ACTSC 445
Quantitative Enterprise
Risk Management




Final Review
：面条
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6HH kykR *ha* 998
6BMH _2pB2r
.2+2K#2` Ry- kykR
*QMi2Mib
R .2T2M/2M+2 JQ/2HHBM; j
RXR 1HHBTiB+H .Bbi`B#miBQMb X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X j
RXRXR JmHiBp`Bi2 LQ`KH .Bbi`B#miBQM X X X X X X X X X X X X X X X X X X X X j
RXRXk JmHiBp`Bi2 t@.Bbi`B#miBQM X X X X X X X X X X X X X X X X X X X X X X X X 9
RXk *QTmH Ĝ "bB+ T`QT2`iB2b X X X X X X X X X X X X X X X X X X X X X X X X X X X X 8
RXj aQK2 AKTQ`iMi *QTmHb X X X X X X X X X X X X X X X X X X X X X X X X X X X X X d
RX9 am`pBpH *QTmHb X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X N
RX8 .2T2M/2M+2 +QM+2Tib M/ K2bm`2b X X X X X X X X X X X X X X X X X X X X X X X Ry
k aBKmHiBQM Q7 JmHiBp`Bi2 .Bbi`B#miBQM rBi? *QTmH Rk
kXR :2M2`H ai2Tb X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Rk
kXk aBKmHiBQM Q7 *QTmH pB *QM/BiBQMH aKTHBM; X X X X X X X X X X X X X X X X Rk
j 1t2`+Bb2b 7Q` .2T2M/2M+2 JQ/2HHBM; R9
9 *`2/Bi _BbF R3
9XR J2`iQM JQ/2H X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X R3
9Xk SQ`i7QHBQ *`2/Bi _BbF X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X ky
9XkXR aBM;H2 6+iQ` JQ/2H X X X X X X X X X X X X X X X X X X X X X X X X X X X ky
9XkXk JmHiB@6+iQ` JQ/2H X X X X X X X X X X X X X X X X X X X X X X X X X X X X kR
8 1t2`+Bb2b 7Q` *`2/Bi _BbF kk
e *TBiH HHQ+iBQM kj
eXR aQK2 tBQKb 7Q` *TBiH HHQ+iBQM X X X X X X X X X X X X X X X X X X X X X X kj
eXk J2i?Q/b 7Q` *TBiH HHQ+iBQM X X X X X X X X X X X X X X X X X X X X X X X X X k9
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考试范围
一共 d 道题，题型包含文字题，计算题，证明题（T`QQ7f/BbT`QQ7）。
Ç *?Ti2` R@k , R 题
Ç *?Ti2` j U_BbF J2bm`2V, k 题
Ç *?Ti2` 9 U1ti`2K2 oHm2 h?2Q`vV, R 题
Ç *?Ti2` 8 U.2T2M/2M+2 JQ/2HBM;V, k 题
Ç *?Ti2` e U*`2/Bi `BbFV, k 题
Ç *?Ti2` d U*TBiH HHQ+iBQMV, R 题
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R .2T2M/2M+2 JQ/2HHBM;
RXR 1HHBTiB+H .Bbi`B#miBQMb
RXRXR JmHiBp`Bi2 LQ`KH .Bbi`B#miBQM
.2}MBiBQM RXR UJmHiBp`Bi2 MQ`KH /Bbi`B#miBQMV X = (X1, X2, . . . , Xd)T ?b KmH@
iBp`Bi2 MQ`KH UQ` :mbbBMV /Bbi`B#miBQM B7
X
d
= µ+ LZ (aiQ+?biB+ _2T`2b2MiiBQM)
r?2`2 Z = (Z1, . . . , Zk)T Bb p2+iQ` Q7 BB/ mMBp`Bi2 biM/`/ MQ`KH `M/QK p`B#H2b
M/ A ∈ Rd×k,µ ∈ RdX
Ç 1[X] = µ+ AZ = µ
Ç *Qp(X) = L · *Qp(Z) · LT = LLT =: Σ U*?QH2bFv /2+QKTQbBiBQMV
S`QTQbBiBQM RXR
X ∼MN(µ,Σ)⇔ aTX ∼ N(aTµ,aTΣa), ∀a ∈ Rd.
这个定理说明对于任意一个 KmHiBp`Bi2 MQ`KH /Bbi`B#miBQM，它的任意 HBM2` +QK#BMiBQM
必须是一个 mMBp`Bi2 MQ`KH /Bbi`B#miBQM。
aKTHBM; MN(µ,Σ)
G2i X ∼MN(µ,Σ) rBi? bvKK2i`B+ Σ M/ TQbBiBp2 /2}MBi2X
RX *QKTmi2 i?2 *?QH2bFv 7+iQ` L Q7 Σ- r?2`2 Σ = LLT X
kX :2M2`i2 Zj ind.∼ N(0, 1), j ∈ {1, . . . , d}X
jX _2im`M X = µ+ LZ- r?2`2 Z = (Z1, . . . , Zd)T X
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RXRXk JmHiBp`Bi2 t@.Bbi`B#miBQM
.2}MBiBQM RXk UJmHiBp`Bi2 MQ`KH /Bbi`B#miBQMV X = (X1, X2, . . . , Xd)T ?b KmH@
iBp`Bi2 t@/Bbi`B#miBQM MT (µ,Σ, ν) B7
X
d
= µ+
Z√
V /ν
(aiQ+?biB+ _2T`2b2MiiBQM)
r?2`2 Z = (Z1, . . . , Zk)T Bb p2+iQ` Q7 BB/ mMBp`Bi2 biM/`/ MQ`KH `M/QK p`B#H2b
M/ A ∈ Rd×k,µ ∈ Rd, V ∼ χ2ν M/ Z M/ V `2 BM/2T2M/2MiX
aKTHBM; MT (µ,Σ, ν)
G2i X ∼MT (µ,Σ, ν) rBi? bvKK2i`B+ Σ M/ TQbBiBp2 /2}MBi2X
RX *QKTmi2 i?2 *?QH2bFv 7+iQ` L Q7 Σ- r?2`2 Σ = LLT X
kX :2M2`i2 Zj ind.∼ N(0, 1), j ∈ {1, . . . , d}X
jX :2M2`i2 `M/QK MmK#2` V 7`QK χ2ν BM/2T2M/2Mi Q7 Zj, j = 1, . . . , d
9X _2im`M X = µ+ Z√
V /ν
- r?2`2 Z = (Z1, . . . , Zd)T X
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RXk *QTmH Ĝ "bB+ T`QT2`iB2b
.2}MBiBQM RXj U*QTmHV +QTmH C Bb /7 rBi? U(0, 1) K`;BMbX
C(u1, . . . , ud) = P(U1 ≤ u1, . . . , Ud ≤ ud)
*?`+i2`BxiBQM, C Bb +QTmH B7 M/ QMHv B7
Ç C Bb ;`QmM/2/- BX2X- C(u1, . . . , ud) = 0 B7 uj = 0 7Q` i H2bi QM2 j ∈ {1, . . . , d}
Ç C ?b mMB7Q`K mMBp`Bi2 K`;BMb- i?i Bb-
C(1, . . . , 1, uj, 1, . . . , 1) = uj
7Q` HH uj ∈ [0, 1] M/ j ∈ {1, . . . , d}X
Ç C Bb d@BM+`2bBM;- i?i Bb 7Q` (a1, . . . , ad), (b1, . . . , bd) ∈ [0, 1]d, ai ≤ bi, i = 1, . . . , d
2∑
i1=1
· · ·
2∑
id=1
(−1)i1+···+idC(u(1)i1 , . . . , u(d)id ) ≥ 0
r?2`2 u(k)1 = ak M/ u(k)2 = bk 7Q` k = 1, . . . , dX
_2K`F RXR A7 d = 2- i?2 d−BM+`2bBM; +QM/BiBQM im`Mb iQ b?Qr
C(a1, a2)− C(a1, b2)− C(b1, a2) + C(b1, b2) ≥ 0.
_2K`F RXk A7 *QTmH C ?b DQBMi T/7 c- i?2M Bi Bb Hrvb d@BM+`2bBM;X
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G2KK RXR US`Q##BHBiv i`Mb7Q`KV A7 X ?b /7 F - r?2`2 F Bb +QMiBMmQmb mMBp`Bi2
/7- i?2M F (X) ∼ U(0, 1)X
G2KK RXk UZmMiBH2 i`Mb7Q`KV A7 U ∼ U(0, 1) ?b biM/`/ mMB7Q`K /Bbi`B#miBQM-
i?2M X d= F←(U) ∼ F - r?2`2 F← /2MQi2 ;2M2`HBx2/ BMp2`b2- BX2X- F←(p) = BM7{x : F (x) ≥
p}X
h?2Q`2K RXR UaFH`Ƕb h?2Q`2KV
URV 6Q` Mv /7 F rBi? K`;BMb F1, . . . , Fd- i?2`2 2tBbib +QTmH C bm+? i?i
F (x1, . . . , xd) = C(F1(x1), . . . , Fd(xd)) URXRV
C Bb mMB[m2Hv /2}M2/ QM Poddj=1_MFj M/ ;Bp2M #v
C(u1, . . . , ud) = F (F
←
1 (u1), . . . , F
←
d (ud))
r?2`2 _MFj /2MQi2b i?2 `M;2 Q7 FjX 6Q` +QMiBMmQmb /Bbi`B#miBQM- F← = F−1X
UkV *QMp2`b2Hv- ;Bp2M Mv +QTmH C M/ mMBp`Bi2 /7b F1, . . . , Fd- F /2}M2/ #v URXRV Bb /7
rBi? K`;BMb F1, . . . , Fd-
_2K`F RXj aFH`Ƕb h?2Q`2K 使得我们可以把研究 DQBMi /Bbi`B#miBQM 的问题分成两部分。
一部分是各自的 K`;BMH /Bbi`B#miBQM，另一部分是相互间的 /2T2M/2M+2 U用 +QTmH刻画V
_2K`F RX9 A7 d = 2- H2i (X, Y ) #2 `M/QK p2+iQ` rBi? DQBMi +/7 H(x, y)X bbmK2 i?2B`
K`;BMH +/7Ƕb F M/ G `2 +QMiBMmQmb M/ bi`B+iHv BM+`2bBM;X G2i F−1 M/ G−1 #2 i?2
BMp2`b2 Q7 F M/ G `2bT2+iBp2HvX h?2M- i?2 +QTmH Q7 (X, Y ) Bb ;Bp2M #v
C(u, v) = H(F−1(u), G−1(v))
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RXj aQK2 AKTQ`iMi *QTmHb
Ç AM/2T2M/2M+2 +QTmH
Π(u) =
d∏
j=1
uj
Ç *QmMi2`KQMQiQMB+Biv +QTmH
W (u1, u2) = Kt{u1 + u2 − 1, 0}
_2K`F RX8
Ĝ (U, 1−U) ∼ W , h?Bb Bb T2`72+iHv M2;iBp2Hv /2T2M/2Mi- Q` +QmMi2`@+QKQMQiQMB+
Ĝ X1 M/ X2 `2 +QmMi2`KQMQiQMB+ B7 M/ QMHv B7
(X1, X2)
d
= (T1(U), T2(1− U))
7Q` `p U rBi? T1 M/ T2 #Qi? bi`B+iHv BM+`2bBM; M/ +QMiBMmQmbX
Ĝ PMHv rQ`Fb 7Q` d = 2
Ç *QKQMQiQMB+Biv +QTmH
M(u) = KBM
1≤j≤d
{uj}.
_2K`F RXe
Ĝ (U, . . . , U) ∼M , h?Bb Bb T2`72+iHv TQbBiBp2Hv /2T2M/2Mi- Q` +QKQMQiQMB+
Ĝ X1, . . . , Xd `2 bB/ iQ #2 +QKQMQiQMB+ B7 M/ QMHv B7
(X1, . . . , Xd)
d
= (T1(U), . . . , Td(U))
7Q` U M/ bi`B+iHv BM+`2bBM; M/ +QMiBMmQmb T1, . . . , Td
h?2Q`2K RXk U6`û+?2i@>Q2z/BM; #QmM/bV G2i W (u) = Kt{∑dj=1 uj−d+1, 0}
M/ M(u) = KBM1≤j≤d{uj}X
URV 6Q` Mv d@/BK2MbBQM +QTmH C-
W (u) ≤ C(u) ≤M(u)
UkV W Bb +QTmH B7 M/ QMHv B7 d = 2
UjV M Bb +QTmH 7Q` HH d ≥ 2X
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Ç :mbbBM +QTmHb
Cρ(u) = P(Φ(X1) ≤ u1, . . . ,Φ(Xd) ≤ ud)
= P(X1 ≤ Φ−1(u1), . . . , Xd ≤ Φ−1(ud))
= Φρ(Φ
−1(u1), . . . ,Φ−1(ud))
h?2 +QMbi`m+iBQM Q7 :mbbBM +QTmH bi`ib 7`QK MN(0,ρ)- ?Qr2p2`- i?2 /2T2M/2M+2
bi`m+im`2 +M #2 2ti2M/2/ iQ Mv K`;BMH /Bbi`B#miBQMbX
Ç `+?BK2/2M +QTmHb
C(u) = φ←(φ(u1) + · · ·+ φ(ud))
r?2`2 φ Bb i?2 ;2M2`iQ` biBb7vBM;
Ĝ φ : [0, 1] → [0,∞) Bb /2+`2bBM;- +QMiBMmQmb- M/ +QMp2t 7mM+iBQM M/ biBb}2b
φ(1) = 0X
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RX9 am`pBpH *QTmHb
Ç A7 U ∼ C- i?2M 1−U ∼ ĈX
Ç
Ĉ(u1, . . . , ud) = P(1− U1 ≤ u1, . . . , 1− Ud ≤ ud)
h?2M r2 ?p2 i?2 B/2MiBiv
F¯ (x1, . . . , xd) = P(X1 > x1, . . . , Xd > xd)
= P(1− F1(X1) ≤ F¯1(x1), . . . , 1− Fd(Xd) ≤ F¯d(xd))
= Ĉ(F¯1(x1), . . . , F¯d(xd))
因为通过 Ĉ 连接的是 DQBMi bm`pBpH 7mM+iBQM 和 K`;BMH bm`pBpH 7mM+iBQM，所以 Ĉ
叫做 bm`pBpH +QTmH。
Ç A7 Ĉ = C- C Bb +HH2/ `/BHHv bvKK2i`B+X
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RX8 .2T2M/2M+2 +QM+2Tib M/ K2bm`2b
GBM2` +Q``2HiBQM
.2}MBiBQM RX9 US2`bQMǶb rV
rXY =
*Qp(X, Y )√o`(X)√o`(Y )
_2K`F RXd Ai +M QMHv +Tim`2 HBM2` /2T2M/2M+2X AiǶb TQbbB#H2 i?i mM+Q``2Hi2/ `M/QK
p`B#H2b `2 /2T2M/2MiX MQi?2` /`r#+F Bb +Q``2HiBQM ρ Kv MQi #2 r2HH@/2}M2/ 7Q`
+2`iBM /Bbi`B#miBQMbX
_MF +Q``2HiBQM
.2}MBiBQM RX8 UE2M/HHǶb imV G2i X ∼ F, Y ∼ G rBi? F,G +QMiBMmQmbX G2i (X ′, Y ′)
#2 M BM/2T2M/2Mi +QTv Q7 (X, Y )X E2M/HHǶb im Bb /2}M2/ #v
τK = E(bB;M((X −X ′)(Y − Y ′)))
= P((X −X ′)(Y − Y ′) > 0)− P((X −X ′)(Y − Y ′) < 0)
Ç P((X −X ′)(Y − Y ′) > 0), i?2 T`Q##BHBiv Q7 +QM+Q`/M+2c
Ç P((X −X ′)(Y − Y ′) < 0), i?2 T`Q##BHBiv Q7 /Bb+Q`/M+2X
S`QTQbBiBQM RXk U6Q`KmH 7Q` E2M/HHǶb imV A7 X M/ Y `2 +QMiBMmQmb `M/QK p`B@
#H2b- i?2M
τK = 4
∫ 1
0
∫ 1
0
C(u, v)dC(u, v)− 1
S`QTQbBiBQM RXj U6Q`KmH 7Q` E2M/HHǶb im U`+?BK2/2M *QTmHVV A7 X M/ Y
`2 +QMiBMmQmb `M/QK p`B#H2b- i?2M
τK = 1 + 4
∫ 1
0
φ(t)
φ′(t)
dt
.2}MBiBQM RXe UaT2`KMǶb `?QV G2i X ∼ F, Y ∼ G rBi? F,G +QMiBMmQmbX aT2`KMǶb
`?Q Bb /2}M2/ #v
ρS = *Q``(F (X), G(Y ))
S`QTQbBiBQM RX9 U6Q`KmH 7Q` bT2`KMǶb `?QV A7 X M/ Y `2 +QMiBMmQmb `M/QK
p`B#H2b- i?2M
ρS = 12
∫ 1
0
∫ 1
0
uvdC(u, v)− 3.
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*Q2{+B2Mib Q7 iBH /2T2M/2M+2
.2}MBiBQM RXd UhBH /2T2M/2M+2V G2i X ∼ F, Y ∼ G rBi? F,G +QMiBMmQmbX S`QpB/2/
i?i i?2 HBKBib 2tBbi- i?2 HQr2` iBH@/2T2M/2M+2 +Q2{+B2Mi λL M/ mTT2` iBH@/2T2M/2M+2
+Q2{+B2Mi λU Q7 X1 M/ X2 `2 /2}M2/ #v
λL = HBK
q→0+
P(X ≤ F−1(q)|Y ≤ G−1(q))
λU = HBK
q→1−
P(X > F−1(q)|Y > G−1(q))
A7 λL = 0(λU = 0)- i?2M X M/ Y `2 HQr2` UmTT2`V iBH BM/2T2M/2MiX
S`QTQbBiBQM RX8 U6Q`KmH 7Q` hBH /2T2M/2M+2V G2i (U, V ) #2 `M/QK p2+iQ` rBi?
C(u, v) b Bib +/7- i?2M
λL = HBK
t→0+
C(t, t)
t
= HBK
t→0+
d
dt
C(t, t)
λU = HBK
t→1−
1− 2t+ C(t, t)
1− t = 2− HBKt→1−
d
dt
C(t, t)
_2K`F RX3 h?2 7QHHQrBM; i#H2 T`QpB/2b iBH /2T2M/2M+2 7Q` +QKKQM +QTmHb,
*QTmH λL λU
:mbbBM y y
t > 0 > 0
:mK#2H 0 > 0
*HviQM > 0 0
6`MF > 0 > 0
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k aBKmHiBQM Q7 JmHiBp`Bi2 .Bbi`B#miBQM rBi? *QTmH
a2imT
Ç Fi, J`;BMH /Bbi`B#miBQM 7Q` `M/QK p`B#H2 Xi, i = 1, . . . , d
Ç C, +QTmH mb2/ iQ /2b+`B#2 i?2 /2T2M/2M+2 bi`m+im`2 KQM; (X1, . . . , Xd)
h?2 K`;BMH /Bbi`B#miBQM M/ DQBMi /Bbi`B#miBQM `2 HBMF2/ #v +QTmH C rBi? 7QHHQrBM;
`2HiBQM,
F (x1, . . . , xd) = C(F1(x1), . . . , Fd(xd))
kXR :2M2`H ai2Tb
ai2T R aBKmHi2 U1, . . . , Ud #b2/ QM +QTmH C
ai2T k G2i Xi = F−1i (Ui) U[mMiBH2 i`Mb7Q`KV
一个自然的问题就是如何生成 7QHHQr 某一个指定 +QTmH C 的 mMB7Q`K p`Bi2b？
kXk aBKmHiBQM Q7 *QTmH pB *QM/BiBQMH aKTHBM;
6Q` bBKTHB+Biv- r2 `2bi`B+i Qm`b2Hp2b iQ #Bp`Bi2 +b2- BX2X- (U, V ) ∼ C(u, v)X "2bB/2b- r2
bbmK2 i?2 +QTmH ?b /2MbBiv 7mM+iBQM h(u, v)- BX2X-
h(u, v) =
∂2
∂u∂v
C(u, v)
G2KK kXR A7 i?2 +QTmH C ?b /2MbBiv 7mM+iBQM- i?2M
C2|1(v|u) = P(V ≤ v|U = u) = ∂∂uC(u, v).
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aBKmHiBQM ai2Tb rBi? *QM/BiBQMH aKTHBM; U"Bp`Bi2V
ai2T R aBKmHi2 U,W iid∼ U(0, 1)
ai2T k G2i V = C−12|1(W |U) UZmMiBH2 i`Mb7Q`KV
ai2T j G2i X = F−1X (U), Y = F−1Y (V ) U[mMiBH2 i`Mb7Q`KV
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j 1t2`+Bb2b 7Q` .2T2M/2M+2 JQ/2HHBM;
S`Q#H2K jXR G2i Z1, Z2, Z3 ind∼ F (x) = 2tT(−1/x), x > 0X 6m`i?2`KQ`2- H2i
X1 = Kt{Z1, Z2},
X2 = Kt{Z1, Z3}.
RX *H+mHi2 i?2 /Bbi`B#miBQM 7mM+iBQM H(x1, x2) = S`(X1 ≤ x1, X2 ≤ x2), x1, x2 ≥ 0- Q7
i?2 `M/QK p2+iQ` (X1, X2)X
kX *H+mHi2 i?2 +QTmH C r?B+? +Q``2bTQM/b iQ H #v aFH`Ƕb h?2Q`2KX a?Qr i?i Bi
BM/22/ ?b U(0, 1) K`;BMbX
jX :Bp2 biQ+?biB+ `2T`2b2MiiBQM Q7 (X1, X2) ∼ H BM i2`Kb Q7 V1, V2, V3 ind∼ U(0, 1)X
9X :Bp2 biQ+?biB+ `2T`2b2MiiBQM Q7 (U1, U2) ∼ C BM i2`Kb Q7 V1, V2, V3 ind∼ U(0, 1)X
8X :Bp2 biQ+?biB+ `2T`2b2MiiBQM 7Q` (Y1, Y2) ∼ G- r?2`2 G Bb /Bbi`B#miBQM 7mM+iBQM
rBi? +QTmH C M/ N(0, 1) K`;BMb BM i2`Kb Q7 V1, V2, V3 ind∼ U(0, 1)X
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S`Q#H2K jXk :Bp2 biQ+?biB+ `2T`2b2MiiBQM Q7 #Bp`Bi2 +QKQMQiQM2 `M/QK p2+iQ`
rBi? N(0, 1) M/ t2.5 K`;BMbX
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S`Q#H2K jXj G2i α ∈ (0, 1) M/ X1, X2 #2 +QKQMQiQM2X h?2M
o_α(X1 +X2) = o_α(X1) + o_α(X2)
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S`Q#H2K jX9 G2i C1 M/ C2 #2 irQ d@/BK2MbBQMH +QTmHb M/ /2}M2- 7Q` γ ∈ [0, 1]-
C(u) = γC1(µ) + (1− γ)C2(µ), µ ∈ [0, 1]d.
a?Qr i?i * Bb HbQ pHB/ +QTmHX
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9 *`2/Bi _BbF
LQiiBQMb
Ç _2+Qp2`v `i2 U__V, T2`+2Mi;2 Q7 i?2 K`F2i pHm2 Q7 i?2 +QMi`+i i?i `2KBMb
7i2` i?2 Q++m``2M+2 Q7 /27mHi
Ç HQbb ;Bp2M /27mHi UG:.V, T2`+2Mi;2 i?i Bb HQbi
G:. = 1− __
Ç S`Q##BHBiv Q7 /27mHi US.V
9XR J2`iQM JQ/2H
a2imT
Ç *QMbB/2` }`K r?B+? }MM+2b Bib2H7 #v 2[mBiv UBX2X- #v BbbmBM; b?`2bV M/ #v /2#i
Ç h?2`2 Bb bBM;H2 /2#i- r?B+? Bb x2`Q@+QmTQM #QM/ rBi? 7+2 pHm2 B M/ Kim`Biv T
Ç LQ /BpB/2M/b `2 TB/ Qmi M/ MQ M2r /2#i Bb Bbbm2/ mT iQ i?2 Kim`Biv /i2 T
Ç h?2 `BbF 7`22 BMi2`2bi `i2 Bb /2i2`KBMBbiB+ M/ 2[mH iQ r ≥ 0
Ç .27mHi Q++m`b B7 i?2 }`K KBbb2b TvK2Mi iQ Bib /2#i ?QH/2`b- r?B+? +M QMHv Q++m`
i iBK2 T
Ç h?2 }`K pHm2 T`Q+2bb (Vt)t≥0 7QHHQrb :2QK2i`B+ "`QrMBM KQiBQM
dVt = µVtdt+ σVtdWt
Ç G2i St M/ Bt `2bT2+iBp2Hv /2MQi2 i?2 pHm2b Q7 i?2 2[mBiv M/ x2`Q@+QmTQM #QM/ i
iBK2 t 7Q` t ∈ [0, T ]
Ĝ h?2 TvQz iQ i?2 #QM/?QH/2`
BT = KBM{VT , B} = B − (B − VT )+
r?B+? Bb /27mHi@7`22 /2#i THmb b?Q`i TQbBiBQM BM Tmi r`Bii2M QM i?2 }`K
pHm2
Ĝ h?2 TvQz iQ i?2 b?`2?QH/2`
ST = Kt{VT − B, 0} = (VT − B)+
r?B+? Bb bBKTHv HQM; +HH TQbBiBQM QM i?2 }`K pHm2
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AKTQ`iMi 6Q`KmHb
6QHHQrBM; 7Q`KmHb `2 /2`Bp2/ #b2/ QM "H+F@a+?QH2b 7Q`KmH- +Qp2`2/ BM *ha* 99eX
hBK2@t pHm2 Q7 i?2 }`K
Vt = V0 2tT{(µ− σ2/2)t+ σWt},
r?2`2 Wt ∼ N(0, t)
hBK2@t pHm2 Q7 i?2 /2#i
Bt = Be
−r(T−t) − PBS(t, Vt; r, σ, B, T )
r?2`2
PBS(t, Vt; r, σ, B, T ) = Be
−r(T−t)Φ(−dt,2)− VtΦ(−dt,1)
M/
dt,1 =
HMVt − HMB + (r + 12σ2)(T − t)
σ
√
T − t , dt,2 = dt,1 − σ
√
T − t
hBK2@t pHm2 Q7 i?2 2[mBiv
St = C
BS(t, Vt; r, σ, B, T )
r?2`2
CBS(t, Vt; r, σ, B, T ) = VtΦ(dt,1)− Be−r(T−t)Φ(dt,2)
.27mHi S`Q##BHBiv,
P(VT < B) = P(V0 2tT{(µ− σ2/2)T + σWT} < B)
= P
⎛⎝WT < HM
(
B
V0
)
−
(
µ− σ22
)
T
σ
√
T
⎞⎠
= Φ
⎛⎝ HM
(
B
V0
)
−
(
µ− σ22
)
T
σ
√
T
⎞⎠
*`2/Bi aT`2/,
+QMbiMi c bm+? i?i B0 = Be−(r+c)T
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a2imT
˙ ei , 1tTQbm‘2 i /27mHi U1.V 7Q‘ bb2i i
˙ +i (0 & +i & 1), GQbb ;Bp2M /27mHi UG:.V7Q‘ bb2i i U1tT2+i2/ T‘QTQ‘iBQM Q7 i?2
2tTQbm‘2 r?B+? Bb HQbi ;Bp2M mTQM i?2 /27mHi 7Q‘ bb2ii V
˙ Yi = 1 { X i & H i } , +‘2/Bi BM/B+iQ‘ 7Q‘ bb2ii X
X i , +‘2/Bi BM/2t- mbmHHv 7QHHQrN (0, 1)
H i , i?‘2b?QH/- mbmHHv /2}M2/ #v /27mHi2/ T‘Q##BHBiv S.i - BX2X-H i = " # 1( S. i )
A7X i & H i - r2 bv /27mHi Q++m‘bX h?2‘27Q‘2-Yi = 1 X
A7X i > H i - MQ /27mHi rQmH/ ?TT2MX h?2‘27Q‘2-Yi = 0
˙ L =
$ n
i =1 ei +i Yi , hQiH TQ‘i7QHBQ HQbb
9XkXR aBM;H2 6+iQ‘ JQ/2H
˙ X i = , i Z
-./0
avbi2KiB+ *QKTQM2Mi
+
1
1 ’ , 2i - i
- ./ 0
A/BQbvM+‘iB+ *QKTQM2Mi
˙ Z , +QKKQM 7+iQ‘
˙ Z, - i , i = 1 , . . . , n iid" N (0, 1)
˙ 6‘QK *ha* jdk- B7 TQ‘i7QHBQ Bb K/2 mT Q7n /Bz2‘2Mi bb2ibX
G2i n ) * - bvbi2KiB+ ‘BbF +MMQi #2 /Bp2‘bB}2/ rv
G2i n ) * - A/BQbvM+‘iB+ ‘BbF rBHH /BbTT2‘
˙ .2}M2 bvbi2KiB+ HQbbb 7QHHQrb,
LS = 1 [L |Z ] =
n
"
i =1
ei +i 1 [Yi |Z ]
=
n
"
i =1
ei +i P(Yi = 1 |Z )
=
n
"
i =1
ei +i P(X i & H i |Z )
=
n
"
i =1
ei +i "
2
"
# 1( S. i ) ’ , i Z!
1 ’ , 2i
3
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SQ‘i7QHBQ o_ Bb
o_ " (L s)
- ./ 0
SQ‘i7QHBQ o_
=
n
"
i =1
ei +i "
2
"
# 1( S. i ) ’ , i " # 1(1 ’ ( )!
1 ’ , 2i
3
- ./ 0
*QMi‘B#miBQM 7‘QK i?2 Bi? T‘i
9XkXk JmHiB@6+iQ‘ JQ/2H
˙ X i =
K
"
k=1
, i,k Zk
- ./ 0
avbi2KiB+ *QKTQM2Mi
+
4556 1 ’
K
"
j =1
,
2
i,k - i
- ./ 0
A/BQbvM+‘iB+ *QKTQM2Mi
˙ Z1, . . . , Zk , +QKKQM 7+iQ‘
˙ Z1, . . . , Zk , - i , i = 1 , . . . , n iid" N (0, 1)
˙ .2}M2 bvbi2KiB+ HQbbb 7QHHQrb,
LS = 1 [L |Z ] =
n
"
i =1
ei +i 1 [Yi |Z ]
=
n
"
i =1
ei +i P(Yi = 1 |Z )
=
n
"
i =1
ei +i P(X i & H i |Z )
=
n
"
i =1
ei +i "
’
( " # 1( S. i ) ’
$ K
k=1 , i,k Zk1
1 ’
$ K
k=1 ,
2
i,k
+
,
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8 1t2`+Bb2b 7Q` *`2/Bi _BbF
S`Q#H2K 8XR *QMbB/2` bBM;H2 7+iQ` KQ/2H
Xi =
√
βiF˜i +
√
1− βiεi, i ∈ {1, . . . ,m}
r?2`2 F˜i, ε1, . . . , εm `2 BM/2T2M/2Mi biM/`/ MQ`KH p`B#H2b M/ 0 ≤ βi ≤ 1 7Q` HH iX
h?2 bvbi2KiB+ p`B#H2b F˜i BM im`M biBb7v
F˜i =
{
F1, i = 1, . . . , n,
ρF1 +
√
1− ρ2F2, i = n+ 1, . . . ,m,
r?2`2 F1, F2 `2 BM/2T2M/2Mi biM/`/ MQ`KH 7+iQ`b- 0 ≤ ρ ≤ 1 M/ 1 < n < mƐ1X P#pB@
QmbHv- i?Bb KQ/2H ?b irQ@;`QmT bi`m+im`2 /2i2`KBMBM; i?2 /2T2M/2M+2 #2ir22M /27mHibX
.2`Bp2 2tT`2bbBQMb 7Q` i?2 rBi?BM@;`QmT bb2i +Q``2HiBQMb U7Q` 2+? Q7 i?2 irQ ;`QmTbV M/
i?2 #2ir22M@;`QmT bb2i +Q``2HiBQMX
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˙ bbmK2 X 1, . . . , Xd ‘2 i?2 d ‘BbFb ‘BbBM; 7‘QKd #mbBM2bb mMBib i?i }‘K THMb iQ
2M;;2 M/ M22/b iQ HHQ+i2 +TBiH
˙ bbmK2 i?2 }‘K@rB/2 2+QMQKB+ +TBiH- 1*- Bb /2i2‘KBM2/ mbBM; i?2 ‘BbF K2bm‘2$-
BX2X- 1*= $(X ) - r?2‘2 X = $ di =1 X i
˙ *TBiH HHQ+iBQM+QMbB/2‘b ?Qr Km+? +TBiH b?QmH/ #2 HHQ+i2/ iQ 2+? #mbBM2bb
mMBi
˙ G2i 1* i /2MQi2b i?2 2+QMQKB+ +TBiH HHQ+i2/ iQ #mbBM2bb mMBii, i = 1 , . . . , d
eXR aQK2 tBQKb 7Q‘ *TBiH HHQ+iBQM
RX6mHH HHQ+iBQM, h?2 iQiH HHQ+i2/ +TBiH b?QmH/ #2 2[mH iQ i?2 2Mi2‘T‘Bb2@rB/2
HHQ+i2/ +TBiH,
1* =
d
"
i =1
1* i
kX LQ lM/2‘+mi , h?2 iQiH +TBiH HHQ+i2/ iQ 2+? mMBi b?QmH/ #2 MQ ;‘2i2‘ i?M
i?2 +TBiH i?i i?2 mMBi rQmH/ ‘2[mB‘2 BM/2T2M/2MiHv
1* i & $(X i ), i = 1 , . . . , d
fi¿i ‘BbF /Bp2‘bB}+iBQM V ‡“
º Ÿ A
˜Bæ mMBi¥ ‘BbF ‚ V?Ñ Vƒ æ
mMBi†ÿi H'‹¥ ‘BbFb
jX avKK2i‘v , A7 irQ mMBib UbvX i M/ X j rBi? i += j V T‘QpB/2 i?2 bK2 K‘;BMH
BM+‘2b2 BM 1* iQ HH TQbbB#H2 bm#b2ib Q7 i?2 }‘K i?i /Q MQi BM+Hm/2 i?2K- i?2M i?2v
b?QmH/ #2 HHQ+i2/ i?2 bK2 +TBiH,
1* i = 1* j B7
$
2
X i +
"
k(K
X k
3
= $
2
X j +
"
k(K
X k
3
, $K , { 1, 2, . . . , d}\{ i, j }
9X *QMbBbi2M+v, amTTQb2 i?2 }‘K +QK#BM2b #mbBM2bb mMBibi M/ j - +‘2iBM; M2r
#mbBM2bb mMBii j X h?2 +QMbBbi2M+v +‘Bi2‘BQM bvb i?i i?2 iQiH HHQ+iBQM iQ i?2 mMBib
b?QmH/ MQi #2 +?M;2/ B7 i?2v ‘2 +QMbB/2‘2/ b irQ mMBib Q‘ QM2X
1* i j = 1* i + 1* j
kj
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eXk J2i?Q/b 7Q` *TBiH HHQ+iBQM
J2i?Q/ R, S`QTQ`iBQMH *TBiH HHQ+iBQM
1*i =
(
ρ(Xi)∑d
j=1 ρ(Xj)
)
1*
Ç Ai biBb}2b i?2 +`Bi2`B Q7 6mHH HHQ+iBQM M/ avKK2i`v- #mi MQi LQ lM/2`+mi M/
*QMbBbi2M+v ;2M2`HHv
Ç A7 ρ Bb bm#//BiBp2- i?2M Bi biBb}2b i?2 +`Bi2`BQM Q7 LQ lM/2`+mi
J2i?Q/ k, .Bb+`2i2 *TBiH HHQ+iBQM
1*1 = ρ(X1)
1*2 = ρ(X1 +X2)− ρ(X1)
XXX
1*d = ρ(X1 +X2 + · · ·+Xd)− ρ(X1 +X2 + · · ·+Xd−1)
Ç h?Bb K2i?Q/ biBb}2b 6mHH HHQ+iBQM- #mi MQi i?2 Qi?2` i?`22 tBQKb BM ;2M2`H
Ç A7 ρ Bb bm#//BiBp2- i?2M Bi biBb}2b i?2 +`Bi2`BQM Q7 LQ lM/2`+mi
J2i?Q/ j, *Qp`BM+2 *TBiH HHQ+iBQM
1*i = 1[Xi] + k*Qp(Xi, X)√o`(X)
Ç bbmK2 }`K mb2b i?2 biM/`/ /2pBiBQM T`BM+BTH2 7Q` ;;`2;i2 2+QMQKB+ +TBiH-
BX2X-
ρ(X) = 1[X] + k
√
o`(X)
h?2M- i?2 +Qp`BM+2 +TBiH HHQ+iBQM K2i?Q/ biBb}2b HH i?2 7Qm` +`Bi2`B, 6mHH
HHQ+iBQM- LQ lM/2`+mi- avKK2i`v- M/ *QMbBbi2M+v
J2i?Q/ 9, *Q@o_ *TBiH HHQ+iBQM
1*i = 1[Xi|X = o_α(X)]
J2i?Q/ 9, *Q@*h1 *TBiH HHQ+iBQM
1*i = 1[Xi|X > o_α(X)]
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