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
To illustrate the content of the asymptotic formula 1.6 or 3.10, we consider in this section the two simplest cases r = 1 and r = 2.
In the case r = 1 the balancing condition 1.5 reduces to εu
ε
→ τ
1
and the asymptotic relation 1.6 can be rewritten in the form
Ψ
ε
u
ε
− exp{−Ru
ε
}ψ exp{−Ru
ε
}ψ → e
−τ
1
a
1
− 1 as ε → 0. 5.1
Let us first consider the asymptotically subcritical case in which the limiting constant α = λ
µ c
1 and so the Lundberg exponent R 0. The limit on the right-hand side of 5.1 is the limit of the relative error of
the Cramér–Lundberg approximation in the large deviation zone u
ε
= oε
−1
or u
ε
= Oε
−1
. If τ
1
= 0 then u
ε
= oε
−1
and the asymptotic relative error is 0. If τ
1
0 then u
ε
= Oε
−1
and the asymptotic error is e
−τ
1
a
1
− 1, which differs from zero. Therefore oε
−1
is the asymptotic bound for the large deviation zone in which the Cramér–Lundberg approximation 1.3 has zero asymptotic relative error and does not require any correction. In
the large deviation zone of the order Oε
−1
the Cramér–Lundberg approximation requires an asymptotic correction and 5.1 yields the value for the corresponding multiplicative correction e
−τ
1
a
1
. Similar comments can be made in the asymptotically critical case in which α
= 1 and hence the Lundberg exponent R = 0. In this case ψ = 1. If τ
1
= 0, then u
ε
= oε
−1
and the trivial approximation of the ruin probability by ψ = 1 has zero asymptotic relative error. However if τ
1
0 then u
ε
= Oε
−1
and there is a non-zero asymptotic error e
−τ
1
a
1
− 1. Therefore oε
−1
is the asymptotic bound for the large deviation zone where this trivial approximation gives zero asymptotic relative error. In the large deviation zone of the order Oε
−1
the asymptotic correction e
−τ
1
a
1
has to be used in order to ensure zero asymptotic relative error. In the case r = 2 the balancing condition 1.5 reduces to ε
2
u
ε
→ τ
2
and the asymptotic relation 1.6 can be rewritten in the form
Ψ
ε
u
ε
− exp{−R + a
1
εu
ε
}ψ exp{−R + a
1
εu
ε
}ψ → e
−τ
2
a
2
− 1 as ε → 0. 5.2
Again we consider first the asymptotically subcritical case α 1 and R 0. Now the limit on the right-hand
side is the limit of the relative error of the corrected Cramér–Lundberg approximation in two large deviation zones of higher order: u
ε
= oε
−2
or u
ε
= Oε
−2
, respectively. If τ
2
= 0, then u
ε
= oε
−2
and the asymptotic relative error is 0. If τ
2
0, then u
ε
= Oε
−2
and the asymptotic error is e
−τ
2
a
2
− 1 6= 0. Therefore oε
−2
is the asymptotic bound for the large deviation zone in which the corrected Cramér–Lundberg approximation has zero asymptotic relative error and does not require any additional
correction. In the large deviation zone of the order Oε
−2
the corrected Cramér–Lundberg approximation requires an additional asymptotic correction and 5.2 yields the corresponding value of the correction e
−τ
2
a
2
.
88 M. Gyllenberg, D.S. Silvestrov Insurance: Mathematics and Economics 26 2000 75–90
In the asymptotically critical case when α = 1 and R = 0 the limit on the right-hand side of 5.2 is the limit
of the relative error of the diffusion see 1.4 type approximation in two large deviation zones u
ε
= oε
−2
or u
ε
= Oε
−2
, respectively. What we found for the subcritical case holds true also for the critical case but for the diffusion type approximation instead of the corrected Cramér–Lundberg approximation. In particular, oε
−2
is the asymptotic bound for the large deviation zone in which the diffusion type approximation has zero asymptotic
relative error and does not require any additional correction. In the large deviation zone of the order Oε
−2
the diffusion type approximation requires an asymptotic correction and 5.2 yields the corresponding asymptotic value
of the correction e
−τ
2
a
2
. As an example let us consider the classical model of diffusion approximation in which the parameter λ does not
depend on ε and in which the claim distribution G
ε
u = Gu is also unperturbed. We take the difference c − c as the parameter ε. Here c
= λµ. The expansion in condition H takes the form c = c + ε.
In this case R = 0, ψ
= 1. Also, M[n] = R
∞
s
n
1 − Gsds = µ
n+1
n + 1, where µ
n
= R
∞
s
n
Gds. The formulas given in Remark 5 take the form
a
1
= 2
λµ
2
, a
2
= − 4
3 µ
3
λ
2
µ
3 2
. 5.3
In the case εu
ε
→ τ
1
∈ [0, ∞ Theorem 2 yields the classical formula 1.4 of diffusion approximation: Ψ
ε
u
ε
→ e
−a
1
τ
1
as ε → 0. 5.4
In the case ε
2
u
ε
→ τ
2
∈ [0, ∞ Theorem 2 yields the relation Ψ
ε
u
ε
e
−a
1
εu
ε
→ e
−a
2
τ
2
as ε → 0. 5.5
Relation 5.5 shows that the diffusion approximation does not require any correction in the large deviation zone u
ε
= oε
−2
but it does require the additional correction e
−a
2
τ
2
in the large deviation zone u
ε
= Oε
−2
. The example considered above shows that even under standard conditions of diffusion approximation, when only
the parameter c is perturbed and the perturbation has the simplest possible linear form, Theorem 2 yields a new improved version of the classical diffusion approximation.
6. Concluding remarks
