for the “smooth” two-sided exit problem (TSE) ([142], Thm. 3) is provided.<fnref n="41" /> The Laplace transform of $W_q$ was computed in (33) from [142]. Also, Theorem 2 from [142] provided the formula of the resolvent density for the process killed outside an interval $[a, b]$.<fnref n="42" />

$$u_q(x, y) = \frac{W_q(x - a)}{W_q(b - a)} W_q(b - y) - W_q(x - y).$$

C) [42, (4)-(7)] introduced the notation $W_q$ and the name scale function for spectrally negative Lévy processes. The central object of the paper is now $W_q$ (instead of Suprun's resolvent). Probabilistic proofs of other problems are provided, by reducing them to smooth TSE. The non-smooth two-sided first passage problem is solved in Corollary 1 from [42], and Theorem 2 from [42] determined the decay parameter $\lambda$ of the process killed upon exiting an interval, and showed that the quasi-stationary distribution is $W_{-\lambda}$. The subsequent landmark textbook [43] offers a comprehensive treatment of Lévy processes, including the beautiful excursion theory.

D) A first treatment of the optimal discounted dividends problem in the classical compound Poisson model can be found in Section 6.4 of Buhlmann (1970) [49]. The resulting formula $\frac{W_q(b)}{W_q(b)}$ for dividends at $b$, when starting from $b$, is a consequence of the fact that the discounted dividends have an exponential law of rate $\frac{W_q(b)}{W_q(b)}$.

E) [111] studies the Gerber-Shiu function (a generalization of the ruin probability) for a compound Poisson process with a constant barrier and discovers the “dividends-penalty” identity connecting it to the scale function, denoted by $h$, and to the Gerber-Shiu function without barrier.

F) [22] introduced the second scale function $Z_q$, initially for relating to $W_q$ the solution of the ruin problem $\Psi_q(x) := \mathbb{E}_x[e^{-qT_0} 1_{\{T_0<\infty\}}] = Z_q(x) - W_q(x) \frac{q}{\Phi_q}$. A case could be made for using $\Psi_q(x)$ rather than $Z_q(x)$ as the second “alphabet letter” in first passage formulas. In fact, the former, being bounded, is more convenient to compute numerically. However, it turned out that $Z_q(x)$ leads often to simpler results and proofs, due to the fact that $e^{-qt}Z_q(X_t)$ is a martingale ([22], Rem. 5), [119].

G) [124, 125] solved in terms of $W, Z$ several first passage problems for reflected processes.

H) [153] remarks that previous excursion theory proofs can often be replaced by simple applications of the strong Markov property, and of “e approximation” arguments in the non compound Poisson case.

I) [94] provided a comprehensive textbook on Lévy processes and applications.

J) [86] solved the TSE for refracted processes (which are skip-free, but not Lévy), in terms of extensions of $W$ and $Z$.

K) [32, 78] introduced the two variables extension $Z_q(x, \theta)$, which is useful for example for computing the Gerber-Shiu function $\Psi_{q,\theta}^b(x) := \mathbb{E}_x[e^{-qT_0+\theta X_{T_0}} 1_{\{T_0<T_{b,+}\}}] = Z_q(x, \theta) - \frac{W_q(x)}{W_q(b)} Z_q(b, \theta)$ see Theorem 6.5 A). The first paper showed also that this function was the unique “smooth” $q$-harmonic extension of $e^{x\theta}$, $x \le 0$.

L) [78, 79] showed that the known formulas on spectrally negative Lévy processes apply for spectrally negative Markov additive processes.

M) [12, 20, 33, 40, 115] ibidem for exponential Parisian processes.

N) [104, 114, 146] ibidem for Omega models (processes with state dependent killing).

O) [19] ibidem for skip-free discrete state-space random walks.

P) [145, 147] ibidem for positive self similar Markov processes with one-sided jumps.

Q) [16, 28, 30, 100] initiate the study of time-homogeneous strong Markov processes with one-sided jumps.

<footnote n="41">⁴¹Informally, $W_q$ may be viewed as an analog of the transfer function for discrete systems.</footnote>

<footnote n="42">⁴²Under the Cramér-Lundberg risk model, [59] derived independently the particular case $q = 0$ of the resolvent formula – see also Gerber and Shiu [72, (6.5-6.6)], who extend Dickson’s resolvent formula to $q > 0$.</footnote>
13. LIST OF NOTATIONS

<table><tr><td>T<sub>a,-</sub>, T<sub>b,+</sub>, τ<sub>d(·)</sub>, Ξ<sub>d(·)</sub>, T<sub>a,-</sub><sup>b</sup>, T<sub>b,+</sub><sup>a</sup></td><td>times of first passage (1.2), drawdown, draw-up (1.10), first passage with reflection (4.3), (6.5)</td></tr><tr><td>Ψ<sub>q</sub><sup>b</sup>(x, a) = E<sub>x</sub> [e<sup>-qT<sub>b,+</sub></sup> 1<sub>{T<sub>b,+</sub> &lt; T<sub>a,-</sub></sub>] = W<sub>q</sub>(x-a) / W<sub>q</sub>(b-a)</td><td>survival probability (1.3), (3.5)</td></tr><tr><td>Ψ<sub>q,θ</sub><sup>b</sup>(x, a) = E<sub>x</sub> [e<sup>-qT<sub>a,-</sub> + θ(X<sub>T<sub>a,-</sub></sub> - a)</sup> 1<sub>{T<sub>a,-</sub> &lt; T<sub>b,+</sub></sub>]</td><td>ruin probability (1.4), (4.1)</td></tr><tr><td>X̄<sub>t</sub> = inf<sub>0≤s≤t</sub> X<sub>s</sub>, X̄<sub>t</sub> = sup<sub>0≤s≤t</sub> X<sub>s</sub>,</td><td>infimum and supremum processes (1.7)</td></tr><tr><td>L<sub>t</sub><sup>a</sup> = -(X̄<sub>t</sub> - a)<sub>-</sub>, U<sub>t</sub> = U<sub>t</sub><sup>b</sup> = (X̄<sub>t</sub> - b)<sub>+</sub></td><td>minimal “Skorokhod regulators” (1.7)</td></tr><tr><td>X<sub>t</sub><sup>a</sup> = X<sub>t</sub> + L<sub>t</sub>, X<sub>t</sub><sup>b</sup> = X<sub>t</sub> - U<sub>t</sub></td><td>regulated processes (1.7)</td></tr><tr><td>Y<sub>t</sub> = X̄<sub>t</sub> - X<sub>t</sub>, Ŷ<sub>t</sub> = X<sub>t</sub> - X̄<sub>t</sub></td><td>drawdown and draw-up processes (1.8), (1.9)</td></tr><tr><td>κ(θ), Φ(q), W<sub>q</sub>(x), Z<sub>q</sub>(x,θ), W<sub>q,λ</sub>(x), Z<sub>q,λ</sub>(x,θ)</td><td>Levy exp. (2.1), its inverse (3.2), sc. functions (3.1), (5.1), (5.2), Parisian sc. functions (8.2)</td></tr><tr><td>ν<sub>q</sub>(s) = W<sub>q</sub><sup>(s+)</sup> / W<sub>q</sub>(s)</td><td>rate of down excursions larger than s (3.6)</td></tr><tr><td>u<sub>q</sub>(x), u<sub>q</sub><sup>a</sup>(x, y), u<sub>q</sub><sup>[a,b]</sup>(x, y), u<sub>q</sub><sup>[a,b]</sup>(x, y), u<sub>q</sub><sup>[a,b]</sup>(x, y)</td><td>resolvents of free and constrained processes</td></tr><tr><td>Z<sub>q</sub><sup>(1)</sup>(x) = ∂Z<sub>q</sub>(x,θ) / ∂θ |<sub>θ=0</sub> = Z̄<sub>q</sub>(x) - κ'(0<sub>+</sub>)W̄<sub>q</sub>(x)</td><td>Gerber-Shiu function for w(x) = x (5.6)</td></tr><tr><td>Ψ<sub>q,θ</sub><sup>b</sup>(x, a)] = E<sub>x</sub><sup>[a</sup> [e<sup>-qtT<sub>b</sub><sup>a</sup> - θL<sub>T<sub>b</sub><sup>a</sup></sub></sup> = Z<sub>q</sub>(x - a, θ) / Z<sub>q</sub>(b - a, θ)</td><td>discounted cumulative bailouts (1.11), (6.6)</td></tr><tr><td>V<sup>b</sup>(x) = E<sub>x</sub><sup>[b</sup> [∫<sub>0</sub><sup>T<sub>b</sub><sup>b</sup></sup> e<sup>-qt</sup>dU<sub>t</sub>] = W<sub>q</sub>(x) / W<sub>q</sub>(b)</td><td>expected discounted dividends until T<sub>0</sub><sup>b</sup> (6.1)</td></tr><tr><td>V<sup>[0,b]</sup>(x) = E<sub>x</sub><sup>[0,b]</sup> [∫<sub>0</sub><sup>∞</sup> e<sup>-qt</sup>dU<sub>t</sub>] = Z<sub>q</sub>(x) / Z<sub>q</sub><sup>(b)</sup></td><td>expected discounted dividends with double reflection (6.2)</td></tr><tr><td>V<sup>b</sup>(x) = G<sub>w</sub>(x) + W<sub>q</sub>(x) (1-G<sub>w</sub><sup>(b)</sup> / W<sub>q</sub><sup>(b)</sup>)</td><td>modified de Finetti objective (9.1)</td></tr><tr><td>δ<sub>q,θ</sub>(x, d, s) = E<sub>x</sub> [e<sup>-qτ<sub>d</sub> - θ(Y<sub>τ<sub>d</sub></sub> - d)</sup>; X̄<sub>τ<sub>d</sub></sub> ∈ ds]</td><td>joint law of maximum and drawdown at drawdown time (6.16)</td></tr><tr><td>δ̃<sub>q,θ</sub>(d) = E<sub>x</sub> [e<sup>-qτ<sub>d</sub> - θ(Y<sub>τ<sub>d</sub></sub> - d)</sup>] = Z<sub>q</sub>(d, θ) - W<sub>q</sub>(d) Z<sub>q</sub><sup>(d,θ)</sup> / W<sub>q</sub><sup>(d)</sup>, ∀x</td><td>drawdown function (6.17)</td></tr><tr><td>DP<sub>q,θ,ϑ</sub><sup>b</sup>(x) := E<sub>x</sub><sup>[b</sup> [e<sup>-qT<sub>0</sub><sup>b</sup> + θX<sub>T<sub>0</sub><sup>b</sup></sub> - ϑU<sub>T<sub>0</sub><sup>b</sup></sub></sup>], DP<sub>q,θ,ϑ</sub><sup>(0,b)</sup>(x)</td><td>dividends-penalty functions (6.19), (8.14)</td></tr><tr><td>DB<sub>q,θ,ϑ</sub><sup>[0,b]</sup>(x) = E<sub>x</sub><sup>[0,b]</sup> [e<sup>-ϑU<sub>e<sub>q</sub></sub> - ϑL<sub>e<sub>q</sub></sub></sup>]</td><td>dividends-bailouts function (6.23)</td></tr><tr><td>B<sup>[0,b]</sup>(x) = E<sub>x</sub><sup>[0,b]</sup> [∫<sub>0</sub><sup>T<sub>b</sub><sup>0</sup></sup> e<sup>-qt</sup>dL<sub>t</sub>] = Z<sub>q</sub>(x) / Z<sub>q</sub>(b) G<sub>q</sub><sup>B</sup>(b) - G<sub>q</sub><sup>B</sup>(x), G<sub>q</sub><sup>B</sup>(x) = Z̄<sub>q</sub>(x) + κ'(0<sub>+</sub>) / q, G<sub>q,λ</sub>(x) = λ / (q+λ) G<sub>q</sub><sup>B</sup>(x)</td><td>expected (Parisian) discounted bailouts until T<sub>b</sub><sup>0</sup> (6.26), (6.28), (8.17), (8.18)</td></tr><tr><td>B<sup>[0,b]</sup>(x) = E<sub>x</sub><sup>[0,b]</sup> [∫<sub>0</sub><sup>∞</sup> e<sup>-qt</sup>dL<sub>t</sub>] = Z<sub>q</sub>(x) / Z<sub>q</sub><sup>(b)</sup> (G<sub>q</sub><sup>B</sup>)'(b) - G<sub>q</sub><sup>B</sup>(x)</td><td>expected discounted bailouts with double reflection (6.27)</td></tr><tr><td>V<sup>[0,b]</sup><sub>S,k</sub>(x) = V<sup>[0,b]</sup>(x) - kB<sup>[0,b]</sup>(x)</td><td>Shreve-Lehoczky-Gaver objective (9.14)</td></tr><tr><td>UbD<sub>q,hat(·)</sub><sup>b</sup>(x) = E<sub>x</sub> [e<sup>-qT<sub>b,+</sub></sup>; T<sub>b,+</sub> ≤ τ<sub>hat(·)</sub>]</td><td>up before drawdown (11.12)</td></tr><tr><td>DbU<sub>q,θ,hat(·)</sub><sup>b</sup>(x) = E<sub>x</sub> [e<sup>-qτ<sub>hat(·)</sub> - θȲ<sub>τ<sub>hat(·)</sub></sub> ; τ<sub>hat(·)</sub> &lt; T<sub>b,+</sub>]</td><td>drawdown before up (11.13)</td></tr></table>
13.1. A summary of asymptotic relations for spectrally negative Lévy processes

1. When $\kappa'(0_+) > 0$, $\Phi_q$ is the asymptotically dominant singularity of $W_q(x) \sim \frac{e^{x\Phi_q}}{\kappa'(\Phi_q)} = \Phi'(q)e^{x\Phi_q}$ as $x \to \infty$. Furthermore, by (3.13) $W_q(x) = \Phi'(q)e^{\Phi_q x} - u_q(-x)$.

2. Recalling $Z_q(x, \theta) = (\kappa(\theta) - q) \int_0^\infty e^{-\theta y} W_q(x + y) dy$ (5.1), it follows that

$$ \lim_{x \to \infty} \frac{Z_q(x, \theta)}{W_q(x)} = (\kappa(\theta) - q) \lim_{x \to \infty} \int_0^\infty e^{-\theta y} \frac{W_q(x + y)}{W_q(x)} dy = \frac{\kappa(\theta) - q}{\theta - \Phi_q}. \quad (13.1) $$

When $\theta = 0$, this yields

$$ \lim_{x \to \infty} \frac{Z_q(x)}{W_q(x)} = \frac{q}{\Phi_q}, \quad \lim_{x \to \infty} \frac{Z_q(x)}{Z_q(x, \theta)} = \frac{\Phi_q - \theta}{q - \kappa(\theta)} \frac{q}{\Phi_q}. \quad (13.2) $$

3. Recalling $Z_q(x, \theta) = \frac{\kappa(\theta)-q}{\theta-\Phi_q} W_q(x) + \Psi_{q,\theta}(x)$ (5.1), it follows that

$$ \lim_{\theta \to \infty} Z_q(x, \theta) \frac{\theta - \Phi_q}{\kappa(\theta) - q} = W_q(x) \quad (13.3) $$

and

$$ \lim_{\lambda \to \infty} Z_q(x, \Phi(q + \lambda)) \frac{\Phi(q + \lambda)}{\lambda} = \lim_{\lambda \to \infty} Z_q(x, \Phi(q + \lambda)) \frac{\Phi(q + \lambda) - \Phi_q}{\lambda} = W_q(x) \quad (13.4) $$

Acknowledgements. Many thanks to Hansjoerg Albrecher, Ester Frostig, Jevgenijs Ivanovs, Bin Li, Ronnie Loeffen, Zbigniew Palmovski, José-Luis Perez, Martijn Pistorius, Matija Vidmar and Xiaowen Zhou for useful discussions, and for their invaluable contributions to this field. D. Grahovac acknowledges the support of University of Osijek grant ZUP2018-31.

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