228 F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238
b. Let us now assume that u=0. By Lemmas 1 and 2, I
n
Y
1
, . . . , Y
n
, ct can be replaced with ct
n
n −
Y
n
nct
n−1
n − 1 =
ct
n
n
− 1
1 − Y
n
ct in 13. This proves the first relation 14. The second relation 14 is obvious. Then the last relation 14 results
from an integration by parts. c. 15 results from 1, 12 and 14.
General expectations such as E[ϕX
1
, . . . , X
n
] are almost impossible to evaluate numerically, fastly and precisely enough, if n is large. In particular, 13 is not useful for the practical evaluation of U
n
t, u. On the contrary, expectations such as E[ϕX
1
+ · · · + X
n
] can be calculated. Indeed, they are reduced to single integrals
E[ϕX
1
+ · · · + X
n
] = Z
I
ϕy dF
∗ n
y. In practice I is a bounded interval and F the discretized, i.e. F is hold back as a long vector in the computer
program. Then F
∗n
can be calculated iteratively and only two successive long vectors F
∗k
and F
∗ k +1
k=1,2, . . . , n must be stored simultaneously in the process at each stage.
If the discretized F has ν components, then the direct calculation of E[ϕX
1
, . . . ,X
n
] = Z
· · · Z
D
ϕx
1
, . . . ,x
n
dF x
1
· · · dF x
n
may need the evaluation of ν
n
terms, to be summed up. In the case of U
n
t, u the evaluation of ϕx
1
, . . . , x
n
goes via relations such as 4. All this must be done for values n=0, 1, 2, . . . , n
and then Ut, u should result from 1. In some cases ν and n
may be pretty large, say ν=1000 and n =
100. Clearly this rudimentary procedure is hopeless, even with the fastest computers available today.
3. Homogeneous model with equalized claim amounts
We now start from an homogeneous model with time interval [0, t], called the initial model, and we replace each claim amount X
k
with the average amount X
k ∼
= X
1
+ · · · + X
N
t
N
t
. This model with equalized claim amounts is also called the associated model. The claims instants T
k
k=1, 2, . . . , N
t
are the same in both models: T
k ∼
= T
k
. The superscript
∼
is used systematically for the components of the associated model with equalized claim amounts. We notice that the risk reserve R
τ ∼
0 ≤ τ ≤ t cannot be determined from observations during time interval [0, τ ]. The complete trajectory R
s
0≤s≤t is necessary in order to fix R
τ ∼
. The distribution of the trajectories R
τ ∼
0 ≤ τ ≤ t results from the distribution of the trajectories R
τ
0≤t≤t. Hence, the associated model is clearly defined. In Figs. 1–4, we represent sample functions of the processes R
τ
full lines and R
τ ∼
stippled lines. All four following cases a, b, c and d can occur.
a. No ruin in the initial model, no ruin in the associated model Fig. 1, b. Ruin in the initial model, ruin in the associated model Fig. 2,
c. No ruin in the initial model, ruin in the associated model Fig. 3, d. Ruin in the initial model, no ruin in the associated model Fig. 4.
Due to compensations between occurrences of these cases, the following question is relevant: is U
∼
t, u close to Ut, u? The most surprising answer is that in case of an initial risk reserve u=0, the compensation is perfect: U
∼
t, 0=Ut, 0 Theorem 2b. This throws a new light on Prabhu’s formula. The numerical investigations of Section 5
show that U
∼
t, u≥Ut, u and that U
∼
t, u is rather close to Ut, u. In fact, the comparison of U
∼
t, u with Ut, u is possible in the classical actuarial model only, i.e. when N
t
is Poisson distributed, because Ut, u can be evaluated
F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238 229
Fig. 1. No ruin in initial model. No ruin in associated model.
Fig. 2. Ruin in initial model. Ruin in associated model.
Fig. 3. No ruin in initial model. Ruin in associated model.
Fig. 4. Ruin in initial model. No ruin in associated model.
230 F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238
Table 1 Uniform F, η=0.05, t=1
u 9
9
∼
0.527 0.527
1 0.0925
0.0920 2
0.00827 0.00832
3 0.00049
0.00050 4
0.00002 0.00002
Table 2 Uniform F, η=0.05, t=5
u 9
9
∼
0.777 0.777
1 0.346
0.320 2
0.113 0.105
3 0.0297
0.0281 4
0.00646 0.00622
5 0.00119
0.00117 6
0.00019 0.00019
Table 3 Uniform F, η=0.05, t=10
u 9
9
∼
0.835 0.835
1 0.473
0.433 2
0.221 0.199
3 0.0900
0.0814 4
0.0321 0.0294
5 0.0101
0.00940 6
0.00284 0.00269
7 0.00072
0.00070 8
0.00016 0.00016
Table 4 Uniform F, η=0.05, t=20
u 9
9
∼
0.875 0.875
1 0.583
0.536 2
0.347 0.309
3 0.190
0.168 4
0.0964 0.0850
5 0.0452
0.0400 6
0.0196 0.0176
7 0.00791
0.00718 8
0.00298 0.00274
9 0.00105
0.00098 10
0.00035 0.00033
F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238 231
Table 5 Uniform F, η=0.25, t=1
u 9
9
∼
0.505 0.505
1 0.0822
0.0810 2
0.00734 0.00685
3 0.00038
0.00040 4
0.00002 0.00002
Table 6 Uniform F, η=0.25, t=5
u 9
9
∼
0.710 0.710
1 0.265
0.239 2
0.0742 0.0670
3 0.0172
0.0158 4
0.00335 0.00316
5 0.00056
0.00054 6
0.00008 0.00008
Table 7 Uniform F, η=0.25, t=10
u 9
9
∼
0.753 0.753
1 0.343
0.300 2
0.130 0.111
3 0.0441
0.0379 4
0.0134 0.0117
5 0.00368
0.00328 6
0.00091 0.00083
7 0.00021
0.00019 8
0.00004 0.00004
Table 8 Uniform F, η=0.25, t=20
u 9
9
∼
0.780 0.779
1 0.400
0.343 2
0.183 0.148
3 0.0791
0.0632 4
0.0323 0.0259
5 0.0124
0.0101 6
0.00450 0.00371
7 0.00153
0.00129 8
0.00049 0.00042
9 0.00015
0.00013 10
0.00004 0.00004
232 F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238
Table 9 Exponential, η=0.05, t=1
u 9
9
∼
0.470 0.470
2 0.122
0.120 4
0.0293 0.0290
6 0.00671
0.00667 8
0.00146 0.00147
10 0.00031
0.00031
Table 10 Exponential, η=0.05, t=5
u 9
9
∼
0.735 0.735
2 0.370
0.325 4
0.168 0.146
6 0.0702
0.0618 8
0.0274 0.0245
10 0.0100
0.00915 12
0.00349 0.00325
14 0.00117
0.00111 16
0.00038 0.00036
Table 11 Exponential, η=0.05, t=10
u 9
9
∼
0.803 0.804
4 0.283
0.233 8
0.0761 0.0631
12 0.0165
0.0141 16
0.00302 0.00265
20 0.00047
0.00043
Table 12 Exponential, η=0.05, t=20
u 9
9
∼
0.852 0.853
4 0.408
0.328 8
0.164 0.129
12 0.0565
0.0447 16
0.0168 0.0136
20 0.00436
0.00368 24
0.00103 0.00089
28 0.00022
0.00020
F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238 233
Table 13 Exponential, η=0.25, t=1
u 9
9
∼
0.445 0.445
2 0.113
0.111 4
0.0268 0.0265
6 0.00606
0.00603 8
0.00132 0.00132
10 0.00028
0.00028 Table 14
Exponential, η=0.25, t=5 u
9 9
∼
0.673 0.674
2 0.310
0.260 4
0.131 0.109
6 0.0518
0.0435 8
0.0194 0.0165
10 0.00688
0.00597 12
0.00231 0.00206
14 0.00075
0.00069 16
0.00024 0.00022
numerically in that case only as far as we know. By arguments of De Vylder and Goovaerts 1999, Section 6 the validity of results in the classical model can often be extended to any homogeneous model.
By the discussion at the end of foregoing Section 2 and by next formula 19 combined with the ∼-version of 1, the nonruin probability U
∼
t, u can be calculated in any homogeneous model.
4. Nonruin probability before ttt in case of equalized claim amounts Theorem 2.
