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†
t
(t) (t)
(t)
{N
t
}
t≥0
t
U(t) = u + ct −
N
t
i=1
Y
i
,
• u
• c
• N
t
t
• {Y
n
}
n∈
P (U(t) < 0 t).
i.i.d.
i.i.d.
G
U
k
= U
k−1
(1 + I
k
) − Z
k
, k = 1, 2, ...
U
0
= u ≥ 0 {Z
k
, k = 1, 2, ...}
(i.i.d.) G(z) = P r(Z
1
≤ z) {I
k
, k = 0, 1, ...}
{Z
k
, k = 1, 2, ...}
Z
k
k
k − 1 k
k
Z
k
= Y
k
− X
k
{Y
k
, k = 1, 2, ...}
(i.i.d.) {X
k
, k = 1, 2, ...}
i.i.d. I
k
k U
K
u k G
R
0
> 0 E[e
R
0
Z
1
] =
1
ψ(u, i
s
) ≤ β
0
E
e
−R
0
u(1+I
1
)
| I
0
= i
s
, u ≥ 0,
β
−1
0
= inf
t≥0
+∞
t
e
R
0
z
dG(z)
e
R
0
t
G(t)
ψ(u, i
s
) i
s
E(Z
1
) < 0 R
0
> 0
E[e
R
0
Z
1
] = 1 ρ
s
> 0
E
e
ρ
s
Z
1
(1+I
1
)
−1
| I
0
= i
s
= 1, s = 0, 1, . . . , N
R
1
= min
0≤ s ≤N
{ρ
s
} ≥ R
0
s = 0, 1, . . . , N
E
e
R
1
Z
1
(1+I
1
)
−1
| I
0
= i
s
≤ 1
ψ(u, i
s
) ≤ e
−R
1
u
, u ≥ 0.
Y
1
α = 1 E(Y
1
) = 1 var(Y
1
) = 1
Y
1
1
2
E(Y
1
) = 1
var(Y
1
) = 2 1, 1
10% Z
k
= Y
k
− 1, 1, k = 1, 2, ...
i
0
= 6%, i
1
= 8% i
2
= 10% P = {P
st
}
P =
0.2 0.8 0
0.15 0.7 0.15
0 0.8 0.2
.
(Ω, F, P ) X : Ω →
(Ω, F, P )
( , B, µ)
∀B ∈ B, µ(B) = P
X
−1
(B)
= P (X ∈ B)
µ
X F
X F
F (x) = µ((−∞, x]) = P (X ≤ x)
X f ( , B)
f(X)
X (Ω, F, P )
( , B, µ)
f
Ω
f(X(w))P (dw) =
f(x)µ(dx) =
f(x)dµ(x)
µ(dx) = P
X
(dx) =
dP
X
(x) = dF
X
(x)
(X, Y ) (Ω, F, P )
(
2
, B
2
, µ
2
) f
Ω
f(X(w), Y (w))P (dw) =
2
f(x, y)µ
2
(dx, dy).
µ
2
(dx, dy) =
P
X,Y
(dx, dy) = dP
X,Y
(x, y) = dF
X,Y
(x, y)
µ
X
F
X
X
E(X) =
xµ
X
(dx) =
xP
X
(dx) =
∞
−∞
xdF
X
(x)
E(f(X)) =
f(x)µ
X
(dx) =
f(x)P
X
(dx) =
∞
−∞
f(x)dF
X
(x)
(Ω, F, P ) A ∈ F
P (A) > 0 P
A
(E) = P (E | A) F
P
A
(E) =
P (E ∩ A)
P (A)
P
A
(•) : F → [0, 1]
A ∈ F
Y
E
A
(Y ) =
Ω
Y (w)P
A
(dw) =
1
P (A)
A
Y (w)P (dw)
X Y y P (Y = y) > 0
P (X = x | Y = y) =
P (X = x, Y = y)
P (Y = y)
X Y = y
X Y = y
F
X|Y
(x|y) = P (X ≤ x | Y = y) =
z≤x
P (X = z | Y = y) ∀x ∈
X Y = y
E(X|Y = y) =
xdF
X|Y
(x|y) =
x
xP (X = x|Y = y)
X Y f(x, y) y
f
X|Y
(x|y) =
f(x,y)
f
Y
(y)
, y f
Y
(y) > 0
0, y f
Y
(y) = 0
X Y = y
X Y = y
F
X|Y
(x|y) =
x
−∞
f
X|Y
(z|y)dz ∀x ∈
X Y = y
E(X|Y = y) =
xdF
X|Y
(x|y) =
xf
X|Y
(x|y)dx
Y
E(X|Y = y) Y E(X|Y = y) = ϕ(y)
E(X|Y = y) E(X|Y ) = ϕ(Y )
X Y
Y (Ω, F, P ) B( ) =
B σ σ Y σ(Y )
σ(Y ) = Y
−1
(B) = {A ∈ F; A = [Y ∈ B], B ∈ B}
σ(Y ) ⊂ F σ E(X|Y ) = ϕ(Y ) ϕ
E(X|Y ) σ
Y σ(Y )
X (Ω, F, P ) E|X| < ∞ G
σ G ⊂ F G σ
X G E(X|G)
E(X|G) G
∀A ∈ G
A
E(X|G)dP =
A
XdP
X
Y (Ω, F, P ) E|X| < ∞ G ⊂ F
X G E(X|G) = X
σ(X) ⊂ σ(Y ) E(X|σ(Y )) = X
E(E(X|G)) = E(X)
G
1
G
2
σ G
1
⊂ G
2
⊂ F
E(E(X|G
1
)|G
2
) = E(X|G
1
) = E(E(X|G
2
)|G
1
)
Z G E|XZ| < ∞
E(XZ|G) = ZE(X|G)
X Y E(X|Y ) = E(X)
Y
1
, Y
2
, . . . , Y
n
(Ω, F, P ) B σ
σ Y
1
, Y
2
, . . . , Y
n
σ(Y
1
, Y
2
, . . . , Y
n
)
σ(Y
1
, Y
2
, . . . , Y
n
) = {A ∈ F; A = [Y
1
∈ B
1
, . . . , Y
n
∈ B
n
], onde B
i
∈ B, i = 1, 2, . . . , n }
X (Ω, F, P ) E|X| < ∞
E(X|σ(Y
1
, Y
2
, . . . , Y
n
)) = E(X|Y
1
, Y
2
, . . . , Y
n
) E(X|Y
1
, Y
2
, . . . , Y
n
) = ϕ(Y
1
, Y
2
, . . . , Y
n
)
ϕ B
n
ϕ
−1
(B) ⊂ B
n
ϕ(y
1
, y
2
, . . . , y
n
) = E(X|Y
1
= y
1
, . . . , Y
n
= y
n
) = E(X|Y
1
, Y
2
, . . . , Y
n
)I
[Y
1
=y
1
,...,Y
n
=y
n
]
I
A
A
E(X|Y
1
= y
1
, . . . , Y
n
= y
n
) =
x∈D
xP (X = x|Y
1
= y
1
, . . . , Y
n
= y
n
) D
X
E(X|Y
1
= y
1
, . . . , Y
n
= y
n
) =
xf
X|Y
1
,...,Y
n
(x|y
1
, . . . , y
n
)dx
f
X|Y
1
,...,Y
n
(x|y
1
, . . . , y
n
) =
f
X,Y
1
,...,Y
n
(x, y
1
, . . . , y
n
)
f
Y
1
,...,Y
n
(y
1
, . . . , y
n
)
E(X) = E(E(X|Y
1
, . . . , Y
n
)) =
n
E(X|Y
1
= y
1
, . . . , Y
n
= y
n
)dF
Y
1
,...,Y
n
(y
1
, . . . , y
n
)
(Ω, F, P ) A ∈ F
Y
E(I
A
) =
Ω
I
A
dP =
Ω∩A
dP =
A
dP = P (A)
G ⊂ F
E(I
A
|G) = P (A|G) ∀A ∈ F
P (A) = E(I
A
) = E(E(I
A
|Y )) =
E(I
A
|Y = y)dF
Y
(y) =
P (A|Y = y)dF
Y
(y)
P (A) =
n
P (A|Y
1
= y
1
, . . . , Y
n
= y
n
)dF
Y
1
,...,Y
n
(y
1
, . . . , y
n
)
P (A|Y
1
= y
1
, . . . , Y
n
= y
n
) = E(I
A
|Y
1
= y
1
, . . . , Y
n
= y
n
)
{X(t), t ∈ T }
(Ω, F, P ) t ∈ T
X(t) = X
t
S
S
t X
t
t T
X
t
(w) = X(t, w) t
w t
{X
k
} , k = 1, 2, . . . S =
{1, 2, 3, . . .} n
i
1
, i
2
, . . . , i
n
S
P (X
n
= i
n
| X
n−1
= i
n−1
, X
n−2
= i
n−2
, . . . , X
1
= i
1
) = P (X
n
= i
n
| X
n−1
= i
n−1
)
{X
k
} , k = 0, 1, 2, . . .
i, j ∈ S
P (X
n
= j | X
n−1
= i) = P (X
n+k
= j | X
n+k−1
= i)
k = −(n − 1), −(n − 2), . . . , −1, 0, 1, 2, . . .
n − 1 i
n j n − 1
i P (X
n
= j | X
n−1
= i) = P
(n−1,n)
ij
i j
n − 1 n P
(n−1,n)
ij
= P
(n+k−1,n+k)
ij
k = −(n − 1), −(n − 2), . . . , −1, 0, 1, 2, . . . P
(n−1,n)
ij
n = 1, 2, . . .
P
(n−1,n)
ij
= P (X
n
= j | X
n−1
= i) = P
ij
{X
k
} S = {1, 2, 3, . . . , n}
n
2
{P
ij
} i = 1, 2, . . . , n j =
1, 2, . . . , n
P = {P
ij
; i, j = 1, 2, . . . , n} P
ij
i
j
P =
P
11
P
12
. . . P
1n
P
21
P
22
. . . P
2n
P
n1
P
n2
. . . P
nn
{Z
n
, n ≥ 1} E(|Z
n
|) <
∞ n
E(Z
n+1
| Z
1
, Z
2
, . . . , Z
n
) = Z
n
Z
n
E
E(Z
n+1
| Z
1
, Z
2
, . . . , Z
n
)
= E(Z
n
)
E(Z
n+1
) = E(Z
n
) E(Z
n
) = E(Z
1
) n
{Z
n
, n ≥ 0}
(Ω, F, P ) {A
n
}
n≥0
σ F Z
n
A
n
{Z
n
, n ≥ 0}
E(Z
n+1
| A
n
) ≥ Z
n
E(Z
n+1
| A
n
) ≤ Z
n
∀n ≥ 0
A
n
= σ(Z
0
, Z
1
, . . . , Z
n
) σ Z
0
, Z
1
, . . . , Z
n
{Z
n
}
σ {A
n
}
n≥0
T
{0, 1, 2, . . .} ∪ {+∞} {T = n} ∈ A
n
∀n ≥ 0 P (T < ∞) = 1
T N ∈ P (T ≤ N) = 1 T
A
T
σ
A A ∩ {T ≤ n} ∈ A
n
∀n ≥ 0
{X
n
, A
n
}
n≥0
T
A
n
X
+
T
X
T
E(X
+
T
) < ∞ E(X
T
| A
0
) ≤ X
0
E(X
T
) ≤ E(X
0
)
{X
n
, A
n
} T
{Y
n
, A
n
} Y
n
= X
T ∧n
T ∧ n = min {T, n}
{X
k
} {Y
k
}
(i .i .d ) X
k
k − 1 k Y
k
u
R
n
= u +
n
k=1
(X
k
− Y
k
), n = 1, 2, 3, . . .
R
0
= u R
n
n
n − 1 n
R
n
= R
n−1
+ X
n
− Y
n
,
R
0
= u.
τ(u)
τ(u) = min {n ≥ 0; R
n
< 0 }
ψ(u)
ψ(u) = P (τ (u) < ∞) = P (R
n
< 0, n < ∞) = P (
∞
k=1
[R
k
< 0])
R
n
γ > 0 E(e
−γ(X
1
−Y
1
)
) = 1
ψ(u) ≤ e
−γu
R
n
= (R
n−1
+ X
n
)(1 + I
n
) − Y
n
R
n
= R
n−1
(1 + I
n
) + X
n
− Y
n
X
k
(k − 1, k)
I
k
X
k
(k − 1, k)
I
n
I
n
= αI
n−1
+ W
n
α > 0 I
0
= i
0
{W
1
, W
2
, . . .} i .i .d .
{X
1
, X
2
, . . .} {Y
1
, Y
2
, . . .}
φ(u, i
0
) ϕ(u, i
0
)
A(u, i
0
) B(u, i
0
) φ(u, i
0
) ≤ A(u, i
0
) ϕ(u, i
0
) ≤ B(u, i
0
)
A(u, i
0
) ≤ B(u, i
0
) ≤ e
−γu
ψ(u) ≤ βe
−γu
β ≤ 1 Y
1
U
k
= U
k−1
(1 + I
k
) − Z
k
, k = 1, 2, ...
U
0
= u ≥ 0 {Z
k
, k = 1, 2, ...}
(i.i.d.)
G(z) = P (Z
1
≤ z) {I
k
, k = 0, 1, ...}
{Z
k
, k = 1, 2, ...} Z
k
k k−1 k
k Z
k
= Y
k
− X
k
{Y
k
, k = 1, 2, ...}
{X
k
, k = 1, 2, ...}
i.i.d. I
k
k U
k
u k G
U
k
= u
k
j=1
(1 + I
j
) −
k
j=1
(Z
j
k
t=j+1
(1 + I
t
)), k = 1, 2, ...
b
t=a
X
t
= 1
b
t=a
X
t
= 0 a > b
k = 1
U
1
= U
0
(1 + I
1
) − Z
1
= u(1 + I
1
) − Z
1
= u
1
j=1
(1 + I
j
) −
1
j=1
(Z
j
1
t=j+1
(1 + I
t
))
k = n
k = n + 1
U
n
= u
n
j=1
(1 + I
j
) −
n
j=1
(Z
j
n
t=j+1
(1 + I
t
))
U
n+1
= U
n
(1 + I
n+1
) − Z
n+1
U
n+1
=
u
n
j=1
(1 + I
j
) −
n
j=1
(Z
j
n
t=j+1
(1 + I
t
))
(1 + I
n+1
) − Z
n+1
U
n+1
= u
n+1
j=1
(1 + I
j
) − (1 + I
n+1
)
n
j=1
(Z
j
n
t=j+1
(1 + I
t
)) − Z
n+1
U
n+1
= u
n+1
j=1
(1 + I
j
) −
n
j=1
(Z
j
n+1
t=j+1
(1 + I
t
)) − Z
n+1
U
n+1
= u
n+1
j=1
(1 + I
j
) −
n+1
j=1
(Z
j
n+1
t=j+1
(1 + I
t
))
k = 1, 2, ...
i.i.d.
i.i.d.
G
{I
n
, n = 0, 1, ...}
n = 0, 1, ..., I
n
J = {i
0
, i
1
, ..., i
N
} n = 0, 1, ...
i
s
, i
t
, i
t
0
, ..., i
t
n−1
P (I
n+1
= i
t
|I
n
= i
s
, I
n−1
= i
t
n−1
, ..., I
0
= i
t
0
) = P (I
n+1
= i
t
|I
n
= i
s
) = P
st
≥ 0,
s, t = 0, 1, ...N
N
t=0
P
st
= 1 s = 0, 1, ..., N
{I
n
, n = 0, 1, ...}
{I
n
, n = 0, 1, ...}
u
I
0
= i
s
ψ
n
(u, i
s
) = P
n
k=1
(U
k
< 0)|I
0
= i
s
ψ(u, i
s
) = P
∞
k=1
(U
k
< 0)|I
0
= i
s
,
U
k
ψ
1
(u, i
s
) ≤ ψ
2
(u, i
s
) ≤ ψ
3
(u, i
s
) ≤ . . . lim
n→∞
ψ
n
(u, i
s
) = ψ(u, i
s
)
A
n
= {w; U
k
(w) < 0 1 ≤ k ≤ n}
A
n
=
n
k=1
{w; U
k
(w) < 0}.
A
n
⊂ A
n+1
∀n ∈
P (A
n
|I
0
= i
s
) ≤ P (A
n+1
|I
0
= i
s
) P
n
k=1
(U
k
< 0)|I
0
= i
s
≤ P
n+1
k=1
(U
k
< 0)|I
0
= i
s
ψ
n
(u, i
s
) ≤ ψ
n+1
(u, i
s
) ∀n ∈ A
n
↑
∞
n=1
A
n
∞
n=1
A
n
=
∞
n=1
n
k=1
{w; U
k
(w) < 0} =
∞
k=1
{w; U
k
(w) < 0} = A.
x ∈
∞
n=1
n
k=1
{w; U
k
(w) < 0} =⇒ ∃n
0
∈ x ∈
n
0
k=1
{w; U
k
(w) < 0} =⇒ ∃k
0
≤
n
0
x ∈ {w; U
k
0
(w) < 0} =⇒ x ∈
∞
k=1
{w; U
k
(w) < 0}
∞
n=1
A
n
⊂
A.
x ∈
∞
k=1
{w; U
k
(w) < 0} =⇒ ∃k
0
∈ x ∈ {w; U
k
0
(w) < 0} =⇒
x ∈
n
k=1
{w; U
k
(w) < 0} n ≥ k
0
=⇒ x ∈
∞
n=1
n
k=1
{w; U
k
(w) < 0}
A ⊂
∞
n=1
A
n
P (A
n
|I
0
= i
s
)
n→∞
−→ P (A|I
0
= i
s
) ⇐⇒ ψ
n
(u, i
s
)
n→∞
−→ ψ(u, i
s
)
I
n
= 0, n = 0, 1, . . .
U
k
= u −
k
t=1
Z
t
, k = 1, 2, . . .
ψ(u)
ψ(u) = P
∞
k=1
k
t=1
Z
t
> u
ψ(u) = P (U
k
< 0 k ∈ )
ψ(u) = P
∞
k=1
U
k
< 0
= P
∞
k=1
u −
k
t=1
Z
t
< 0
= P
∞
k=1
k
t=1
Z
t
> u
ψ(u)
E(Z
1
) < 0 R
0
> 0
E[e
R
0
Z
1
] = 1,
ψ(u) ≤ e
−R
0
u
, u ≥ 0.
I
n
≥ 0 n = 0, 1, . . .
ψ(u, i
s
) ≤ ψ(u), u ≥ 0.
U
k
= U
k−1
(1 + I
k
) − Z
k
U
0
= u
U
k
= u −
k
t=1
Z
k
U
k
≥ U
k
∀k ∈
k = 1 U
1
= U
0
(1 + I
1
) − Z
1
≥ U
0
− Z
1
=
u − Z
1
= U
1
k = n
U
n
≥ U
n
k = n + 1
U
n+1
= U
n
(1 + I
n+1
) − Z
n+1
≥ U
n
− Z
n+1
≥ U
n
− Z
n+1
= u −
n
t=1
Z
t
− Z
n+1
U
n+1
≥ u −
n+1
t=1
Z
t
= U
n+1
A =
∞
k=1
U
k
< 0
⊂
∞
k=1
U
k
< 0
= B P
· | I
0
= i
s
P
∞
k=1
U
k
< 0
| I
0
= i
s
≤ P
∞
k=1
U
k
< 0
| I
0
= i
s
U
k
I
0
U
k
P
∞
k=1
U
k
< 0
| I
0
= i
s
≤ P
∞
k=1
U
k
< 0
ψ(u, i
s
) ≤ ψ(u)
ψ(u, i
s
)
{I
n
, n = 0, 1, . . .} ψ(u, i
s
) ≤ ∆(u, i
s
) u ≥ 0 ∆(u, i
s
)
∆(u, i
s
) ≤ e
−R
0
u
, u ≥ 0.
ψ(u, i
s
)
B B(x) = 1 − B(x) I
A
n = 1, 2, . . . u ≥ 0
ψ
n+1
(u, i
s
) =
N
t=0
P
st
G(u(1 + i
t
)) +
u(1+i
t
)
−∞
ψ
n
(u(1 + i
t
) − z, i
t
)dG(z)
ψ
1
(u, i
s
) =
N
t=0
P
st
G(u(1 + i
t
))
ψ(u, i
s
) =
N
t=0
P
st
G(u(1 + i
t
)) +
u(1+i
t
)
−∞
ψ(u(1 + i
t
) − z, i
t
)dG(z)
Z
1
= z I
1
= i
t
U
1
= h − z
h = u(1 + i
t
) z > h
P
U
1
< 0 | Z
1
= z, I
0
= i
s
, I
1
= i
t
= E
I
[U
1
< 0]
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
I
u(1+I
1
)−Z
1
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
I
Z
1
> u(1+I
1
)
| Z
1
= z, I
0
= i
s
, I
1
= i
t
= E
1 | Z
1
= z, I
0
= i
s
, I
1
= i
t
= 1
A =
U
1
< 0
⊂
n+1
k=1
U
k
< 0
= B I
A
≤ I
B
P
n+1
k=1
U
k
< 0
|Z
1
= z, I
0
= i
s
, I
1
= i
t
= 1
0 ≤ z ≤ h
P
U
1
< 0|Z
1
= z, I
0
= i
s
, I
1
= i
t
= E
I
Z
1
> u(1+I
1
)
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
0 | Z
1
= z, I
0
= i
s
, I
1
= i
t
= 0 w ∈ Ω I
1
(w) = i
t
I
0
(w) = i
s
Z
1
(w) = z Z
1
(w) ≤ u(1 + I
1
(w)) z ≤ h
I
Z
1
> u(1+I
1
)
(w) = 0
P
n+1
k=1
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
I
n+1
k=1
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
I
U
1
< 0
n+1
k=2
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
I
U
1
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
+
+E
I
n+1
k=2
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
−
−E
I
U
1
< 0
n+1
k=2
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
I
n+1
k=2
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
I
n+1
k=2
u
k
j=1
(1 + I
j
) −
k
j=1
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
I
n+1
k=2
u(1 + I
1
)
k
j=2
(1 + I
j
) − Z
1
k
λ=2
(1 + I
λ
) −
k
j=2
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| Z
1
=
z, I
0
= i
s
, I
1
= i
t
=
= E
I
n+1
k=2
(u(1 + I
1
) − Z
1
)
k
j=2
(1 + I
j
) −
k
j=2
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| Z
1
= z, I
0
=
i
s
, I
1
= i
t
=
= E
I
n+1
k=2
(u(1 + i
t
) − z)
k
j=2
(1 + I
j
) −
k
j=2
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| Z
1
= z, I
0
=
i
s
, I
1
= i
t
=
= P
n+1
k=2
(h − z)
k
j=2
(1 + I
j
) −
k
j=2
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
{I
n
, n = 0, 1, . . .} {Z
n
, n = 1, 2, . . .} {Z
k
, k = 2, 3, . . .}
Z
1
P
n+1
k=1
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= P
n+1
k=2
(h − z)
k
j=2
(1 + I
j
) −
k
j=2
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| I
0
= i
s
, I
1
= i
t
{I
n
, n = 0, 1, ...}
P
n+1
k=1
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= P
n+1
k=2
(h − z)
k
j=2
(1 + I
j
) −
k
j=2
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| I
1
= i
t
=
= E
I
n+1
k=2
(h − z)
k
j=2
(1 + I
j
) −
k
j=2
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| I
1
= i
t
Z
2
, . . . , Z
n+1
, I
2
, . . . , I
n+1
X = (Z
2
, . . . , Z
n+1
, I
2
, . . . , I
n+1
)
P
n+1
k=2
(h − z)
k
j=2
(1 + I
j
) −
k
j=2
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| I
1
= i
t
=
=
Ω
ϕ(X)dP (w|I
1
= i
t
) =
=
n
×J
n
ϕ(z
2
, . . . , z
n+1
, i
2
, . . . , i
n+1
)dF
X
(z
2
, . . . , z
n+1
, i
2
, . . . , i
n+1
|I
1
= i
t
)
{I
k
, k = 0, 1, ...}
{Z
k
, k = 1, 2, ...} (i.i.d.)
P
n+1
k=2
(h − z)
k
j=2
(1 + I
j
) −
k
j=2
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| I
1
= i
t
=
=
n
×J
n
ϕ(x)dF
Z
2
,...,Z
n+1
(z
2
, . . . , z
n+1
|I
1
= i
t
)dF
I
2
,...,I
n+1
(i
2
, . . . , i
n+1
|I
1
= i
t
) =
=
n
×J
n
ϕ(x)dF
Z
2
,...,Z
n+1
(z
2
, . . . , z
n+1
)dF
I
2
,...,I
n+1
(i
2
, . . . , i
n+1
|I
1
= i
t
) =
=
n
×J
n
ϕ(x)dF
Z
2
(z
2
). . . . .dF
Z
n+1
(z
n+1
).P
I
2
= i
2
, . . . , I
n+1
= i
n+1
|I
1
= i
t
=
n
×J
n
ϕ(x)dF
Z
1
(z
2
). . . . .dF
Z
n
(z
n+1
).P
I
1
= i
2
, . . . , I
n
= i
n+1
|I
0
= i
t
=
=
n
×J
n
ϕ(x)dF
Z
1
,...,Z
n
(z
2
, . . . , z
n+1
).dF
I
1
,...,I
n
(i
2
, . . . , i
n+1
|I
0
= i
t
) =
=
n
×J
n
ϕ(x)dF
Z
1
,...,Z
n
(z
2
, . . . , z
n+1
|I
0
= i
t
).dF
I
1
,...,I
n
(i
2
, . . . , i
n+1
|I
0
= i
t
) =
=
n
×J
n
ϕ(x)dF
Z
1
,...,Z
n
,I
1
,...,I
n
(z
2
, . . . , z
n+1
, i
2
, . . . , i
n+1
|I
0
= i
t
) =
=
Ω
ϕ(Y )dP (w|I
0
= i
t
), Y = Z
1
, ..., Z
n
, I
1
, ..., I
n
P
n+1
k=1
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
=
= E
I
n
k=1
(h − z)
k
j=1
(1 + I
j
) −
k
j=1
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| I
0
= i
t
=
= P
n
k=1
(h − z)
k
j=1
(1 + I
j
) −
k
j=1
(Z
j
k
λ=j+1
(1 + I
λ
)) < 0
| I
0
= i
t
P
n+1
k=1
U
k
< 0
| Z
1
= z, I
0
= i
s
, I
1
= i
t
= ψ
n
(h − z, i
t
) = ψ
n
(u(1 + i
t
) − z, i
t
)
ψ
n+1
(u, i
s
) = P
n+1
k=1
U
k
< 0
|I
0
= i
s
= E
I
n+1
k=1
[U
k
< 0]
| I
0
= i
s
=
=
1
P
I
0
= i
s
.
(I
0
=i
s
)
I
n+1
k=1
[U
k
< 0]
dP
{w; I
0
(w) = i
s
} ∈ σ(I
0
, I
1
, Z
1
)
ψ
n+1
(u, i
s
) =
1
P
I
0
= i
s
.
(I
0
=i
s
)
E
I
n+1
k=1
[U
k
< 0]
| I
0
, I
1
, Z
1
dP =
I
0
, I
1
e Z
1
ψ
n+1
(u, i
s
) =
1
P
I
0
= i
s
.
(I
0
=i
s
)
ξ(I
0
, I
1
, Z
1
)dP =
=
1
P
I
0
= i
s
.
{w; I
0
(w)=i
s
, I
1
(w)∈J , Z
1
(w)∈ }
ξ(I
0
, I
1
, Z
1
)dP =
=
1
P
I
0
= i
s
.
j∈J
i∈{i
s
}
ξ(z, j, i)dP
(Z
1
,I
1
,I
0
)
(z, j, i)
ψ
n+1
(u, i
s
) =
1
P
I
0
= i
s
.
j∈J
ξ(z, j, i
s
)dP
(Z
1
,I
1
,I
0
)
(z, j, i
s
) =
=
1
P
I
0
= i
s
.
j∈J
ξ(z, j, i
s
)dP
I
1
,I
0
(j, i
s
)dP
Z
1
(z) =
=
1
P
I
0
= i
s
.
N
t=0
ξ(z, i
t
, i
s
)P
I
1
= i
t
, I
0
= i
s
)dG(z) =
=
N
t=0
P
st
ξ(z, i
t
, i
s
)dG(z) =
=
N
t=0
P
st
+∞
−∞
E
I
n+1
k=1
[U
k
< 0]
| I
0
= i
s
, I
1
= i
t
, Z
1
= z
dG(z) =
=
N
t=0
P
st
u(1+i
t
)
−∞
P
n+1
k=1
U
k
< 0
|I
0
= i
s
, I
1
= i
t
, Z
1
= z
dG(z)+
+
+∞
u(1+i
t
)
P
n+1
k=1
U
k
< 0
|I
0
= i
s
, I
1
= i
t
, Z
1
= z
dG(z)
=
=
N
t=0
P
st
u(1+i
t
)
−∞
ψ
n
u(1 + i
t
) − z, i
t
dG(z) +
+∞
u(1+i
t
)
dG(z)
=
=
N
t=0
P
st
u(1+i
t
)
−∞
ψ
n
u(1 + i
t
) − z, i
t
dG(z) + 1 − G
u(1 + i
t
)
=
ψ
n+1
(u, i
s
) =
N
t=0
P
st
u(1+i
t
)
−∞
ψ
n
u(1 + i
t
) − z, i
t
dG(z) + G
u(1 + i
t
)
n −→ ∞
lim
n→∞
ψ
n+1
(u, i
s
) = lim
n→∞
N
t=0
P
st
u(1+i
t
)
−∞
ψ
n
u(1 + i
t
) − z, i
t
dG(z) + G
u(1 + i
t
)
ψ(u, i
s
) =
N
t=0
P
st
G
u(1 + i
t
)
+ lim
n→∞
u(1+i
t
)
−∞
ψ
n
u(1 + i
t
) − z, i
t
dG(z)
{ψ
n
} lim
n→∞
ψ
n
= ψ
ψ(u, i
s
) =
N
t=0
P
st
G
u(1 + i
t
)
+
u(1+i
t
)
−∞
lim
n→∞
ψ
n
u(1 + i
t
) − z, i
t
dG(z)
=
=
N
t=0
P
st
G
u(1 + i
t
)
+
u(1+i
t
)
−∞
ψ
u(1 + i
t
) − z, i
t
dG(z)
ψ
1
(u, i
s
) = P
Z
1
> u(1 + I
1
) | I
0
= i
s
= E
I
Z
1
>u(1+I
1
)
| I
0
= i
s
=
1
P (I
0
= i
s
)
.
(I
0
=i
s
)
I
Z
1
>u(1+I
1
)
dP
{w; I
0
(w) = i
s
} ∈ σ(I
0
, I
1
, Z
1
)
ψ
1
(u, i
s
) =
1
P (I
0
= i
s
)
.
(I
0
=i
s
)
E
I
Z
1
>u(1+I
1
)
| I
0
, I
1
, Z
1
dP
=
1
P (I
0
= i
s
)
.
(I
0
=i
s
)
φ(I
0
, I
1
, Z
1
)dP
=
1
P (I
0
= i
s
)
.
×J×{i
s
}
φ(i, i
t
, z)dP
I
0
,I
1
,Z
1
(i, i
t
, z)
=
1
P (I
0
= i
s
)
.
j∈J
φ(i
s
, j, z)dF
I
0
,I
1
(i
s
, j)dF
Z
1
(z)
J
ψ
1
(u, i
s
) =
1
P (I
0
= i
s
)
.
N
t=0
φ(i
s
, i
t
, z)P
I
1
= i
t
, I
0
= i
s
dG(z)
=
N
t=0
P
st
+∞
−∞
φ(i
s
, i
t
, z)dG(z)
=
N
t=0
P
st
+∞
−∞
E
I
Z
1
>u(1+I
1
)
| I
0
= i
s
, I
1
= i
t
, Z
1
= z
dG(z)
=
N
t=0
P
st
u(1+i
t
)
−∞
φ(i
s
, i
t
, z)dG(z) +
+∞
u(1+i
t
)
φ(i
s
, i
t
, z)dG(z)
=
N
t=0
P
st
+∞
u(1+i
t
)
dG(z) =
N
t=0
P
st
G
u(1 + i
t
)
I
n
≥ 0 n = 0, 1, . . .
ψ(u, i
s
)
R
0
> 0
ψ(u, i
s
) ≤ β
0
E
e
−R
0
u(1+I
1
)
| I
0
= i
s
, u ≥ 0,
β
−1
0
= inf
t≥0
+∞
t
e
R
0
z
dG(z)
e
R
0
t
G(t)
x ≥ 0
G(x) = G(x).e
R
0
x
.e
−R
0
x
.
+∞
x
e
R
0
z
dG(z)
−1
.
+∞
x
e
R
0
z
dG(z)
=
+∞
x
e
R
0
z
dG(z)
−1
G(x).e
R
0
x
−1
.e
−R
0
x
.
+∞
x
e
R
0
z
dG(z)
β
−1
0
= inf
t≥0
+∞
t
e
R
0
z
dG(z)
e
R
0
t
G(t)
≤
+∞
x
e
R
0
z
dG(z)
e
R
0
x
.G(x)
, ∀x ≥ 0
β
0
≥
+∞
x
e
R
0
z
dG(z)
e
R
0
x
.G(x)
−1
G(x) ≤ β
0
.e
−R
0
x
.
+∞
x
e
R
0
z
dG(z)
≤ β
0
.e
−R
0
x
.
+∞
−∞
e
R
0
z
dG(z)
≤ β
0
.e
−R
0
x
u ≥ 0 i
s
≥ 0
ψ
1
(u, i
s
) =
N
t=0
P
st
G(u(1 + i
t
))
≤
N
t=0
P
st
β
0
e
−R
0
u(1+i
t
)
≤ β
0
N
t=0
e
−R
0
u(1+i
t
)
P (I
1
= i
t
| I
0
= i
s
)
= β
0
E
e
−R
0
u(1+I
1
)
| I
0
= i
s
u ≥ 0
i
s
≥ 0
ψ
n
(u, i
s
) ≤ β
0
E
e
−R
0
u(1+I
1
)
| I
0
= i
s
0 ≤ z ≤ u(1 + i
t
) u i
s
u(1 + i
t
) − z i
t
I
1
≥ 0
ψ
n
(u(1 + i
t
) − z, i
t
) ≤ β
0
E
e
−R
0
(u(1+i
t
)−z)(1+I
1
)
| I
0
= i
t
0 ≤ z ≤ u(1 + i
t
) u(1 + i
t
) − z ≥ 0 R
0
> 0 −R
0
< 0
−R
0
u(1 + i
t
) − z
≤ 0 I
1
≥ 0 1 + I
1
≥ 1
−R
0
u(1 + i
t
) − z
(1 + I
1
) ≤ −R
0
u(1 + i
t
) − z
e
−R
0
u(1+i
t
)−z
(1+I
1
)
≤ e
−R
0
u(1+i
t
)−z
E
e
−R
0
u(1+i
t
)−z
(1+I
1
)
| I
0
= i
t
≤ E
e
−R
0
u(1+i
t
)−z
| I
0
= i
t
e
−R
0
u(1+i
t
)−z
E
e
−R
0
u(1+i
t
)−z
(1+I
1
)
| I
0
= i
t
≤ e
−R
0
u(1+i
t
)−z
ψ
n
(u(1 + i
t
) − z, i
t
) ≤ β
0
e
−R
0
(u(1+i
t
)−z)
ψ
n+1
(u, i
s
) =
N
t=0
P
st
G(u(1 + i
t
)) +
u(1+i
t
)
−∞
ψ
n
(u(1 + i
t
) − z, i
t
)dG(z)
≤
N
t=0
P
st
β
0
.e
−R
0
(u(1+i
t
))
.
+∞
u(1+i
t
)
e
R
0
z
dG(z) +
u(1+i
t
)
−∞
β
0
e
−R
0
(u(1+i
t
)−z)
dG(z)
≤
N
t=0
P
st
β
0
.e
−R
0
(u(1+i
t
))
.
+∞
u(1+i
t
)
e
R
0
z
dG(z) + β
0
.e
−R
0
(u(1+i
t
))
u(1+i
t
)
−∞
e
R
0
z
dG(z)
ψ
n+1
(u, i
s
) ≤
N
t=0
P
st
β
0
.e
−R
0
(u(1+i
t
))
+∞
−∞
e
R
0
z
dG(z)
≤
N
t=0
P
st
β
0
.e
−R
0
u(1+i
t
)
E(e
R
0
Z
1
)
≤
N
t=0
P
st
β
0
.e
−R
0
u(1+i
t
)
≤ β
0
E
e
−R
0
u(1+I
1
)
| I
0
= i
s
n = 1, 2, . . . ,
n −→ ∞
0 ≤ β
0
≤ 1 t ≥ 0
+∞
t
e
R
0
z
dG(z) ≥
+∞
t
e
R
0
t
dG(z) = e
R
0
t
+∞
t
dG(z)
+∞
t
e
R
0
z
dG(z)
e
R
0
t
G(t)
≥
e
R
0
t
+∞
t
dG(z)
e
R
0
t
G(t)
=
1 −
t
−∞
dG(z)
G(t)
=
1 − G(t)
G(t)
= 1
β
0
E
e
−R
0
u(1+I
1
)
| I
0
= i
s
≤ β
0
E
e
−R
0
u
| I
0
= i
s
= β
0
e
−R
0
u
≤ e
−R
0
u
, u ≥ 0
I
n
= 0, ∀n = 0, 1, . . .
β
0
e
−R
0
u
0 ≤ β
0
≤ 1
k p > 0 Y
k
Z
k
= Y
k
−p
k = 1, 2, . . .
ψ(u, i
s
)
F x ≥ 0, y ≥ 0
+∞
x+y
F (t)dt ≥
F (x)
+∞
y
F (t)dt
t > 0 x ≥ 0
∞
0
e
ty
dF (y) < ∞ F (y)
inf
0≤z≤x,F (z)>0
∞
z
e
ty
dF (y)
e
tz
F (z)
=
∞
0
e
ty
dF (y) = E(e
tY
)
Z
k
= Y
k
− p k = 1, 2, . . . Y
1
, Y
2
, . . .
p > E(Y
k
)
R
0
> 0 E(e
R
0
Z
1
) = 1
ψ(u, i
s
) ≤ β.E
e
−R
0
u(1+I
1
)
| I
0
= i
s
, u ≥ 0,
β
−1
= inf
t≥0
+∞
t
e
R
0
y
dF (y)
e
R
0
t
F (t)
,
F Y
k
k = 1, 2, . . .
ψ(u, i
s
) ≤
E(e
R
0
Y
1
)
−1
.E
e
−R
0
u(1+I
1
)
| I
0
= i
s
, u ≥ 0
G(z) = P (Y
1
− z ≤ p) = F (z + p)
+∞
t
e
R
0
z
dG(z)
e
R
0
t
G(t)
=
+∞
t
e
R
0
z
dF (z + p)
e
R
0
t
F (t + p)
=
+∞
t+p
e
R
0
(y−p)
dF (y)
e
R
0
t
F (t + p)
=
+∞
t+p
e
R
0
y
e
−R
0
p
dF (y)
e
R
0
t
F (t + p)
=
+∞
t+p
e
R
0
y
dF (y)
e
R
0
(t+p)
F (t + p)
β
−1
0
= inf
t≥0
+∞
t
e
R
0
z
dG(z)
e
R
0
t
G(t)
= inf
t≥0
+∞
t+p
e
R
0
y
dF (y)
e
R
0
(t+p)
F (t + p)
= inf
t≥0
g(t + p)
g(x) =
+∞
x
e
R
0
y
dF (y)
e
R
0
x
F (x)
inf
t≥0
g(t + p) ≥ inf
t≥0
g(t) = β
−1
β
−1
0
≥ β
−1
β
0
≤ β
ψ(u, i
s
) ≤ β
0
E
e
−R
0
u(1+i
t
)
| I
0
= i
s
≤ β.E
e
−R
0
u(1+i
t
)
| I
0
= i
s
F β
−1
= E(e
R
0
Y
1
)
{V
n
, n = 1, 2, . . .}
V
k
U
k
k U
k
I
k
k − 1
I
k−1
k − 2
V
k
ψ
n
(u, i
s
) = P
n
k=1
U
k
< 0
| I
0
= i
s
= P
n
k=1
V
k
< 0
| I
0
= i
s
V
k
= U
k
k
j=1
(1 + I
j
)
−1
, k = 1, 2, . . .
A =
n
k=1
{w ∈ Ω; U
k
(w) < 0} B =
n
k=1
{w ∈ Ω; V
k
(w) < 0}
x ∈ A =⇒ ∃k
0
∈ N, k
0
≤ n U
k
0
(x) < 0 V
k
0
(x) = U
k
0
(x)
k
0
j=1
(1 + I
j
(x))
−1
I
j
(x) ≥ 0, ∀j V
k
0
(x) < 0 k
0
≤ n x ∈
n
k=1
{w ∈ Ω, V
k
(w) < 0} =
B
x ∈ B =⇒ ∃k
0
∈ N, k
0
≤ n, tal que V
k
0
(x) < 0 I
j
(x) ≥ 0, ∀j
U
k
0
(x) < 0 k
0
≤ n x ∈
n
k=1
{w ∈ Ω, U
k
(w) < 0} = A
A = B
e
−R
0
U
n
, n = 1, 2, . . . ,
r > 0
e
−rU
n
, n = 1, 2, . . .
r > 0
e
−rV
n
, n = 1, 2, . . .
E(Z
1
) < 0 R
0
> 0
ρ
s
> 0
E
e
ρ
s
Z
1
(1+I
1
)
−1
| I
0
= i
s
= 1, s = 0, 1, . . . , N
R
1
= min
0≤ s ≤N
{ρ
s
} ≥ R
0
s = 0, 1, . . . , N
E
e
R
1
Z
1
(1+I
1
)
−1
| I
0
= i
s
≤ 1
s = 0, 1, . . . , N
l
s
(r) = E
e
rZ
1
(1+I
1
)
−1
| I
0
= i
s
− 1
l
s
(r) =
1
P (I
0
= i
s
)
(I
0
=i
s
)
e
rZ
1
(1+I
1
)
−1
dP − 1
r
l
s
(r) =
1
P (I
0
= i
s
)
d
dr
(I
0
=i
s
)
e
rZ
1
(1+I
1
)
−1
dP
=
1
P (I
0
= i
s
)
(I
0
=i
s
)
d
dr
e
rZ
1
(1+I
1
)
−1
dP
=
1
P (I
0
= i
s
)
(I
0
=i
s
)
e
rZ
1
(1+I
1
)
−1
Z
1
(1 + I
1
)
−1
dP
l
s
(r) =
1
P (I
0
= i
s
)
(I
0
=i
s
)
e
rZ
1
(1+I
1
)
−1
Z
2
1
(1 + I
1
)
−2
dP l
s
(r) ≥ 0 ∀r
l
s
(0) = E
e
0Z
1
(1+I
1
)
−1
| I
0
= i
s
− 1 = 0
l
s
(0) =
1
P (I
0
= i
s
)
(I
0
=i
s
)
Z
1
(1 + I
1
)
−1
dP = E
Z
1
(1 + I
1
)
−1
| I
0
= i
s
l
s
(0) = E(Z
1
)E
(1 + I
1
)
−1
| I
0
= i
s
E(Z
1
) < 0 (1 + I
1
)
−1
> 0
E
(1 + I
1
)
−1
| I
0
= i
s
> 0 l
s
(0) < 0
ρ
s
l
s
(r) = 0 (0, +∞)
E
e
R
0
Z
1
(1+I
1
)
−1
| I
0
= i
s
=
1
P (I
0
= i
s
)
(I
0
=i
s
)
e
R
0
Z
1
(1+I
1
)
−1
dP
=
1
P (I
0
= i
s
)
{w,I
0
(w)=i
s
,I
1
(w)∈J ,Z
1
(w)∈ }
e
R
0
Z
1
(1+I
1
)
−1
dP
=
1
P (I
0
= i
s
)
R
j∈J
e
R
0
z(1+j)
−1
dP
Z
1
(z)dP
I
1
,I
0
(j, i
s
)
=
1
P (I
0
= i
s
)
j∈J
e
R
0
z(1+j)
−1
dG(z)P (I
1
= j, I
0
= i
s
)
=
N
t=0
P
st
e
R
0
z(1+i
t
)
−1
dG(z)
=
N
t=0
P
st
E
e
R
0
Z
1
(1+i
t
)
−1
Y = e
R
0
Z
1
ϕ(y) = y
(1+i
t
)
−1
ϕ
(y) =
(1 + i
t
)
−2
− (1 + i
t
)
−1
y
(1+i
t
)
−1
−2
(1 + i
t
)
2
= (1 + i
t
)(1 + i
t
) ≥ (1 + i
t
) (1 + i
t
)
−2
− (1 + i
t
)
−1
≤ 0
y
(1+i
t
)
−1
−2
y ϕ
(y) ≤ 0 ϕ(y)
E[ϕ(Y )] ≤ ϕ
EY
=⇒ E
e
R
0
Z
1
(1+i
t
)
−1
≤
EY
(1+i
t
)
−1
=
E(e
R
0
Z
1
)
(1+i
t
)
−1
E
e
R
0
Z
1
(1+I
1
)
−1
| I
0
= i
s
=
N
t=0
P
st
E
e
R
0
Z
1
(1+i
t
)
−1
≤
N
t=0
P
st
E(e
R
0
Z
1
)
(1+i
t
)
−1
≤
N
t=0
P
st
= 1
l
s
(R
0
) = E
e
R
0
Z
1
(1+I
1
)
−1
| I
0
= i
s
− 1 l
s
(R
0
) ≤ 0 R
0
≤ ρ
s
R
1
= min
0≤ s ≤N
ρ
s
≥ R
0
s = 0, 1, . . . , N R
1
= min
0≤ t ≤N
ρ
t
≤ ρ
s
l
s
(R
1
) ≤ 0
E
e
R
1
Z
1
(1+I
1
)
−1
| I
0
= i
s
− 1 ≤ 0
E
e
R
1
Z
1
(1+I
1
)
−1
| I
0
= i
s
≤ 1
s = 0, 1, . . . , N
ψ(u, i
s
) ≤ e
−R
1
u
, u ≥ 0
{U
k
}
V
k
= U
k
k
j=1
(1 + I
j
)
−1
V
k
=
u
k
t=1
(1 + I
t
) −
k
t=1
(Z
t
k
λ=t+1
(1 + I
λ
))
k
j=1
(1 + I
j
)
−1
= u
k
t=1
(1 + I
t
)
k
j=1
(1 + I
j
)
−1
−
k
j=1
(1 + I
j
)
−1
k
t=1
(Z
t
k
λ=t+1
(1 + I
λ
))
= u
k
j=1
(1 + I
j
)(1 + I
j
)
−1
−
k
t=1
(Z
t
k
j=1
(1 + I
j
)
−1
k
λ=t+1
(1 + I
λ
))
= u −
k
t=1
(Z
t
t
j=1
(1 + I
j
)
−1
k
j=t+1
(1 + I
J
)
−1
k
λ=t+1
(1 + I
λ
))
= u −
k
t=1
(Z
t
t
j=1
(1 + I
j
)
−1
S
n
= e
−R
1
V
n
S
n+1
= e
−R
1
u −
n+1
t=1
(Z
t
t
j=1
(1 + I
j
)
−1
)
= e
−R
1
u −
n
t=1
(Z
t
t
j=1
(1 + I
j
)
−1
) − Z
n+1
n+1
j=1
(1 + I
j
)
−1
= e
−R
1
V
n
e
R
1
Z
n+1
n+1
j=1
(1 + I
j
)
−1
= S
n
e
R
1
Z
n+1
n+1
j=1
(1 + I
j
)
−1
n ≥ 1 S
n
σ(I
1
, . . . , I
n
, Z
1
, . . . , Z
n
)
E(S
n+1
| Z
1
, . . . , Z
n
, I
1
, . . . , I
n
) = E
S
n
e
R
1
Z
n+1
n+1
j=1
(1 + I
j
)
−1
| Z
1
, . . . , Z
n
, I
1
, . . . , I
n
= S
n
E
e
R
1
Z
n+1
n+1
j=1
(1 + I
j
)
−1
| Z
1
, . . . , Z
n
, I
1
, . . . , I
n
= S
n
E
e
R
1
Z
n+1
(1+I
n+1
)
−1
n
j=1
(1 + I
j
)
−1
| Z
1
, . . . , Z
n
, I
1
, . . . , I
n
{Z
n
, n = 1, 2, . . .} {I
n
, n = 1, 2, . . .}
E(S
n+1
| Z
1
, . . . , Z
n
, I
1
, . . . , I
n
) = S
n
E
e
R
1
Z
n+1
(1+I
n+1
)
−1
n
j=1
(1 + I
j
)
−1
| I
1
, . . . , I
n
Y = e
R
1
Z
n+1
(1+I
n+1
)
−1
ϕ(y) = y
n
j=1
(1 + I
j
)
−1
ϕ
E(S
n+1
| Z
1
, . . . , Z
n
, I
1
, . . . , I
n
) ≤ S
n
E
e
R
1
Z
n+1
(1+I
n+1
)
−1
| I
1
, . . . , I
n
n
j=1
(1 + I
j
)
−1
≤ S
n
E
e
R
1
Z
n+1
(1+I
n+1
)
−1
| I
1
, . . . , I
n
n
j=1
(1 + I
j
)
−1
{I
n
, n = 1, 2, . . .}
E(S
n+1
| Z
1
, . . . , Z
n
, I
1
, . . . , I
n
) ≤ S
n
E
e
R
1
Z
n+1
(1+I
n+1
)
−1
| I
n
n
j=1
(1 + I
j
)
−1
e
R
1
Z
n+1
(1+I
n+1
)
−1
Z
n+1
e I
n+1
ξ(Z
n+1
, I
n+1
) = e
R
1
Z
n+1
(1+I
n+1
)
−1
E(ξ(Z
n+1
, I
n+1
) | I
n
= j) =
Ω
ξ(Z
n+1
, I
n+1
)dP (w | I
n
= j)
=
×J
ξ(z, i)dF
Z
n+1
,I
n+1
(z, i | I
n
= j)
{Z
n
} {I
n
}
{Z
k
}
k≥1
E(ξ(Z
n+1
, I
n+1
) | I
n
= j) =
×J
ξ(z, i)dP
Z
n+1
(z | I
n
= j)dP
I
n+1
(i | I
n
= j)
=
R×J
ξ(z, i)dP
Z
n+1
(z)P (I
n+1
= i | I
n
= j)
=
×J
ξ(z, i)dP
Z
1
(z)P (I
1
= i | I
0
= j)
=
×J
ξ(z, i)dP
Z
1
(z | I
0
= j)dP
I
1
(i | I
0
= j)
=
×J
ξ(z, i)dP
Z
1
,I
1
(z, i | I
0
= j)
=
Ω
ξ(Z
1
, I
1
)dP (w | I
0
= j)
= E
e
R
1
Z
1
(1+I
1
)
−1
| I
0
= j
j ∈ J
E(ξ(Z
n+1
, I
n+1
) | I
n
) = E(ξ(Z
1
, I
1
) | I
0
)
E
S
n+1
| Z
1
, . . . , Z
n
, I
1
, . . . , I
n
≤ S
n
E
e
R
1
Z
1
(1+I
1
)
−1
| I
0
n
j=1
(1 + I
j
)
−1
≤ S
n
{S
n
, n = 1, 2, . . .}
σ(I
1
, . . . , I
n
, Z
1
, . . . , Z
n
) T
s
= min {k; V
k
< 0 | I
0
= i
s
} V
k
T
s
n ∧ T
s
= min {n, T
s
}
E(S
n∧T
s
) ≤ E(S
0
) = E(e
−R
1
V
0
) = E(e
−R
1
u
) = e
−R
1
u
e
−R
1
u
≥ E(S
n∧T
s
) ≥ E(S
n∧T
s
I
T
s
≤n
) = E(S
T
s
I
T
s
≤n
) = E(e
−R
1
V
T
s
I
T
s
≤n
)
V
T
s
< 0 T
s
= min {k; V
k
< 0 | I
0
= i
s
} −R
1
V
T
s
≥ 0
e
−R
1
V
T
s
≥ e
0
= 1
e
−R
1
u
≥ E(I
T
s
≤n
) = P (T
s
≤ n)
e
−R
1
u
≥ P
[V
1
< 0] ∪ [V
2
< 0] ∪ . . . ∪ [V
n
< 0] | I
0
= i
s
≥ P
n
k=1
[V
k
< 0] | I
0
= i
s
ψ
n
(u, i
s
) ≤ e
−R
1
u
n −→ +∞
e
−R
1
u
≤ e
−R
0
u
, u ≥ 0
I
n
= 0 n = 0, 1, . . . R
1
= R
0
k Y
k
Y
1
1
2
E(Y
1
) = 1 V ar(Y
1
) = 2
p = 1, 1 10% Z
k
= Y
k
− 1, 1
k = 1, 2, . . .
i
0
= 6%, i
1
= 8% e i
2
= 10% P = {P
st
}
P =
0, 2 0, 8 0
0, 15 0, 7 0, 15
0 0, 8 0, 2
8%
R
0
E(e
R
0
Z
1
) = 1 =⇒ E(e
R
0
(Y
1
−1,1)
) = 1 =⇒ E(e
R
0
Y
1
e
−1,1R
0
) = 1 =⇒ e
−1,1R
0
E(e
R
0
Y
1
) = 1
M
Y
(t) = E(e
tY
1
) = (1 − 2t)
−1
2
e
−1,1R
0
(1 − 2R
0
)
−1
2
= 1 =⇒ −1, 1R
0
−
1
2
ln(1 − 2R
0
) = 0
1, 1R
0
+
1
2
ln(1 − 2R
0
) = 0 =⇒ R
0
= 0, 08807
s = 0
ψ(u, i
0
) ≤
E(e
R
0
Y
1
)
−1
E(e
−R
0
u(1+I
1
)
| I
0
= i
0
), u ≥ 0
ξ(I
1
, I
0
) = e
−R
0
u(1+I
1
)
ψ(u, i
0
) ≤ ((1 − 2R
0
)
−1/2
)
−1
1
P (I
0
= i
0
)
(I
0
=i
0
)
ξ(I
1
, I
0
)dP
ψ(u, i
0
) ≤ (1 − 2R
0
)
1/2
1
P (I
0
= i
0
)
I
{i
0
}
ξ(i
t
, i)dF
I
1
,I
0
(i
t
, i)
≤ (1 − 2R
0
)
1/2
1
P (I
0
= i
0
)
I
ξ(i
t
, i
0
)dF
I
1
,I
0
(i
t
, i
0
)
≤ (1 − 2R
0
)
1/2
1
P (I
0
= i
0
)
2
t=0
ξ(i
t
, i
0
)P (I
1
= i
t
, I
0
= i
0
)
≤ (1 − 2R
0
)
1/2
2
t=0
P
0t
ξ(i
t
, i
0
)
≤ (1 − 2R
0
)
1/2
P
00
ξ(i
0
, i
0
) + P
01
ξ(i
1
, i
0
) + P
02
ξ(i
2
, i
0
)
≤ (1 − 2R
0
)
1/2
0, 2e
−1,06R
0
u
+ 0, 8e
−1,08R
0
u
+ 0e
−1,1R
0
u
ψ(u, i
0
) ≤ (1 − 2R
0
)
1/2
0, 2e
−1,06R
0
u
+ 0, 8e
−1,08R
0
u
, u ≥ 0.
ψ(u, i
s
) s = 1
s = 2
R
1
s = 0, 1, 2
E
e
ρ
s
Z
1
(1+I
1
)
−1
| I
0
= i
s
= 1
E
e
ρ
s
(Y
1
−1,1)(1+I
1
)
−1
| I
0
= i
s
= 1
E
e
(ρ
s
Y
1
−1,1ρ
s
)(1+I
1
)
−1
| I
0
= i
s
= 1
Ω
e
(ρ
s
Y
1
−1,1ρ
s
)(1+I
1
)
−1
dP (w | I
0
= i
s
) = 1
J
e
(ρ
s
y−1,1ρ
s
)(1+i
t
)
−1
dP
Y
1
,I
1
(y, i
t
| I
0
= i
s
) = 1
J
e
(ρ
s
y−1,1ρ
s
)(1+i
t
)
−1
dP
Y
1
(y | I
0
= i
s
)dP
I
1
(i
t
| I
0
= i
s
) = 1
J
e
(ρ
s
y−1,1ρ
s
)(1+i
t
)
−1
dP
Y
1
(y)dP
I
1
(i
t
| I
0
= i
s
) = 1
2
t=0
e
(ρ
s
y−1,1ρ
s
)(1+i
t
)
−1
dP
Y
1
(y)P (I
1
= i
t
| I
0
= i
s
) = 1
2
t=0
P
st
e
−1,1ρ
s
(1+i
t
)
−1
e
ρ
s
y(1+i
t
)
−1
dP
y
1
(y) = 1
2
t=0
P
st
e
−1,1ρ
s
(1+i
t
)
−1
M
Y
1
ρ
s
1 + i
t
= 1
s = 0
P
00
e
−1,1ρ
0
(1+i
0
)
−1
M
Y
1
(
ρ
0
1 + i
0
)+P
01
e
−1,1ρ
0
(1+i
1
)
−1
M
Y
1
(
ρ
0
1 + i
1
)+P
02
e
−1,1ρ
0
(1+i
2
)
−1
M
Y
1
(
ρ
0
1 + i
2
) = 1
0, 2(1 −
2ρ
0
1, 06
)
−1
2
e
−1,1ρ
0
1,06
+ 0, 8(1 −
2ρ
0
1, 08
)
−1
2
e
−1,1ρ
0
1,08
+ 0(1 −
2ρ
0
1, 1
)
−1
2
e
−1,1ρ
0
1,1
= 1
ρ
0
= 0, 09475
s = 1
P
10
e
−1,1ρ
1
(1+i
0
)
−1
M
Y
1
(
ρ
1
1 + i
0
)+P
11
e
−1,1ρ
1
(1+i
1
)
−1
M
Y
1
(
ρ
1
1 + i
1
)+P
12
e
−1,1ρ
1
(1+i
2
)
−1
M
Y
1
(
ρ
1
1 + i
2
) = 1
0, 15(1 −
2ρ
1
1, 06
)
−1
2
e
−1,1ρ
1
1,06
+ 0, 7(1 −
2ρ
1
1, 08
)
−1
2
e
−1,1ρ
1
1,08
+ 0, 15(1 −
2ρ
1
1, 1
)
−1
2
e
−1,1ρ
1
1,1
= 1
ρ
1
= 0, 09509
s = 2
P
20
e
−1,1ρ
2
(1+i
0
)
−1
M
Y
1
(
ρ
2
1 + i
0
)+P
21
e
−1,1ρ
2
(1+i
1
)
−1
M
Y
1
(
ρ
2
1 + i
1
)+P
22
e
−1,1ρ
2
(1+i
2
)
−1
M
Y
1
(
ρ
2
1 + i
2
) = 1
0(1 −
2ρ
2
1, 06
)
−1
2
e
−1,1ρ
2
1,06
+ 0, 8(1 −
2ρ
2
1, 08
)
−1
2
e
−1,1ρ
2
1,08
+ 0, 2(1 −
2ρ
2
1, 1
)
−1
2
e
−1,1ρ
2
1,1
= 1
ρ
2
= 0, 09545
R
1
= min {ρ
0
, ρ
1
, ρ
2
} = 0, 09475
ψ(u, i
s
) ≤ e
−R
1
u
, u ≥ 0
k Y
k
Y
1
λ = 1
E(Y
1
) = 1 var(Y
1
) = 1 Y
1
1
2
E(Y
1
) = 1 var(Y
1
) =
2 1, 1
10% Z
k
= Y
k
− 1, 1, k = 1, 2, ...
i
0
= 6%, i
1
= 8% i
2
= 10%
P = {P
st
}
P =
0.2 0.8 0
0.15 0.7 0.15
0 0.8 0.2
.
E(Z
1
) < 0 R
0
e
R
0
Z
1
= 1 ψ(u) ≤
e
−R
0
u
, u ≥ 0
Z
1
= Y
1
− 1.1 ⇒ E(Z
1
) = E(Y
1
) − 1.1 = 1 − 1.1 = −0.1 < 0
e
R
0
Z
1
= 1 ⇔ e
R
0
(Y
1
−1.1)
= 1 ⇔ e
R
0
Y
1
e
−1.1R
0
= 1
e
R
0
Y
1
=
1
1−R
0
R
0
e
R
0
Z
1
= 1 ⇔
1
1 − R
0
e
−1.1R
0
= 1.
>
ψ(u) ≤ e
−R
0
u
= e
−0.1761341436u
, u ≥ 0
Z
k
= Y
k
− p, k = 1, 2, ... Y
1
, Y
2
, ...
p > E(Y
1
) R
0
ψ(u, i
s
) ≤ βE(e
−R
0
u(1+I
1
)
|I
0
= i
s
), u ≥ 0
β
−1
= inf
t≥0
∞
t
e
R
0
y
dF (y)
e
R
0
t
F (t)
λ f(y) =
λe
−λy
, y ≥ 0 f(y) = 0, y < 0 λ = R
0
∞
t
e
R
0
y
dF (y)
e
R
0
t
F (t)
=
∞
t
e
R
0
y
λe
−λy
dy
e
R
0
t
∞
t
λe
−λy
dy
=
λ
∞
t
e
−y(λ−R
0
)
dy
e
R
0
t
e
−λt
=
λ
λ−R
0
e
−t(λ−R
0
)
e
−t(λ−R
0
)
=
λ
λ − R
0
.
t ≥ 0
β
−1
=
λ
λ−R
0
λ = 1
R
0
= 0.1761341436
>
> %
β
E(e
−R
0
u(1+I
1
)
|I
0
= i
s
)
E(e
−R
0
u(1+I
1
)
|I
0
= i
s
) =
2
k=0
e
−R
0
u(1+i
k
)
P (I
1
= i
k
|I
0
= i
s
) =
2
k=0
e
−R
0
u(1+i
k
)
P
sk
.
P
sk
P i
0
, i
1
, i
2
>
P :=
0.2 0.8 0
0.15 0.7 0.15
0 0.8 0.2
.
>
>
>
E(e
−R
0
u(1+I
1
)
|I
0
= i
s
) =
2
k=0
e
−R
0
u(1+i
k
)
P
sk
i
s
= i
0
= i[0]
>
u
u u
ψ(u, i
0
) ≤ βE(e
−R
0
u(1+I
1
)
|I
0
= i
s
) = 0.8238658562
2
k=0
e
−0.1761341436u(1+i
k
)
P
0k
>
E(Z
1
) < 0 ρ
s
, s = 0, 1, 2, ..., N
E(e
ρ
s
Z
1
(1+I
1
)
−1
|I
0
= i
s
) = 1, s = 0, 1, ..., N
R
1
= min
0≤s≤N
{ρ
s
} ≥ R
0
φ(u, i
s
) ≤ e
−R
1
u
, u ≥ 0.
ρ
0
, ρ
1
, ρ
2
R
1
= min
0≤s≤2
{ρ
s
}
φ(u, i
s
) ≤ e
−R
1
u
, u ≥ 0.
E(e
ρ
s
Z
1
(1+I
1
)
−1
|I
0
= i
s
) = 1
E(e
ρ
s
Z
1
(1+I
1
)
−1
|I
0
= i
s
) = 1 ⇔ E(e
ρ
s
(Y
1
−1.1)(1+I
1
)
−1
|I
0
= i
s
) = 1 ⇔
E(e
ρ
s
Y
1
(1+I
1
)
−1
−1.1ρ
s
(1+I
1
)
−1
|I
0
= i
s
) = 1 ⇔ E(e
ρ
s
Y
1
(1+I
1
)
−1
e
−1.1ρ
s
(1+I
1
)
−1
|I
0
= i
s
) = 1 ⇔
E(e
ρ
s
Y
1
(1+I
1
)
−1
e
−1.1ρ
s
(1+I
1
)
−1
|I
0
= i
s
) =
j∈J
∞
0
e
ρ
s
y(1+j)
−1
e
−1.1ρ
s
(1+j)
−1
P
Y
1
,I
1
(y, j|I
0
= i
s
)
j∈J
∞
0
e
ρ
s
y(1+j)
−1
e
−1.1ρ
s
(1+j)
−1
P
Y
1
(y)P
I
1
(j|I
0
= i
s
)
=
j∈J
∞
0
e
ρ
s
y(1+j)
−1
P
Y
1
(dy)
e
−1.1ρ
s
(1+j)
−1
P
i
s
,j
=
j∈J
M
Y
1
(ρ
s
(1 + j)
−1
)e
−1.1ρ
s
(1+j)
−1
P
i
s
,j
=
j∈J
1
1 − ρ
s
(1 + j)
−1
e
−1.1ρ
s
(1+j)
−1
P
i
s
,j
> →
→
1
1−x
> →
→ e
−1.1x
ρ
0
, ρ
1
, ρ
2
>
ρ
1
>
ρ
2
>
ρ
3
>
ψ(u, i
s
) ≤ e
−R
1
u
, u ≥ 0
u
u
i
s
= 6% i
s
= 8% i
s
= 10%
Y
1
E(Z
1
) < 0 R
0
e
R
0
Z
1
= 1 ψ(u) ≤
e
−R
0
u
, u ≥ 0
Z
1
= Y
1
− 1.1 ⇒ E(Z
1
) = E(Y
1
) − 1.1 = 1 − 1.1 = −0.1 < 0
e
R
0
Z
1
= 1 ⇔ e
R
0
(Y
1
−1.1)
= 1 ⇔ e
R
0
Y
1
e
−1.1R
0
= 1
e
R
0
Y
1
= (1 − 2R
0
)
−
1
2
1
2
R
0
e
R
0
Z
1
= 1 ⇔ (1 − 2R
0
)
−
1
2
e
−1.1R
0
= 1.
> ˆ
ψ(u) ≤ e
−R
0
u
= e
−0.08806707182u
, u ≥ 0
Z
k
= Y
k
− p, k = 1, 2, ... Y
1
, Y
2
, ...
p > E(Y
1
) R
0
ψ(u, i
s
) ≤ βE(e
−R
0
u(1+I
1
)
|I
0
= i
s
), u ≥ 0
β
−1
= E(e
R
0
Y
1
) F
β
−1
= E(e
R
0
Y
1
) = (1 − 2R
0
)
−
1
2
.
> ˆ
β := .9076705660
E(e
−R
0
u(1+I
1
)
|I
0
= i
s
)
E(e
−R
0
u(1+I
1
)
|I
0
= i
s
) =
2
k=0
e
−R
0
u(1+i
k
)
P (I
1
= i
k
|I
0
= i
s
) =
2
k=0
e
−R
0
u(1+i
k
)
P
sk
.
P
sk
P i
0
, i
1
, i
2
>
P :=
0.2 0.8 0
0.15 0.7 0.15
0 0.8 0.2
.
>
>
>
E(e
−R
0
u(1+I
1
)
|I
0
= i
s
) =
2
k=0
e
−R
0
u(1+i
k
)
P
sk
i
s
= i
0
= i[0]
>
u
u u
ψ(u, i
0
) ≤ βE(e
−R
0
u(1+I
1
)
|I
0
= i
s
) = 0.9076705660
2
k=0
e
−0.08806707182u(1+i
k
)
P
0k
>
E(Z
1
) < 0 ρ
s
, s = 0, 1, 2, ..., N
E(e
ρ
s
Z
1
(1+I
1
)
−1
|I
0
= i
s
) = 1, s = 0, 1, ..., N
R
1
= min
0≤s≤N
{ρ
s
} ≥ R
0
φ(u, i
s
) ≤ e
−R
1
u
, u ≥ 0.
ρ
0
, ρ
1
, ρ
2
R
1
= min
0≤s≤2
{ρ
s
}
φ(u, i
s
) ≤ e
−R
1
u
, u ≥ 0.
E(e
ρ
s
Z
1
(1+I
1
)
−1
|I
0
= i
s
) = 1
E(e
ρ
s
Z
1
(1+I
1
)
−1
|I
0
= i
s
) = 1 ⇔ E(e
ρ
s
(Y
1
−1.1)(1+I
1
)
−1
|I
0
= i
s
) = 1 ⇔
E(e
ρ
s
Y
1
(1+I
1
)
−1
−1.1ρ
s
(1+I
1
)
−1
|I
0
= i
s
) = 1 ⇔ E(e
ρ
s
Y
1
(1+I
1
)
−1
e
−1.1ρ
s
(1+I
1
)
−1
|I
0
= i
s
) = 1
E(e
ρ
s
Y
1
(1+I
1
)
−1
e
−1.1ρ
s
(1+I
1
)
−1
|I
0
= i
s
) =
j∈J
∞
0
e
ρ
s
y(1+j)
−1
e
−1.1ρ
s
(1+j)
−1
P
Y
1
,I
1
(y, j|I
0
= i
s
)
=
j∈J
∞
0
e
ρ
s
y(1+j)
−1
e
−1.1ρ
s
(1+j)
−1
P
Y
1
(y)P
I
1
(j|I
0
= i
s
)
=
j∈J
∞
0
e
ρ
s
y(1+j)
−1
P
Y
1
(dy)
e
−1.1ρ
s
(1+j)
−1
P
i
s
,j
=
j∈J
M
Y
1
(ρ
s
(1 + j)
−1
)e
−1.1ρ
s
(1+j)
−1
P
i
s
,j
=
j∈J
(1 − 2ρ
s
(1 + j)
−1
)
−
1
2
e
−1.1ρ
s
(1+j)
−1
P
i
s
,j
> → ˆ
→
1
√
1−2x
> →
→ e
−1.1x
ρ
0
, ρ
1
, ρ
2
>
ρ
1
>
ρ
2
>
ρ
3
>
ψ(u, i
s
) ≤ e
−R
1
u
, u ≥ 0
u
u
i
s
= 6% i
s
= 8% i
s
= 10%
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