Дронов С.В. Конспект лекций по теории случайных процессов
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k n ≥ n(k)
M sup |ξ
n(k)
(t) − µ(t)| ≤
r
M sup |ξ
n(k)
(t) − µ(t)|
2
<
1
k
2
,
n(k)
∞
X
k=1
M sup |ξ
n(k)
(t) − µ(t)| = M
∞
X
k=1
sup |ξ
n(k)
(t) − µ(t)| < ∞.
ξ
n(k)
µ(t)
µ
σ
µ(t)
σ
ξ(t) : [a, b] → R
F
t
, t ∈ [a, b]
t [a, t]
σ(F
t
× B[a, t])
t ξ(t)
F
t
ξ(t) F
t
F
t
, t ∈
[a, b].
a = t
0
< t
1
(n) < ... < t
n
(n) = b
n → ∞.
ξ
n
(t) =
n
X
j=1
ξ(t
j
(n))1
{t
j−1
(n)≤t<t
j
(n)}
.
n → ∞ t, ω ξ
n
(t) → ξ(t).
ξ
n
(t, ω) σ(F
t
×
B[a, t])
µ(t), t ∈ [0, b]
A
a(t)
µ
2
(t) − A(t)
A(t) =
Z
t
0
a(s) ds.
µ(t)
σ
t > s
M((µ(t) − µ(s))
2
/F
s
) = M(µ
2
(t)/F
s
) − µ
2
(s) =
= M((A(t) − A(s))/F
s
).
a = t
0
< t
1
< ... < t
N
= b
η(t) =
N
X
j=1
η
j
1
{t
j−1
<t≤t
j
}
,
η
j
F
t
j
, j =
1, ..., N
η ◦ µ(t) ≡
R
t
a
η(s)dµ(s) =
=
P
k≤m(t)
η
k
(µ(t
k+1
) − µ(t
k
)) + η
m(t)
(µ(t) − µ(t
m(t)
),
m(t) = max{k| t
k
≤ t}.
η ◦µ(t)
µ t
k
≥ s
M(η
k
(µ(t
k+1
) − µ(t
k
))/F
s
) =
= M(M(η
k
(µ(t
k+1
) − µ(t
k
))/F
t
k
)/F
s
) =
= M(η
k
M((µ(t
k+1
) − µ(t
k
))/F
t
k
)/F
s
) = 0.
t(k) < s < t
M(η
k
(µ(t) − µ(t
k
))/F
s
) = η
k
(µ(s) − µ(t
k
).
M(η ◦ µ(t)/F
s
) = η ◦ µ(s), s < t.
s < t < u < v, η F
s
¯η
F
u
M(η(µ(t) − µ(s))¯η(µ(v) − µ(u))/F
s
) =
= M(M(η(µ(t) − µ(s))¯η(µ(v) − µ(u))/F
u
)/F
s
) =
M(η(µ(t) − µ(s))¯ηM((µ(v) − µ(u))/F
u
)/F
s
) = 0.
M((η ◦ µ(t))
2
/F
s
) − (η ◦ µ(s))
2
= M
Z
t
s
η
2
(u)a(u) du/F
s
!
,
η ◦µ
η(t)
η
n
(t) =
n−1
X
k=0
η
a + (b − a)
k
n
!
1
{
a+(b−a)
k
n
< t < a+(b−a)
k+1
n
}
,
η ◦ µ(t) = lim
n→∞
η
n
◦ µ(t).
t η
n
◦µ(t)
M(η
n
◦ µ(b) − η
m
◦ µ(b))
2
= M((η
n
− η
m
) ◦ µ(b))
2
=
= M
R
b
a
(η
n
(m) − η
m
(u))
2
a(u) du −→ 0.
η(t)
M
Z
b
a
(η
n
(m) − η
m
(u))
2
a(u) du −→ 0.
M
Z
b
0
η
2
(t)a(t) dt < ∞.
η ◦ µ(t) = lim
n→∞
lim
u→∞
η
n,u
◦ µ(t),
η
n,u
(t) =
Z
t
0
max{η(s), u}exp{−(s − t)n}n ds.
σ
ξ(t)
µ(t) w(t)
µ([a, b]) = w(b) − w(a)
∆
1
∆
2
µ(∆
1
) µ(∆
2
)
ξ(a) µ([a, b])
b
Z
a
ξ(t) dw(t)
ξ(t)
t
Z
a
ξ(s)dw(s) =
n→∞
n
X
k=1
ξ(t
k−1
)(w(t
k
) − w(t
k−1
)),
t
k
, k = 0, ..., n [a, t]
∆ = max
k
(t
k
− t
k−1
) → 0, n → ∞.
ξ(t) 6= 0
h
t+h
Z
t
ξ(s)dw(s) = ξ(t)(w(t + h) − w(t)) + o(
√
h).
R
t
0
w(s)dw(s)
2w(t
k−1
)(w(t
k
)−w(t
k−1
)) = (w
2
(t
k
)−w
2
(t
k−1
))−(w(t
k
)−w(t
k−1
))
2
.
n
X
k=1
w(t
k−1
)(w(t
k
)−w(t
k−1
)) =
1
2
w
2
(t)−
1
2
n
X
k=1
(w(t
k
)−w(t
k−1
))
2
.
n→∞
n
X
k=1
(w(t
k
) − w(t
k−1
))
2
= t.
δ
k
= (w(t
k
) − w(t
k−1
))
2
− (t
k
− t
k−1
), k = 1, ..., n.
n
X
k=1
(w(t
k
) − w(t
k−1
))
2
− t =
n
X
k=1
δ
k
,
δ
k
Mδ
k
= 0
Dδ
k
= M(w(t
k
) − w(t
k−1
))
4
− (t
k
− t
k−1
)
2
= 2(t
k
− t
k−1
)
2
,
ξ
σ
2
Mξ
4
= 3σ
4
M(
n
X
k=1
δ
k
)
2
= 2
n
X
k=1
(t
k
− t
k−1
)
2
≤ 2∆
n
X
k=1
(t
k
− t
k−1
) = 2∆t → 0,
t
Z
0
w(s)dw(s) =
1
2
(w
2
(t) − t
2
).
ξ(t)
dξ(t) = a(t)dt + b(t)dw(t),
a(t) = a(t, ξ(t)) b(t) = b(t, ξ(t))
ξ(t) = ξ(t
0
) +
t
Z
t
0
a(s)ds +
t
Z
t
0
b(s)dw(s), t ≥ t
0
,
s+h
Z
s
q
Ma
2
(t) dt → 0,
s+h
Z
s
q
Mb
2
(t) dt → 0
h → 0 s
a(t, ξ(t)) b(t, ξ(t))
ξ(t), t ≥ t
0
ξ(s) = x
h
a(s, x)h + o(h)
o(
√
h)
b(s, x) a(t)
b(t)
ξ(t)
dw
2
(t) = dt + 2w(t)dt.
w
2
(t) = w(0) +
1
2
t
2
+ 2
t
Z
0
w(s)dw(s),
η
F (x) ξ(t) = η + t
ξ η η
P(η = 0) = 0
X(t) = t
2
+ tη + ξ
t > 0
ξ(t)
ξ(0) = 0 ξ(1/2) −ξ(1/3)
ξ(t)
K(t, s) = min{t, s}, t, s > 0
K(t, s) =
t(1 − s), t, s ∈ [0, 1]
ξ(t) = X sin ωt ω
MX = 1, DX = 0, 2
ξ(t) = X
2
e
−t
2
/2
X
X
MX = 2, DX = 1
U, V
MU = 0, 5, MV = 0, 5; DU = 1, DV = 0, 05.
X(t) = Ut + V t
2
ξ(t) = 2U sin ωt + 3V t
2
+ 5,
U, V MU = 1, MV = 2, DU =
0, 1, DV = 0, 5, ρ(U, V ) = −0, 3
X(t)
X(t) = t − 3 cos t + U(t + cos t) + V cos 2t,
DU = 1, DV = 2 MX(t), K(t, s), DX(t)
λ