Кошкин Г.М. Основы страховой математики
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◦
e
x:ne
< n;
n = ω x = 0,
◦
e
0:ωe
=
◦
e
◦
= ω/2;
x = ω −n,
◦
e
(ω−n):ne
= n/2.
D (T (x), n) =
n
3
3(ω −x)
−
n
4
4(ω −x)
2
.
n = ω x = 0
D (T (x), ω) =
ω
2
12
,
(0, ω).
ω = 90 n = 5
n = 10
n = 5, 10
x
◦
e
x:5e
D (T (x), 5)
◦
e
x:10e
D (T (x), 10)
T (x)
K(x) = [T (x)],
T (x) = 10 9 = 10, 75 K(x) = 10
K(x)
P(K(x) = k), k = 0, 1, 2, . . .
P(K(x) = k) = P(k ≤ T (x) < k + 1).
T (x)
P(T (x) = k) = P(T(x) = k + 1) = 0,
P(K(x) = k) = P(k < T (x) ≤ k + 1) =
=
s(x + k) − s(x + k + 1)
s(x)
=
k
p
x
−
k+1
p
x
.
X = T (0),
K(0) = [X] :
P(K(0) = k) =
s(k) − s(k + 1)
s(0)
= s(k) − s(k + 1) =
l
k
− l
k+1
l
◦
=
d
k
l
◦
.
d
x
≈ l
◦
f(x),
P(K(0) = k) ≈ f (k),
f(x) X,
f(k)·1 ,
P(K(0) = k) ≈ f (k) · 1 ,
K(x)
e
x
= E K(x),
e
x
=
∞
X
k=1
k P(K(x) = k).
P(K(x) = k) =
s(x + k) − s(x + k + 1)
s(x)
∞
X
k=1
k[s(x + k) − s(x + k + 1)] = 1 · s(x + 1) + 2 · s(x + 2) + ···
··· − 1 · s(x + 2) − 2 · s(x + 3) − ··· =
∞
X
k=1
s(x + k),
e
x
=
1
s(x)
∞
X
k=1
s(x + k).
E [K(x)]
2
=
∞
X
k=0
k
2
P(K(x) = k) =
=
1
s(x)
(
∞
X
k=0
k
2
s(x + k) −
∞
X
k=0
k
2
s(x + k + 1)
)
=
=
1
s(x)
(
∞
X
k=0
k
2
s(x + k)−
−
∞
X
k=0
(k + 1)
2
s(x + k + 1) +
∞
X
k=0
(2k + 1) s(x + k + 1)
)
=
=
1
s(x)
∞
X
k=1
(2k −1) s(x + k) =
2
s(x)
∞
X
k=1
k s(x + k) − e
x
,
∞
X
k=0
k
2
s(x + k) −
∞
X
k=0
(k + 1)
2
s(x + k + 1) +
∞
X
k=0
(2k + 1) s(x + k + 1) =
= 1
2
· s(x + 1) + 2
2
· s(x + 2) + 3
2
· s(x + 3) + ···
··· − 1
2
· s(x + 1) − 2
2
· s(x + 2) − 3
2
· s(x + 3) − ···
··· + 1 · s(x + 1) + 3 · s(x + 2) + 5 · s(x + 3) + ··· =
∞
X
k=1
(2k − 1)s(x + k).
D K(x) = E [K(x)]
2
− e
2
x
=
2
s(x)
∞
X
k=1
k s(x + k) − e
x
− e
2
x
.
e
x
=
1
l
x
∞
X
k=1
l
x+k
, e
◦
=
1
l
◦
∞
X
k=1
l
k
,
E [K(x)]
2
=
2
l
x
∞
X
k=1
k l
x+k
− e
x
, E [K(0)]
2
=
2
l
◦
∞
X
k=1
k l
k
− e
◦
.
e
89
, e
88
, e
84
;
DK(89), DK(88).
e
89
=
290
1449
= 0, 2, DK(89) =
2 · 290
1449
− 0, 2 − (0, 2)
2
= 0, 16;
e
88
=
1449 + 290
3623
= 0, 48,
DK(88) =
2(1 · 1449 + 2 · 290)
3623
− 0, 48 − (0, 48)
2
= 0, 41;
e
84
=
9063 + 7546 + 6037 + 3623 + 1449 + 290
10735
= 2, 6.
e
89
=
806
4030
= 0, 2, DK(89) =
2 · 806
4030
− 0, 2 − (0, 2)
2
= 0, 16;
e
88
=
4030 + 806
10075
= 0, 48,
DK(88) =
2(1 · 4030 + 2 · 806)
10075
− 0, 48 − (0, 48)
2
= 0, 41;
e
84
=
24265 + 20988 + 16791 + 10075 + 4030 + 806
27665
= 2, 75.
f(x) =
x
a
2
e
−x/a
, x ≥ 0.
F
x
(t)
T (x) = X − x.
f
x
(t) =
d
dt
F
x
(t)
1
a
e
−t/a
t
a
2
e
−t/a
.
P (T (x) > t), lim
x→∞
P (T (x) > t)
x.
p
21
,
5
p
21
, q
21
,
5
q
21
,
1|
q
21
,
3|
q
25
,
3|4
q
29
,
q
60
q
20
,
q
80
q
20
.
t|u
q
x
=
t
p
x
·
u
q
x+t
.
(20)
n = 5 n = 10
s
1
(x) =
x + a
a
e
−x/a
, x ≥ 0; s
2
(x) =
r
1 −
x
110
, 0 ≤ x ≤ 110.
◦
e
◦
◦
e
x
DT (x)
◦
e
x:ne
D{min(T (x), n)}
e
◦
, e
10
, e
21
, e
40
,
e
70
;
DK(0), DK(21),
DK(70), DK(84).
x
x, 0, 1, 2, . . . .
x.
s(x)
s(x)
s(x) = a
n
+ b
n
x n ≤ x ≤ n + 1 s(n) s(n + 1)
a
n
+ b
n
n = s(n)
a
n
+ b
n
(n + 1) = s(n + 1)
a
n
b
n
b
n
= s(n + 1) − s(n),
a
n
= s(n)(n + 1) − s(n + 1)n.
b
n
< 0.
n ≤ x ≤ n + 1 s(x)
s(x) = (n + 1 − x)s(n) + (x − n)s(n + 1),
n ≤ x ≤ n + 1. f(x)
µ
x
f(x) = −s
0
(x) = s(n) − s(n + 1), n < x < n + 1,
µ
x
=
f(x)
s(x)
=
s(n) − s(n − 1)
(n + 1 − x)s(n) + (x − n)s(n + 1)
=
=
s(n) − s(n + 1)
(n + 1)s(n) − ns(n + 1) − x[s(n) − s(n + 1)]
.
f(x) = −b
n
= s(n) − s(n + 1),
n < x < n + 1.
q
n
=
s(n) − s(n + 1)
s(n)
n
µ
x
=
s(n) − s(n + 1)
s(n) + (x − n)(s(n + 1) − s(n))
=
q
n
1 − (x − n)q
n
,
n < x < n + 1.
µ
x
=
q
n
1 − (x − n)q
n
n < x < n+1
f(x) = f(x) µ
x
q
n
> s(n) − s(n + 1) s(n) < 1.
s(x) n ≤ x ≤ n + 1
a
n
e
−b
n
x
a
n
e
−b
n
n
= s(n)
a
n
e
−b
n
(n+1)
= s(n + 1),
−b
n
=
s(n + 1)
s(n)
, a
n
= s(n)e
b
n
n
= s(n)
s(n + 1)
s(n)
−n
,
b
n
= −ln p
n
, a
n
= s(n)p
−n
n
,
p
n
n
s(x) = a
n
e
−b
n
x
= s(n)p
−n
n
e
(ln p
n
)x
= s(n)p
x−n
n
, n ≤ x ≤ n + 1,
s(x) = s(n)p
x−n
n
, n ≤ x ≤ n + 1,
f(x) = −s
0
(x) = −s(n)p
x−n
n
ln p
n
, µ
x
=
f(x)
s(x)
= −ln p
n
,
n < x < n + 1, µ
x
=
−ln p
n
> −s(n) ln p
n
s(n) < 1
s(x)
s
−1
(x). s(x)
1
s(x)
1
s(x)
=
n + 1 − x
s(n)
+
x − n
s(n + 1)
, n ≤ x ≤ n + 1,
s(x) =
s(n)s(n + 1)
(n + 1 − x)s(n + 1) + (x − n)s(n)
=
s(n + 1)
(n + 1 − x)p
n
+ x − n
=
=
s(n + 1)
p
n
+ (x − n)q
n
, n ≤ x ≤ n + 1,
f(x) = −s
0
(x) =
s(n)s(n + 1)(s(n) − s(n + 1))
(s(n + 1)(n + 1 − x) + (x − n)s(n))
2
=
=
s(n + 1)q
n
(p
n
+ (x − n)q
n
)
2
, n < x < n + 1,