Кошкин Г.М. Основы страховой математики
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γ
N
=
(−1)
ν
h
ν
N
ν!
∞
Z
−∞
h
f
(ν)
(x + (−1)
ν
uh
N
θ) − f
(ν)
(x)
i
u
ν
K(u)du,
0 < θ < 1.
∞
Z
−∞
|u
ν
K(u)|du < ∞, x ∈ R
1
N → ∞
f
(ν)
(x + (−1)
ν
uh
N
θ) → f
(ν)
(x)
|γ
N
| = o (h
ν
N
) .
T
i
= 0, i =
1, . . . , ν − 1 ♠
b (f
N
(x))
K(u) K(u) ∈ A
ν
, ν ≥ 4.
K(u) ∈ A
2
,
sup
u
K(u) < ∞, K(u) = K(−u)
Z
u
2
K(u)du < ∞.
K(u) ∈ A
4
,
K(u) =
15(3 − 10u
2
+ 7u
4
)/2
5
, |u| ≤ 1,
0, |u| > 1.
K(u) ∈ A
6
,
K(u) =
105(5 − 35u
2
+ 63u
4
− 33u
6
)/2
8
, |u| ≤ 1,
0, |u| > 1.
ρ(u) = {1 − u
2
, |u| ≤ 1; 0, |u| > 1} K(u) ∈ A
ν
,
K(u) = ρ(u)
ν−2
X
j=0
p
j
(0)p
j
(u)(2j + 3)(j + 2)/8(j + 1),
p
j+2
(u) =
j + 3
j + 4
2j + 5
j + 2
up
j+1
(u) − p
j
(u)
,
p
0
(u) = p
0
(0) = 1, p
1
(u) = 2u, p
1
(0) = 0.
K(u) ∈ A
ν
, ν ≥ 4,
K(u) ≥ 0
K(u) ∈ A
ν
, ν ≥ 4,
K(u)
t
N
t
O
N
−α
, α > 0 t
N
∈ V
N
−α
,
u
2
(t
N
)
lim
N→∞
N
α
u
2
(t
N
) = C, 0 < C < ∞.
α
N
β
N
α
N
∼ β
N
lim
N→∞
|α
N
/β
N
| = 1.
b (f
N
(x))
u
2
(f
N
)
h
N
∈ H
2
N → ∞
h
N,o
= argmin
h
N
>0
u
2
(f
N
(x)) ∼
A
2νB
2
N
1
2ν+1
,
A = f(x)
∞
Z
−∞
K
2
(u)du, B =
f
(ν)
T
ν
ν!
,
u
2
f
N
(x)|
h
N
=h
N,o
= u
2
(f
N,o
(x)) ∼ O
n
−
2ν
2ν+1
.
u
2
(f
N
(x)) =
A
Nh
N
+ B
2
h
2ν
N
+ o
1
Nh
N
+ h
2ν
N
.
h
N
h
N,o
∼
A
2νB
2
N
1
2ν+1
= O
N
−
1
2ν+1
.
u
2
(f
N,o
(x)) ∼
A
N
2νB
2
N
A
1
2ν+1
+ B
2
A
2νB
2
N
2ν
2ν+1
=
=
A
N
2ν
2ν+1
B
2
2ν+1
"
(2ν)
1
2ν+1
+
1
2ν
2ν
2ν+1
#
= O
N
−
2ν
2ν+1
. ♠
ν → ∞
f
N,o
(x) ∈ V
N
−
2ν
2ν+1
→ V
N
−1
.
f
N,o
(x)
F
N
(x)
f
F
N
(x) s
N
(x) fs
N
(x)
h
N,o
(X
1
, . . . , X
N
) X
1
, . . . , X
N
h
N,o
f
N,o
(x) = f
N
(x)|
h
N
=h
N,o
K(u)
f
N,o
(x)
h
N,o
f(x)
ν f
(ν)
(x)
h
N,o
A B
J
N
= E
Z
R
1
(f
N
(x) − f (x))
2
dt
K(u)
K(u) ∈ A
2
f(x)
Z
R
1
(f
00
(x))
2
dx <
∞ h
N
∈ H
2
J
N
∼ (Nh
N
)
−1
Z
R
1
K
2
(u)du +
1
4
h
4
N
Z
R
1
u
2
K(u)du
2
Z
R
1
(f
00
(x))
2
dt.
h
N,o
=
C
N
Z
R
1
(f
00
(x))
2
dx
1/5
,
C =
Z
R
1
K
2
(u)du
Z
R
1
u
2
K(u)du
2
K(u)
Z
R
1
(f
00
(x))
2
dx
3/
8
√
πσ
5
Z
R
1
(f
00
(x))
2
dx
Z
R
1
(f
00
(x))
2
dx
f(x)
f(x)
h
N
h
N,o
h
N,o
= h
N,o
(X
1
, . . . , X
N
) = argmax
h>0
"
N
Y
i=1
f
N−1
(X
i
)
#
,
f
N−1
(X
i
) =
1
(N − 1)h
N−1
X
j=1,j6=i
K
X
i
− X
j
h
,
h
N,o
f
N,o
(x) = f
N
(x)|
h
N
=h
N,o
◦
e
◦
,
◦
e
x
,
◦
e
x:ne
,
◦
e
x
1
:x
2
D X, D T (x)
◦
e
◦
.
◦
e
◦
= E X =
Z
∞
0
x dF (x) =
Z
∞
0
s(x) dx,
b
◦
e
◦
=
Z
∞
0
x dF
N
(x)
b
b
◦
e
◦
=
Z
∞
0
s
N
(x) dx.
b
◦
e
◦
=
Z
∞
0
x
1
N
N
X
i=1
δ(x − X
i
) dx =
1
N
N
X
i=1
X
i
= x,
b
b
◦
e
◦
=
Z
∞
0
1
N
N
X
i=1
I(X
i
> x) =
1
N
N
X
i=1
Z
X
i
dx = x.
D X = E(X−
◦
e
x
)
2
=
Z
∞
0
(x−
◦
e
◦
)
2
dF (x),
D X = 2
Z
∞
0
x s(x) dx − (
◦
e
◦
)
2
,
d
DX =
1
N
N
X
i=1
(X
i
− x)
2
,
d
d
DX = 2
Z
∞
0
x
1
N
N
X
i=1
I(X
i
> x) dx − (x)
2
=
=
1
N
N
X
i=1
X
2
i
− (x)
2
=
1
N
N
X
i=1
(X
i
− x)
2
.
◦
e
x
=
1
s(x)
Z
∞
0
s(x + t) dt
b
◦
e
x
=
1
s
N
(x)
Z
∞
0
1
N
N
X
i=1
I(X
i
> x + t) dt =
=
1
s
N
(x) N
N
X
i=1
Z
(X
i
−x)I(X
i
−x>0)
0
dt =
=
N
X
i=1
(X
i
− x)I(X
i
− x > 0)
,
N
X
i=1
I(X
i
− x > 0).
[X
(N)
, ∞)
b
◦
e
x
˜
◦
e
x
=
N
X
i=1
(X
i
− x)I(X
i
− x > 0)
,
N
X
i=1
S
x − X
i
a
N
.
◦
e
x:ne
=
1
s(x)
Z
n
0
s(x+t) dt, D T (x) D{min(T (x), n)}
d
◦
e
x:ne
=
1
s
N
(x)
Z
n
0
1
N
N
X
i=1
I(X
i
> x + t) dt =
=
1
s
N
(x) N
N
X
i=1
Z
min(n,X
i
−x)I(X
i
−x>0)
0
dt =
=
N
X
i=1
min(n, X
i
− x)I(X
i
− x > 0)
,
N
X
i=1
I(X
i
− x > 0),
d
d
D T (x) =
N
X
i=1
(X
i
− x)
2
I(X
i
− x > 0)
N
X
i=1
S
x − X
i
a
N
− (
ˆ
◦
e
x
)
2
,
d
d
D min(T (x), n) =
N
X
i=1
(min(n, X
i
− x))
2
I(X
i
− x > 0)
N
X
i=1
S
x − X
i
a
N
− (
ˆ
◦
e
x
)
2
.
◦
e
x
1
:x
2
d
◦
e
x
1
:x
2
=
1
s
N
(x
1
, x
2
)
Z
∞
0
ˆ
P{min(X − x
1
, Y − x
2
) > t}dt =
=
1
s
N
(x
1
, x
2
)
Z
∞
0
1
N
N
X
i=1
I{min(X
i
− x
1
, Y
i
− x
2
) > t}dt =
=
1
s
N
(x
1
, x
2
) N
N
X
i=1
Z
min(X
i
−x
1
,Y
i
−x
2
)
0
dt =
=
1
s
N
(x
1
, x
2
) N
N
X
i=1
min(X
i
− x
1
, Y
i
− x
2
)I{min(X
i
− x
1
, Y
i
− x
2
) > 0},
s
N
(x
1
, x
2
) =
1
N
N
X
i=1
I{X
i
− x
1
> 0}I{Y
i
− x
2
> 0).
X Y
b
N
= (b
1N
, . . . , b
sN
)
C(b
n
) =
"
cov(b
1N
, b
1N
) . . . cov(b
1N
, b
sN
)
. . . . . . . . .
cov(b
sN
, b
1N
) . . . cov(b
sN
, b
sN
)
#
.
f
N
, fs
N
f(x) ∈
N
0,1
(R) , sup
x∈R
1+
f(x) < ∞, K(u) ∈ A,
h
N
∈ H
2
,
lim
N→∞
Nh
N
C (f
N
(x), fs
N
(x)) = f(x)
"
R
R
1
K
2
(u)du 0
0 0
#
= ¯σ
λ
.
cov ( fs
N
(x), fs
N
(x)) = D fs
N
(x) =
1
N
s(x) (1 − s(x)) + o
1
N
,
cov (f
N
(x), f
N
(x)) = Df
N
(x) =
f(x)
Nh
N
∞
Z
−∞
K
2
(u)du + o
1
Nh
N
,
lim
N→∞
nh
N
cov (f
N
(x), f
N
(x)) = f(x)
∞
Z
−∞
K
2
(u)du,
lim
N→∞
Nh
N
cov ( fs
N
(x), fs
N
(x)) = lim
N→∞
h
N
s(x) (1 − s(x)) = 0.
cov (f
N
(x), fs
N
(x)) =
1
Nh
N
∞
Z
0
K
x − y
h
N
S
x − y
a
N
f(y)dy−
−
∞
Z
0
K
x − y
h
N
f(y)dy
∞
Z
0
S
x − y
a
N
f(y)dy
.
S(·) sup
x∈R
1+
S(x) ≤ C < ∞
I
1
=
1
Nh
N
∞
Z
0
K
x − y
h
N
S
x − y
a
N
f(y)dy ≤
≤ C
1
Nh
N
∞
Z
0
K
x − y
h
N
f(y)dy.
N → ∞ I
1
= O
N
−1
.
N → ∞
1
Nh
N
∞
Z
0
K
x − y
h
N
f(y)dy
∞
Z
0
S
x − y
a
N
f(y)dy = O(N
−1
),
cov (f
N
(x), fs
N
(x)) = cov ( fs
N
(x), f
N
(x)) = O
N
−1
.
fs
N
(x)
f
N
(x)
λ
N
(x) =
f
N
(x)
fs
N
(x)
.
λ
N
(x)
λ
N
(x)
p
Nh
N
(f
N
(x) − f (x), fs
N
(x) − s(x)) .
{ξ
j,N
, η
j,N
}
N
j=1
, N = 1, 2, . . .
(ξ
j,N
, η
j,N
) N σ
N
= NE (ξ
1,N
, η
1,N
)
T
(ξ
1,N
, η
1,N
) ,
T
k(ξ, η)k =
p
ξ
2
+ η
2
.
f(z) ∈ N
ν,1
(R) ;
sup
x∈R
1
f
(m)
(x)
< ∞, m = 0, ν; K(u) ∈ A
ν
;
1 − K(x) = o
x
−ν
x → ∞;
h
N
∈ H
2
, lim
N→∞
p
Nh
N
(a
N
+ h
ν
N
) = 0.
p
Nh
N
(f
N
(x) − f (x), fs
N
(x) − s(x))
N
2
{(0, 0), ¯σ
λ
}, ¯σ
λ
p
Nh
N
(f
N
(x) − f (x), fs
N
(x) − s(x)) =
=
p
Nh
N
(f
N
(x) − Ef
N
(x), fs
N
(x) − E fs
N
(x)) +
+
p
Nh
N
(b (f
N
(x)) , b (fs
N
(x))) .
N → ∞
p
Nh
N
(b (f
N
(x)) , b (fs
N
(x))) =
p
Nh
N
(O (h
ν
N
) , O (a
N
)) =
=
O
q
Nh
2ν+1
N
, O
p
Nh
N
a
N
−→ (0, 0).