Кошкин Г.М. Основы страховой математики
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ξ
j,N
=
1
√
Nh
N
K
x − X
j
h
N
− h
N
Ef
N
(x)
,
η
j,N
=
√
h
N
√
N
S
x − X
j
a
N
− ES
x − X
j
a
N
.
f
N
(x) − Ef
N
(x) =
1
√
Nh
N
N
X
j=1
ξ
j,N
, fs
N
(x) − E fs
N
(x) =
1
√
Nh
N
N
X
j=1
η
j,N
.
E (ξ
j,N
, η
j,N
) = (0, 0)
E kξ
j,N
, η
j,N
k
2
=
1
Nh
N
DK
x − X
j
h
N
+
h
N
N
DS
x − X
j
a
N
< ∞.
fs
N
(x) f
N
(x)
E kξ
j,N
, η
j,N
k
2
≤
1
N
C
1
∞
Z
−∞
K
2
(u)du +
h
N
N
C
2
< ∞,
0 ≤ C
1
, C
2
< ∞
lim
N→∞
NE (ξ
1,N
, η
1,N
)
T
(ξ
1,N
, η
1,N
) = ¯σ
λ
.
τ > 0
β
N
= NE
kξ
1,N
, η
1,N
k
2
; kξ
1,N
, η
1,N
k > τ
<
N
τ
E kξ
1,N
, η
1,N
k
3
=
=
N
τ
E
ξ
2
1,N
+ η
2
1,N
3/2
≤
CN
τ
h
E |ξ
1,N
|
3
+ E |η
1,N
|
3
i
=
=
CN
(Nh
N
)
3/2
"
E
K
x − X
1
h
N
− EK
x − X
1
h
N
3
+
+h
3
N
E
S
x − X
1
h
N
− ES
x − X
1
h
N
3
#
≤
≤
C
(Nh
N
)
1/2
"
1
h
N
(
E
K
x − X
1
h
N
3
+ h
3
N
E
3
f
N
(x)
)
+
+h
2
N
(
E
S
x − X
1
h
N
3
+ E
3
S
x − X
1
h
N
)#
.
K(·), S(·) R
1
β
N
= O
1
√
Nh
N
−→ 0
N → ∞
x
N
= (x
N1
, . . . , x
Ns
) x
Nj
=
x
Nj
(x) = x
Nj
(t; X
1
, . . . , X
N
), j = 1, s
H(z) : R
s
→ R
1
H
(1)
(z) = ∇H(z) = (H
1
(z), . . . , H
s
(z)) , H
j
(z) = ∂H(z)/∂z
j
d
N
N
φ = φ(x) = (φ
1
(x), . . . , φ
s
(x))
x
N
λ
N
λ(x), s(x) 6= 0,
p
Nh
N
(λ
N
(x) − λ(x)) =⇒ N
1
0,
λ(x)
R
R
1
K
2
(u)du
s(x)
.
d
N
=
√
Nh
N
, λ
N
= H(x
N
) = H (x
N1
, x
N2
) =
x
N1
x
N2
=
f
N
fs
N
,
λ = H(φ) = H (φ
1
, φ
2
) =
φ
1
φ
2
=
f
s
,
H
(1)
(φ) =
1
φ
2
, −
φ
1
φ
2
2
=
1
s
, −
f
s
2
, µ
1
= µ
2
= 0, σ = ¯σ
λ
¯σ
λ
H
(1)
j
(φ) µ
j
σ
jp
λ
N,2
(x) = f
N
(x)/ (1 − F
N,2
(x)) , f
N
(x) =
f
N
(x, h
N
) f
N
(x)
F
N,2
(x) =
x
Z
0
f
N
(t, a
N
)dt
λ
N
(x)
{T
N
, N = 1, 2, . . .}
√
q
N
[T
N
− θ] =⇒ N
1
0, σ
2
(θ)
,
q
N
↑ ∞
√
q
N
[g (T
N
) − g(θ)] =⇒ N
1
n
0, [g
0
(θ)σ(θ)]
2
o
,
g g
0
(θ) 6= 0
g
g
0
(θ)σ(θ) = c,
c θ
g(T
N
) θ
g g =
Z
cdθ
σ(θ)
.
g =
Z
cdθ
σ(θ)
=
Z
p
s(x)dλ(x)
p
λ(x)
q
R
R
1
K
2
(u)du
=
Z
cdλ(x)
p
λ(x)
.
g(x) =
√
x.
2
√
Nh
N
p
s(x)
s
∞
R
−∞
K
2
(u)du
h
p
λ
N
(x) −
p
λ(x)
i
=⇒ N
1
{0, 1}.
q
N
= Nh
N
, T
N
= λ
N
, θ = λ g(λ) =
√
λ
g
0
(λ
N
)σ(λ
N
) =
s
∞
R
−∞
K
2
(u)du
2
p
s(x)
,
|b (fs
N
(x))| = O(a
N
)
lim
N→∞
a
N
p
Nh
N
= 0 a
N
= o
n
−1/2
p
s(x)
p
fs
N
(x)
2
√
Nh
N
p
fs
N
(x)
s
∞
R
−∞
K
2
(u)du
h
p
λ
N
(x) −
p
λ(x)
i
=⇒ N
1
{0, 1}.
(1 − α) λ(x)
2
√
Nh
N
p
fs
N
(x)
s
∞
R
−∞
K
2
(u)du
p
λ
N
(x) −
p
λ(x)
< U
1−α/2
,
U
1−α/2
1 −
α
2
p
λ
N
(x) −
s
∞
R
−∞
K
2
(u)du
2
√
Nh
N
p
fs
N
(x)
U
1−α/2
2
< λ(x) <
<
p
λ
N
(x) +
s
∞
R
−∞
K
2
(u)du
2
√
Nh
N
p
fs
N
(x)
U
1−α/2
2
.
λ
N
(x)
λ
N
(x)
λ
N
(x)
Ψ (λ
N
, δ) =
e
Ψ (x, δ) =
λ
N
(x)
[1 + δ |λ
N
(x)|
τ
]
ρ
,
τ > 0, ρ > 0, ρτ ≥ 1, δ > 0.
M
ν
ky
N
k = E ky
N
− φk
ν
ν
y
N
(x) φ(x) x
(τ, k, m) k, m
T (m) = {(τ, k) : τ ≥ τ(m) = 2k/(m − k − 1) > 0,
m ≥ m
0
= [3, k = 1; 2k, k ≥ 2]}
Ψ
λ
(r)
N
, δ
r
e
Ψ(x, δ)
fs
N
(x), f
N
(x)
a
N
= o
N
−1/2
,
N → ∞
M
4
kfs
N
(x)k = E (fs
N
(x) − s(x))
4
= O
N
−2
.
E (fs
N
(x) − s(x))
4
= E [(fs
N
(x) − E fs
N
(x)) + b (fs
N
(x))]
4
.
p = 4 m = 2
m
X
i=1
|a
i
|
!
p
≤ m
p−1
m
X
i=1
|a
i
|
p
, p > 1,
E (fs
N
(x) − s(x))
4
≤ 2
3
h
E (fs
N
(x) − E fs
N
(x))
4
+ b
4
(fs
N
(x))
i
4
.
b (fs
N
(x)) = O (a
N
) b
4
(fs
N
(x)) = o
N
−2
E
f
F
N
(x) − E
f
F
N
(x)
4
= O
N
−2
,
E (fs
N
(x) − E fs
N
(x))
4
= O
N
−2
,
f
N
(x) f(x)
f(x) ∈ N
2,1
(R); sup
x∈R
1
f
(m)
(x)
< ∞, m = 0, 2;
K(u) ∈ A
2
; x → ∞ : 1 − K(x) = o
x
−2
;
lim
N→∞
h
N
+
1
Nh
N
= 0;
N → ∞
M
4
kf
N
(x)k = E (f
N
(x) − f (x))
4
= O
1
Nh
N
+ h
4
N
2
!
.
E (f
N
(x) − f (x))
4
=
O
(Nh
N
)
−2
, ν = 2 :
b (f
N
(x)) = O
h
2
N
Ψ (λ
N
, δ)
λ(x) 6=
0, s(x) 6= 0; δ =
1
Nh
N
+ h
4
N
.
(τ, 2) ∈ T (m) N → ∞
E [Ψ (λ
N
, δ) − λ(x)]
2
−
λ(x)
s(x)Nh
N
∞
Z
−∞
(K(u))
2
du−
−
T
2
2
h
4
N
4
f
(2)
(x)
2
s
2
(x)
!
= O
1
Nh
N
+ h
4
N
3/2
!
.
s = 2, z = (z
1
, z
2
) , H(z) = z
1
/z
2
, y
N
= (y
1N
, y
2N
) = (f
N
, fs
N
) , φ =
(φ
1
, φ
2
) = (f(x), s(x)) , d
N
= O
Nh
N
+ h
−4
N
, k = 2 m = m
0
=
4 M
4
kf
N
, fs
N
k = O
d
−2
N
M
4
kf
N
, fs
N
k ≤
2 [M
4
kf
N
k + M
4
kfs
N
k]
s(x) 6= 0 z
2
6= 0
z
1
/z
2
∈ N
2,2
(f(x), s(x)) ,
Ψ (λ
N
, δ) = Ψ (λ
N
, δ)
τ ≥ τ
0
= 4.
P
N
(A)
q
x
t
q
x
,
t
p
x
,p
x
,
t|u
q
x
,
t|
q
x
.
F(u)
F(u) =
0, u ≤ −
1
2
,
1
2
+ u, −
1
2
< u ≤
1
2
,
1, u >
1
2
;
F(u) =
1
2
+
1
π
u, −∞ < u < ∞;
F(u) =
1
1 + e
−u
, −∞ < u < ∞;
F(u) = 1 −
2
π
e
−u
, −∞ < u < ∞;
F(u) = e
−e
−u
, −∞ < u < ∞;
x
