class="bi x0 y0 w1 h1"
class="bi x0 y1 w1 h2"
class="bi x0 y2 w2 h3"
X(t)
γ > 0
lim sup
t→∞
1
t
log |X
t
|  −γ,
dX
t
= −
p
1 + t
X
t
dt +
1
(1 + t)
p
X
t
dB
t
t  0 X
0
= x
0

2
),
dX
t
= (
1
2(1 + t)
+
q
(1 + t) log(1 + t)
)X
t
dt + (1 + t)
−1/2
(log(1 + t))
−q
dB
t
.
X
t
= (x
0
+ B
t
)(1 + t)
−
1
2
(log(1 + t))

W = {W
t
}
t0
{F
t
}
t0
.
0  a  b < ∞. M
2
([a, b]; R)
f = {f(t)}
t0
(F
t
)
f
2
a,b
= E

b
a
|f(t)|
2
dt < ∞.
f f M
2
([a, b]; R) f −f

atb
[a, b] a = t
0
< t
1
< . . . <
t
k
= b ξ
i
, 0  i  k − 1 ξ
i
F
t
i
g(t) = ξ
0
I
[t
0
,t
1
]
(t) +
k−1

i=1
ξ
i
I

g
M
0
([a, b]; R). g
{W
t
}

b
a
g(t)dW
t
=
k−1

i=0
ξ
i
(W
t
i+1
− W
t
i
).
f ∈ M
2
([a, b]; R)
{g
n

(t)dW
t




2
= E





b
a
[g
n
(t) − g
m
(t)]dW
t




2
= E

b
a


b
a
f(t)dW
t
= lim
n→∞

b
a
g
n
(t)dW
t
,
{g
n
} M
0
([a, b]; R)
lim
n→∞
E

b
a
|f(t) − g
n
(t)|
2




2
= E

b
a
|f(t)|
2
dt;

b
a
[αf(t) + βg(t)]dW
t
= α

b
a
f(t)dW
t
+ β

b
a
g(t)dW
t
.
T > 0 f ∈ M

0  a < b < c  T.
f ∈ M
2
([a, b]; R) t ∈ [a, b]
I(a) = 0; I(t) =

t
a
f(τ)dW
τ
, ∀t ∈ (a, b],
f
W
t

t
a
f(τ)dW
τ
f ∈ M
2
([0, T ]; R) {I(t)}
0tT
{F
t
}.
E

sup
0tT

t
0
g(s)dB
s
,
f ∈ L
1
(R
+
; R) g ∈ L
2
(R
+
; R). x(t)
dx(t)
dx(t) = f(t)dt + g(t)dB
t
.
C
2,1
(R
n
× R
+
; R
+
) V (x, t) :
R
n
×R

V
xx
=

∂
2
V
∂x
i
∂x
j

d×d
=





∂
2
V
∂x
1
∂x
1
···
∂
2
V

=
∂V
∂x
; V
xx
=
∂
2
V
∂x
2
.
{x(t)}
t0
dx(t) = f(t)dt + g(t)dB
t
,
f ∈ L
1
(R
+
; R) g ∈ L
2
(R
+
; R). V ∈ C
2,1
(R × R
+
; R

1
(R
+
; R
d
) g ∈ L
2
(R
+
; R
d×m
). V ∈ C
2,1
(R
d
× R
+
; R)
V (x(t), t)
dV (x(t), t) =

V
t
(x(t), t)+V
x
(x(t), t)f(t)+
1
2
trace


F
t
0
E|x
0
|
2
< ∞. f : R
d
×[t
0
, T ] → R
d
g : R
d
×[t
0
, T ] → R
d×m
dx(t) = f(x(t), t)dt + g(x(t), t)dB(t), t
0
 t  T
x(t
0
) = x
0
x(t) = x
0
+


, T ]; R
d×m
),
t ∈ [t
0
, T ]
[t
0
, T ]
X(t)
X(t)
P {X(t) = X(t), t ∈ [t
0
, T ]} = 1.
K K
x, y ∈ R
d
t ∈ [t
0
, T ]
|f(x, t) − f(y, t)|
2
∨ |g(x, t) − g(y, t)|
2
 K|x − y|
2
,
(x, t) ∈ R
d
× [t


sup
0tT


t
0
g(s)dB(s) −
α
2

t
0
|g(s)|
2
ds

> β

 e
−αβ
.
T > 0 c  0. u(·)
[0, T ] v(·)
[0, T ].
u(t)  c +

t
0
v(s)u(s)ds , 0  t  T,


t
0
g(s)ds
− 1)h
0
t ∈ [0, T ] (e

t
0
g(s)ds
− 1)h
0
< 1.
h(t), u(t) ∈ C([0, T ], R
+
)
ω(t) [0, T ]
0  α < 1. h(t)  ω(t) +

t
0
u(s)h
α
(s)ds, 0  t < T
h(t)  ω(t)
1−α
+ (1 − α)

t



dX(t) = f(X(t), t)dt + g(X(t), t)dW (t), t  0
X
0
= x
0
.
f(x, t) = (f
1
, , f
n
)
T
: R
n
× R
+
→ R
n
,
g = (g
ij
)
n×m
: R
n
× R
+
→ R

)
LV QV
LV (x, t) =
∂
∂t
V (x, t)
n

i=1
f
i
(x, t)
∂
∂x
i
V (x, t)
+
1
2
n

i,j=1
m

k=1
g
ik
(x, t)g
jk
(x, t)

log λ(t) {t : t  T }
τ  0
lim sup
t→∞
log log t
log λ(t)
 τ;
λ(s), λ(t)  λ(s + t), s, t  T.
λ(t), γ > 0
lim sup
t→∞
log X
t
(x
0
)
log λ(t)
 −γ, .
x
0
F
0
λ(t) = e
t
λ(t) = log(t)
λ(t).
V (x, t) ∈ C
2,1
(R
n

t→∞
log(

t
0
ψ
1
(s)ds)
log λ(t)
 ν; lim sup
t→∞

t
0
ψ
2
(s)ds)
log λ(t)
 θ;
lim inf
t→∞
log ξ(t)
log λ(t)
 −µ.
lim sup
t→∞
log |X
t
(x
0

V (x, t)f
i
(x, t)
+
1
2
n

i,j=1
m

k=1
∂
2
∂x
i
∂x
j
V (x, t)g
ik
(x, t)g
jk
(x, t)

dt
+
n

i=1
m


t
0
n

i=1
m

k=1
∂
∂x
i
V (x, s)g
ik
(x, s)dW (s).
log λ(t)  > 0
N = N() k
1
= k
1
()
k−1
2
N
 t 
k
2
N
k  k
1


k=1
g
ik
(X
s
, s)
∂
∂x
i
V (X
s
, s)dW (s) −

t
0
u
2
QV (X
s
, s)ds]  v

 e
−uv
.
u = 2ξ(
k
2
N
), v = ξ(

s
, s)
∂
∂x
i
V (X
s
, s)dW (s)  ξ(
k
2
N
)
−1
log(
k − 1
2
N
)
+ξ(
k
2
N
)

t
0
QV (X
s
, s)ds,
0  t 

QV (X
s
, s)ds
 V (X
0
, 0) +

t
0
LV (X
s
, s)ds + ξ(
k
2
N
)
−1
log(
k − 1
2
N
) +

t
0
ξ(s)QV (X
s
, s)ds
 V (x
0

log(
k − 1
2
N
) +

t
0

ψ
1
(s) + ψ
2
(s)V (X
s
, s)

ds,
0  t 
k
2
N
, k  k
0
(, ω) ∨ k
1
().
V (X
t
, t) 

0  t 
k
2
N
, k  k
0
(, ω) ∨ k
1
().
log V (X
t
, t)  log

V (x
0
, 0) + ξ(
k
2
N
)
−1
log(
k − 1
2
N
) +

t
0
ψ

0
ψ
1
(s)ds

+

t
0
ψ
2
(s)ds
 log

log V (x
0
, 0)ξ(
k
2
N
)
−1
log(
k − 1
2
N
)

t
0

k − 1
2
N
+

t
0
ψ
2
(s)ds
 log

V (x
0
, 0) − (µ + ) log λ(
k
2
N
)
−1
+ (ν + ) log λ(t)

+ log log
k − 1
2
N
+ (θ + ) log λ(t)
 log

V (x

2
N
+ (θ + ) log λ(t)
 log

V (x
0
, 0) + e
(µ+)
.λ(t)
(µ+)
+ λ(t)
(ν+)

+ log log
k − 1
2
N
+ (θ + ) log λ(t)
k−1
2
N
 t 
k
2
N
, k  k
0
(, ω) ∨ k
1

log[λ(t)
−m
V (X
t
, t)]
log λ(t)
−
m − [ν ∨ µ + τ + θ]
p
.
V (x, t) ∈ C
2,1
(R
n
× R
+
; R
+
) ψ
1
(t), ψ
2
(t), ψ
3
(t)
p m
µ, ν, θ, p 0  α  1 ξ(t) > 0
(x, t) ∈ R
n
× R

ψ
2
(s)ds)
log λ(t)
 ν(1 −α);
lim inf
t→∞
log ξ(t)
log λ(t)
 −µ; lim sup
t→∞
log(

t
0
ψ
3
(s)ds)
log λ(t)
 ρ(1 − α).
lim sup
t→∞
log |X
t
(x
0
)|
log λ(t)
 −
m − [ν + τ + (µ ∨ θ ∨ρ)]

s
, s)

ds
 V (x
0
, 0) +
log(
k−1
2
N
)
ξ

k
2
N

+

t
0

ψ
1
(s) + ψ
2
(s)V (X
s
, s) + ψ

2
N
) +

t
0
ψ
1
(s)ds
+

t
0
ψ
3
(s)V (X
s
, s)
α
ds

exp


t
0
ψ
2
(s)ds


t
0
ψ
1
(s)ds

1−α
exp


t
0
ψ
2
(s)ds

+(1 − α) exp(

t
0
ψ
2
(s)ds)

t
0
ψ
3
(s)ds


+

t
0
ψ
1
(s)ds]
1−α
exp(

t
0
ψ
2
(s)ds)
1−α
+ (1 − α) exp(

t
0
ψ
2
(s)ds)

t
0
ψ
3
(s)ds}



t
0
ψ
2
(s)ds.
t t > e log t > 1.
log V (X
t
, t) 
1
1 − α
log{[V (x
0
, 0) + e
(µ+)
.λ(t)
(µ+)
+ λ(t)
(θ+)
]
1−α
+ λ(t)
(1−α)(ρ+)
}
+ (ν + ) log λ(t) + log log(
k − 1
2
N
) + .

t
, t)]
log λ(t)
 −
m − [µ ∨ θ ∨ ρ + τ + ν]
p
.
X
t
(x
0
) =
0, t  0 x
0
= 0 V (x, t) ∈ C
2,1
((R
n
− {0}) × R
+
; R
+
)
ψ
1
(t) ∈ R
1
, ψ
2
(t) ∈ R

t→∞

t
0
ψ
1
(s)ds)
log λ(t)

µα
2
; lim sup
t→∞
log t
log λ(t)
 θ
lim inf
t→∞

t
0
ψ
2
(s)ds
log λ(t)

2γ
1 − α
.
lim sup

V (X
s
, s)
−
QV (X
s
, s)
2V
2
(X
s
, s)

ds,
M(t) =

t
0
1
V (X
s
, s)
n

i=1
m

k=1
g
ik

− log λ(t)




 .
u, v, ν
P

ω : sup
0tω

M(t) −

t
0
u
2V
2
(X
s
, s)
QV s(X
s
, s)ds

> ν

 e
−uv

, s)
V
2
(X
s
, s)
ds
0  t 
k
2
N
, k  k
0
(, ω) ∨ k
1
().
k
2
(, ω)
log V (X
t
, t)  log V (x
0
, 0) + 2α
−1
log(
k − 1
2
N
)

log |X
t
|
log λ(t)

log V (X
t
, t)
p log λ(t)
−
m
p

1
p log λ(t)

log V (x
0
, 0) + 2α
−1
log(
k − 1
2
N
)
+ (θ + ) log λ(t) −
1
2
(1 − α)(1 − α)
−1

+
; R
+
) ψ
1
(t) ∈
R, ψ
2
(t) ∈ R
+
, t ∈ R
+
x = 0
t  0 p, m γ  0 θ ∈ R
|x|
p
 V (x, t), (x, t) ∈ (R
n
− {0}) × R
+
;
LV (x, t)  ψ
1
(t)V (x, t), x ∈ R
n
, t ∈ R
+
;
QV (x, t)  ψ
2

t
(x
0
)|
t
 −
γ − θ
p
,
x
0
F
0