Краткие сведения по стохастической финансовой математике. Барабанов А.Е. - 17 стр.

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A =
1 a
11
··· a
1q
···
1 a
N1
··· a
Nq
, a
0
=
a
0
(A
1
)
a
0
(A
N
)
, ¯ρ =
ρ(A
1
)
ρ(A
N
)
, ζ =
ζ(A
1
)
ζ(A
N
)
, f =
f
1
f
q
.
¯ρ = a
0
+ Af + ζ d
∗
A = 0 d
∗
a
0
= |d|
2
N À q A
d
∗
A = 0
d = (I −A(A
∗
A)
−1
A
∗
)γ
γ q × q γ = a
0
B = A(A
∗
A)
−1
A
∗
B = B
∗
= B
2
d
∗
a
0
= a
∗
0
(I −B)a
0
= a
∗
0
(I − B)
2
a
0
= |d|
2
,
Ba
0
= Aλ
λ = (A
∗
A)
−1
A
∗
a
0
= (λ
0
, λ
1
, . . . , λ
q
)
∗
.
d = a
0
− Ba
0
= a
0
− λ
0
1
N
−
q
X
k=1
λ
k
a
k
,
a
k
= (a
k
(A
1
), a
k
(A
2
), . . . , a
k
(A
N
))
∗
1
N
= (1, 2, . . . , 1)
∗
d
|d|
2
=
N
X
i=1
Ã
a
0
(A
i
) − λ
0
−
q
X
k=1
λ
k
a
k
(A
i
)
!
2
.
|d| → ∞ N → ∞
θd θ = |d|
−4/3
ρ(θd) = θρ(d)
Eρ(θd) = θd
∗
a
0
= θ|d|
2
= |d|
2/3
→ ∞, (N → ∞)
Dρ(θd) = θ
2
N
X
i,j=1
d
i
d
j
σ
2
ij
.
ζ(A
i
) σ
ij
= δ
ij
Dρ(θd) = θ
2
|d|
2
= |d|
−
2
/
3
→ 0
N → ∞

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