(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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