Hệ động lực ngẫu nhiên trên thang thời gian

d−
R
T. T
R, Z, N, N
0
, [0, 1] ∪ [2, 3], [0, 1] ∪ N, ,
Q, R \ Q, (0, 1),
T σ : T → T
σ(t) = inf{s ∈ T : s > t},
T. ρ : T → T
ρ(t) = sup{s ∈ T : s < t},
T.
inf ∅ = sup T σ(M) = M T
M sup ∅ = inf T ρ(m) = m T
m
T t ∈ T
σ(t) = t
σ(t) > t ρ(t) = t
ρ(t) < t t
a, b ∈ T [a, b] {t ∈ T : a  t  b}
(a, b]; (a, b); [a, b)
{t ∈ T : a < t  b}; {t ∈ T : a < t < b}; {t ∈ T : a  t < b}
T
a
= {t ∈ T : t  a}
k

T =





T min T = −∞
T \ [m, σ(m)) min T = m,
T
k
=





T max T = +∞
T \ (ρ(M), M] max T = M.
I
1
= {t : t }, I
2
= {t : t }, I = I
1
∪ I
2
.
I
T
T µ : T

k
→ R
+
µ(t) = σ(t) − t,
T.
ν : T → R
+
ν(t) = t − ρ(t),
T.
T = R ρ(t) = t = σ(t), µ(t) = ρ(t) = 0;
T = Z ρ(t) = t − 1, σ(t) = t + 1, µ(t) = ν(t) = 1.
h T = hZ
hZ = {kh : k ∈ Z} = {· · · − 3h, −2h, −h, 0, h, 2h, 3h, · · · },
ρ(t) = t − h, σ(t) = t + h, µ(t) = ν(t) = h.
f : T → R f
f
rd− rd− f
rd−
C
rd
C
rd
(T, R).
ld− ld− f
ld−
C
ld
C
ld
(T, R).

f : T → R T
f
ρ
: T → R f
ρ
= f
◦
ρ f
ρ
(t) = f (ρ(t))
t ∈
k
T lim
σ(s)↑t
f(s) f(t
−
) f
t
−
t f
t
−
= f
ρ
(t)
f : T → R T
f f rd− ld−
f rd− f
σ rd−
ρ ld−

f ld− f
ρ
ld−
f T R
f ∇−
t ∈
k
T f
∇
(t) ∈ R ε > 0
U t
|f(ρ(t)) − f(s) − f
∇
(t)(ρ(t) − s)|  ε|ρ(t) − s| s ∈ U.
f
∇
(t) ∈ R ∇− f
f ∇− t ∈
k
T f
∇− T
T = R f
∇
(t) ≡ f

(t)
T = Z f
∇
(t) = f (t) − f (t − 1)
f : T → R T t ∈

k
T
f ∇− t f t
f t f ∇− t
f
∇
(t) =
f(t) − f(ρ(t))
ν(t)
.
t f ∇− t
lim
s→t
f(t) − f(s)
t − s
,
f
∇
(t) = lim
s→t
f(t) − f(s)
t − s
.
f ∇− t
f
ρ
(t) = f (t) − ν(t)f
∇
(t).
f, g : T → R T

∇− t ∈
k
T
f + g : T → R ∇− t
(f + g)
∇
(t) = f
∇
(t) + g
∇
(t).
fg : T → R ∇− t
(fg)
∇
(t) = f
∇
(t)g(t) + f
ρ
(t)g
∇
(t) = f (t)g
∇
(t) + f
∇
(t)g
ρ
(t).
g(t)g
ρ
(t) = 0

f
g
∇− t

f
g

∇
(t) =
f
∇
(t)g(t) − f(t)g
∇
(t)
g(t)g
ρ
(t)
.
p T
1 + µ(t)p(t) = 0, t ∈ T
k
.
R = {p : T → R : p rd − 1 + µ(t)p(t) = 0}.
R
+
= {p : T → R : p rd − 1 + µ(t)p(t) > 0}.
A T. M
1
=
{(a; b] : a, b ∈ T} T

M
1
T m
1
M
1
m
1
((a, b]) = A
b
− A
a
.
m
1
M
1
. µ
A
∇
m
1
M
1
∇
A
− A T
t
0
∈

k
T {t
0
} ∇
A
−
µ
A
∇
({t}) = A
t
− A
t
−
.
a, b ∈ T a  b
µ
A
∇
((a, b)) = A
b
−
− A
a
; µ
A
∇
([a, b)) = A
b
−

− A
a
−
; µ
A
∇
([a, b]) = A
b
− A
a
−
.
E ⊂
k
T µ
A
∇
− f : T → R
µ
A
∇
−

E
f
τ
∇A
τ
f
µ

A
∇
E ∇
A
− A(t) = t
t ∈ T µ
A
∇
∇− T

E
f
τ
∇τ ∇−

b
a
f(τ)∇τ

(a,b]
f(τ)∇τ.
a, b, c ∈ T, α ∈ R f : T → R, g : T → R
ld−

b
a
(f(τ) + g(τ ))∇τ =

b
a

f(τ)∇τ +

b
a
g(τ)∇τ ;

b
a
αf(τ )∇τ = α

b
a
f(τ)∇τ ;

a
b
f(τ)∇τ = −

b
a
f(τ)∇τ ;

c
a
f(τ)∇τ +

b
c
f(τ)∇τ =


b
a
f(τ)∇τ ;

b
a
f(ρ(τ))g
∇
(τ)∇τ = f (b)g(b) − f (a)g(a) −

b
a
f
∇
(τ)g(τ )∇τ ;

b
a
f(τ)g
∇
(τ)∇τ = f (b)g(b) − f (a)g(a) −

b
a
f
∇
(τ)g(ρ(τ ))∇τ.
a, b ∈ T, f : T → R T
ld−
T = R


b
a
f(τ)∇τ =

b
a
f(τ)dτ.
T

b
a
f(τ)∇τ =
















t∈(a,b]

f(t)ν(t) a < b
0 a = b
−

t∈(b,a]
f(t)ν(t) a > b.
∆−
∇−
∇− ∆−
f : T → R T, b ∈ T
k
,
a ∈
k
T, a < b.

b
a
f(τ
−
)∇τ =

b
a
f(τ)∆τ.
p(t)
e
p
(t, t
0

)
y(t) = 1 +

t
a
p(τ)y(τ )∆τ,





y
∇
(t) = p(t
−
)y(t
−
) ∀ t ∈ T
a
y(a) = 1;
h
k
: T × T → R; k ∈ N
0
h
0
(t, s) = 1 h
k+1
(t, s) =


t
s
h
k
(τ, s)∆τ k ∈ N
0
.
h
k
(t, s) t
h
k+1
(t, s) =

t
s
h
k
(τ
−
, s)∇τ.
0  h
k
(t, s) 
(t − s)
k
k!
,
k ∈ N t > s
u(t)

t ∈ T
a
u
a
, p ∈ R
+
u(t)  u
a
+ p

t
a
u(τ
−
)∇τ ∀ t ∈ T
a
,
u(t)  u
a
e
p
(t, a) ∀ t ∈ T
a
.
A = {A
t
}
t∈T
a
A

A
a
= 0 A = (A
t
) (F
t
)−
A t T
a
A = {A
t
}
t∈T
a
EA
t
< ∞, ∀ t ∈ T
a
.
A M
t ∈ T
a
EM
t
A
t
= E

t
a

M
τ
∇A
τ
.
π
(n)
[a, t]
π
(n)
: a = t
(n)
0
< t
(n)
1
< · · · < t
(n)
k
n
= t,
max
i
(ρ(t
(n)
i+1
) − t
(n)
i
)  2

−n
.
(n)
t
(n)
i
N
π
(n)
s
:=
k
n

i=1
M
t
i
1
(t
i−1
,t
i
]
(s).
M
M
s
= lim
n→∞

N
π
(n)
s
∀ s ∈ (a, t].
E

t
a
M
τ
∇A
τ
= E

lim
n→∞

t
a
N
π
(n)
τ
∇A
τ

= lim
n→∞
E


k
n

i=1
M
t
i
(A
t
i
− A
t
i−1
)

= lim
n→∞
E

M
t
A
t
+
k
n

i=1
A

t
i−1
(M
t
i
− M
t
i−1
)

= EM
t
A
t
.
A = (A
t
)
t∈T
a
A
EM
t
A
t
= E

t
a
M

τ
−
∇A
τ
.
(A
t
)
t∈T
a
A = (A
t
) A
t
F
t
−
−
t ∈ I ∩ T
a
A
t
A = A
t
A = (A
t
)
t∈T
a
(F

t
−
)−
A = {A
t
} µ
A
∇
{t} = 0
t ∈ T
a
\ I
1
M = {M
t
} t
M
t
−
= M
t

(a,t]\I
1
(M
τ
− M
τ
−
)∇A

τ
= 0, .
E

t
a
(M
τ
− M
τ
−
)∇A
τ
= E

(a,t]\I
1
(M
τ
− M
τ
−
)∇A
τ
+ E

I
1
∩(a,t]
(M

τ
− M
τ
−
)∇A
τ
= E


s∈I
1
∩(a,t]
(M
s
− M
s
−
)(A
s
− A
s
−
)

.
A
s
F
s
−

s ∈ I
1
∩ (a, t]
E

(M
s
− M
s
−
)(A
s
− A
s
−
)

= E

E(M
s
− M
s
−
)(A
s
− A
s
−
)|F

s
−

= E

(A
s
− A
s
−
)E{(M
s
− M
s
−
)|F
s
−
}

= 0.
E

t
a
(M
τ
− M
τ
−

)∇A
τ
= 0.
E

t
a
M
τ
−
∇A
τ
= E

t
a
M
τ
∇A
τ
= EM
t
A
t
,
(A
t
)
A = (A
t

) A
t
F
t
−
− t ∈ T
a
M
t
T
a
a  s < t
E

t
s
M
τ
−
∇A
τ
= E

t
a
M
τ
−
∇A
τ

− E

s
a
M
τ
−
∇A
τ
= EM
t
A
t
− EM
s
A
s
.
lim
σ(s)↑t
E

t
s
M
τ
−
∇A
τ
= EM

t
−
(A
t
− A
t
−
).
EM
t
−
(A
t
− A
t
−
) = EM
t
A
t
− EM
t
−
A
t
−
,
E(M
t
− M

t
−
)A
t
= 0.
E(M
t
− M
t
−
)E[A
t
| F
t
−
] = 0.
E(M
t
− M
t
−
)(A
t
− E[A
t
| F
t
−
]) = 0.
M

τ
:=





E [A
t
| F
τ
] τ < t
A
t
τ  t.
(M
τ
) (F
τ
)−
E

A
t
− E[A
t
| F
t
−
]


2
= E(M
t
− M
t
−
)(A
t
− E[A
t
| F
t
−
]) = 0.
A
t
− E[A
t
| F
t
−
] = 0
(A
t
) T.
T = N A
t
A
t

F
t−1
− ∀ t = 1, 2, . . .
T = R (A
t
)
X = (X
t
)
t∈T
a
(DL)
M A
X
t
= M
t
+ A
t
∀ t ∈ T
a
M M

A A

X
t
= M
t
+ A

t
= M

t
+ A

t
∀ t ∈ T
a
.
B
t
= A
t
− A

t
= M

t
− M
t
π
(n)
[a, t]
B
π
(n)
s
:= B

a
1
{a}
+
k
n
−1

i=0
B
t
i
1
(t
i
,t
i+1
]
.
B
s
−
= lim
n→∞
B
π
(n)
s
∀ s ∈ [a, t].
EB

t
(A
t
− A

t
) = E

t
a
B
τ
−
∇A
τ
− E

t
a
B
τ
−
∇A

τ
= lim
n→∞
E

k

n

i=1
B
t
i−1
(B
t
i
− B
t
i−1
)

= 0.
E(A
t
− A

t
)
2
= E[B
t
(A
t
− A

t
)] = 0 A

t
− A

t
= 0
t ∈ T
a
. A
t
= A

t
t ∈ T
a
M A
M A
[a; b] b ∈ T
a
X
a
= 0.
π
(n)
: a = t
(n)
0
< t
(n)
1
< · · · < t

(n)
k
n
= b [a, b]
max
i
(ρ(t
(n)
i+1
) − t
(n)
i
) 
1
2
n
π
(n)
⊂ π
(n+1)
X
(n)
= (X
t
j
)
t
j
∈π
(n)

X
t
j
= M
(n)
t
j
+ A
(n)
t
j
, j = 0, 1, , k
n
,
A
(n)
t
j
=
j

i=1
E[X
t
i
− X
t
i−1
|F
t

i−1
]
{F
t
j
}
k
n
j=0
− M
(n)
t
j
= X
t
j
− A
(n)
t
j
M
(n)
t
j
= E(M
(n)
b
|F
t
j

) = E(X
b
− A
(n)
b
|F
t
j
).
X (DL) {A
(n)
b
}
n∈N
(A
(n
k
)
b
)
k∈N
{A
(n)
b
}
n∈N
A
b
. M
A

M
t
= E(X
b
− A
b
|F
t
); A
t
= X
t
− M
t
; ∀ t ∈ [a, b].
M
t
A
t
− lim
k→∞
A
(n
k
)
b
= A
b
, − lim
k→∞

M
(n
k
)
b
= M
b
.
− lim
k→∞
E(M
(n
k
)
b
|G) = E(M
b
|G),
G σ− σ− F
Π =

n∈N
π
(n)
a  s  t  b s, t ∈ Π
A
t
− A
s
= X

t
− X
s
− [E(M
b
|F
t
) − E(M
b
|F
s
)]
= X
t
− X
s
− lim
k→∞

E(M
(n
k
)
b
|F
t
) − E(M
(n
k
)

b
|F
s
)

= lim
k→∞

X
t
− X
s
− E(M
(n
k
)
b
|F
t
) + E(M
(n
k
)
b
|F
s
)

= lim
k→∞


X
t
− X
s
− M
(n
k
)
t
+ M
(n
k
)
s

= lim
k→∞

A
(n
k
)
t
− A
(n
k
)
s


 0 .
Π [a, b] A A
t
 A
s
t > s A
A ξ
ξ
π
(n)
s
:=
k
n

i=1
ξ
t
i−1
1
(t
i−1
,t
i
]
(s).
ξ
s
−
= lim

n→∞
ξ
π
(n)
s
∀ s ∈ (a, b].
E

b
a
ξ
s
−
∇A
s
= lim
n→∞
E

b
a
ξ
π
(n)
s
∇A
s
= lim
n→∞
E


k
n

i=1
ξ
t
i−1
(A
t
i
− A
t
i−1
)

.
n m
k
↑ ∞
E

b
a
ξ
π
(n)
s
∇A
s

= lim
m
k
→∞
E

k
n

i=1
ξ
t
i−1
(A
(m
k
)
t
i
− A
(m
k
)
t
i−1
)

.
A
(m

k
)
E

k
n

i=1
ξ
t
i−1
(A
(m
k
)
t
i
− A
(m
k
)
t
i−1
)

= Eξ
b

k
n


i=1
(A
(m
k
)
t
i
− A
(m
k
)
t
i−1
)

= E

ξ
b
A
(m
k
)
b

.
E

b

a
ξ
s
−
∇A
s
= lim
n→∞
E

b
a
ξ
π
(n)
s
∇A
s
= E

ξ
b
A
b

.
E

b
a

ξ
s
−
∇A
s
= E

ξ
b
A
b

,
A = (A
t
)
M ∈ M
2
M
2
M = (M
t
)
t∈T
a
M
2
t
− M
t

M
t
M
L (φ
t
)
t∈T
a
T
a
× Ω T
a
(F
ρ(t)
)−
P σ− T
a
× Ω
L P {(s, t] × F :
s, t ∈ T
a
, s < t, F ∈ F
s
}
σ− P
φ σ−
P.
i) T = N φ
t
φ

t
F
t−1
−
ii) T = R φ
t
σ−
Φ
φ : T
a
× Ω → R
Φ φ φ ∈ L;
{φ
n
} ⊂ Φ lim
n→∞
φ
n
= φ
Φ.
Φ
M ∈ M
2
L
2
(M)
φ = {φ
t
}
t∈T

a
,
φ
2
T,M
= E

T
a
φ
2
τ
∇M
τ
< ∞, ∀ T > a.
b > a L
2
((a, b]; M) L
2
(M)
(a, b] L
2
((a, b]; M)
φ
2
b,M
= E

b
a

φ
2
τ
∇M
τ
.
φ, φ ∈ L
2
((a, b]; M) φ − φ
b,M
= 0
φ [a, b]
π : a = t
0
< t
1
< · · · < t
n
= b [a, b]
{f
i
} f
i
F
t
i−1
− i = 1, n
φ(t) =
n


i=1
f
i
1
(t
i−1
,t
i
]
(t); t ∈ (a, b].
L
0
L
0
L
2
((a, b]; M)
d(φ, ϕ)
2
= φ − ϕ
2
b,M
= E

b
a
|φ
τ
− ϕ
τ

|
2
∇M
τ
.
L
0
⊂ L
2
((a, b]; M) φ ∈ L
2
((a, b]; M)
φ
K
(t, ω) := φ(t, ω)1
[−K,K]
(φ(t, ω)).
φ
K
∈ L
2
((a, b]; M) φ − φ
K

b,M
→ 0 K → +∞.
φ ∈ L
2
((a, b]; M)
φ

(n)
∈ L
0
, n = 1, 2, · · · , φ − φ
(n)

b,M
→ 0 n → ∞.
Υ = {φ ∈ L
2
((a, b]; M) : φ φ
(n)
∈ L
0
φ − φ
(n)

b,M
→ 0 n → ∞}.
Υ φ
(n)
∈ Υ, φ
(n)
 < K K > 0
φ
(n)
↑ φ φ ∈ Υ. φ ∈ L
φ
(n)
(t) := φ(σ(t

i
)), t ∈ (t
i
, t
i+1
] i =
0, k
n
− 1,
{t
i
} [a, b] max
i
(ρ(t
i+1
) − t
i
)  2
−n
.
φ
(n)
∈ L
0
φ
(n)
− φ
b,M
→ 0 n → ∞.
Υ

Υ = L
2
((a, b]; M).
φ L
0
,

b
a
φ
τ
∇M
τ
:=
k
n

i=1
f
i
(M
t
i
− M
t
i−1
),
∇− φ ∈ L
0
M (a, b].

∇−

b
a
φ
τ
∇M
τ
F
b
−
φ L
0
α, β
E

b
a
φ
τ
∇M
τ
= 0,
E


b
a
φ
τ

∇M
τ

2
= E


b
a
φ
2
τ
∇M
τ

,

b
a
[αφ
τ
+ βξ
τ
]∇M
τ
= α

b
a
φ

τ
∇M
τ
+ β

b
a
ξ
τ
∇M
τ
.
φ ∈ L
2
((a, b]; M) {φ
(n)
} ⊂ L
0
φ − φ
(n)

b,M
→ 0 n → ∞.
E


b
a
φ
(n)

τ
∇M
τ
−

b
a
φ
(m)
τ
∇M
τ

2
= φ
(m)
− φ
(n)

2
b,M
,
{

b
a
φ
(n)
(τ)∇M
τ

} {

b
a
φ
(n)
(τ)∇M
τ
}
ξ L
2
(Ω, F, P)
ξ = L
2
− lim
n→∞

b
a
φ
(n)
τ
∇M
τ
.
ξ {φ
(n)
}
φ ∈ L
2

((a, b]; M) ∇−
φ M ∈ M
2
(a, b]

b
a
φ
τ
∇M
τ

b
a
φ
τ
∇M
τ
= L
2
− lim
n→∞

b
a
φ
(n)
τ
∇M
τ

,
{φ
(n)
} L
0
lim
n→∞
E

b
a
|φ
τ
− φ
(n)
τ
|
2
∇M
τ
= 0.
i) T = N φ ∈ L
2
((a, b]; M) (φ
n
)
(F
n−1
)−


b
a
φ
τ
∇M
τ
=
b

i=a+1
φ
i
(M
i
− M
i−1
).