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V ar(Y ) = E(Y Y t ) − E(Y )E(Y t )
= E(At X(At X)t ) − E(At X)E((At X)t )
= E(At XX t A) − E(At X)E(X t A)
= At E(XX t )A − At E(X)E(X t )A
= At (E(XX t ) − E(X)E(X t ))A
= At V ar(X)A
= At ΣA.
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Σ = P ΛP t ,
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λi , i = 1, 2, · · · , n=
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0 0 · · · λn
T
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P1n
λ1 0 · · · 0
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P2n 
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((  ×  ((
((
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V ar(Y ) = V ar(At X)
= At ΣA = At P ΛP t A
= P t P ΛP t P
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Var(Y1 ) = Var(Pt1 X) = λ1 '
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((
(
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. Y1 , Y2 , · · · , Yn /%& 8#4.9.#4.#$./(
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tr(Σ) =
n
X
σi2 ,
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i=1
. 40 9-&9-8.404. 4& $-0E& 4. >0$-8I./ A104-040/ tr(AB) = tr(BA) /. &7$5>' 9&- )*(!(S,'
tr(Σ) = tr(P ΛP t ) = tr(ΛP t P ) = tr(ΛP −1 P )
n
X
= tr(Λ) =
λi , i = 1, · · · , n,
i=1
Pn
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t
2
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t
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)-I*%*24*/5 N/4& -"$,-.#
E(dzi (t1 ), dzi (t2 )) = 0.
BE5 5OC
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BE5 5EC
E(dzi (t), dzj (t)) = σi σj ρij (t),
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2&4#'(&C3
cov(dfi , dfj ) = E(dfi , dfj ) − E(dfi )E(dfj ),
BE5 5JC
"#/
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BE5 5PC
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BE5 5SC
cov(dfi , dfj ) = E(dfi , dfj ),
* )* BE5 5EC * BE5 5SC /*Q!* 9!*
E(dfi , dfj ) = σi σj ρij dt,
se i 6= j
E(dfi , dfi ) = V ar(dfi ) = σi2 dt,
e
ρii = 1.
BE5 5TC
BE5 5UC
K&)*=/* 4%#2/I&%"#% G3 & 0*4&% .&"$&/4& $*,#/ 0#%-<0*-/ gi )* BE5 5 C3 *" !" 0*4&% Y 3
Y = At G,
BE5 5VC
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!
n
X
aks akr = 0
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para s 6= r,
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n
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n
X
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At A = I ⇔ At = A−1 ,
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"#$ $ D&
X
"#$ $ E&
8'-' Y = At G+ 1* '=2<-C
'9
gj =
ajs ys .
s
F* "#$ $G& * "#$ $H&C
X
cov(dfi , dfj ) = E(dfi , dfj ) = E
ais ys
s
*

var(dfi ) = E(dfi )2 = E 
X
s
!
X
!2 
ais ys  = E
ajr yr
r
!!
XX
s
r
= σi σj ρij dt
ais air ys yr
!
.
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J-/ ,(2*0.0*2/34' 5*'-<20,8/+ 8'(1,)*0* 9- 7*2'0 Vg )* ),-*(14' n )* 8'-.'(*(2*1 V gi

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
V g1
σ1 dz1
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
 

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V gn
σn dzn
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!
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=
X
X
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j
j
2
E(dfj ) = E
j
=E
X#
X
XXX
s
ajs ajr ys yr
r
XX X
s
r
XX
=E
(dfj2 )
j
j
=E
E(gj2 ) − E(gj )2
s
!
(ajs ajr ys yr )
j
X
ys yr
r
(ajs ajr
j
!!
!!
.!/ / 01
#"$ -2 &(2*&'34(2 +- "&*")"&$-5'+-+( +- $-*&'6 A +-+-2 ($ .!/ / 71 ( .!/ / 81 L(V g) , 2'$9
%5':#-+- %-&!
X
X
X
L(V g) = E
yj yj =
E(yj2 ) =
σj2 .
.!/ /871
j
j
j
;(<- ys =
ys = µs dt + σys dzs ,
paras = 1, 2, · · · , n,
.!/ /8 1
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

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 0 λ2 · · · 0 
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
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.!/ /881
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0
0
···
λn
")+( λi , - G-&'I)#'- +( yi %-&- i = 1, 2, · · · , n/ ( " #"$%&'$()*" +( V y , +-+" %"&L
L(V y) =
n
X
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j=1
M"5*-)+" %-&- .!/ /871= 2(J?( @?(
L(V g) =
n
X
j=1
n
X
λj
j=1
σj2
=
n
X
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f (t, v) dv = −[logB(t, T ) − logB(t, t)] = −logB(t, T ),
345 5C7
t
*&$# 1# .1( * A('* ;.# B(t, t) = 15 D*-'(&'*2
Z T
B(t, T ) = exp −
f (t, v) dv ,
0 ≤ t ≤ T ≤ T.
t
345 547
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r(t) = f (t, t).
345 5N7
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Z t
f (t, T ) = f (0, T ) +
α(u, T ) du +
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34565 7
0
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df (t, T ) = α(t, T ) dt + σ(t, T ) dW (t),
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3456567
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34565C7
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<=6@6C?
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<=6@6D?
t
T
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t
T
f (t, v) dv
t
<=6@6>?
t
T
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<=6@6F?
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2
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dD(t) = −r(t)D(t) dt,
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z
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4567689
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45676?9
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45676N9
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45676O9
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df (t, T ) = α(t, T ) dt + σ(t, T ) dW (t)
σ(t, T )σ ∗ (t, T ) dt + σ(t, T ) [θ(t) + dW (t)]
f t,
σ(t, T )σ ∗ (t, T ) dt + σ(t, T ) dW
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j = 1, 2, 3.
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r
1 # 1
1
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σ (Ti )ξ1 + σ 2 (Ti )ξ2 + σ 3 (Ti )ξ3 ,
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252
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252
252
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σ 1 (T1 ) σ 2 (T1 ) σ 3 (T1 )
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