Risk process with stochastic income and two-step premium rate

Abstract

In this paper we deal with the risk reserve process with stochastic premium function. We assume that the premiums sizes have exponential distribution with the rate depending on some threshold level. The representation for the discounted defective joint density of surplus and deficit at ruin is obtained.

Introduction

We consider risk reserve process

$$U(t)=u+C(t)-S(t)\text{,}$$

where u ⩾ 0 is the initial capital, S(t) is the aggregate claim amount up to time t, C(t) is the premium income (see [6] for the study of related model). We assume that C(t), S(t) are independent compound Poisson processes with intensities λ₁ and λ₂, correspondingly. The claim sizes have common cumulative distribution function P(x) with finite mean ${\int }_0^{\infty }\overline{P}(x)\mathit{dx}<\infty$ and the density p(x). Premiums have exponential distribution with the parameter c > 0.

Let $T=\inf \{t>0:U(t)<0\}$ be the time of ruin. Note that if ruin occurs, ∣U(T)∣ is the deficit at ruin and U(T−) is the surplus immediately prior to ruin. Denote by

$$f_{\delta }(\mathit{dx}\text{,}\mathit{dy}|u)=\text{E}\left[e^{-\delta T}I\{U(T-)\in \mathit{dx}\text{,}|U(T)|\in \mathit{dy}\text{,}T<\infty \}|U(0)=u\right]\text{,}$$

the discounted defective joint density. Where δ ⩾ 0 is the discounted factor and I{ · } is the indicator function. Then Gerber–Shiu discounted penalty function is represented by

$${\varphi }_{\delta }(u)={\int }_0^{\infty }{\int }_0^{\infty }w(x\text{,}y)f_{\delta }(\mathit{dx}\text{,}\mathit{dy}|u)\text{,}$$

where w(x, y) > 0 is the penalty when ruin occurs. Following [3], ϕ_δ(u) can be interpreted as expectation of the discounted penalty when ruin occurs. If w expresses the benefit amount of an insurance payable at time of ruin, then ϕ_δ(u) is the single premium of the insurance. Also ϕ_δ(u) has some interpretation in financial mathematics (for detail see [3], [13]).

The representation of f_δ was obtained in [12], see also [1]. For the classical case when C(t) is the linear function ϕ_δ(u) have been studied in [3]. For some another model with random income see [7].

With w(x, y) = 1 function ϕ_δ(u) will be denoted by ψ_δ(u). Note that ψ_δ(0−) = 1 and as $\delta \to 0:{\psi }_{\delta }(u)\to \psi (u)=P\{T<\infty |U(0)=u\}$. If $m={\lambda }_1/c-{\lambda }_2{\int }_0^{\infty }\overline{P}(y)\mathit{dy}>0$, then ψ(u) < 1. Function ψ_δ(u) can be defined by generalized Beekman’s convolution formula

$${\psi }_{\delta }(u)=\left(1-{\int }_0^{\infty }{g}_{+}(x)\mathit{dx}\right)\sum _{n=0}^{\infty }{\overline{G}}_{+}^{\ast n}(u)\text{,}$$

where $G_{+}^{\ast n}$ is the n-fold convolution of G₊ and G₊ is given by the defective density on (0, ∞)

$$g_{+}(x)=\frac{{\lambda }_2}{\delta +{\lambda }_1+{\lambda }_2}\left(p(x)+(c-\rho ){\int }_x^{\infty }{e}^{\rho (x-y)}p(y)\mathit{dy}\right)\text{,}$$

ρ = ρ(δ) is the unique non-negative solution to the Lundberg fundamental equation

$${\lambda }_1(c(c-\xi {)}^{-1}-1)+{\lambda }_2(\hat{p}(\xi )-1)=\delta \text{,}$$

$\hat{p}$ is the Laplace transform of p.

For the study of Beekman’s convolution formula in the classical case see [14] and references there, further treatments for the model with random income can be found in [6]. Formula (1) is also referred to as Pollaczeck–Khinchine formula, see for example [2, Th. 2.1]. A further relevant reference related to Pollaczeck–Khinchine formula is [15]. For our process formula (1) can be obtained by inverting second factorization identity [12, Eq. (2.30)]. Formula for g₊(x) is due to [12, Corollary 6.2, Eq. (6.51)].

In this paper we consider modification of the risk process with a two-step premium rate (see [2]). For the modified model, the insurer’s risk process we denote by $\stackrel{\sim }{U}(t)$. In the model, two-step premium rate $\tilde{c}$ is of the following expression:

$$\tilde{c}=\tilde{c}(\stackrel{\sim }{U}(t))=\left\{\begin{array}{cc}c\text{,}\hfill & \stackrel{\sim }{U}(t)⩽b\text{,}\hfill \\ c_{\ast }\text{,}\hfill & \stackrel{\sim }{U}(t)>b.\hfill \end{array}\right.$$

The premium rate is c if the surplus is not greater than b, otherwise premium rate is c_∗. The aim of this paper is to study the joint density of surplus and deficit at ruin for the risk process with random income and two-step premium rate

$$f_{\delta }^b(\mathit{dx}\text{,}\mathit{dy}|u)=\text{E}\left[e^{-\delta \stackrel{\sim }{T}}I\{\stackrel{\sim }{U}(\stackrel{\sim }{T}-)\in \mathit{dx}\text{,}|\stackrel{\sim }{U}(\stackrel{\sim }{T})|\in \mathit{dy}\text{,}\stackrel{\sim }{T}<\infty \}|\stackrel{\sim }{U}(0)=u\right]$$

which would result in an expression of Gerber–Shiu function ($\stackrel{\sim }{T}$ is the time to ruin for the modified process).

For the study of joint density $f_{\delta }^b$ we use approach which is similar to [4] (compare with [5], [8], [9]). We represent the distributions and moment generating functions (m.g.f.) of all functionals in term of corresponding Lundberg’s root, initial parameters of the process and Laplace transform of the time to ruin ψ_δ(u) = ϕ_δ(u)∣_w(x,y) = 1.

The paper organized as follows. Section 2 recalls some results obtained in [1], [11], [12]. In particular, we recall the representations for the joint density f_δ and for the m.g.f. of two-sided boundary functionals. In Section 3 we find the representation for the density $f_{\delta }^b$ and consider the case with the inert zone (see [10]). Also using analog of the Cramer–Lundberg approximation we performed asymptotic analysis of ruin probabilities.