The Gerber-Shiu discounted penalty function: A review from practical perspectives⁰⁰footnotetext: This work was partially supported by JSPS Grants-in-Aid for Scientific Research 19H01791, 20K03758, 20K22301, 21K03347 and 21K03358.

In the formulation (2.4), the penalty function is a function of the surplus prior to ruin $U_{\tau-}$ and the deficit at ruin $|U_{\tau}|$. Here, we review other random elements that sometimes appear as part of the generalization of the Gerber-Shiu function. Below, we consider $U$ with downward jumps of finite activities as in the surplus process (2.1).

(1)

Define the total discounted claim costs until ruin:

$$\sum_{k=1}^{N_{\tau}}e^{-\delta T_{k}}\kappa(Y_{k})$$

where $\kappa:(0,+\infty)\to(0,+\infty)$ quantifies the cost of each claim. With this random variable, a variant of the Gerber-Shiu function is investigated:

$$\phi(u)=\mathbb{E}_{u}\left[e^{-n_{1}\delta_{0}\tau}\left(\sum_{k=1}^{N_{\tau}}e^{-\delta T_{k}}\kappa(Y_{k})\right)^{n_{2}}w\left(\left|U_{\tau}\right|\right)\mathbbm{1}\left(\tau<+\infty\right)\right],$$

(2.10)

where $\delta_{0}\geq 0$ is a constant force of interest and $n_{1}$ and $n_{2}$ are non-negative integers, with $\delta_{0}=\delta$ in [43], and $n_{1}=1$ in [130].

(2)

On the event $\{U_{\tau}<0,\tau<+\infty\}$ where $U$ downcrosses zero by a jump, it necessarily holds that $\tau=T_{N_{\tau}}$ and thus $U_{\tau}=U_{T_{N_{\tau}}}$. The surplus immediately after the second to last claim before ruin $U_{T_{N_{\tau}-1}}$ is often of interest [21, 44, 47, 192]. With this as the third argument, the resulting Gerber-Shiu function is given by

$$\phi(u)=\mathbb{E}_{u}\left[e^{-\delta\tau}w\left(U_{\tau-},\left|U_{\tau}\right|,U_{T_{N_{\tau}-1}}\right)\mathbbm{1}\left(\tau<+\infty\right)\right].$$

(2.11)

(3)

Another variable that sometimes appears in the context of the Gerber-Shiu function is the last minimum of the surplus before ruin

$$\underline{U}_{\tau-}:=\inf_{t\in[0,\tau)}U_{t},$$

(2.12)

as in [25, 100]. The corresponding Gerber-Shiu function is given by $\phi(u)=\mathbb{E}_{u}[e^{-\delta\tau}w(U_{\tau-},|U_{\tau}|,\underline{U}_{\tau-})\mathbbm{1}(\tau<+\infty)]$. Moreover, with this further introduced in (2.11), a generalized Gerber-Shiu function

$$\phi(u)=\mathbb{E}_{u}\left[e^{-\delta\tau}w\left(U_{\tau-},\left|U_{\tau}\right|,U_{T_{N_{\tau}-1}},\underline{U}_{\tau-}\right)\mathbbm{1}\left(\tau<+\infty\right)\right],$$

(2.13)

is investigated in [48]. In a similar manner, the maximum surplus level prior to ruin $\overline{U}_{\tau-}:=\sup_{t\in[0,\tau)}U_{t}$ is also incorporated in [46], instead of the running minimum $\underline{U}_{\tau-}$, while a two-sided truncated Gerber–Shiu function is investigated in [196] in the form of

$$\phi(u)=\mathbb{E}_{u}\left[e^{-\delta\tau}w\left(U_{\tau-},\left|U_{\tau}\right|,U_{T_{N_{\tau}-1}}\right)\mathbbm{1}\left(\overline{U}_{\tau-}<b,\,\underline{U}_{\tau-}\geq a,\,\tau<+\infty\right)\right],\quad u\in(a,b],$$

for some $0\leq a<b<+\infty$.

(4)

The number of claims arriving until ruin $N_{\tau}$ is incorporated into the Gerber-Shiu function in [70], as follows:

$$\phi(u)=\mathbb{E}_{u}\left[e^{-\delta\tau}z^{N_{\tau}}w\left(\left|U_{\tau}\right|\right)\mathbbm{1}\left(\tau<+\infty\right)\right],$$

for some $|z|<1$. By setting $\delta=0$ and $w(x)=1$, it reduces to $\mathbb{E}_{u}[z^{N_{\tau}}\mathbbm{1}(\tau<+\infty)]$, that is, the probability generating function of the number of claims until ruin.

(5)

Further to the formulation (2.10), the joint moment of the total discounted dividends until ruin $D_{\delta}(\tau):=\int_{0}^{\tau}e^{-\delta t}dD_{t}$ and aggregate discounted claim costs until ruin $Z_{\delta}(\tau):=\sum_{k=1}^{N_{\tau}}e^{-\delta T_{k}}f(Y_{k})$ is incorporated in [42], in the form of

$$\phi(u)=\mathbb{E}_{u}\left[e^{-\delta_{1}\tau}D_{\delta_{2}}^{n}(\tau)Z_{\delta_{3}}^{m}(\tau)w\left(U_{\tau-},\left|U_{\tau}\right|\right)\mathbbm{1}\left(\tau<+\infty\right)\right],\quad u\in[0,b],$$

for $m,n\in\mathbb{N}_{0}$, some discount rates $\delta_{1},\delta_{2},\delta_{3}$ and dividend barrier $b$. Moreover, we refer the reader to [41] for the Gerber-Shiu function with the joing moment of the total discounted gains and losses in the renewal risk model with two-sided jumps.

Another line of the generalization is formulated by replacing the constant discounting $\delta t$ in (2.4) (that is, linear in time) by a stochastic process (say, $\{\delta_{t}:\,t\geq 0\}$). One such example is given in [182] with $\delta_{t}=\delta t+\alpha M^{(1)}_{t}+\beta M^{(2)}_{t}$ for non-negative constants $\delta$, $\alpha$ and $\beta$, a Poisson process $\{M^{(1)}_{t}:\,t\geq 0\}$ and a Gaussian process $\{M^{(2)}_{t}:\,t\geq 0\}$, independent of the surplus process $U$. In a different direction, the cases where $\delta$ is dependent on the surplus $U$ have also been of interest. For example, the so-called omega-killing model [108, 115] considers the stochastic discounting $\delta_{t}=\int_{0}^{t}\omega(U_{s})ds$ for a non-negative measurable function $\omega$, in relation to the omega-model [82]. In the Markov additive model (Section 2.3.4), the discount factor is allowed to be dependent on the state of the background Markov chain [88]. As an alternative to the classical exponential discounting as in (2.4), a version of discounting $\delta_{1}^{N_{t}}$ is considered in [189] with respect to the number of claims $N_{t}$ for $\delta_{1}\in(0,1]$.

The computation of the Gerber-Shiu function in a finite time horizon setting is often of realistic interest, as the risk management is usually planned for a short term [73]. As an extension of the early version (2.5), a Gerber-Shiu type function is given by

$$\phi(u,T)=\mathbb{E}_{u}\left[w\left(U_{\tau}\right)\mathbbm{1}\left(\tau\leq T\right)+P\left(U_{\tau}\right)\mathbbm{1}\left(\tau>T\right)\right],$$

(2.14)

for a finite time period $T>0$ [20, 62], where $w(\cdot)$ models the penalty in the case of ruin whereas $P(\cdot)$ models the reward for survival. This is further generalized with two variables [107, 121, 122], as in

$$\phi(u,T)=\mathbb{E}_{u}\left[e^{-\delta\tau}w\left(U_{\tau-},\left|U_{\tau}\right|\right)\mathbbm{1}\left(\tau\leq T\right)\right],\quad\overline{\phi}(u,T)=\mathbb{E}_{u}\left[e^{-\delta\tau}w\left(U_{\tau-},\left|U_{\tau}\right|\right)\mathbbm{1}\left(\tau>T\right)\right].$$

(2.15)

The former of (2.15), but with only $|U_{\tau}|$ in the penalty function, is discussed in [202]. Its extension with the running minimum $\underline{U}_{\tau-}$ as defined in (2.12) as the third argument is considered in [100]. An alternative formulation of the finite-time Gerber-Shiu function

$$\phi(u,T)=\mathbb{E}_{u}\left[e^{-\delta\tau^{T}}w\left(U_{\tau^{T}-},U_{\tau^{T}}\right)\right],\quad\tau^{T}:=\tau\land T,$$

(2.16)

is studied in [73].

The expected total discounted dividends up to ruin is, strictly speaking, not a finite-time problem but can be thought of as its approximation, first discussed in [31] in the form of $\mathbb{E}_{u}[\int_{0}^{\tau}e^{-\delta t}dD_{t}]$, where $D_{t}$ denotes the accumulated dividend paid up to time $t$. Subsequently, the expected discounted total cost up to default is formulated [67], as:

$$\mathbb{E}_{u}\left[\int_{0}^{\tau}e^{-\delta t}l(U_{t})dt\right]=\frac{1}{\delta}\mathbb{E}_{u}\left[e^{-e_{\delta}}l(U_{e_{\delta}})\mathbbm{1}\left(e_{\delta}<\tau\right)\right],$$

(2.17)

where $l:[0,\infty)\to\mathbb{R}$ and an independent exponential random variable $e_{\delta}$ with rate $\delta=1/T$ so that the mean of $e_{\delta}$ is equal to $T$.

The most classical risk process, known as the Cramér-Lundberg model introduced in [55, 137], is given by

$$U_{t}=u+ct-L_{t},\quad t\geq 0,$$

(2.18)

which is within the framework (2.1) with $L$ being a compound Poisson process. More precisely, the process $L$ is a special case of (2.2) where $N$ is a Poisson process with rate $\lambda>0$ (so that the interclaim times $\{S_{k}\}_{k\in\mathbb{N}}$ are exponentially distributed with mean $\lambda^{-1}$) and individual claim sizes $\{Y_{k}\}_{k\in\mathbb{N}}$ are independent and identically distributed positive random variables, independent of the interclaim times $\{S_{k}\}_{k\in\mathbb{N}}$. We note that the two sequences $\{S_{k}\}_{k\in\mathbb{N}}$ and $\{Y_{k}\}_{k\in\mathbb{N}}$ are sometimes assumed dependent [27, 205].

Regarding the claim size $Y$, the most common choices (because they often lead to analytical expressions) are exponential [9, 104, 116, 130, 205], hyper-exponential [36, 127, 181] and Erlang distributions [7, 192], which are in the class of phase-type distributions (Section 2.3.4). Note that analytical expressions are not available for general claim size $Y$, whereas renewal theoretic arguments often provide asymptotic results (Section 3.7) that differ depending on whether the claim size distribution is light- or heavy-tailed [16].

The Lévy process is a continuous-time stochastic process with independent and stationary increments with càdlàg paths. The class of Lévy processes is a lot richer than the Cramér-Lundberg model (2.18) in the sense that it can accommodate jumps of infinite activity and/or paths of unbounded variation.

In our context, a convenient and concise characterization of the Lévy process can be given in terms of the Laplace exponent. For any Lévy process $X=\{X_{t}:\,t\geq 0\}$, we denote its Laplace exponent by

$\displaystyle\Psi_{X}(\theta):=\log\mathbb{E}[e^{\theta X_{1}}],$

(2.19)

for all $\theta$ such that the expression is finite valued. The characteristic exponent (with $\theta$ replaced with $i\theta$) always exists and is thus often preferred for the study of general Lévy processes. However, for the study of the Gerber-Shiu function, which usually considers the surplus process with only downward jumps (spectrally negative Lévy processes), it is sufficient in many instances to use the Laplace exponent since then it is well defined for every $\theta\geq 0$ and uniquely characterizes the marginal law just as the characteristic exponent does. As discussed later in Section 3.4, the so-called scale function plays an important role in the recent study of the Gerber-Shiu function on spectrally negative Lévy processes.

When the surplus process $U$ is a spectrally negative Lévy process with a finite first moment, it follows from the Lévy-Khintchine formula that

$$\Psi_{U}(\theta)=\log\mathbb{E}[e^{\theta U_{1}}]=\theta a+\frac{1}{2}\sigma^{2}\theta^{2}+\int_{0+}^{+\infty}\left(e^{-\theta x}-1+\theta x\right)\Pi(dx),\quad\theta\geq 0,$$

(2.20)

where $a\in\mathbb{R}$, $\sigma\geq 0$, and the Lévy measure $\Pi$ (of the dual process $-U$) satisfies the integrability condition $\int_{(0,+\infty)}(1\wedge x^{2})\Pi(dx)<+\infty$. We note that the term “$\theta x$” in the integrand is usually cut off, like “$\theta x\mathbbm{1}(x<1)$,” when the marginal $U_{1}$ does not have a finite first moment (that is, $\mathbb{E}[U_{1}]=-\infty$). Also, the first moment $\mathbb{E}[U_{1}]$ is often assumed to be strictly positive in the relevant literature, since then the surplus diverges ($\lim_{t\to+\infty}U_{t}=+\infty$, $a.s.$) and thus ruin does not always occur ($\mathbb{P}(\tau<+\infty)<1$).

A spectrally negative Lévy process $U$ is of bounded variation if and only if $\sigma=0$ and $\int_{0+}^{1}x\Pi(dx)<+\infty$, with the exponent (2.20) reduces to

$$\Psi_{U}(\theta)=c\theta+\int_{0+}^{+\infty}\left(e^{-\theta x}-1\right)\Pi(dx),\quad\theta\geq 0,$$

with $c:=a+\int_{0+}^{+\infty}x\Pi(dx)$. If, moreover, $\Pi(0,+\infty)<+\infty$, then it is of finite activity and admits the form (2.1); otherwise ($\Pi(0,+\infty)=+\infty$), it is said to be of infinite activity.

A Lévy process $\{L_{t}:\,t\geq 0\}$ is called a subordinator if its paths are almost surely non-decreasing. With no drift term, its Laplace exponent is then given by

$\displaystyle\Psi_{L}(\theta)=\log\mathbb{E}[e^{\theta L_{1}}]=\int_{0+}^{+\infty}\left(e^{\theta x}-1\right)\Pi(dx),$

(2.21)

which is well defined at least for all $\theta\leq 0$. In particular, if $\lambda=\Pi(0,+\infty)<+\infty$, a subordinator reduces to a compound Poisson process, like in the Cramér-Lundberg model (2.18). In this case, with $F_{Y}$ the distribution of the claim size $Y$, one can write $\Pi(dx)=\lambda F_{Y}(dx)$ on $(0,+\infty)$ and thus $\Psi_{L}(\theta)=\lambda\int_{0+}^{+\infty}(e^{\theta x}-1)F_{Y}(dx)=\lambda(\mathbb{E}[e^{\theta Y_{1}}]-1)$. Moreover, the Laplace exponent of the surplus process $U$ in the Cramér-Lundberg model (2.18) can then be written as

$\displaystyle\Psi_{U}(\theta)=\log\mathbb{E}\left[e^{\theta U_{1}}\right]=c\theta+\Psi_{L}(-\theta),$

(2.22)

which is well defined at least for all $\theta\geq 0$. It is worth mentioning that the negative of a subordinator is usually not categorized in the class of spectrally negative Lévy processes.

As a generalization of the Cramér-Lundberg model (2.18), its diffusion perturbation has long attracted a great deal of interest in modeling small perturbations caused by measurement errors, deviation of the premiums and claims [16], given by

$$U_{t}=u+ct-L_{t}+\sigma W_{t},\quad t\geq 0,$$

with a standard Wiener process $W=\{W_{t}:\,t\geq 0\}$ (see, among others, [20, 50, 122, 123, 135, 180]). This generalization carries over the diffusion component of (2.20) on top of the expression (2.22), with its Laplace exponent given by $\Psi_{U}(\theta)=c\theta+\frac{1}{2}\sigma^{2}\theta^{2}+\Psi_{L}(-\theta)$ for $\theta\geq 0$.

The Cramér-Lundberg model (2.18) has also been generalized by employing a more general claim arrival process, rather than using an ordinary Poisson process. One important example in this direction is the Sparre-Andersen risk model [11], given by a modification of (2.18), with the Poisson process $\{N_{t}:\,t\geq 0\}$ replaced by a suitable renewal process, whose interclaim times $\{S_{k}\}_{k\in\mathbb{N}}$ form a sequence of iid positive random variables, rather than iid exponential random variables. Again, the inter-claim times $\{S_{k}\}_{k\in\mathbb{N}}$ and claim sizes $\{Y_{k}\}_{k\in\mathbb{N}}$ are usually assumed independent, with some exceptions [21, 141].

The Sparre-Andersen model is in general not analytically tractable, whereas if the interclaim times are of phase-type, then semi-analytical results are often available. A phase-type distribution is concerned with the first absorption time of a finite-state continuous-time Markov chain on a state space, consisting of a finite number of transient states $E:=\{1,2,\cdots,n\}$ and a single absorbing state. Its initial distribution on the state space $E$ is given by an $n$-dimensional row vector ${\bf\alpha}=(\alpha_{1},\cdots,\alpha_{n})$, while the transition matrix is given by

$$\left(\begin{array}[]{ll}{\bf T}&{\bf t}\\ 0^{\top}&0\end{array}\right),$$

where ${\bf t}:=-{\bf T}{\bf 1}$ with the column vector of ones $\bf 1$. The exponential distribution with rate $\lambda>0$ is its special case with $E=\{1\}$, ${\bf T}=-\lambda$ and ${\bf t}=\lambda$. The Erlang distribution with shape parameter $n$ can be written as a phase-type distribution with $E=\{1,\cdots,n\}$ with

$${\bf T}_{ij}=\left\{\begin{array}[]{ll}-\lambda,&\text{if }i=j,\\ \lambda,&\text{if }j=i+1,\\ 0,&\textrm{otherwise,}\end{array}\right.\quad\textrm{and}\quad{\bf t}_{i}=\left\{\begin{array}[]{ll}0,&\text{if }i\neq n,\\ \lambda,&\textrm{otherwise.}\end{array}\right.$$

The hyper-exponential distribution, or a mixture of exponential distributions with rate $\lambda_{i}$ for $i\in\{1,\cdots,n\}$, is also of phase-type with

$${\bf T}_{ij}=\left\{\begin{array}[]{ll}-\lambda_{i}&\text{if }i=j\\ 0&\textrm{otherwise}\end{array}\right.\quad\textrm{and}\quad{\bf t}_{i}=\lambda_{i}.$$

The Coxian distribution is another important class of the phase-type distribution, modeled by a Markov chain with transition matrix of the form

$${\bf T}_{ij}=\left\{\begin{array}[]{ll}-\lambda_{i},&\text{if }i=j,\\ p_{i}\lambda_{i},&\text{if }j=i+1,\\ 0,&\textrm{otherwise,}\end{array}\right.\quad\textrm{and}\quad{\bf t}_{i}=\left\{\begin{array}[]{ll}(1-p_{i})\lambda_{i},&\text{if }i\neq n,\\ \lambda_{i},&\textrm{otherwise.}\end{array}\right.$$

It reduces to the (generalized) Erlang and hyperexponential distribution when $p_{i}\equiv 1$ and $p_{i}\equiv 0$, respectively. It is known to be almost as general as the phase-type distribution itself, in the sense that every phase-type distribution that can be represented with acyclic Markov chains has an equivalent Coxian representation. A variety of the Sparre-Andersen models in the literature are built with phase-type distributed interclaim times. For example, the Erlang(n) case has been studied in [79, 117, 220].

As an extension, in the stationary Sparre-Andersen model, claims arrive according to a stationary renewal process $N$, so that the law of the process $\{N_{t+s}-N_{t}:\,s\geq 0\}$ is invariant in $t$ [15, Section V.3]. The Gerber-Shiu function on the stationary Sparre-Andersen model can often be written in terms of that on the ordinary Sparre-Andersen model [191]. Another stationary Sparre-Andersen model can be found in [148] in the discrete setting. Other variants of the Sparre-Andersen model include a delayed Sparre-Andersen model [190] and its extensions where the premium $ct$ in (2.18) is replaced by an independent Poisson process [22] and with a multi-layer dividend strategy [57]. We refer the reader to the monograph [193] for more detail on the Sparre-Andersen model in the context of the Gerber-Shiu function.

Renewal processes with phase-type interclaim times are often mathematically tractable owing to its equivalent representation as a Markov-modulated Poisson process [15]. To be more precise, a renewal process can be represented as a Markovian arrival process whose background continuous-time Markov chain is a variant of the one appeared in Section 2.3.4, here with transition matrix ${\bf T}+{\bf t\bf\alpha}$ where ${\bf T}$ describes the state transition without arrivals and ${\bf t\bf\alpha}$ with arrivals. By generalizing Markov-modulated Poisson processes, the Sparre-Andersen process can be written as a Markov additive process (MAP), which is a bivariate Markov process $(U,J)$ where the increments of the surplus process $U$ are governed by the continuous-time Markov chain $J$ in a sense. Conditionally on the state of the Markov chain (say, $J=i\in\mathcal{E}$), the surplus process $U$ evolves as a Lévy process (say, $U^{(i)}$), until the Markov chain makes a transition to another state ($j\in\mathcal{E}$), at which instant an independent jump specified by the pair $(i,j)$ is introduced into the surplus process $U$. In the case of the Sparre-Andersen process in the form (2.18) with phase-type interclaim times, the corresponding MAP $(U,J)$ can be characterized by the transform

$$\mathbb{E}\left[e^{\theta(U_{t}-U_{0})}\mathbbm{1}(J_{t}=j)\Big{|}\,J_{0}=i\right]=\left(e^{{\bf F}(\theta)t}\right)_{i,j},\quad{\bf F}(\theta):=c\theta\mathbb{I}_{n}+{\bf T}+\mathbb{E}\left[e^{-\theta Y_{1}}\right]{\bf t\alpha},$$

for $i,j\in\mathcal{E}$, $t\geq 0$ and $\theta\geq 0$, where $\mathbb{I}_{n}$ denotes the identity matrix of order $n$. MAPs are first introduced in [13] to model surplus in risk theory. The MAP risk model is flexible enough to handle cases where the claim size as well as the discount factor depend on the state transition (that is, $i$ and $j$) [88]. The Gerber-Shiu function on Markov-modulated Poisson processes has been studied in [16, 119, 135]. A variant of the Gerber-Shiu function incorporating the maximum surplus prior to ruin in a MAP risk model is investigated in [46]. A discrete-time counterpart, the so-called Markov additive chain, has recently been introduced in [146] for addressing the discrete-time Gerber-Shiu function.

We remark that the surplus process can also be analyzed by embedding a fluid queue process [14, 152], based on which the Gerber-Shiu function can be computed when the claims occur according to a Markovian arrival process and the claim size is phase-type distributed [2].

The Cramér-Lundberg model (2.18) has been generalized by superimposing a stochastic process $\{Z_{t}:\,t\geq 0\}$ onto the surplus process $\{U_{t}:\,t\geq 0\}$, like $U_{t}=u+ct-L_{t}+Z_{t}$, for instance, for modelling additional random income or claims. In fact, the diffusion perturbation (Section 2.3.3) can also be interpreted in this way with $Z=\sigma W$. Other examples in the literature include Poisson [22, 23] and compound Poisson processes [33, 71, 105, 229]. In the presence of upward jumps in the additional component, the resulting surplus process is no longer spectrally negative, while it can still be written as a Markov additive process (Section 2.3.5) as long as the positive jump size distribution is of phase-type [17]. The risk model containing by-claims induced by main claims can be modeled by setting this additional component to the negative of the running sum of by-claims [71, 204, 235].

In the Cramér-Lundberg framework, if the surplus earns interest at a constant rate $r$, then the risk process can be described [30, 160, 181, 197] as

$$U_{t}=ue^{rt}+c\int_{0}^{t}e^{rs}ds-\int_{0}^{t}e^{r(t-s)}dL_{s}.$$

(2.23)

If the insurer is allowed to borrow at a constant force of debit interest $r_{2}$ when the surplus becomes negative while interest is earned at a constant rate $r_{1}$ otherwise, then the risk process can be written in the form of stochastic differential equation [120]:

$$dU_{t}=\left(c+\left(r_{1}\mathbbm{1}\left(U_{t}\geq 0\right)+r_{2}\mathbbm{1}\left(U_{t}<0\right)\right)U_{t}\right)dt-dL_{t}.$$

(2.24)

The constant force of interest $r$ is replaced by a suitable stochastic rate in [20, 62, 213]. Tax payment at a fixed rate $\gamma$ is incorporated [142] together with a constant debit interest, where the surplus process is modelled as

$$dU_{t}=\left(c-c\gamma\mathbbm{1}\left(U_{t}=\sup\nolimits_{s\in[0,t]}U_{s},\,U_{t}\geq 0\right)+r_{2}\mathbbm{1}(U_{t}<0)U_{t}\right)dt-dL_{t}.$$

(2.25)

A constant dividend strategy [33, 46, 81, 119, 123, 129, 214], which is only slightly outside the realm of the Cramér-Lundberg framework (2.18), is formulated via the surplus process

$$dU_{t}=c\mathbbm{1}\left(U_{t}<b\right)dt-dL_{t},$$

(2.26)

where any excess of the surplus above a certain level $b$ is to be paid as dividend. In such dividend strategies, the present value of the total dividends paid up to ruin can be expressed as (2.17), where $l(U_{t})$ here represents the dividend rate depending on the surplus $U_{t}$ [32, 67, 80]. More generally, a reflected process $U_{t}=X_{t}-\sup_{s\in[0,t]}(X_{s}-b)$ with a general Lévy process $X$ has attracted attention in the context of de Finetti’s optimal dividend problem, which is concerned with the optimality of such barrier dividend strategies. As another generalization, reflected MAPs have also been investigated [88].

The surplus process (2.26) has been further generalized with layers by replacing the constant barrier $b$ with a linear barrier $b(t)=b_{0}+at$ [10], where the dynamics is given by $dU_{t}=(c\mathbbm{1}(U_{t}<b_{0}+at)+a\mathbbm{1}(U_{t}\geq b_{0}+at))dt-dL_{t}$. Other dividend strategies include impulsive dividend strategies [132], resulting in a lower level of the surplus $a<b$ through a dividend payment when it reaches the constant barrier level $b$, and the multi-threshold dividend strategy [9, 57, 126, 205, 207, 217], which models the premium rate by a step function of the risk surplus in order to justify the ratio of earnings retention and dividend payments. In the latter strategy, the surplus process is given by $dU_{t}=\sum_{k=1}^{M}c_{k}\mathbbm{1}(U_{t}\in[b_{k-1},b_{k}))dt-dL_{t}$, with threshold layers $0=b_{0}<b_{1}<\cdots<b_{M-1}<b_{M}=+\infty$. A variant driven by a Lévy process is studied in [56] as a generalization of the refracted Lévy process [102]. A related three-barrier model is investigated recently in [216].

We add that in the Sparre-Andersen framework (Section 2.3.4), those formulations have been studied extensively in [39] along with a variety of examples for the generalized Gerber-Shiu function (2.11) where the premium rate depends on the surplus subject to threshold dividend strategy, credit or debit interests, and the absolute ruin problem.

The Gerber-Shiu function is closely related to the structural approach of credit risk analysis, which is a major subfield of mathematical and corporate finance, where the default of a firm is defined as the first down-crossing time of the asset price dynamics [161], in a similar manner to how the ruin time $\tau$ is defined in (2.3). In particular, the fair premium of a credit default swap (CDS) contract can be expressed in terms of a combination of finite-time Gerber-Shiu functions [83] in a simple case where the recovery rate is a constant, and is further generalized as well when the recovery rate depends on the firm value at bankruptcy.

Consider a defaultable bond with maturity $T$ of a firm whose asset value dynamics is given by $V_{t}=V_{0}\exp(X_{t})$. The driving process $X_{t}(=\log(V_{t}/V_{0}))$ in the exponent is often set to a Brownian motion or a Lévy process in the relevant literature. We assume that the firm defaults at the moment when the asset value goes below a certain level, say $d(<V_{0})$. By setting $U_{t}=u+X_{t}$ with $u=\log(V_{0}/d)$, we have that $V_{t}<d$ if and only if $U_{t}<0$, and hence the time of default can be written exactly the same as the ruin time (2.3) in the context of the Gerber-Shiu function. That is, the structural model of credit risk has now been reformulated into the Gerber-Shiu framework by interpreting the log asset value as the surplus process.

The main objective of the credit risk research is to evaluate the fair premium, so that the expected cash flow between the protection buyer and seller is zero. This quantity of interest depends on the recovery rate, that is often assumed to be a function of the asset value upon bankruptcy, say $R(U_{\tau})$. A protection buyer, aiming to hedge default risk, buys a CDS with constant premium rate $P\in(0,1)$ to receive the $1-R(U_{\tau})$ fraction of the face value (called default payment) upon default, if it occurs. The counterpart, a protection seller, on the other hand, pays the default payment when default occurs while receiving the premium up to maturity $T$ or default, whichever comes first. To be fair to both parties, the premium $P$ should be set in such a way that the net present values of the expected payoffs coincide, that is,

$$\mathbb{E}\left[\int_{0}^{\tau\land T}e^{-\delta s}P\,ds\right]=\mathbb{E}\left[e^{-\delta\tau}(1-R(U_{\tau}))\mathbbm{1}(\tau\leq T)\right],$$

with a constant interest rate $\delta>0$. By solving this equation for $P$, one finds the fair premium rate $P^{*}$ in terms of two types of finite-time Gerber-Shiu functions (2.15) and (2.16):

$$P^{*}=\frac{\mathbb{E}[e^{-\delta\tau}(1-R(U_{\tau}))\mathbbm{1}(\tau\leq T)]}{\delta^{-1}\mathbb{E}[1-e^{-\delta(\tau\land T)}]}.$$

It is worth noting that the study of the Gerber-Shiu function has also been very close to other types of research in finance. For example, a risk measure is investigated in [167] in the framework of the Gerber-Shiu function for the evaluation of solvency and default risks in a variety of financial contexts.

Another practical application of the Gerber-Shiu function can be found in the study involving capital injections, which aims to maintain the solvency of the portfolio [65, 145, 159, 211]. Given a surplus process $U$ and $\tau$, the first time $U$ goes below zero (which corresponds to the time of ruin in the Gerber-Shiu framework), the capital injection model is designed to inject the capital $|U_{\tau}|$ (or more), taken out from an external source, to the company in order to avoid its bankruptcy. By repeatedly receiving capital injection, the company avoids ruin permanently.

Let $Z=\{Z_{t}\,:\,t\geq 0\}$ be a process that aggregates capital injection in such a way that bankruptcy never occurs, that is, $U_{t}+Z_{t}\geq 0$, $a.s.$ The primary objective here is to minimize the net present value of the total capital injection $\mathbb{E}_{u}[\int_{0}^{+\infty}e^{-\delta t}dZ_{t}]$ over all injection strategy $Z$ under the condition $U_{t}+Z_{t}\geq 0$, $a.s.$, that is, $f^{*}(u):=\inf_{Z}\mathbb{E}_{u}[\int_{0}^{+\infty}e^{-\delta t}dZ_{t}]$. The minimizer should first inject exactly $|U_{\tau}|$ to bring the surplus back up to zero, so that the jump size $dZ_{t}$ then can be kept as small as possible. If the surplus process is time-homogeneous and strongly Markovian, then it holds by the renewal argument that the minimum amount of injection after the first ruin is $f^{*}(0)$, and thus moreover,

$$f^{*}(u)=\mathbb{E}_{u}\left[e^{-\delta\tau}\left(f^{*}(0)+|U_{\tau}|\right)\mathbbm{1}(\tau<+\infty)\right],$$

which is nothing but a combination of the two formulations (2.6) and (2.7) of the Gerber-Shiu function [159].

Closely related to capital injection is reinsurance. For instance, in [145], a capital injection strategy with positive barrier $k$ is formulated as a minimization problem of the probability of ultimate ruin, modeled by the event that the deficit exceeds $k$ (and hence the pre-capital-injection surplus downcrosses zero). This formulation is considered as a type of reinsurance contract that restores the insurer’s surplus to a level $k$. In a more standard reinsurance policy problem, the objective is set to select an optimal reinsurance contract, or a mapping from the claim size $Y_{k}$ to the portion the first insurer needs to cover. The drift term there is modified by the reinsurance premium which depends on the contract. Also, in [150], a problem of minimizing the Gerber-Shiu function is addressed over a set of reinsurance strategies in the Cramér-Lundberg model.

The Gerber-Shiu function has also attracted attention in the context of ruin-related risk measures, naturally because bankruptcy is the risk that insurance companies wish to avoid most. For instance, risk measures are developed based on ruin probabilities and applied to the construction of reinsurance strategies, such as a VaR-type risk measure [179] on the Cramér-Lundberg model (2.18): for $\epsilon\in(0,1)$,

$$\rho_{\epsilon}[Y]:=\inf_{u\in(0,+\infty)}\left\{\mathbb{P}_{u}(\tau<+\infty)\leq\epsilon\right\}=\inf_{u\in(0,+\infty)}\left\{\mathbb{P}\left(\sup\nolimits_{t\in[0,+\infty)}\left\{L_{t}-ct\right\}>u\right)>1-\epsilon\right\},$$

(2.27)

with respect to the law of the claim size $Y$, which returns the minimum initial reserve that keeps the ruin probability less than $\epsilon$. It is known [179] that the map (2.27) satisfies the definition of a risk measure in the financial context [12]. This approach has been tailored to the Gerber-Shiu framework [65, 157, 159] for measuring the minimum initial reserve that controls “Gerber-Shiu type” risks, such as total capital injections (Section 2.4.3), whereas those developments do not meet all requirements to be a risk measure in the sense of [12], that is, it is not entirely clear what the “risk” measured (that is, the law of the claim size $Y$) represents. Most recently, now in line with the framework [12], a dynamic version is developed [167] based on the finite-time Gerber-Shiu function (Section 2.2.3) in the form of the so-called Gerber-Shiu process $\{\phi_{t}(u,T):\,t\in[0,T]\}$, defined by

$$\phi_{t}(u,T):=\begin{cases}\mathbb{E}_{u}[e^{-\delta(\tau\wedge T)}w(X_{(\tau\land T)-},|X_{\tau\wedge T}|)|\,\mathcal{F}_{t}],&\text{on }\{\tau>t\},\\ +\infty,&\text{on }\{\tau\leq t\},\end{cases}$$

(2.28)

where $\{X_{t}:\,t\in[0,T]\}$ denotes a general surplus process with initial state $X_{0}=u$ and its natural filtration $(\mathcal{F}_{t})_{t\in[0,T]}$. The Gerber-Shiu risk measure is then proposed with respect to time, as follows:

$${\rm GS}_{t,T}^{\epsilon}(X):=\inf_{z\in\mathbb{R}}\left\{\phi_{t}(u+z,T)<\epsilon\right\},$$

which represents the minimum amount to be adjusted at time $t$ to keep the Gerber-Shiu process less than $\epsilon$ all the way up to maturity $T$. That is, this map measures a dynamic risk in time on the path space of the trajectory $X$ (for instance, the Skorokhod space $\mathbb{D}$ of càdlàg functions). Despite this dynamic risk measure is rich enough to accommodate various practical problems, the computation of the conditional expectation involved in the Gerber-Shiu process (2.28) is not a trivial task as of yet. On the whole, the risk measure in the Gerber-Shiu framework would be an important future line of research deserving of separate investigation.

The Laplace transform is closely related to the (integro-)differential equation, which can be derived in various manners. One approach is to use the fact that the discounted and killed process $\{e^{-\delta(t\wedge\tau)}\phi(U_{t\wedge\tau}):\,t\geq 0\}$ is a local martingale. If the Gerber-Shiu function $\phi$ is sufficiently smooth for the Ito formula, the Lundberg equation (3.1) yields the identity $\mathcal{A}\phi(u)=\delta\phi(u)$, where $\mathcal{A}$ is the infinitesimal generator of the Lévy process $U$. This approach relies on the differentiability of the Gerber-Shiu function $\phi$ (typically depending on the path variation), which holds true in many instances [16, Chapter XII.2].

As for the Cramér-Lundberg model (2.18), a more intuitive derivation is available. By conditioning on the time and amount of the first claim before time $h(>0)$ (provided that there is such a claim), we obtain

$\displaystyle\phi(u)=e^{-(\delta+\lambda)h}\phi(u+ch)+\int_{0}^{h}\lambda e^{-(\lambda+\delta)t}\left[\int_{0}^{u+ct}\phi(u+ct-x)p_{Y}(x)dx+\int_{u+ct}^{+\infty}w(u+ct,x-u-ct)p_{Y}(x)dx\right]dt,$

(3.2)

where $c$ and $\lambda$ denote the constant premium rate and the unit-time intensity of the homogeneous Poisson process in (2.18). Hereafter, we assume that the claim size $Y$ admits a density $p_{Y}$ so that $F_{Y}(dx)=p_{Y}(x)dx$ on $(0,+\infty)$ for convenience. By differentiating (3.2) with respect to $h$ at $0+$, we obtain

$$-(\lambda+\delta)\phi(u)+c\phi^{\prime}(u)+\lambda\int_{0}^{u}\phi(u-x)p_{Y}(x)dx+\lambda\int_{u}^{+\infty}w(u,x-u)p_{Y}(x)dx=0.$$

(3.3)

Taking the Laplace transform on both sides of the integro-differential equation (3.3) yields the Laplace transform of the Gerber-Shiu function $\mathcal{L}\phi(s)=\int_{0}^{+\infty}e^{-su}\phi(u)du$. In particular, we obtain, for $s>\Phi(\delta)$,

$$\mathcal{L}\phi(s)=\frac{\lambda\left(\mathcal{L}\omega(\Phi(\delta))-\mathcal{L}\omega(s)\right)}{cs+\lambda(\mathbb{E}[e^{-sY_{1}}]-1)-\delta},$$

where $\omega(s):=\int_{0}^{+\infty}e^{-su}\int_{u}^{+\infty}w(u,x-u)p_{Y}(x)dxdu$. Here, we recall that $\Phi(\delta)$ is the unique non-negative solution to the Lundberg equation (3.1), or more precisely in this case,

$$\lambda\left(\mathbb{E}\left[e^{-\Phi(\delta)Y_{1}}\right]-1\right)+c\Phi(\delta)-\delta=0.$$

(3.4)

We note that the Dickson-Hipp operator [60, 116] has often been involved in the computation of the Laplace transform of the Gerber-Shiu function.