(Ω, F, P)
(F
n
) = (F
n
)
n≥0
σ
F
0
⊆ F
1
⊆ ··· ⊆ F
n
⊆ ··· ⊆ F.
(X
n
) (F
n
) n
X
n
F
n
d
(F
n
)
(X
n
) F
n
X
n
N H
T ω
1
ω
2
. . . ω
N
ω
n
= H T ω = ω
1
ω
2
. . . ω
n
H
T n A
ω
ων ν = ω
n+1
ω
n+2
. . . ω
N
H T
N − n N =
4 A
T H
= {T HHH, T HHT, T HT H, T HT T }
A
ω
σ F
0
F
0
= {∅, Ω}
H T
σ F
1
{∅, Ω, A
H
, A
T
}
ω
1
ω
2
= HH HT T H T T
σ F
2
∅ Ω
A
ω
1
ω
2
(F
n
)
N
n=0
F
n
n ≥ 1 σ
A
ω
N
F
N
Ω
X
n
n
(X
n
) (F
n
)
{X
4
= 2} A
HHT T
A
HT HT
A
HT T H
A
T T HH
A
T HT H
A
T HHT
F
4
A
HT HT
X
4
= 2 X
1
= X
2
= 1
X
3
= 2
(X
n
)
n≥0
n
F
X
n
= σ(X
j
: j ≤ n)
σ {X
j
∈ J} j ≤ n J
(F
X
n
)
(X
n
) (X
n
)
(X
n
)
X = (X
n
)
(F
n
) X
n
F
n−1
n ≥ 1
d
X
n
F
n
X
n
F
n−1
n −1
n
(F
n
)
n X
n
F
n−1
(X
n
)
Y
n
n
(Y
n
)
S S =
(S
n
)
N
n=0
(Ω, F, P) S
0
F = F
S
N
(F
S
n
) S
S
0
F
S
0
σ {∅, Ω} (F
S
n
)
[0, N]
B i
S B
S
B = (B
n
)
N
n=0
B B
n
= (1 + i)
n
(B, S)
(φ, θ) = ((φ
n
, θ
n
))
N
n=1
(Ω, F, P)
φ
n
θ
n
B
S n V
n
n
V
0
= φ
1
+ θ
1
S
0
, V
n
= φ
n
B
n
+ θ
n
S
n
, n = 1, 2, . . . , N.
V = (V
n
)
N
n=0
V
0
φ
1
B θ
1
S
S
0
n ≥ 1
S
n
φ
n
B
n−1
+ θ
n
S
n−1
.
S
n
φ
n
B
n
+ θ
n
S
n
.
S B
S
0
S
1
. . . S
n
φ
n+1
θ
n+1
F
S
n
(n, n + 1)
φ
n+1
B
n
+ θ
n+1
S
n
.
n + 1
S B
∆x
n
:= x
n+1
− x
n
,
(φ, θ)
B
n
∆φ
n
+ S
n
∆θ
n
= 0, n = 1, 2, . . . , N − 1.
X = (X
n
)
N
n=0
˜
X
˜
X
n
= (1 + i)
−n
X
n
, n = 0, 1, . . . , N.
˜
X X
B
(φ, θ) V
(φ, θ)
∆V
n
= φ
n+1
∆B
n
+ θ
n+1
∆S
n
n = 0, 1, . . . N − 1
V
V
n+1
= θ
n+1
[S
n+1
− (1 + i)S
n
] + (1 + i)V
n
= θ
n+1
S
n+1
+ (1 + i)[V
n
− θ
n+1
S
n
], n = 0, 1, . . . , N − 1;
∆
˜
V
n
= θ
n+1
∆
˜
S
n
n = 0, 1, . . . , N − 1
φ
n
= V
0
−
P
n−1
j=0
˜
S
j
∆θ
j
n = 1, 2, . . . , N θ
0
:= 0
n = 0, 1, . . . , N − 1
Y
n
= B
n
∆φ
n
+ S
n
∆θ
n
= φ
n+1
B
n
+ θ
n+1
S
n
− V
n
.
V
n
= φ
n+1
B
n
+ θ
n+1
S
n
− Y
n
∆V
n
= φ
n+1
B
n+1
+ θ
n+1
S
n+1
− (φ
n+1
B
n
+ θ
n+1
S
n
− Y
n
)
= φ
n+1
∆B
n
+ θ
n+1
∆S
n
+ Y
n
.
Y
n
= 0 n
φ
n+1
B
n
= Y
n
+ V
n
− θ
n+1
S
n
B
n+1
= (1 + i)B
n
V
n+1
= φ
n+1
B
n+1
+ θ
n+1
S
n+1
= (1 + i) [Y
n
+ V
n
− θ
n+1
S
n
] + θ
n+1
S
n+1
= θ
n+1
[S
n+1
− (1 + i)S
n
] + (1 + i)(V
n
+ Y
n
).
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