
$$
where
$$X_0$$
, taking values in
$${\mathbb Z}$$
, is the initial capital,
$$c \in {\mathbb N}$$
is the premium rate and the claims
$${C}_i $$
,
$$i\in \mathbb{N}$$
, take values in
$$\mathbb{N}_0$$
and are independent, identically distributed random variables with probability mass function
$$p_k=\mathbb{P}({C}_1=k)$$
for
$$k\in \mathbb{N}_0$$
.
One advantage of the discrete setup over the more popular continuous-time models is the possibility to replace the Wiener–Hopf factorization by the conceptually simpler factorization of Laurent series (see, for example, [Reference Banderier and Flajolet8] and [Reference Xin37]); another advantage is that one has access to Panjer recursions for computing compound distributions.
The results simplify considerably for the upwards skip-free compound binomial model obtained when
$$c=1$$
(see [Reference Brown, Peköz and Ross11], [Reference Choi and Patie13, Section 4.1], [Reference Marchal26], [Reference Quine29], [Reference Spitzer31], among others),
$$
\begin{equation}
X_n =X_0 + n - \sum_{i=1}^n {C}_i, \qquad n\in {\mathbb N}_0,
\label{eqn1}
\end{equation}$$
that we now consider as having been fixed and to which we specialize all discussion henceforth. Ignoring its interpretation as an actuarial model,
$$X=(X_n)_{n\in \mathbb{N}_0}$$
is nothing but a right-continuous random walk, and while our motivation for this investigation comes chiefly from risk models in the insurance context, and some actuarial-motivated vocabulary has (e.g. claims) and will (e.g. ruin probabilities) be used, the results presented are completely general and, hence, more widely applicable.
We insist throughout that
$$p_0>0$$
and we let
$$
\tilde{p}( z)\,{:\!=}\,\mathbb{E}[z^{{C}_1}]=\sum_{k=0}^\infty p_k z^k, \qquad z\in (0,1],
$$
denote the probability generating function of the claims. Then, for
$$n\in
\mathbb{N}$$
, (in the obvious notation)
$$\smash{\mathbb{E}[z^{\sum_{i=1}^n
{C}_i}]}=[\tilde p(z)]^n=(p_0 + (1-p_0)\ \tilde{p}_{{C}|{C}\ \geq\ 1}(
z))^n$$
, which makes it manifest that
$$\smash{\sum_{i=1}^n {C}_i}$$
, the total claims arising from n time periods, has a compound binomial distribution, explaining the name compound binomial model: at each instant in discrete time, a positive claim either occurs or not, with probability
$$1-p_0$$
and
$$p_0$$
, respectively, independently of the sizes of the positive claims.
Remark 1. By the independence of the claims we may also write, for
$$n\in \mathbb {N}_0$$
,
$$
\begin{equation*}
\mathbb{E} [z^{\sum_{i=1}^n ({C}_i-1)}]= \bigg(\frac{\tilde{p}( z)}{z}\bigg)^n\Longrightarrow \sum_{m=0}^\infty v^m \mathbb{E} [z^{\sum_{i=1}^m
({C}_i-1)}]
= \frac 1{1-v \tilde{p}( z)/z}, \qquad v\in (0,{z}/{\tilde{p}(z)}).
\end{equation*}$$
The last expression, called the ‘unrestricted generating function’ in [8, Equation (8)], identifies already potential singularities as the roots of the Lundberg equation [Reference Lundberg25]
$${\tilde{p}(
z)}/{z}=v^{-1}$$
. The smallest (positive) root of this equation plays a central role in our story; see the next section.
Next, let
$$
\begin{equation}
\tau_b^-=\inf\{t\geq 0\colon X_t\leq b\} \quad\text{and}\quad
\tau_b^+=\inf\{t\geq 0\colon X_t\geq b\}
\label{eqn2}
\end{equation}$$
respectively denote the first passage times below and above a level b (with
$$\inf \emptyset = \infty $$
).
Remark 2. Note that this differs slightly from the usual definition of these quantities for a spectrally negative Lévy process, say U. Here one replaces
$$t\geq 0$$
by
$$t>0$$
and
$$\leq b$$
(
$$\geq b$$
) by
$$\lt b$$
(
$$\gt b$$
); and, of course, X by U. When considering
$$\tau^{\pm}_b$$
for a spectrally negative Lévy process U, we shall mean these quantities with the latter replacements in effect.
Lastly, for convenience, we assume that a family of measures
$$(\mathbb {P}_x)_{x\in \mathbb {Z}}$$
is given with corresponding expectation operators
$$(\mathbb{E}_x)_{x\in \mathbb{Z}}$$
, for which: (i)
$$\mathbb {P}_x[X_0=x]=1$$
for all
$$x\in {\mathbb Z}$$
; and (ii) the
$$C_i$$
,
$$i\in \mathbb{N}$$
, have the same law under all the
$$\mathbb{P}_x$$
,
$$x\in {\mathbb Z}$$
, as they do under
$$\mathbb{P}=\mathbb{P}_0$$
.
Remark 3. The discrete-time, discrete-space compound binomial model is embedded into continuous time via subordination (time-change) by an independent homogeneous Poisson process N. In precise terms, allowing also a scaling of space, we have the following correspondence between the right-continuous random walk X of (1) and the upwards skip-free Lévy chain of [Reference Vidmar35, Section 2], which we denote by Y,
$$X\quad \quad Y\;:\quad {Y_t}\;: = \;h{X_{{N_t}}},\qquad t \in [0,\infty ),$$
where
$$h\in (0,\infty)$$
is space scaling. In particular, denoting the intensity of N by
$$\gamma$$
, the Lévy measure
$$\lambda$$
of Y is given by
$$\lambda=\smash{\gamma\sum_{i\in \mathbb{N}_0\backslash
\{1\}}p_i\delta_{h(1-i)}}$$
, and if we denote the Laplace exponent of Y by
$$\psi$$
(so
$$\psi(\beta)={\log \mathbb{E}[\mathrm{e}^{\beta Y_t}]}/{t}$$
for
$$\beta\in
[0,\infty)$$
) then
$$\psi(\beta)=\gamma[\mathrm{e}^{\beta h}\tilde{p}(\mathrm{e}^{-\beta
h})-1]$$
. Note that the mass of the Lévy measure
$$\lambda$$
is
$$\gamma(1-p_1)$$
, which may be strictly less than
$$\gamma$$
.
Remark 4. In the following, when the
$$\tau_b^\pm$$
appear in the context of the upwards skip-free Lévy chain Y, they are to be interpreted in the sense of (2) with Y replacing X.
The remainder of the paper is organised as follows. In Sections 2, 3, and 4, we respectively review (with v indicating discounting) the smooth (i.e. upwards; hence, skip-free) one-sided first passage problem, which introduces the Lundberg root
$$\varphi_v$$
(analogue of
$$\Phi(q)$$
from the Lévy theory); the nonsmooth (i.e. downwards) one-sided first passage problem, which involves the ruin and survival probabilities
$$\Psi_v$$
and
$$\smash{\bar{\Psi}_v}$$
; the smooth (i.e. exiting at the upper boundary) two-sided first passage problem, where the fundamental scale function
$$W_v$$
first appears.
We then turn to original contributions in Section 5, computing the generating function (z-transform) of the second hero of the first passage theory: the
$$Z_v(\cdot,w)$$
scale function. This is introduced via the problem of position on the nonsmooth two-sided exit. We provide the analogue (12) of the following two-sided exit identity for a spectrally negative Lévy process U (in standard notation, see below):
$$
\mathbb{E}_x [\mathrm{e}^{-q \tau^{-}_0 +\theta U(\tau^-_0)} ; \tau^-_{0}
\lt \tau_b^+]=Z^{(q)}(x,\theta)-\frac{W^{(q)}(x)}{W^{(q)}(b)}Z^{(q)}(b,\theta).
\label{bea}
$$
Ivanovs and Palmowski [Reference Ivanovs and Palmowski20, Corollary 3] provide its beautiful probabilistic interpretation. We also determine the analogue (17) of the formula [Reference Kyprianou23, Equation (8.9)] (again for a spectrally negative Lévy process, in standard notation, see below)
$$
\mathbb{E}_x [\mathrm{e}^{-q \tau^{-}_0} ; \tau^-_{0} \lt \infty]= Z^{(q)}(x) - \frac q{\Phi(q)}
W^{(q)} (x), \qquad q>0,
$$
which is interesting, for example, since it reveals that the two protagonists of the ‘reflected’ and ‘absorbed’ smooth passage problems,
$$\smash{Z^{(q)}}$$
and
$$\smash{W^{(q)}}$$
, have the same asymptotics at
$$\infty$$
, up to a constant. (For definiteness, let us mention that, in the preceding formulae, for
$$q\in [0,\infty)$$
:
$$\Phi(q)$$
is the largest root of
$$\kappa-q$$
, where
$$\kappa$$
is the Laplace exponent of the underlying Lévy process U;
$$\smash{W^{(q)}}$$
is the unique function mapping
$$\mathbb{R}\to \mathbb{R}$$
, vanishing on
$$({-}\infty,0)$$
, continuous on
$$[0,\infty)$$
, and having Laplace transform
$$\smash{\int_0^\infty \mathrm{e}^{-\lambda x}W^{(q)}(x)}\,\mathrm{d}
x=(\kappa(\lambda)-q)^{-1}$$
for
$$\lambda\in (\Phi(q),\infty)\vphantom{\int_q}$$
; finally,
$$\smash{Z^{(q)}(x,\theta)}=\mathrm{e}^{\theta
x}+(q-\kappa(\theta))\smash{\int_0^x\mathrm{e}^{\theta(x-y)}W^{(q)}(y)}\,\mathrm{d} y$$
for
$$x\in \mathbb{R}$$
.) A distinguishing element of the scale functions
$$W_v$$
and
$$Z_v(\cdot,w)$$
, in the present context, are explicit recursions available for their computation, see (9) and (13), respectively.
For applications of this theory to combined capital injections-dividend payout-penalty at ruin optimization problems, see an earlier version of this paper [Reference Avram and Vidmar3, Sections 6 and 7].
2 Smooth one-sided first passage problem: the Lundberg equation
The first key observation is that, for the first passage upwards, the stationary independent increments and skip-free properties imply a multiplicative structure. For integer
$$x \leq b$$
and
$$v\in (0,1]$$
, we have
$$
\begin{equation}
\mathbb{E}_x [v^{\tau_b^+}; \tau_b^+\lt \infty]=\varphi_v^{b-x},
\label{eqn3}
\end{equation}$$
where
$$
\varphi_v\ {:\!=}\ \mathbb{E} [v^{\tau_1^+};\tau_1^+\lt \infty ]= \sum_{k=1}^\infty v^k \mathbb{P}[\tau_1^+=k] \in
(0,v].
$$
Conditioning at time 1, we obtain
$$
\varphi_v = v\, \mathbb{E} [\mathbb{E}_{ 1 -{C}_1}[ v^{\tau_1^+};\tau_1^+\lt \infty]]=v
\sum_{k=0}^\infty p_k \varphi_v^{k}=v \tilde{p}(\varphi_v),
$$
which reveals that
$$\varphi_v$$
in (3) satisfies the Lundberg equation [Reference Cheng, Gerber and Shiu12, Equation (3.3)], [Reference Gerber, Shiu and Yang18, Equation (6.8)]
$$
\begin{equation}
\frac{\varphi_v}{\tilde{p}(\varphi_v)}=v.
\label{eqn4}
\end{equation}$$
Alternatively, this relation may be derived by looking for exponential martingales of the form
$$(v^t \xi^{-X_t})_{t\in {\mathbb N}_0}$$
, for fixed v, and
$$\xi$$
from (0,1]:
$$(v^t\xi^{-X_t})_{t\in {\mathbb N}_0}$$
is a martingale if and only if
$${\xi}/{\tilde{p}(\xi)}=v$$
; and then applying optional sampling.
Remark 5. The function
$$(0,1]\ni \xi\mapsto \tilde{p}(\xi)/\xi=\mathbb{E}[\xi^{{C}_1-1}]$$
is strictly convex, equal to 1 at 1, and tending to
$$\infty$$
at 0. It follows that the equation (in
$$\xi\in (0,1]$$
)
$${\tilde{p}(\xi)}/{\xi}=v^{-1}$$
has as its unique solution
$$\varphi_v\in (0,1)$$
, when
$$v\lt 1$$
(furthermore, in this case,
$$\varphi_v \lt v$$
), whereas in the
$$v=1$$
case, this equation has one or two solutions (one of which is always 1), according as to whether
$$\mathbb{E}[{C}_1]\leq 1$$
or
$$\mathbb{E}[{C}_1]>1$$
. In the latter case X drifts to
$$-\infty$$
, and
$$\varphi_1 \in (0,1)$$
is the smallest solution to
$$\xi=\tilde{p}(\xi)$$
(in
$$\xi\in (0,1]$$
). Altogether, this gives a continuous strictly increasing bijection
$$\varphi\colon(0,1]\to (0,\varphi_1 ]$$
.
Remark 6. If, for
$$q\in [0,\infty)$$
, we let
$$\Phi(q)$$
be the largest zero of
$$\psi-q$$
, then we see from Remark 3 that
$$\varphi_v=\mathrm{e}^{-h\Phi(\gamma(v^{-1}-1))}$$
for all
$$v\in (0,1]$$
.
Remark 7. Note that (4) identifies
$$\tau_1^+$$
as a Lagrangian type distribution [Reference Consul and Famoye14]. Indeed the distribution of
$$\tau_1^+$$
may be obtained using the Lagrange inversion formula [Reference Consul and Famoye14, Paragraph 1.2.6]
$$
\varphi_v=\sum_{n=1}^\infty \frac{v^n}{n!}\bigg[\bigg(\frac{\mathrm{d}}{\mathrm{d} w}\bigg)^{n-1}
\tilde{p}(w)^n\bigg]_{w=0} =\sum_{n=1}^\infty
\frac{v^n}{n}p^{n*}(n-1), %\label{eq:LI}
$$
where, for
$$n\in \mathbb{N}$$
,
$$p^{n*}$$
is the n-fold convolution of the distribution p with itself (the second equality follows from the fact that, for
$$n\in \mathbb{N}$$
,
$$\tilde{p}^n$$
is the probability generating function of the probability mass function
$$p^{n*}$$
, viz. of the sum of n independent random variables, all distributed as
$$C_1$$
). More generally, for
$$b\in \mathbb{N}$$
,
$$
\varphi_v^b=b\sum_{n=b}^\infty \frac{v^n}{n}p^{n*}(n-b),
$$
yielding Kemperman’s formula [Reference Kemperman21] for the distribution of
$$\tau_b^+$$
,
$$\mathbb{P}[\tau _{b}^{+}=n]=\frac{b}{n}{{p}^{n*}}(n-b)=\frac{b}{n}\mathbb{P}[{{X}_{n}}=b],\qquad n\in {{\mathbb{N}}_{\ge b}}.$$
3 Nonsmooth one-sided first passage problem: ruin and survival probabilities; the Lundberg recurrence
For initial capital
$$x\in {\mathbb Z}$$
, the finite time and eventual ruin probabilities are defined by
$$
\Psi(n;\ x)\ {:\!=}\ \mathbb{P}_x [\tau_{-1}^- \leq n]\quad\text{for } n\in {\mathbb N}_0, \qquad
\Psi(x)\ {:\!=}\ \lim_{n\to\infty}\Psi(n;\ x)=\mathbb{P}_x [\tau_{-1}^- \lt \infty];
$$
similarly, we introduce the finite time and perpetual survival probabilities
$$
\bar{\Psi}(n;\ x)\ {:\!=}\ \mathbb{P}_x [\tau_{-1}^- > n]\quad\text{for } n\in {\mathbb N}_0, \qquad
\bar{\Psi}(x)\ {:\!=}\ \lim_{n \to \infty} \bar{\Psi}(n;\ x)=\mathbb{P}_x [\tau_{-1}^- =\infty].
$$
Of course,
$$\Psi(n;\ x)+\bar{\Psi}(n;\ x)=1$$
,
$$\Psi(x)+\bar{\Psi}(x)=1$$
, and we have the recursions, valid for all integers
$$x \geq 0$$
,
$$ n \geq 1$$
,
$$
\begin{gather}
\bar{\Psi}(n;\ x)=\sum_{i=0}^{x + 1} p_i \bar{\Psi}(n-1;\ x+ 1 -i),\qquad
\bar{\Psi}(0;\ x)=1,\label{eqn5}\end{gather}$$
$$\Psi (n;\;x) = \sum\limits_{i = 0}^{x + 1} {{p_i}} \Psi (n - 1;\;x + 1 - i) + \sum\limits_{k = x + 2}^\infty {{p_k}} ,\qquad \Psi (0;\;x) = 0.$$
These two recurrences may, for a sequence of functions
$$f_n\colon {\mathbb Z}\to
[0,1]$$
, standing in lieu of
$$\Psi(n;\cdot)$$
and
$$\bar{\Psi}(n;\cdot)$$
, be written symbolically as
$$
f_n= K\ \tilde{p}(K^{-1}) f_{n-1} \quad\text{on }\mathbb{N}_0,
$$
which passes to the limit (as
$$n\to\infty$$
)
$$f = K\;\tilde p({K^{ - 1}})f\quad {\rm{on }}{_0},$$
where K is the translation operator,
$$Kg(x)\ {:\!=}\ g(x+1)$$
, and
$$f(x)\ {:\!=}\ \lim_{n\to\infty}f_n(x) $$
. This limiting recurrence (satisfied by the eventual ruin and perpetual survival probabilities
$$\Psi$$
and
$$\bar{\Psi}$$
) may be called the ‘Lundberg recurrence’. It constitutes a linear difference equation for f, whose characteristic equation is (in
$$x\ne
0$$
)
$$1=x\tilde{p}(1/x)$$
. The latter is (formally) just the Lundberg equation (4) with
$$v=1$$
upon substituting
$$x^{-1}$$
for
$$\varphi_1$$
. When the distribution p has a finite support, then from the theory of finite-order linear difference equations with constant coefficients, this implies that f, in particular the ultimate ruin and perpetual survival probabilities, may be expressed as combinations of powers of the roots of the characteristic equation (in
$$x\ne 0$$
)
$$1 = x{\mkern 1mu} \tilde p({1 \over x}).$$
Classical ruin theory proceeds by computing double (generating function) transforms, briefly reviewed in Appendix A. For example, one useful result, similar to the Pollaczek–Khinchine formula for the Cramér–Lundberg model, is
$$
\begin{equation}
\tilde{\bar{\Psi}}(z)\ {:\!=}\ \sum_{x=0}^\infty z^x \bar{\Psi}(x)=\frac{(1-\mathbb{E} [{C}_1])\lor
0}{\tilde{p}(z)-z},\qquad z\in (0,1);
\label{eqn6}
\end{equation}$$
see [Reference Willmot36, Equation (3.5)]. Another is
$$
\begin{align}
\tilde{\Psi}_v(z)&\ {:\!=}\ \sum_{x=0}^\infty \sum_{n=0}^\infty z^x v^n \Psi(n;\ x) \nonumber
\\
&\hskip2.9pt= \frac{1 }{{z}-v {\tilde{p}(z)}}\bigg(\frac{v(z-\tilde{p}(z))}{(1-v)(1-z)}
+\frac{\varphi_v}{1-\varphi_v}\bigg), \qquad v, z \in (0,1),z\ne\varphi_v.
\label{eqn7}\end{align}$$
We will follow next an alternate approach, which focuses on the two-sided exit problem from an interval.
4 Smooth two-sided first passage problem: the W scale functions
In the context of Lévy processes, the
$$\smash{W^{(q)}}$$
scale function is often defined first for
$$q=0$$
, in the case when the underlying process drifts to
$$\infty$$
, by proportionality to the survival probability, and then in the remainder of the cases by an Esscher transform/approximation [Reference Bertoin9, Section VII.2], [Reference Kyprianou23, Section 8.2], [Reference Vidmar35, Section 4.2].
In our setting of the right-continuous random walk X, we introduce, for
$$v\in (0,1]$$
, the discrete-time analogue
$$W_v$$
of
$$\smash{W^{(q)}}$$
, by setting
$$W_v(y)\ {:\!=}\ \smash{(p_0 \mathbb{E}[v^{ \tau_y^+}; \tau_y^+ \lt \tau_{-1}^-])^{-1}}$$
for
$$y\in \mathbb{N}_0$$
and
$$W_v(y)=0$$
for
$$y\in -\mathbb{N}$$
. For integer
$$0\leq x \leq N$$
, the Markov property at the time
$$\tau_x^+$$
and the skip-free property (yielding
$$\smash{X_{\tau_x^+}}=x$$
on
$$\smash{\{\tau_x^+\lt \infty\}}$$
) then imply that
$$
\begin{align*}
\mathbb{E}[v^{ \tau_N^+};\ \tau_N^+ \lt \tau_{-1}^-] &=\mathbb{E}[v^{
\tau_x^+}\mathbb{E}_{X_{\tau_x^+}}[v^{\tau^+_N};\tau_N^+\lt \tau_{-1}^-];\ \tau_x^+ \lt
\tau_{-1}^-]
\\
&=\mathbb{E}_x [v^{ \tau_N^+};\ \tau_N^+ \lt \tau_{-1}^-]\mathbb{E}[v^{ \tau_x^+};\ \tau_x^+ \lt
\tau_{-1}^-],
\end{align*}$$
i.e. the ‘gambler’s winning’ relation [Reference Gerber, Lin and Yang17], [Reference Marchal26]
$$
\begin{equation}
\mathbb{E}_x [v^{ \tau_N^+};\ \tau_N^+ \lt \tau_{-1}^-]=\frac{ W_v(x)}{ W_v(N)},
\label{eqn8}
\end{equation}$$
which is valid for all integers
$$N\geq 0$$
and
$$x\leq N$$
(it is trivial for
$$x\lt 0$$
as both sides are then 0).
We call
$$W_v$$
the v-scale function and we simply write W for the 1-scale function
$$W_1$$
. (The choice of the normalization
$$W_v(0)=1/p_0$$
is somewhat arbitrary, though it is guided by obtaining the simplest possible form for the z-transform of
$$W_v$$
(see (10) below); by comparison to the W scale function of [Reference Vidmar35] (see Remark 10 below); and the simplicity of subsequent formulae in which
$$W_v$$
features.)
Remark 8. We use the subscript notation
$$W_v$$
for the scale functions of X, reserving the superscript version
$$W^{(q)}$$
for the corresponding quantities from the Lévy setting. When only W appears, it will be clear from the context which of the two is meant. We will adhere to a similar convention with respect to the scale functions
$$\smash{Z^{(q)}(\cdot, \theta)}$$
,
$$\smash{Z^{(q)}\ {:\!=}\ Z^{(q)}(\cdot, 0)}$$
and (hence the notation) their discrete-time analogues
$$Z_v(\cdot,w)$$
,
$$Z_v$$
.
Conditioning on the first jump, (8) implies the harmonic recursion
$$
\begin{equation}
W_v(x)=v \sum_{y=-1}^{x} W_v(x-y) p_{y+1}, \qquad x\in \mathbb{N}_0;
\label{eqn9}
\end{equation}$$
see [Reference Marchal26, Equation (3.1)]
Taking the z-transform yields
$$
\begin{equation}
\tilde W_v(z)\ {:\!=}\ \sum_{x=0}^\infty z^x W_v(x)=\frac {1}{\tilde{p}(z)-{z}/{v}}, \qquad z \in
(0,\varphi_v);
\label{eqn10}
\end{equation}$$
see [Reference Marchal26, Equation (3.2)]
Since the z-transform (10) of
$$W_v$$
is known, the computation of the scale function
$$ W_v$$
finally reduces to Taylor coefficient extraction of (10) expanded in a power series.
Remark 9. It is seen from (9), or directly from (8), that we have
$$W_v/v={^vW}$$
, where
$$^vW$$
is the 1-scale function of the process X geometrically killed with probability
$$1-v$$
, i.e. of the process which has, ceteris paribus, the sub-probability mass function
$$(vp_k)_{k\in \mathbb{N}_0}$$
governing the sizes of the
$${C}_n$$
,
$$n\in
\mathbb{N}$$
.
Remark 10. For X embedded into continuous time as an upwards skip-free Lévy chain, i.e. for the process Y of Remark 3, (9) and (10) become, respectively, [Reference Vidmar35, Equations (20) and (16)]. This is seen through the identification
$$\smash{W^{(q)}(mh)}=({1}/{\gamma h})W_{{\gamma}/{(\gamma+q)}}(m)$$
for
$$m\in \mathbb{N}_0$$
,
$$q\in [0,\infty)$$
, where
$$\smash{W^{(q)}}$$
is the q-scale function of [Reference Vidmar35]. Note also that the normalization
$$W_v(0)=p_0^{-1}$$
is consistent with
$$W^{(q)}(0)=1/(h\lambda(\{h\}))=1/(\gamma hp_0)$$
of [Reference Vidmar35, Proposition 5]. On the other hand, in the spectrally negative case, there is no direct analogue of recursion (9), though one can consider the heuristic relation (it is rigorous in the upwards skip-free case [Reference Vidmar35, Remark 8])
$$\smash{(L-q)W^{(q)}}=0$$
on
$$(0,\infty)$$
[Reference Kuznetsov, Kyprianou and Rivero22, p. 136], L being the infinitesimal generator of the underlying Lévy process, to be a close relative. (10) has the Laplace transform equivalent [Reference Kuznetsov, Kyprianou and Rivero22, Equation (8.8)] that formally differs from [35, Equation (16)] only by the factor
$$(\mathrm{e}^{\beta h}-1)/(\beta h)\,{\to}\, 1\text{ as }h\downarrow 0$$
(with
$$\beta$$
the argument of the Laplace transform).
Remark 11. An alternative form of recursion 9 is
$$
W_v(n+1)=W_v(0)+\sum_{k=1}^{n+1}\frac{{1}/{v}-
\sum_{l=0}^kp_l}{p_0}W_v(n+1-k),\qquad n\in \mathbb{N}_0;
$$
see [Reference Vidmar35, Equation (23)]. In particular, we see via induction that, for each fixed integer x, the map
$$[1,\infty)\ni
\xi\mapsto W_{1/\xi}(x)$$
extends to a polynomial function defined on the whole complex plane.
Remark 12. When X drifts to
$$\infty$$
, i.e. when
$$\mathbb{E}[{C}_1]\lt 1$$
, then with
$$v=1$$
, (10) coincides up to a multiplicative constant with the perpetual survival transform (6). We conclude that
$$
\bar{\Psi}(x)=(1- \mathbb{E}[{C}_1])W(x).
$$
Remark 13. It follows from (10), i.e. by inspecting the equality of generating functions, that
$$_vW(x)=({\varphi_v}/{v})W_v(x)\varphi_v^x$$
, where
$$_vW$$
is the 1-scale function of the Esscher transformed process in which
$${C}_1$$
has the geometrically tilted probability mass function
$$\mathbb{N}_0\ni k\mapsto ({v}/{\varphi_v})p_k\varphi_v^k$$
(and so the probability generating function
$$(0,1]\ni z\mapsto
({v}/{\varphi_v})\ \tilde{p}\ (z\varphi_v)$$
). Furthermore, we see that
$$
\begin{align*}
\lim_{x\to \infty}W_v(x)\varphi_v^{x+1} &= v\lim_{x\to\infty}{}_vW(x)
\\
&={}_vW(\infty)v
\\
&=v\lim_{z\uparrow 1}\sum_{x=0}^\infty(1-z) z^x{}_vW(x)
\\
&=v\lim_{z\uparrow 1}(1-z)\tilde{_vW}(z)
\\
&=v\lim_{z\uparrow 1} \frac{1-z}{({v}/{\varphi_v})\tilde{p}(z\varphi_v)-z}
\\
&=\frac{v}{1-v\tilde{p}'(\varphi_v-)},
\end{align*}$$
where we understand
$$1/0=\infty$$
(the equality
$${}_vW(\infty)=\lim_{z\uparrow 1}\smash{\sum_{x=0}^\infty(1-z)
z^x{}_vW(x)}$$
is seen to hold, for instance, by monotone convergence, because one can view
$$\smash{\sum_{x=0}^\infty(1-z) z^x{}_vW(x)}$$
as the expectation of
$${}_vW(g_z)$$
, where
$$g_z$$
is a geometric random variable with success parameter
$$1-z$$
, and the
$$g_z$$
can be defined on a common probability space so as to be increasing to
$$\infty$$
as
$$z\uparrow 1$$
). This confirms [Reference Vidmar35, Proposition 6(1)]. For a more detailed study of the behaviour of
$$W_1$$
in the case when X oscillates, i.e. when
$$\tilde{p}'(1-)=1$$
and
$$\varphi_1=1$$
, and so when the preceding does not yield the precise asymptotics of W at infinity, see [Reference Vidmar35, Proposition 6(2)].
Remark 14. We note the following interesting observation of [Reference Marchal26] that the scale function is essentially a determinant. For an arbitrary homogeneous Markov chain
$$(V_n)_{n\in
\mathbb{N}_0}$$
on a countable state space, let
$$(V_n')_{n\in
\mathbb{N}_0}$$
denote the chain killed outside a finite nonempty set M, and let Q denote the corresponding restriction of the transition matrix to M. For
$$v\in (0,1)$$
, denote by
$$D_v$$
the determinant of the matrix
$$I-v Q$$
. Then the killed resolvent expresses as
$$
\sum_{n=0}^\infty \mathbb{P}_i[V_n'=j] v^n=((I-v
Q)^{-1})_{ij}=\frac{N_{ij}(v)}{D_v},\qquad \{i,j\}\subset M,
$$
where
$$N_{ij}(v)$$
are the entries of the adjoint matrix
$$\mathrm{adj}(I-v
Q)$$
(see, for example, [Reference Marchal26, Corollary 2.2]). Restricting now to the upwards skip-free case (while [Reference Marchal26] considers the downwards skip-free case), for
$$v\in (0,1],$$
let
$$D_v(N), N \in{\mathbb N}$$
, denote the determinant corresponding (in the above sense) to the restriction of X to
$$\{0,1,2,\ldots,N-1\}$$
, and set
$$D_v(0)\ {:\!=}\ 1$$
. From [Reference Marchal26, Proposition 3.3],
$$%\label {twos}
\mathbb{E}_i[v^{\tau_N^+}, \tau_N^+\lt \tau_{-1}^-]=(p_0
v)^{N-i}\frac{D_v(i)}{D_v(N)}, \qquad \{i,N\}\subset \mathbb{N}_0,\; v\in
(0,1).
$$
It follows that
$$
W_v(i)=p_0^{-1}(p_0v)^{-i}D_v(i) \quad\text{for all }i\in \mathbb{N}_0,\,
v\in (0,1].
$$
Remark 15. For
$$N\in \mathbb{N}$$
, the resolvent of the process X killed on exiting
$$I_N:=\{0,\ldots,N-1\}$$
, denoted by X’, is given by
$$
\sum_{n=0}^\infty \mathbb{P}_i[X_n'=j] v^n=v^{-1}\bigg(\frac{W_v(N-1-j)
W_v(i)}{W_v(N)} -W_v(i-j-1)\bigg),
$$
where
$$\{i,j\}\subset I_N,\; v\in (0,1]$$
; see [Reference Marchal26, Proposition 3.2]. For the analogue of the latter in the spectrally negative case, see, e.g. [Reference Kyprianou23, Theorem 8.7].
We conclude this section with the following important observation.
Proposition 1. For every
$$v\in (0,1]$$
,
$$\smash{( v^{n\land \tau_{-1}^-} W_v(X_{n\land
\tau_{-1}^-}))_{n\in \mathbb{N}_0}}$$
is a martingale under each
$$\mathbb{P}_x$$
,
$$x\in \mathbb{Z}$$
.
Proof. This follows from the harmonic recurrence (9).
Remark 16. The analogue of Proposition 1 in the setting of upwards skip-free Lévy chains are the martingales, for
$$q\in
[0,\infty), \smash{(\mathrm{e}^{-q(t\land \tau_{-h}^-)}W^{(q)}}(Y_{t\land
\tau_{-h}^-}))_{t\in [0,\infty)}$$
[Reference Vidmar35, Corollary 2]. In the case of a spectrally negative Lévy process U,
$$\smash{(\mathrm{e}^{-q(t\land \tau_0^-)}}W^{(q)}(U_{t\land
\tau_0^-}))_{t\in [0,\infty)}$$
is a local martingale with localizing sequence
$$(\tau_n^+)_{n\in \mathbb{N}}$$
[Reference Kyprianou23, Example 8.12]. There are no issues with integrability in the discrete space case, because thanks to the skip-free property,
$$\mathbb{P}_x$$
-a.s. for any
$$x\in \mathbb{Z}$$
, by any deterministic time, the stopped process
$$\smash{X^{\tau_{-1}^-}}$$
is automatically bounded from above (and, for the upwards skip-free Lévy chain Y, the further subordination by the independent homogeneous Poisson process N does not ruin this).
Corollary 1. For each
$$v\in (0,1]$$
and integer
$$x\leq N$$
,
$$b\leq N$$
,
$$
\mathbb{E}_x[W_v(X_{\tau_{b-1}^-})v^{\tau_{b-1}^-};\tau_{b-1}^-\lt \tau_N^+]
=W_v(x)-\frac{W_v(x-b)}{W_v(N-b)}W_v(N).
$$
In particular,
$${\mathbb{E}_x[W_v(X_{\tau_{b-1}^-})v^{\tau_{b-1}^-};\tau_{b-1}^-\lt \infty]}
=W_v(x)-W_v(x-b) \varphi_v^b.
$$
Proof. For any integer x, by optional sampling, the skip-free property and spatial homogeneity
$$
\begin{align*}
W_v(x)&=\mathbb{E}_x[W_v(X(\tau_{b-1}^-))v^{\tau_{b-1}^-};
\tau_{b-1}^-\lt\tau_N^+]+\mathbb{E}_x[W_v(X(\tau_N^+))v^{\tau_N^+};\tau_N^+
\lt \tau_{b-1}^-]
\\
&=\mathbb{E}_x[W_v(X(\tau_{b-1}^-))v^{\tau_{b-1}^-};\tau_{b-1}^-\lt \tau_N^+]
+W_v(N)\mathbb{E}_x[v^{\tau_N^+};\tau_N^+\lt \tau_{b-1}^-]
\\
&=\mathbb{E}_x[W_v(X(\tau_{b-1}^-))v^{\tau_{b-1}^-};\tau_{b-1}^-\lt \tau_N^+]
+W_v(N)\mathbb{E}_{x-b}[v^{\tau_{N-b}^+};\tau_{N-b}^+\lt \tau_{-1}^-].
\end{align*}$$
The first identity then follows from (8). In particular, letting
$$N\uparrow \infty$$
and using Remark 13, we obtain the second identity (for instance, first for
$$v\lt1$$
and then taking the limit
$$v\uparrow 1$$
).
Remark 17. For the analogue of Corollary 1 in the spectrally negative Lévy setting, see [Reference Loeffen, Renaud and Zhou24, Lemma 2.1, Equation (19) and Lemma 2.2(i)].
5 Nonsmooth two-sided first passage problem: the
$$\mathbf{\mathit{Z}}$$
scale functions
Let
$$v\in (0,1]$$
,
$$w\in (0,1]$$
. For integer
$$x\leq b$$
,
$$b\geq 0$$
, by the Markov property at time
$$\tau_b^+$$
and the skip-free property (yielding
$$X_{\tau_b^+}=b$$
on
$$\{\tau_b^+\lt \infty\}$$
),
$$
\begin{align*}
&\mathbb{E}_x [v^{ \tau^-_{-1}} w^ {-X(\tau^-_{-1})} ;\ \tau^-_{-1} \lt \tau_b^+]
\\
&\qquad= \mathbb{E}_x [v^{ \tau^-_{-1}} w^ {-X(\tau^-_{-1})} ;\ \tau^-_{-1} \lt \infty]- \mathbb{E}_x
[v^{ \tau^-_{-1}} w^ {-X(\tau^-_{-1})} ;\ \tau^+_b \lt \tau_{-1}^-\lt \infty]
\\
&\qquad=\mathbb{E}_x [v^{ \tau^-_{-1}} w^ {-X(\tau^-_{-1})} ;\ \tau^-_{-1} \lt \infty]-
\mathbb{E}_x[v^{\tau_b^+};\ \tau_b^+\lt \tau_{-1}^-]\mathbb{E}_b [v^{ \tau^-_{-1}} w^
{-X(\tau^-_{-1})} ;\ \tau_{-1}^-\lt \infty].
\end{align*}$$
Setting
$$\Psi_v(x,w)\, {:\!=}\, {\mathbb{E}_x [v^{ \tau^-_{-1}} w^ {-X(\tau^-_{-1})} ;\ \tau^-_{-1} \lt \infty]}$$
, we then have from the preceding and using (8), the neat identity
$${\mathbb{E}_x [v^{ \tau^-_{-1}} w^ {-X(\tau^-_{-1})} ;\ \tau^-_{-1} \lt \tau_b^+]}=\Psi_v(x,w)-({W_v(x)}/{W_v(b)})\Psi_v(b,w)$$
. We introduce now, for some
$$\alpha_v(w)\in [0,\infty)$$
that we shall specify later on,
$$
\begin{equation}
Z_v(x,w)\ {:\!=}\ \Psi_v(x,w)+\alpha_v(w) W_v(x),
\label{eqn11}
\end{equation}$$
a slightly modified
$$\Psi_v(\cdot,w)$$
, which also satisfies the identity
$$
\begin{equation}
\mathbb{E}_x [v^{ \tau^-_{-1}} w^ {-X(\tau^-_{-1})} ;\ \tau^-_{-1} \lt \tau_b^+]=
Z_v(x,w)-\frac{W_v(x)}{W_v(b)}Z_v(b,w)
\label{eqn12}
\end{equation}$$
(easy to check). The first motivation for preferring to use
$$Z_v(\cdot,w)$$
with a suitable choice of
$$\alpha_v(w)$$
instead of
$$\Psi_v(\cdot,w)$$
appears below in (14); many other formulae where the analogue of
$$Z_v(\cdot,w)$$
is preferable are known in the literature on spectrally negative Lévy processes, see, for example, [Reference Avram, Grahovac and Vardar-Acar4] and [Reference Ivanovs and Palmowski20].
Remark 18. Note that
$$Z_v(x,w)=\Psi_v(x,w)=w^{-x} \text{ for all integer } x \leq
-1$$
.
We compute now the z-transform of Z. Conditioning on the first jump, we obtain from (11) and the definition of
$$\Psi_v(\cdot,w)$$
, via (9), the recurrence relation
$$
\begin{equation}
\frac{Z_v(x,w)}{v} =\sum_{k=-1}^x p_{k+1}Z_v(x-k,w) + \sum_{k=x+1}^\infty
w^{k-x} p_{k+1},\qquad x\in \mathbb{N}_0.
\label{eqn13}
\end{equation}$$
Hence, the generating function
$$
{\tilde Z_v(z,w)\ {:\!=}\ \sum_{x=0}^\infty z^x
Z_v(x,w)}
$$
satisfies, for
$$z\in (0,\varphi_v)\backslash \{w\}$$
,
$$
\begin{align*}
&\frac{\tilde Z_v(z,w)}{v}
\\
&\!\qquad =p_0 \frac{\tilde Z_v(z,w)-Z_v(0,w)} z + \sum_{x=0}^\infty z^x \sum_{k=0}^x
Z_v(x-k,w) p_{k+1} + \sum_{x=0}^\infty z^x \sum_{k = x+1}^\infty w^{k-x}
p_{k+1}
\\
&\!\qquad =p_0 \frac{\tilde Z_v(z,w)-Z_v(0,w)} z + \sum_{k=0}^\infty p_{k+1} z^k
\sum_{x=k}^\infty z^{x-k} Z_v(x-k,w)
\\
&\!\qquad\quad\, + \sum_{k=1}^\infty p_{k+1} w^k \sum_{x=0}^{ k-1} \bigg(\frac z
w\bigg)^{x}
\\
&\!\qquad =p_0 \frac{\tilde Z_v(z,w)-Z_v(0,w)} z + \tilde Z_v(z,w) \sum_{k=0}^\infty p_{k+1}
z^k + \sum_{k=1}^\infty p_{k+1} w^k \frac{1-( z/w)^{k}}{1- z/w}
\\
&\!\qquad =p_0 \frac{\tilde Z_v(z,w)-Z_v(0,w)} z + \tilde Z_v(z,w) \frac{\tilde{p}(z)-p_0}{z}
+\frac{{(\tilde{p}(w)-p_0)}/{w}-{(\tilde{p}(z)-p_0)}/{z}}{1- z/w},
\end{align*}$$
i.e. in view of (10),
$$
\tilde Z_v(z,w)= -p_0 (1-Z_v(0,w))\tilde W_v(z)+ \frac{z\tilde p (w)- w \tilde p(z)}{ (z
-w)(\tilde p(z)-{z}/{v})}.
$$
Recall now that in the Lévy case,
$$\smash{Z^{(q)}(0,\theta)}$$
is chosen so as to ensure a ‘smooth fit’ [Reference Avram, Palmowski and Pistorius7, Definition 5.8] to the boundary condition
$$\mathrm{e}^{x \theta}$$
for
$$x\in ({-}\infty,0)$$
. The analogue in the discrete case is to insist on
$$Z_v(0,w)=1$$
, which we may do by choosing (cf. (11))
$$\alpha_v(w)=p_0(1-\Psi_v(0,w))$$
. Furthermore, this choice (that we assume henceforth) leads to the simple expression
$$
\begin{equation}
\tilde Z_v(z,w)=\frac 1 {\tilde p(z)-{z}/{v}} \frac{z\tilde p (w)- w \tilde p(z)}{ z
-w},\qquad z\in (0,\varphi_v), \; v\in (0,1],\; w\in (0,1]
\label{eqn14}
\end{equation}$$
(where the quotient must be understood in the limiting sense when
$$z=w$$
).
Extracting the coefficients of the z-power series yields finally an expression similar to that of the Dickson–Hipp type representation in the Lévy case (see [Reference Ivanovs and Palmowski20])
$${{{Z_v}(x,w) = (\tilde p(w) - {w \over v})\sum\limits_{k = 0}^\infty {{w^k}} {W_v}(x + k),\qquad w \in (0,{\varphi _v}),\;v \in (0,1],\;x \in {_0}} \over {}}$$
(it is easy to check that this expression has z-transform (14)).
In the special case
$$w=1$$
we set
$$Z_v(x)\ {:\!=}\ Z_v(x,1)$$
, (14) simplifies to
$$
\begin{equation}
\tilde Z_v(z)\ {:\!=}\ \sum_{x=0}^\infty z^xZ_v(x)=\frac{ \tilde p(z)-z}{ (\tilde
p(z)-{z}/{v})(1-z)},\qquad z\in (0,\varphi_v),\, v\in (0,1],
\label{eqn15}
\end{equation}$$
and we have the representation
$$
\begin{equation}
Z_v(x)=1+\bigg(\frac{1}{v}-1\bigg)\sum_{y=0}^{x-1}W_v(y), \qquad v\in
(0,1],\; x\in \mathbb{N}_0.
\label{eqn16}
\end{equation}$$
Remark 19. Using (7) in the form
$${\Psi _v}(z) = {1 \over {z/v - \tilde p(z)}}({{z - \tilde p(z)} \over {(1 - v)(1 - z)}} + {{{\varphi _v}} \over {v(1 - {\varphi _v})}}),\qquad v,z \in (0,1),\;z \ne {\varphi _v},$$
it follows from (10) and (15) that
$$
{\Psi}_v(x)\ {:\!=}\ \sum_{n=0}^\infty v^n
\Psi(n;\ x)=\frac{1}{1-v}Z_v(x)-\frac{\varphi_v}{v(1-\varphi_v)}W_v(x),
$$
i.e.
$$
\begin{align}\nonumber\\[-24pt]
\mathbb{E}_x[v^{\tau_{-1}^-};\ \tau_{-1}^-\lt\infty]&=
Z_v(x)-\frac{\varphi_v(1-v)}{v(1-\varphi_v)}W_v(x)\nonumber
\\
&= Z_v(x)-\alpha_v W_v(x),\qquad x\in\mathbb{N}_0,\, v\in
(0,1),
\label{eqn17}\end{align}$$
where we have set
$$\alpha_v\ {:\!=}\ \alpha_v(1)$$
(recall that we have chosen
$$\alpha_v(1)$$
so that
$$Z_v(0)=$$
$$1=\smash{\mathbb{E}[v^{\tau_{-1}^-};\ \tau_{-1}^-\lt \infty]+\alpha_v(1)W_v(0)}$$
). Passing to the limit
$$v\uparrow 1$$
, we find that
$$\mathbb{P}_x(\tau_{-1}^-\lt \infty)=1-W(x)(1-\tilde{p}'(1-)\land 1)$$
.
Remark 20. It is seen from (16), Remark 10, and [Reference Vidmar35, Definition 4] that we have the identification
$$Z^{(q)}(mh)=Z_{{\gamma}/{(\gamma+q)}}(m)$$
for
$$q\in
[0,\infty)$$
,
$$m\in {\mathbb Z}$$
, where
$$Z^{(q)}$$
is the Z q-scale function of [Reference Vidmar35]. Then (14), (12) and (13), with
$$w=1$$
, become [Reference Vidmar35, Equation (19), Proposition 8, and Equation (21)], respectively; (17) becomes [Reference Vidmar35, Equation (4.8)]. For an alternative form of (13) (when
$$w=1$$
), see [Reference Vidmar35, Equation (18)].
Proposition 2. For all
$$v\in (0,1]$$
,
$$w\in (0,1]$$
, the process
$$\smash{(v^{n\land
\tau_{-1}^-}Z_v(X_{n\land \tau_{-1}^-},w))_{n\in \mathbb{N}_0}}$$
is a martingale.
Proof. This follows, for instance, by linearity, from Proposition 1, and from the definition of
$$Z_v(\cdot,w)$$
via the Markov property and the terminal time property of
$$\tau_{-1}^-$$
.
Remark 21. For the
$$w=1$$
case, the analogue of Proposition 2 in the setting of upwards skip-free Lévy chains are the martingales
$$\smash{(\mathrm{e}^{-q(t\land
\tau_{-h}^-)}Z^{(q)}(Y_{t\land \tau_{-h}^-}))_{t\in [0,\infty)}}$$
for
$$q\in [0,\infty)$$
[Reference Vidmar35, Corollary 2]. In the case of a spectrally negative Lévy processes U,
$${(\mathrm{e}^{-q(t\land \tau_0^-)}Z^{(q)}(U_{t\land \tau_0^-}))_{t\in [0,\infty)}}$$
is a local martingale with localizing sequence
$$(\tau_n^+)_{n\in \mathbb{N}}$$
[Reference Kyprianou23, Example 8.12]. See also [Reference Avram, Palmowski and Pistorius7], in which Gerber–Shiu functions are defined as solutions to martingale problems [Reference Avram, Palmowski and Pistorius7, Definition 5.1], and the
$$\smash{Z^{(q)}(\cdot,\theta)}$$
function is the Gerber-Shiu function with boundary condition
$$\mathrm{e}^{\theta
x}$$
for
$$x\in ({-}\infty,0)$$
[Reference Avram, Palmowski and Pistorius7, Definition 5.8].
Remark 22. Assume that
$$\mathbb{E}[{C}_1]\lt \infty$$
. Let
$$v\in (0,1]$$
,
$$x\in \mathbb{Z}$$
. We can obtain the expected undershoot at ruin by differentiating (12) with respect to w from the left at 1. Setting
$$
Z_{1,v}(x)\ {:\!=}\ -\frac{\partial Z_{v}(x,w)}{\partial w}\bigg\vert_{w=1-},
$$
we find that, for
$$b\in \mathbb{N}_0$$
,
$${_x}[X(\tau _{ - 1}^ - )\;{v^{\tau _{ - 1}^ - }};\;\tau _{ - 1}^ - \lt \tau _b^ + ] = {Z_{1,v}}(x) - {{{W_v}(x)} \over {{W_v}(b)}}{Z_{1,v}}(b),\qquad x \le b.$$
The generating function transform of
$$Z_{1,v}$$
is given by
$${Z_{1,v}}(z)\;: = \;\sum\limits_{k = 0}^\infty {{z^k}} {Z_{1,v}}(k) = {z \over {1 - z}}{1 \over {\tilde p(z) - z/v}}({{\tilde p(z) - z} \over {1 - z}} - (1 - \tilde p'(1 - ))),\qquad z \in (0,{\varphi _v}).$$
Setting for
$$f\colon \mathbb{N}_0\to \mathbb{R}$$
and
$$y\in \mathbb{N}_0$$
,
$$\bar f(y)\ {:\!=}\ \sum_{z=0}^{y-1}f(z)$$
(in particular,
$$\bar f(0)=0$$
), and using
$$
\sum_{k=0}^\infty z^k\overline{f}(k)=\bigg(\frac{z}{1-z}\bigg)
\sum_{k=0}^\infty z^kf(k) \quad\text{for }z\in (0,1],
$$
we see that this coincides with the generating function of
$$
\mathbb{N}_0\ni x\mapsto \bar Z_{v}(x)-(1-\tilde p'(1-))\bar W_{v}(x),
$$
i.e.
$$
\begin{equation}
Z_{1,v}(x)=\bar Z_{v}(x)-(1-\tilde p'(1-))\bar W_{v}(x), \qquad x\in
\mathbb{N}_0,
\label{eqn18}
\end{equation}$$
Note also that when
$$x \lt 0$$
,
$$Z_{1,v}(x)=x$$
.
Appendix A. Double (generating function) transforms of ruin probabilities
Recall the notation of Section 3. From [Reference Willmot36, Equations (2.7) and (2.13)], we can deduce the double transform
$$
\begin{align}
\tilde{\bar{\Psi}}_v(z)&\ {:\!=}\ \sum_{n=0}^\infty v^n \bar{\Psi}_z(n)\nonumber
\\
&\ {:\!=}\ \sum_{n=0}^\infty v^n \bigg ( \sum_{x=0}^\infty z^x \bar{\Psi}(n;x)\bigg )\nonumber
\\
&\hskip2.9pt =\frac{{z}/{(1-z)}-{\varphi_v}/{(1-\varphi_v)} }{{z}-v {\tilde{p}(z)}},
\qquad v, z \in (0,1),\, z\ne\varphi_v,
\label{eqn19}\end{align}$$
where
$$\varphi_v \in (0,1)$$
is the Lundberg root (4 (note that
$$z=\varphi_v$$
is a removable singularity). Indeed, from (5), for all
$$n\in \mathbb{N}$$
,
$$
z\bar{\Psi}_z(n)=\tilde p(z) \bar{\Psi}_z(n-1) -p_0 \bar{\Psi}(n-1;\ 0)
$$
(see [Reference Willmot36, Equation (2.3)]), and summing over n after multiplication by
$$v^n$$
yields
$$
\begin{align*}
z(\tilde{\bar{\Psi}}_v(z)-(1-z)^{-1}) &= v \tilde p(z) \tilde{\bar{\Psi}}_v(z)- p_0 v
\sum_{n=0}^\infty v^n \bar{\Psi}(n;\ 0)
\\
&\Rightarrow(z-v \tilde p(z))\tilde{\bar{\Psi}}_v(z)
\\
&=\frac{z}{1-z}- p_0 v \sum_{n=0}^\infty v^n \bar{\Psi}(n;\ 0)
\end{align*}$$
(see [Reference Willmot36, Equation (2.7)]), from where (19) is obtained by requiring that the root
$$z=\varphi_v$$
on the left-hand side annihilates also the right-hand side.
Equation (19) implies the transform (for
$$v, z \in (0,1)$$
,
$$z\ne\varphi_v$$
)
$$
\begin{align*}
\tilde{\Psi}_v(z)&\ {:\!=}\ \sum_{x=0}^\infty \sum_{n=0}^\infty z^x v^n \Psi(n;\ x)
\\
&\hskip2.9pt= \frac 1{(1-z)(1-v)}-\tilde{\bar{\Psi}}_v(z)
\\
&\hskip2.9pt=\frac{1 }{{z}-v {\tilde{p}(z)}}\bigg (\frac{v(z-\tilde{p}(z))}{(1-v)(1-z)}
+\frac{\varphi_v}{1-\varphi_v}\bigg ).
\end{align*}$$
Remark 23. Note the single transforms
$$\matrix{{\tilde {\bar {\Psi}} (z)\;: = \;\sum\limits_{x = 0}^\infty {{z^x}} {\bar {\Psi}} (x) = \mathop {\lim }\limits_{v \uparrow 1} (1 - v){{\tilde {\bar {\Psi}} }_v}(z) = {{(1 - [{C_1}]) \vee 0} \over {\tilde p(z) - z}},\qquad z \in (0,1),} \cr
{\tilde \Psi (z)\;: = \;\sum\limits_{x = 0}^\infty {{z^x}} \Psi (x) = {1 \over {1 - z}} - {{(1 - [{C_1}]) \vee 0} \over {\tilde p(z) - z}},\qquad z \in (0,1)} \cr} $$
(see [Reference Willmot36, Equation (3.5)]), which are similar to the Pollaczek–Khinchine formulae of the Cramér–Lundberg model. We also have
$$
\begin{equation*}
{\bar{\Psi}}(0)=\lim_{z\downarrow 0}\tilde{\bar{\Psi}}(z)=\frac{(1-\mathbb{E}[{C}_1])\lor
0}{p_0};
\end{equation*}$$
see [Reference Shiu30, Equation (2.14)].
Appendix B
See Table 1 for a summary of the features of the
$$\Phi,W,Z$$
theory for the three types of processes (i)-(ii)-(iii) as discussed in the introduction.
Table 1: Summary table.
Remark 24. Every spectrally negative Lévy process may be seen as a (weak) limit of a net
$$Y^h$$
of upwards skip-free Lévy chains, as
$$h\downarrow 0$$
[Reference Mijatovi, Vidmar and Jacka27]. This means that a great many relations in the spectrally negative Lévy setting may be obtained (at least naively) by simply passing to the limit
$$h\downarrow 0$$
(formally, one must of course pay attention to whether or not the relevant functional is continuous with respect to such a weak limit).
Remark 25. One of the important contributions of having a unified
$$\Phi,W,Z$$
theory developed in all three settings featured in Table 1 is that whenever a result is available for one of them, it may often be simply ‘guessed’ in the others, by ‘translating’ one set of quantities into the other (though ultimately it still needs to be proved). We have seen this time and again in the results of this paper.
Acknowledgement
MV acknowledges financial support from the Slovenian Research Agency (research core funding number P1-0222). Both authors thank two anonymous referees whose comments have led to improvements in the presentation of this paper.
