Ruin Problems and Gerber–Shiu Theory

Abstract

A natural generalisation of the classical Cramér–Lundberg insurance risk model is a spectrally negative Lévy process; also called a Lévy insurance risk process. In this chapter, we shall return to the first-passage problem for Lévy processes, which has already been studied in Chap. 7, and look at the role it plays in a family of problems which have proved to be an extensive topic of research in the actuarial literature. Many of the problems we shall consider are inspired by the longstanding collaborative contributions of Hans Gerber and Elias Shiu, thereby motivating the title of this chapter.

We shall start by reviewing classical results that have already been treated implicitly, if not explicitly, earlier in this book. Largely, this concerns the exact and asymptotic distributions of overshoots and undershoots of the Lévy insurance risk process at ruin. Thereafter, we shall turn our attention to more complex models of insurance risk in which dividends or tax are paid out of the insurance risk process, thereby adjusting its trajectory. In this setting, a number of identities concerning ruin of the resulting adjusted process, as well as the dividends or tax paid out until ruin, are investigated.

Exercises

10.1

Using an exponential change of measure together with (10.3), show that, for x,q≥0,

$$\begin{aligned} \mathbb{E}_x\bigl(\mathrm{e}^{-q\tau^-_0} ; X_{\tau^-_0}=0\bigr) =& \mathbb {E}_x\bigl(\mathrm{e}^{-q\tau^-_0} ; X_{\tau^-_0}=X_{\tau^-_0 -} = \underline {X}_{\tau^-_0} = 0\bigr) \\ =& \frac{\sigma^2}{2} \bigl\{ W^{(q)\prime}(x) - \varPhi(q)W^{(q)}(x) \bigr\} , \end{aligned}$$

where the right-hand side is understood to be zero when σ=0.

10.2

Find an expression for the Gerber–Shiu measure in Theorem 10.1 for the case that X has paths of bounded variation and x=0.

10.3

The following exercise is based on results found in Huzak et al. (2004b). Suppose that X is a Lévy insurance risk process. In particular, we will assume that $X = \sum_{i=1}^{n} X^{(i)}$, where each of the X ⁽ⁱ⁾ are independent spectrally negative Lévy processes with respective Lévy measures, Π ⁽ⁱ⁾, concentrated on (−∞,0). One may think of them as competing risk processes.

• (i) With the help of the compensation formula, show that, for x≥0,y>0,u<0 and i=1,…,n,

  $$\begin{aligned} &\mathbb{P}_x\bigl(X_{\tau^-_0}\in{\mathrm{d}}u,\,X_{\tau^-_0 - }\in{\mathrm{d}}y, \, \Delta X_{\tau^-_0} = \Delta X^{(i)}_{\tau^-_0}\bigr) \\ &\quad{}= r(x,y) \varPi^{(i)}(-y + \mathrm{d}u)\mathrm{d}y, \end{aligned}$$

  where r(x,y) is the potential density of the process killed on first passage into (−∞,0) given in Corollary 8.8.

• (ii) Suppose now that x=0 and each of the processes X ⁽ⁱ⁾ is of bounded variation. Recall that any such spectrally negative Lévy process is the difference of a linear drift and a driftless subordinator. Let δ be the drift of X. Show that for y>0,u<0,

  $$\begin{aligned} &\mathbb{P}\bigl(X_{\tau^-_0}\in\mathrm{d}u,\, X_{\tau^-_0 - }\in \mathrm{d}y,\, \Delta X_{\tau^-_0} = \Delta X^{(i)}_{\tau^-_0}\bigr) \\ &\quad{}= \frac{1}{\delta}\varPi^{(i)}(-y + \mathrm{d}u)\mathrm{d}y. \end{aligned}$$

• (iii) For each i=1,…,n, let δ _i be the drift of X ⁽ⁱ⁾. Note that, necessarily, ${\delta}= \sum_{i=1}^{n} {\delta}_{i}$. Suppose further that for each i=1,…,n, $\mu_{i} : = \delta_{i} -\mathbb{E}(X^{(i)}_{1}) <\infty$. Show that the probability that ruin occurs as a result of a claim from the i-th process when x=0 is equal to μ _i /δ.

10.4

Suppose that S={S _t :t≥0} is a subordinator, with Laplace exponent $\varPhi(q) = t^{-1}\log\mathbb{E}(\exp\{-q S_{t}\})$, t≥0, and e _κ is an independent exponentially distributed random variable with rate κ>0.

• (i) Use the ideas in the proof of Theorem 10.3 to deduce that

  $$\mathbb{E} \biggl[ \biggl(\int_0^{\mathbf{e}_\kappa} \mathrm{e}^{- qS_t}\mathrm{d} t \biggr)^n \biggr] = n!\prod _{k=1}^n\frac{1}{\kappa+ \varPhi(qk)} . $$

• (ii) Explain how part (i) above can be used to rephrase the proof of Theorem 10.3.

10.5

Suppose that X is a Cramér–Lundberg process with premium rate c>0, compound Poisson arrival rate λ>0 and claim distribution F with mean value μ. In the notation of Theorem 10.3, define

$$V_a = \mathbb{E}_a \biggl[ \int_0^{\sigma^a} \mathrm{e}^{-qt } \mathrm{d}\xi^a_t \biggr]. $$

• (i) By conditioning on the first jump of X, show that

  $$V_a = \frac{c}{\lambda+ q} + V_a\frac{\lambda}{\lambda+ q}\int _{(0,a]}\frac{W^{(q)} (a-y)}{W^{(q)}(a)}F(\mathrm{d} y). $$

• (ii) Show by means of taking Laplace transforms that, for all q≥0 and a>0,

  $$cW^{(q)\prime}_+(a) = (\lambda+ q)W^{(q)}(a) - \lambda\int_{(0,a]} W^{(q)}(a-y) F(\mathrm{d} y). $$

• (iii) Use parts (i) and (ii) to prove Theorem 10.3 in the case that n=1.

10.6

Use reasoning similar to that of the proof of (8.34) to deduce the following result. Let a>0, x∈[0,a], q≥0 and f,g be positive, bounded measurable functions. Further suppose that either X has no Gaussian component or it has a Gaussian component and f(0)g(0)=0. Then

$$\begin{aligned} &\mathbb{E}_x\bigl(\mathrm{e}^{-q\tau^-_0} f(X_{\tau^-_0}) g(X_{\tau ^-_0 -})\mathbf{1}_{\{\tau^-_0<\tau^+_a\}} \bigr) \\ &=\int_0^a\int_{(-\infty, -y)}f(y+ \theta)g(y) \biggl\{ \frac{W^{(q)} (x) W^{(q)}(a-y)}{W^{(q)}(a)} - W^{(q)}(x-y) \biggr\} \varPi( \mathrm{d} \theta)\mathrm{d}y. \end{aligned}$$

10.7

Show that, for q≥0 and 0≤x,b≤a, we have for the refracted process (10.20),

$$\mathbb{E}_x \bigl(\mathrm{e}^{-q\kappa^-_0}\mathbf{1}_{\{\kappa ^-_0<\kappa^+_a\}} \bigr) = z^{(q)}(x) - z^{(q)}(a)\frac{w^{(q)}(x;0)}{w^{(q)}(a;0)}. $$

10.8

Suppose that X is a spectrally negative Lévy process with bounded variation paths satisfying (10.1). Write, as usual, ψ for its Laplace exponent and Φ for the right inverse of ψ. Thinking of X as a Lévy insurance risk process, we may have the following adjusted definition of ruin. Every time the process X becomes negative, an independent and exponentially distributed clock is started with parameter q≥0. If the process X recovers and enters (0,∞) before this clock rings, then the insurance company may continue without becoming ruined. If, however, the process X spends longer below zero than it takes the associated exponential clock to ring, then the process is declared ruined.

• (i) Explain why the probability of ruin (according to the new definition) may now be written 1−V where

  $$V:= \mathbb{E} \bigl(\mathrm{e}^{-q\int_0^\infty\mathbf{1}_{\{X_s<0\}} \mathrm{d}s} \bigr),\quad x\geq0. $$

• (ii) Show that

  $$V = \mathbb{E} \bigl(\mathbf{1}_{\{\tau^-_0<\infty\}} g(X_{\tau ^-_0}) \bigr) V + \mathbb{P}\bigl(\tau^-_0 = \infty\bigr), $$

  where, for x≤0, $g(x) = \mathbb{E}_{x}(\mathrm{e}^{-q\tau^{+}_{0}})$. Hence deduce that

  $$V = \psi'(0+)\frac{\varPhi(q)}{q}. $$

• (iii) Now suppose that the drift term of X is denoted δ as in (10.2) and let U be the associated refracted process as in Sect. 10.4, where the threshold for refraction is b and α is the rate of refraction. Using ideas similar to those found in the previous parts of this question, show that, when ψ′(0+)>α and q≥0,

  $$\mathbb{E}_b \bigl(\mathrm{e}^{- q\int_0^\infty\mathbf{1}_{\{U_s<b\}} \mathrm{d}s} \bigr) = \frac{(\psi'(0+) - \alpha)\varPhi(q)}{q-\alpha\varPhi(q)}. $$

10.9

Consider the perturbed spectrally negative Lévy process (10.40). Suppose that we write $\overleftarrow{U}_{t} = \sup_{s\leq t}U_{s}$, t≥0, in the light-perturbation regime and $\overrightarrow {U}_{t} = \sup_{s\geq t}U_{s}$, t≥0, in the heavy-perturbation regime. Show that both of these processes agree with the definition of the process A in (10.42).

10.10

This exercise reproduces the results of Albrecher and Hipp (2007) and Albrecher et al. (2008) for the case of constant light-perturbation. Suppose that U is a perturbed spectrally negative Lévy process with constant tax rate γ∈(0,1). Show that, for all a≥x,

$$\mathbb{E}_x \bigl[\mathrm{e}^{-q T^+_a} \mathbf{1}_{\{T_a^+ < T_0^-\}} \bigr]= \biggl(\frac{W^{(q)}(x)}{W^{(q)}(a)} \biggr)^{1/(1-\gamma)} $$

and give an expression for the probability of ruin. Show also that

$$\mathbb{E}_x \biggl[\int_0^{T_0^-} \mathrm{e}^{-qu}\gamma(\overline{X}_u)\, \mathrm{d} \overline{X}_u \biggr]=\frac{\gamma}{1-\gamma}\int_x^\infty \biggl(\frac{W^{(q)}(x)}{W^{(q)}(u)} \biggr)^{1/(1-\gamma)}\mathrm{d}u $$

and derive an expression for the last expectation in the case that γ is a constant in (1,∞).

10.11

This exercise is based on computations found in Kyprianou and Ott (2012). Consider the perturbed process U for the heavy-perturbation regime with U ₀=x>0. Assume that s ^∗(x)<∞.

• (i) Suppose that γ is a continuous function. Show that U exhibits type II creeping under $\mathbb{P}_{x}$ if and only if X has paths of bounded variation.

• (ii) Fix c>0. Define for s∈[0,c],

  $$\gamma(s) = 1+ \frac{1}{2}(c - s)^{-\frac{1}{2}}. $$

  Show that there exists an x>0 such that $\bar{\gamma}_{x}(s) = x - \int_{x}^{s} \gamma(y)\mathrm{d}y = (c-s)^{1/2}$ with s ^∗(x)=c. Let U be the associated perturbed process such that the underlying Lévy process has a non-zero Gaussian component and U ₀=x. Show that type II creeping can occur.