M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 85
This yields d
= λ
c ,
d
n
= λ
n
− P
n−1 l=0
d
l
c
n−l
c ,
n = 1, . . . , k. 3.16
After this, the moments m
ε
[n] can be expanded in an asymptotic series as m
ε
[n] = d M
n
+ d
1
M
n
ε + · · · + d
k
M
n
ε
k
+ oε
k
, n = 0, . . . , k. 3.17
Therefore the perturbation condition E holds with b
ln
= d
l
M
n
, l, n = 0, . . . , k.
3.18 A direct application of Theorem 1 to the renewal equation 3.5 completes the proof.
Remark 4. It is possible that all the coefficients a
1
, . . . , a
k
in the asymptotic expansion 3.10 equal 0. This happens if the coefficients c
1
, . . . , c
k
, λ
1
, . . . , λ
k
in the perturbation conditions H and I equal 0. The first non-zero coefficient in the sequence a
1
, . . . , a
k
, if such a coefficient exists, can be either positive or negative in case R 0. However, if R = 0, then condition G implies that the first non-zero coefficient has to be positive. To see
this, assume that such a coefficient exists and that it is negative. Then the first non-zero coefficient in the sequence d
1
, . . . , d
k
exists and is positive. This implies that α
ε
1 for ε small enough. This contradicts condition G.
Remark 5. The first two coefficients a
1
and a
2
are given by a
1
= λ
c
1
− λ
1
c λ
c M
M
1
, a
2
= −λ
2
λ c
+ λ
2 1
c − λ
1
λ c
1
+ λ
2
c
2
λ
2
c M
M
1
− 1
2 λ
c
1
− λ
1
c
2
λ
2
c
2
M
2
M
2
M
3 1
, where the moment coefficients M
n
are determined by formula 3.13.
4. Cramér–Lundberg approximation for risk processes with perturbed claim distributions
In this section we consider the general model in which not only the premium rate c = c
ε
and the intensity of claim flow λ = λ
ε
but also the claim distribution Gu = G
ε
u and so the mean µ = µ
ε
are perturbed. The perturbation conditions H and I for c
ε
and λ
ε
were formulated in Section 3. The generating distribution in the renewal equation 3.5 is F
ε
u = α
ε
G
ε
u and its total variation is α
ε
= λ
ε
µ
ε
c
ε
. The forcing function is q
ε
u = α
ε
1 − G
ε
u. We assume the following analogue of condition A:
Condition J: G
ε
converges weakly to G as ε → 0. G
is not concentrated at the origin. Let,
K
ε
β = Z
∞
e
βs
1 − G
ε
s ds. 4.1
The following condition is an analogue of C: Condition K: There exists a δ 0 such, that
1. lim
ε→0
K
ε
δ ∞, 2.
λ c
K δ ∈ 1, ∞.
86 M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90
The condition K guarantees the existence of a unique solution of the characteristic equation λ
c Z
∞
e
R s
1 − G s ds = 1.
4.2 We also define the constant ψ
analogously with 3.4: ψ
= R
∞
e
R s
1 − G s ds
R
∞
se
R s
G ds
. 4.3
The moments m
ε
[n] take the form m
ε
[n] = λ
ε
c
ε
M
ε
[n], n = 0, . . . , k,
4.4 where
M
ε
[n] = Z
∞
s
n
e
R s
1 − G
ε
s ds, n = 0, . . . , k. 4.5
The analogue of the perturbation condition E is now formulated in terms of the moment coefficients M
ε
[n]: Condition L: There exists a positive integer k such that M
ε
[n] = e
0n
+ e
1n
ε + · · · + e
kn
ε
k
+ oε
k
for n = 0, . . . , k, where |e
ln
| ∞, l, n = 0, . . . , k and 0 e
0n
∞, n = 0, . . . , k. Note that conditions J and K imply that the moments M
ε
[n] → M [n] as ε → 0. This implies that
e
0n
= M [n], n = 0, . . . , k.
The perturbation condition L is a very natural one. When one considers a specific class of claim distributions a smooth polynomial perturbation of the parameters involved usually implies condition L. In most cases it is an easy
task to translate the expansions of the parameters into the corresponding expansions for the moment coefficients M
ε
[n] in L. We are now ready to formulate the main result of this paper.
Theorem 3. Let conditions G–L be satisfied and assume that u
ε
→ ∞ such that ε
r
u
ε
→ τ
r
∈ [0, ∞ as 0 ε → 0 for some 1 ≤ r ≤ k. Then the following asymptotic relation holds:
Ψ
ε
u
ε
exp{−R + a
1
ε + · · · + a
r−1
ε
r−1
u
ε
} → e
−τ
r
a
r
ψ as 0 ε → 0,
4.6 where a R
is the Lundberg exponent determined from the Eq. 4.2; b ψ is the limiting constant defined
in 4.3; c a
1
, . . . , a
k
are determined by formulas 2.16 with the coefficients b
ln
, l, n = 0, . . . , k determined by formula 4.8.
Remark 6. It is possible that all the coefficients a
1
, . . . , a
k
in 4.6 are equal to 0. This happens if the corresponding coefficients in the perturbation conditions H, I and L equal 0. The first non-zero coefficient in the sequence
a
1
, . . . , a
k
, if such a coefficient exists, can be either positive or negative in the case R 0. However see Remark
3, in the case R = 0 condition G implies that the first non-zero coefficient has to be positive.
Remark 7. The balancing condition used in Theorems 2 and 3 can be weakened and replaced by the condition lim
0ε→0
ε
r
u
ε
= τ
r
∈ [0, ∞ see Remark 3.
Proof of Theorem 3. We apply Theorem 1 to Eq. 3.5. As follow from the remarks made above, conditions H–L imply the conditions A–D.
We must also check that conditions H–L imply the perturbation condition E. Since it was already proven that the functions λ
ε
c
ε
can be expanded in an asymptotic series 3.14 and that the moment functionals M
ε
[n] can
M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90 87
be expanded according to condition L, the required expansion for their product m
ε
[n] also exists, and is given by
m
ε
[n] = b
0n
+ b
1n
ε + · · · + b
kn
ε
k
+ oε
k
, n = 0, . . . , k, 4.7
where the coefficients b
ln
can be calculated from the simple discrete convolution formula b
ln
=
l
X
m=0
d
m
e
l−m,n
, l, n = 0, . . . , k. 4.8
Hence condition E holds. Now we can apply Theorem 1 and complete the proof.
5. Some important special cases
