Insurance: Mathematics and Economics 26 2000 75–90
Cramér–Lundberg approximation for nonlinearly perturbed risk processes
q
Mats Gyllenberg
a,∗
, Dmitrii S. Silvestrov
a,b,c
a
Department of Mathematics, University of Turku, Turku, Finland
b
Department of Mathematical Statistics, Umeå University, Umeå, Sweden
c
Department of Mathematics and Physics, Mälardalen University, SE-72123 Västerås, Sweden Received April 1999; accepted September 1999
Abstract
An extension of the classical Cramér–Lundberg approximation for ruin probabilities to a model of nonlinearly perturbed risk processes is presented. We introduce correction terms for the Cramér–Lundberg and diffusion type approximations,
which provide the right asymptotic behaviour of relative errors in a perturbed model. The dependence of these correction terms on relations between the rate of perturbation and the speed of growth of an initial capital is investigated. Various types
of perturbations of risk processes are discussed. The results are based on a new type of exponential asymptotics for perturbed renewal equations. ©2000 Elsevier Science B.V. All rights reserved.
MSC: 60K05; 60K30; 90A46 Keywords: Risk process; Cramér–Lundberg approximation; Diffusion approximation; Large deviations; Renewal equation
1. Introduction
The aim of this paper is to present an extension of the classical Cramér–Lundberg approximation of ruin proba- bilities to a model of nonlinearly perturbed risk processes. In traditional risk theory a risk process
Xt = ct −
N t
X
k=1
Z
k
, t ≥ 0 1.1
is used to model the business of an insurance company. In 1.1 the positive constant c is the gross premium rate, N t is a Poisson process with parameter λ counting the number of claims on the company in the time-interval
q
This work has been supported by The Academy of Finland and The Swedish Academy of Sciences Grant 1413.
∗
Corresponding author. Tel.: +358-23336567; fax: +358-23336595. E-mail addresses: matsgylutu.fi M. Gyllenberg, dmitrii.silvestrovmdh.se D.S. Silvestrov.
0167-668700 – see front matter ©2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 6 6 8 7 9 9 0 0 0 4 3 - 8
76 M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90
[0, t ] and Z
k
is a sequence of non-negative i.i.d. random variables with a finite mean µ which are independent on N t . Z
k
is the amount of the kth claim. An important object of study is the ruin probability Ψ u = P {u + inf
t ≥0
Xt 0} 1.2
for a company which has an initial capital u. Of special interest is the asymptotics of the ruin probabilities for large values of initial capital u.
The loading claim rate of the company is characterized by a constant α = λµc. Only the subcritical case α 1 is non-trivial if α ≥ 1, then the ruin probability is identically equal to 1: Ψ u ≡ 1. The classical result, known as
the Cramér–Lundberg approximation, gives under certain conditions for the claim distribution G so-called Cramér type conditions, the asymptotics of the ruin probability in the form
Ψ u e
−Ru
→ ψ as u → ∞, 1.3
where the constant R, known as the Lundberg exponent, and the limiting constant ψ are determined by the constant α and the claim distribution G.
We refer to the paper by Cramér 1955 for a survey of basic results in the area. Later developments are reviewed in the books by Gerber 1979 and Grandell 1991 and in the paper by Thorin 1982.
The Cramér–Lundberg approximation describes the asymptotics of ruin probabilities for fixed values of α 1. On the other hand, it does not give a detailed description of the asymptotics for the ruin probability Ψ u for large
values of u and values of α which are less than but close to 1. For example, it does not give the asymptotic behaviour of ruin probabilities when the initial capital u → ∞ and simultaneously the gross premium rate c ↓ λµ. Here the
so-called diffusion approximation gives the answer. The asymptotics depend on the relation between the speeds at which c tends to λµ and u tends to infinity. Under the condition that u → ∞ such that c − λµu → τ
1
the diffusion approximation of risk processes yields the asymptotics of the ruin probability in the form
Ψ u → e
−a
1
τ
1
as u → ∞, 1.4
where the constant a
1
is determined by the parameter λ and the claim distribution G. Taking the asymptotic relations 1.3 and 1.4 as the starting point we formulate the problem in a more general
way. We consider a whole family X
ε
t , t ≥ 0 of risk processes depending on a small parameter ε ≥ 0. The process X
ε
t is considered as a perturbation of the process X t and therefore we assume some weak continuity
conditions for the characteristic quantities c = c
ε
, λ = λ
ε
and the distributions G = G
ε
as functions of ε at point ε = 0. Moreover, we admit nonlinear perturbations which means that the characteristic quantities of the perturbed
risk processes are nonlinear functions of ε. However, we restrict our attention to the case of a smooth perturbation and hence assume that these functions have k derivatives at ε = 0, i.e., these functions can be expanded in a power
series with respect to ε up to and including the order k. The object of our study is the asymptotic behaviour of the ruin probabilities Ψ
ε
u when the initial capital u → ∞ and the perturbation parameter ε → 0. The balance between the speeds at which ε tends to 0 and the initial capital u tends to ∞ has a delicate influence
upon the results. Without loss of generality it can be assumed that u = u
ε
is a function of the parameter ε. The balance between the rate of perturbation and the rate of growth of the initial capital is characterized by the following
balancing condition ε
r
u
ε
→ τ
r
as ε → 0 1.5
for some positive integer r and τ
r
∈ [0, ∞. Under the assumptions described above and some natural Cramér type condition for the claim distributions G
ε
we obtain the asymptotic relation Ψ
ε
u
ε
exp{−R + a
1
ε + · · · + a
r−1
ε
r−1
u
ε
} → e
−τ
r
a
r
ψ as ε → 0, 1.6
M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 77
where R and ψ are the same as in 1.3 and an explicit recursive algorithm can be given for calculating the coefficients a
1
, . . . , a
r
as rational functions of the coefficients in the expansions for the characteristic quantities of the perturbed risk processes.
The ruin probability Ψ
ε
u
ε
= P {inf
t ≥0
X
ε
t −u
ε
} can be interpreted as a tail probability for the infimum of the risk process X
ε
t . With this interpretation the asymptotic relation 1.6 takes the form of a large deviation theorem. Depending on whether τ
r
equals 0 or is positive, the balancing condition 1.5 states that either u
ε
= oε
−r
or u
ε
= Oε
−r
. Following the standard terminology of large deviation theory we refer to this asymptotic behaviour of u
ε
as two different large deviation zones. Our approach is based on renewal arguments as developed by Feller 1971. We use results obtained by Silvestrov
1976, 1978, 1979 concerning an extension of the renewal theory to a model of perturbed renewal equation and follow the lines of recent works by Silvestrov 1995 and Gyllenberg and Silvestrov 2000a concerning exponential
asymptotics for perturbed renewal equations. The main improvement is that the new version of exponential asymp- totics obtained in the present paper covers not only the case where the total variation of the distribution generating
the limiting renewal equation is equal to 1 see Silvestrov, 1995; Gyllenberg and Silvestrov, 2000a but also the cases in which this variation is less than or greater than 1. This new result plays a key role in the extension of the
Cramér–Lundberg approximation to the model of perturbed risk processes.
The paper is organized as follows. In Section 2 we present the results concerning exponential asymptotics for perturbed renewal equation, which we think are interesting in their own right. We present our main results concerning
the Cramér–Lundberg approximation for nonlinearly perturbed risk processes in Sections 3 and 4 and in Section 5 we apply these to some important special cases. In Section 6 we give some concluding remarks.
2. Exponential asymptotics for perturbed renewal equations
