Дронов С.В. Конспект лекций по теории случайных процессов
Подождите немного. Документ загружается.
π
¯
t¯s
(λ)(t
j
) = 0 j 6∈ ¯s
ϕ
¯
t
(π
¯
t¯s
(λ)) = exp
n
−
1
2
P
i,j∈
¯
t
B
i,j
π
¯
t¯s
(λ)(t
i
)π
¯
t¯s
(λ)(t
j
)
o
=
= exp
n
−
1
2
P
i,j∈¯s
B
i,j
λ(t
i
)λ(t
j
)
o
= ϕ
¯s
(λ),
w(t, ω)
w(0) = 0, w(t) −
w(s) N(0, t − s) t > s
0 ≤ t
1
< t
2
< t
3
< t
4
w(t
2
) − w(t
1
)
w(t
4
) − w(t
3
)
t
1
, ..., t
n
≥ 0
t
1
< ... < t
n
−→
w = (w
1
, ...w
n
) w
j
=
w(t
j
), j = 1, 2, ..., n
X = (w
1
, w
2
−w
1
, ..., w
n
−w
n−1
)
P
X
(x
1
, ..., x
n
) =
n
Y
i=1
[2π(t
i
− t
i−1
)]
−1/2
exp
−
x
2
i
2(t
i
− t
i−1
)
,
t
0
= 0
−→
w = AX A
A =
1 0 0 ... 0
1 1 0 ... 0
... ... ... ... ...
1 1 1 ... 1
.
P
−→
w
(X) = |det A|
−1
P
X
(A
−1
X).
det A = 1
A
−1
=
1 0 0 ... 0 0
−1 1 0 ... 0 0
0 −1 1 ... 0 0
... ... ... ... ... ...
0 0 0 ... −1 1
,
A
−1
X = (x
1
, x
2
− x
1
, ..., x
n
− x
n−1
)
P
−→
w
(x
1
, ..., x
n
) =
n
Y
i=1
exp{−(x
i
− x
i−1
)
2
/2(t
i
− t
i−1
)}
q
2π(t
i
− t
i−1
)
,
x
0
= 0
w
0
(t) = w(t) − tw(1), t ∈ [0, 1].
w
0
(0) = w
0
(1) = 0
t = 1
t ≥ s
K(t, s) = M(w(t) − tw(1))(w(s) − sw(1)) = Mw(t)w(s) −
−tMw(1)w(s) − sMw(1)w(t) + stMw
2
(1).
a Mw
2
(a) = a h > u
Mw(h)w(u) = M(w(h) − w(u))w(u) + Mw
2
(u) = u,
K(t, s) = s − ts − st + st = s(1 − t), t ≥ s.
a
{ξ(t) > a} T
a
a T N
a
τ
a
ξ(t) t
Mτ
a
= π
σ
x
σ
v
exp
−
(a − m
x
)
2
2σ
2
x
1 − Φ
a − m
x
σ
x
!!
,
m
x
= Mξ(t), σ
2
x
= Dξ(t) t
σ
2
v
MT
a
= T Mτ
a
MN
a
=
T σ
v
2πσ
x
exp
−
(a − m
x
)
2
2σ
2
x
.
T = [a, b] ξ(t), t ∈ T
k
t
1
< ... < t
k
∈ T
ξ(t
1
) − ξ(a), ξ(t
2
) − ξ(t
1
), ..., ξ(b) − ξ(t
k
)
P
a
ξ(a) P
s,t
ξ(t) −ξ(s), t > s.
ξ(t
j
) = ξ(a) +
j
X
i=1
(ξ(t
i
) − ξ(t
i−1
)), (t
0
= a),
ξ
¯
t
ξ
0
+ ξ
1
, ξ
0
+ ξ
1
+ ξ
2
, ..., ξ
0
+ ... + ξ
n
, ξ
0
=
ξ(a), ξ
i
= ξ(t
i
) − ξ(t
i−1
) i ≥ 1
ξ(t)
s, t, h ξ(t) − ξ(s) ξ(t + h) − ξ(s + h)
ξ(t)
s → t ξ(s)
P
−→ ξ(t)
ξ(t)
P
a
P
t,s
a = 0 ϕ
t
(λ)
P
t,0
ξ(t) − ξ(0) =
n
X
i=1
ξ
it
n
!
− ξ
(i − 1)t
n
(∀t, n, λ) ϕ
t
(λ) = ϕ
n
t
n
(λ).
λ ϕ
t
(λ)
n → ∞
ξ
t
n
!
− ξ(0)
P
−→ 0,
ϕ
t/n
(λ) n → ∞
Ψ
t
(λ)
ϕ
t
(λ) = exp {Ψ
t
(λ)}.
(∀t, λ) Ψ
t
(λ) = tΨ
1
(λ) ≡ tΨ(λ).
t =
1
n
t = 1
ϕ
1
(λ) = ϕ
n
1/n
(λ) =⇒ exp Ψ(λ) = exp{nΨ
1/n
(λ)},
t =
1
n
ξ
m
n
!
− ξ(0) =
m
X
j=1
ξ
j
n
!
− ξ
j − 1
n
!!
,
ϕ
m/n
(λ) = ϕ
m
1/n
(λ)
Ψ
m
n
(λ) = mΨ
1
n
(λ) =
m
n
Ψ(λ).
t > 0
x
n
t
ξ(x
n
) − ξ(0)
P
−→ ξ(t) − ξ(0),
ϕ
x
n
(λ) → ϕ
t
(λ) x
n
Ψ(λ) → Ψ
t
(λ)
x
n
Ψ(λ) → tΨ(λ)
P
0,1
Ψ(λ)
t Ψ
t
ϕ
t
P
0,t
ξ
¯
t
= (ξ(t
1
), ..., ξ(t
n
))
ϕ
¯
t
(
~
λ) = ϕ
a
(Λ
n
) · exp
n
X
j=1
(t
j
− t
j−1
)Ψ(Λ
n
− Λ
j−1
)
,
ϕ
a
P
a
Λ
k
=
k
X
j=1
λ
j
, Λ
0
= 0.
ξ(t)
F
≤t
= σ({ξ(s), s ≤ t}), F
=t
= σ(ξ(t)), F
≥t
= σ({ξ(s), s ≥ t}).
ξ(t)
(∀t)(∀B ∈ F
≥t
) P(B/F
≤t
) = P(B/F
=t
).
F
≤t
F
=t
F
≥t
ξ(s) −ξ(t) ξ(t) −ξ(u)
u < t < s ξ(t)
{ ξ(t), t ∈ T }
ξ(t)
k n
1
, ..., n
k+1
∈ N∪{0}
t
1
< ... < t
k+1
P(ξ(t
k+1
) = n
k+1
/ ξ(t
1
) = n
1
, ..., ξ(t
k
) = n
k
) =
= P(ξ(t
k+1
) = n
k+1
/ ξ(t
k
) = n
k
) = P
n
k
,n
k+1
(t
k+1
, t
k
).
∆t = t
k+1
− t
k
P
m,n
(∆t) = P(ξ(t + ∆t) = n / ξ(t) = m)−
m n
∆t T = N
ξ(t), t ∈ T
{0, 1, 2, ...}
n, m
P
n,n+1
(∆t) = λ
n
∆t + o(∆t);
P
n,n−1
(∆t) = µ
n
∆t + o(∆t);
P
n,n
(∆t) = 1 − (λ
n
+ µ
n
)∆t + o(∆t);
P
n,m
(0) = δ
n,m
δ
n,m
−
λ
n
, µ
n
, n = 0, 1, ...
P
m,n
(∆t) = o(∆t), |n − m| ≥ 2.
n
n − 1, n, n + 1
λ, µ
(∀n) λ
n
= nλ, µ
n
= nµ.
T
n
n
n F
n
(t) = P(T
n
≥
t)
F
n
(t + ∆t) = F
n
(t)P
n,n
(∆t) =
= F
n
(t) − F
n
(t)(λ
n
+ µ
n
)∆t + o(∆t),
F
n
(t + ∆t) − F
n
(t)
∆t
= (λ
n
+ µ
n
)F
n
(t) + o(1).
∆t → 0
F
n
(0) = 1
F
n
(t) = exp{−(λ
n
+ µ
n
)t},
n
λ
n
+ µ
n
P(T
n
< t) = 1 − e
−((λ
n
+µ
n
)t
, t ≥ 0.
k ≥ 1
P
n,k
(t + h) = P
n,k−1
(t)P
k−1,k
(h) + P
n,k
(t)P
k,k
(h)+
+P
n,k+1
(h)P
k+1,k
(h) + o(h), n > 0,
P
n,0
(t + h) = P
n,0
(t)P
0,0
(h) + P
n,1
(t)P
1,0
(h) + o(h).
P
j,k
(h), j = k − 1, k, k + 1
h
P
0
n,0
(t) = −λ
0
P
n,0
(t) + µ
1
P
n,1
(t),
P
0
n,k
(t) = µ
k+1
P
n,k+1
(t) − (λ
k
+ µ
k
)P
n,k
(t)+
+λ
k−1
P
n,k−1
(t), k > 0.