Homogeneous risk model on a bounded time interval [0, ttt]
Insurance: Mathematics and Economics 26 2000 223–238
Homogeneous risk models with equalized claim amounts q
F. De Vylder
a
, M. Goovaerts
a,b,∗
a
Universiteit van Amsterdam, Amsterdam, Netherlands
b
Katholieke Universiteit Leuven, CRIR, Huid Eygen Heerd, Minderbroederstraat 5, 3000 Leuven, Belgium Received 1 June 1998; received in revised form 1 September 1999; accepted 24 November 1999
Abstract
We consider an homogeneous risk model on a fixed bounded time interval [0, t] and we denote by N
t
the number of claims in that interval. The claim amounts are X
1
, X
2
, . . . , X
N
t
. The homogeneous model is an extension of the classical actuarial risk model with N
t
not necessarily Poisson distributed. In the model with equalized claim amounts, each amount X
k
is replaced with X
∼ k
= X
1
+ · · · + X
N
t
N
t
. Let 9t, u be the ruin probability before t in the homogenous model, corresponding to the initial risk reserve u≥0 and let 9
∼
t, u be the corresponding ruin probability evaluated in the associ- ated model with equalized claim amounts. The essence of the classical Prabhu formula is that 9t, 0=9
∼
t, 0. By rather systematic numerical investigations in the classical risk model, we verify that 9
∼
t, u≤9t, u for any value of u≥0 and that 9
∼
t, u is an excellent approximation of 9t, u. Then these conclusions must be valid in any homogeneous model and this is an interesting observation because 9
∼
t, u can be calculated numerically, whereas no algorithms are yet avail- able for the numerical evaluation of 9t, u in general homogeneous risk models. © 2000 Elsevier Science B.V. All rights
reserved.
Keywords: Risk model; Homogeneous risk model; Ruin probability; Prabhu’s formula
1. Homogeneous risk model on a bounded time interval [0, ttt]
N
t
is the number of claims in [0, t]. The claim instants process T
1
, T
2
, . . . , T
N
t
is any homogeneous point process on the fixed interval [0, t]. Its distribution is completely specified by the probabilities PN
t
= n≥0 n=0, 1,
2, . . . such that 6
n≥0
PN
t
= n=1. The latter probabilities may be any numbers satisfying the indicated relations
see Appendix of De Vylder and Goovaerts 1999. The claim amounts are X
1
, X
2
, . . . , X
N
t
. It is assumed that X
1
, X
2
, . . . are i.i.d. random variables with distribution function F concentrated on [0, ∞ and that these amounts are independent from the claim instants. The risk reserve process is R
τ
= u+cτ −S
τ
0≤τ ≤t, where u≥0 is the initial risk reserve, c0 the premium income rate and S
τ
the total claim amount in [0, τ ], i.e. S
τ
= X
1
+ · · · + X
N
t
, where N
τ
is the number of claims in [0, τ ]. Of course, S
τ
= 0, if N
τ
= 0.
q
Presented at the Second International IME Congress, University of Lausanne, Switzerland, July 1998.
∗
Corresponding author. Tel.: +32-16-323-746; fax: +32-16-323-740. E-mail address: marc.goovaertsecon.kuleuven.ac.be M. Goovaerts
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 5 5 - 4
224 F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238
We denote by Ut, u the probability of nonruin before t corresponding to the initial risk reserve u≥0, and by U
n
t, u the corresponding conditional probability of nonruin for fixed N
t
= n. Then
U t, u = X
n≥0
P N
t
= nU
n
t, u. 1
The homogeneous model with fixed N
t
= n is defined by the probabilities PN
t
= n=1 and PN
t
= m=0 m6=n.
Then U
n
t, u is the nonconditional probability of ruin in the homogeneous model with fixed N
t
= n. In the
homogeneous model with fixed N
t
= n, the claim instants are T
1
T
2
· · · T
n
and the random vector T
1
, . . . , T
n
has a constant density on the subset W
tn
= { t
1
, . . . , t
n
|0 t
1
t
2
· · · t
n
t } of R
n
. The Lebesgue volume of W
tn
is t
n
n. Hence the density of T
1
, . . . , T
n
equals nt
n
on W
tn
. Then, by the foregoing assumptions, the distribution of the random vector T
1
, . . . , T
n
, X
1
, . . . , X
n
is completely specified in the homogeneous model with fixed N
t
= n.