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.
a. In the homogeneous risk model with fixed N
t
= n and equalized claim amounts,
U
n ∼
t ,u = nct
− n
E I
n
1S
n
n −
u
+
, 2S
n
n −
u
+
, . . . , nS
n
n −
u
+
, ct I S
n
≤ u+ct
= nct
− n
Z
[0,u+ct]
I
n
1x n
− u
+
, 2x
n −
u
+
, . . . , nx
n −
u
+
, ct dF
∗ n
x, 19
where S
n
= X
1
+ · · · + X
n
. b. Interpretation of Prabhu’s formula. In the general homogenous model on[0, t] and its associated model with
equalized claim amounts, U
∼
t , 0 =U t , 0 = ct
− 1
Z
[0,ct]
F
t
x dx. 20
Proof.
a. The second relation 19 is obvious because F
∗n
is the distribution function of S
n
. For the first relation 19 it is enough to repeat the proof of Theorem 1a. Now x
1
+· · ·+ x
k
must everywhere be replaced with kx
1
+· · ·+ x
n
n k=1, . . . , n. Hence, y
k
must be replaced with kx
1
+ · · · + x
n
n − u
+
and Y
k
with kS
n
n−u
+
.
234 F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238
Table 15 Exponential, η=0.25, t=10
u 9
9
∼
0.729 0.729
4 0.204
0.153 8
0.0463 0.0351
12 0.00886
0.00698 16
0.00148 0.00120
20 0.00022
0.00018 Table 16
Exponential, η=0.25, t=20 u
9 9
∼
0.764 0.765
4 0.272
0.187 8
0.0859 0.0568
12 0.0241
0.0163 16
0.00603 0.00427
20 0.00137
0.00101 24
0.00028 0.00022
28 0.00005
0.00004 Table 17
Pareto, η=0.05, t=1 u
9 9
∼
0.531 0.531
10 0.0124
0.0124 20
0.00275 0.00277
30 0.00119
0.00118 40
0.00066 0.00065
b. It is sufficient to consider the model with fixed N
t
= n and equalized claim amounts and to prove that
U
n ∼
t , 0 = Z
[0,ct]
h 1 −
x ct
i dF
∗ n
x 21
because then 20 results from 15, 1 and the ∼-version of 1. Now kxn−u
+
= kxn because u=0. Then,
by 5, I
n
1x n
, 2x
n , . . . ,
nx n
,ct =
ct
n
n −
xnct
n−1
n − 1 and then 21 results from 19.
We denote by 9t, u=1−Ut, u and 9
∼
t, u=1−U
∼
t, u the ruin probabilities before time t corresponding to the initial risk reserve u≥0. Sometimes different values of the premium income rate c are considered simultaneously.
Then we write more explicitly 9t, u, c and 9
∼
t, u, c. 4.1. Conjecture
In any homogeneous risk model with time interval [0, t] and its associated model with equalized claim amounts, Ψ
∼
t, u≤Ψ t, u.
F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238 235
Table 18 Pareto, η=0.05, t=5
u 9
9
∼
0.745 0.745
10 0.0798
0.0742 20
0.0175 0.0166
30 0.00690
0.00662 40
0.00364 0.00352
50 0.00224
0.00219 60
0.00152 0.00149
70 0.00110
0.00108 80
0.00083 0.00082
Table 19 Pareto, η=0.05, t=10
u 9
9
∼
0.802 0.803
20 0.0409
0.0365 40
0.00802 0.00741
60 0.00323
0.00305 80
0.00173 0.00166
100 0.00108
0.00104 120
0.00074 0.00072
Table 20 Pareto, η=0.05, t=20
u 9
9
∼
0.845 0.847
20 0.0908
0.0758 40
0.0185 0.0161
60 0.00707
0.00634 80
0.00369 0.00338
100 0.00226
0.00211 120
0.00153 0.00144
140 0.00110
0.00106 160
0.00084 0.00080
Table 21 Pareto, η=0.25, t=1
u 9
9
∼
0.496 0.496
10 0.0117
0.0117 20
0.00270 0.00270
30 0.00117
0.00116 40
0.00064 0.00065
236 F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238
Table 22 Pareto, η=0.25, t=5
u 9
9
∼
0.674 0.675
10 0.0631
0.0562 20
0.0151 0.0138
30 0.00625
0.00587 40
0.00338 0.00322
50 0.00212
0.00204 60
0.00146 0.00141
70 0.00106
0.00103 80
0.00081 0.00079
Table 23 Pareto, η=0.25, t=10
u 9
9
∼
0.720 0.721
20 0.0307
0.0256 40
0.00694 0.00612
60 0.00294
0.00269 80
0.00162 0.00151
100 0.00102
0.00097 120
0.00071 0.00068
Table 24 Pareto, η=0.25, t=20
u 9
9
∼
0.751 0.753
20 0.0566
0.0415 40
0.0138 0.0109
60 0.00585
0.00490 80
0.00322 0.00279
100 0.00204
0.00181 120
0.00140 0.00127
140 0.00103
0.00094 160
0.00079 0.00073
Table 25 Uniform, η=0.25, t=10
u 9
sqrt
9
∼
0.743 0.743
1 0.366
0.308 2
0.126 0.128
3 0.0637
0.0561 4
0.0292 0.0227
5 0.0108
0.00735 6
0.00296 0.00170
7 0.00055
0.00026
F. De Vylder, M. Goovaerts Insurance: Mathematics and Economics 26 2000 223–238 237
Table 26 Exponential, η=0.25, t=10
u 9
sqrt
9
∼
0.719 0.719
2 0.369
0.325 4
0.167 0.166
6 0.0953
0.0864 8
0.0522 0.0444
10 0.0273
0.0221 12
0.0136 0.0105
14 0.00639
0.00478 16
0.00285 0.00206
18 0.00121
0.00085 20
0.00048 0.00033
Table 27 Pareto, η=0.25, t=10
u 9
sqrt
9
∼
0.711 0.711
10 0.109
0.103 20
0.0332 0.0292
30 0.0135
0.0121 40
0.00710 0.00652
50 0.00436
0.00407 60
0.00295 0.00279
70 0.00214
0.00204 80
0.00162 0.00155
90 0.00127
0.00122 100
0.00102 0.00099
110 0.00084
0.00082
This Conjecture is based on the numerical study of Section 6. By 1 and its ∼-version, it is equivalent to the proposition: 9
n ∼
t, u ≤ 9
n
t, u for all n=1, 2, . . . . Of course, 9
1 ∼
t, u = 9
1
t, u. The proof of 9
2 ∼
t, u ≤ 9
2
t, u is rather direct by 13 and 19 and by the arguments of the proof of Lemma 2. The proof of the general case is still missing.
5. Numerical illustrations
