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Year: 2016
The dynamics of insurance prices
Henriet, Dominique ; Klimenko, Nataliya ; Rochet, Jean-Charles
Abstract: We develop a continuous-time general-equilibrium model to rationalise the dynamics of in-surance prices in a competitive insurance market with financial frictions. Insurance companies chooseunderwriting and financing policies to maximise shareholder value. The equilibrium price dynamics areexplicit, which allows simple numerical simulations and generates testable implications. In particular,we find that the equilibrium price of insurance is (weakly) predictable and the insurance sector alwaysrealises positive expected profits. Moreover, rather than true cycles, insurance prices exhibit asymmetricreversals caused by the reflection of the aggregate capacity process at the dividend and recapitalisationboundaries.
DOI: https://doi.org/10.1057/grir.2015.5
Posted at the Zurich Open Repository and Archive, University of ZurichZORA URL: https://doi.org/10.5167/uzh-130214Journal ArticleAccepted Version
Originally published at:Henriet, Dominique; Klimenko, Nataliya; Rochet, Jean-Charles (2016). The dynamics of insurance prices.The Geneva Risk and Insurance Review, 41(1):2-18.DOI: https://doi.org/10.1057/grir.2015.5
②♥♠s ♦ ♥sr♥ Prs
♦♠♥q ♥rt t② ♠♥♦ ♥rs ♦t
♦ ♥tr rs r réér ♦♦tr rs r♥ ♠
♦♠♥q♥rt♥tr♠rsr❯♥rst② ♦ ❩r Ptt♥strss ❩r t③r♥ ♠ ♥t②♠♥♦③
❯♥rst② ♦ ❩r ♥ Ptt♥strss ❩r t③r♥ ♠
♥rsr♦t③
strt
❲ ♦♣ ♦♥t♥♦st♠ ♥rqr♠ ♠♦ t♦ rt♦♥③ t ②♥♠s ♦ ♥sr
♥ ♣rs ♥ ♦♠♣tt ♥sr♥ ♠rt t ♥♥ rt♦♥s ♥sr♥ ♦♠♣♥s ♦♦s
♥rrt♥ ♥ ♥♥♥ ♣♦s t♦ ♠①♠③ sr♦r qr♠ ♣r ②♥♠s
s ①♣t ♦s s♠♣ ♥♠r s♠t♦♥s ♥ ♥rts tst ♠♣t♦♥s ♥ ♣rt
r ♥ tt t qr♠ ♣r ♦ ♥sr♥ s ② ♣rt ♥ t ♥sr♥ st♦r
②s r③s ♣♦st ①♣t ♣r♦ts ♦r♦r rtr t♥ tr ②s ♥sr♥ ♣rs ①t
s②♠♠tr rrss s ② t rt♦♥ ♦ t rt ♣t② ♣r♦ss t t ♥ ♥
r♣t③t♦♥ ♦♥rs
♥tr♦t♦♥
♣♥♦♠♥♦♥ ♦ ♥rrt♥ ②s ♣♣rs t♦ ♣r♦♠♥♥t tr ♦ t ♥sr♥
♥str② ♥sr♥ ♣r♠♠s ♥ ♣r♦ts rs ♥ r ♠rts ♣ss ♥ ♥ s♦t ♠rts
♣ss t s♦♠ rrt② ♦r t♠ ♥ s♦t ♠rts ♣rs ♥ ♣r♦ts r ♦ ♥ ♥sr♥
♣t② s rs ♥ r ♠rts ♣rs ♥ ♣r♦ts r ♥sr♥ ♣t② s rstrt
♥ ♣♦② ♥t♦♥s ♦r ♥♦♥ r♥s r rq♥t r ♠rts ♥ ♣♣r r② s♥② ♥
t② r ♦t♥ rrr t♦ s t② rss s ♦r ♥st♥ t♥ ♥ ♥ ♣r♠♠
r♥s ♥r② tr♣ ♦r s♦② rs♥ ♥ t ♥①t
r sts t ①st♥ ♦ s ♥rrt♥ ②s ♥ ♣r♦♣rt②t② ♥sr♥
♦r t③r♥ t ❯ ♣♥ ♥ ❲st r♠♥② ♦r ♣r♦ ♦ ②rs ❯s♥
t♦rrss ♠♦s ♦ ♦rr s ♥s ②s ♦r t ❯ ❲st r♠♥ ♥ ss ♠rts
rs ♠♦st s♣t♦♥s ♦r ♣♥ ♦ ♥♦t r ②s ♥ t s♠ ② r ♥ tr
s♦ ♠♣r ♥ ♦ ♥ ♥rrt♥ ② ♥ ♣r♦♣rt②t② ♥sr♥ ♦r r♥
❲ t♥ rt♥ ♦②r ♦r ♦♥♥ ♦② ♥ r♥ç♦s tr Ptr ❩ ♥ P♦rs♦r ♣ ♦♠♠♥ts t② ♠♥♦ ♥ ♥rs ♦t rt② ♥♦ ♥♥ s♣♣♦rt r♦♠ tss ♥♥ ♥sttt ♥ r♦♣♥ sr ♦♥ ♥r t r♦♣♥ ❯♥♦♥s ♥t r♠♦r Pr♦r♠P r♥t r♠♥t
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r♠♥② ♥ t③r♥ ♥ ♦r t r♦♣♥ r♥sr♥ ♥str② t♥ ♥
♦r ♠♦r r♥t st② ② ♦②r qr ♥ ❱♥ ♦r♥ sts ♦t ♦♥ t rt②
♦ ②s r r♠♥t s tt st♥r st♠t♦♥ t♥qs ♦rstt t ♦♦ ♦
♥ ② s ♦ t ♥♦♥ ♥rt② ♦ t ♥ t♥ ♣r♠trs ♥ t
♦ t ♣r♦ ② ♦♥ tt t ♦sr ♥rrt♥ tt♦♥s ♥♥♦t ♦♥sr s
tr② ②
r♦♠ t♦rt ♣rs♣t t s r t♦ ①♣♥ ②s ② t ♦♥♥t♦♥ ♦r♥
t♦ ♥sr♥ ♣r♠♠s s♦ ♥♦r♠t♦♥② ♥t ♣rt♦rs ♦ t ♣rs♥t ♦
♣♦② ♠s ♥ ①♣♥ss ♦r s rtr ♦r♥t t♦rs ♥rrt♥ ②s ♦♠ r②
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s ♦ ♥rrt♥ ②s ② ♠♥ t ♥str②s ♣t② t♠♣♦rr② ♣♥♥t ♦♥ s♦rttr♠
♣r♦ts ♥ ♦sss ♦r♥ t♦ ts ♣♣r♦ ② ♥s ♥ ♥srrs r♠t② r
♣t② tr r ♦sss tt ♣t tr rsrs tr s♦♠ t♠ rsrs r rst♦r
♥ ♣t② ♥rss ♥ t t ♥rs ♥ ♥rrt♥ ♣t② ♥rss ♦♠♣tt♦♥
♥ tr♥ rs ♣r♠♠s ♦♥ ❲♥ ♥srrs ♥r rt♦♥ ♥ ♣t t♦ ♥①♣t
♦sss t② r rt♥t t♦ ss ♥ qt② t♦ ss♥ ♦sts s ♠♣s tt s♦s
t♦ ♥srrs ♣t t t ♣r ♥ q♥tt② ♦ ♥sr♥ s♣♣ ♥ t s♦rt r♥ ♥ s
♦♥t①t ♣rs r ♥♦t r② ♣r♦ t ♦♥② rt ♣st ♦sss r♦♠ tt ♣♦♥t ♦ ②
s t♠ ♣s t♦ r♦r ♣t②
♥ ts ♣♣r ♣r♦♣♦s ② ②♥♠ ♠♦ tt r♦♥s rtr ♦r♥t ♥
♣t② ♦♥str♥t t♦rs ♥ ♦r ♠♦ ♥srrs r ♠①♠③rs ♥ ♦♠♣tt ♦r
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♥sr♥ ♦♠♣♥s strt ♥s ♥ t st♦r ♣t② s ♥ t ①♣t ♣r♦ts
r ♦ ②♠♠tr② r♣t③t♦♥s t ♣ ♥ ♣t② ♦♠s t♦♦ s♠ rt♦♥
♣r♦♣rt② ♦ t ♣t② ♣r♦ss t t♦ ♦♥rs ss ♦st♦♥s tt t♦ qs②
♣ttr♥ qr♠ ♣r ♦ ♥sr♥ s rs♥ ♥t♦♥ ♦ t ♥str② ♣t② ♥
ts ♠①♠♠ s ♥rs♥ t t ♠♥t ♦ t ♥♥ rt♦♥s
♥ ♥trst♥ ♣rt♦♥ ♥rt ② ♦r ♠♦ s tt ♥rrt♥ ②s ♥ ♥r
s②♠♠tr ❲ ♠sr t r rt♦♥ ♦ t s♦t ♠rt ♣s ② t ①♣t t♠ t
ts ♦r t ♣r ♦ ♥sr♥ t♦ r ts ♠♥♠♠ strt♥ t ts ♠①♠♠ ♥
♥♦t♦♥ tt ①tr♥ qt② ♥♥♥ s ♠♦r ♦st② t♥ ♥♥♥ rt♥ r♥♥s s sss ♥❲♥tr ♥ t ♠♣r s r♦♥ ♥ s ♥ ♥ ♦ t ♠♣♦rt♥t ♦sts ♦ rs♥ ①tr♥♣t ♦r t ♣r♦♣rt②st② ♥srrs
s♠r s♦♥ t r rt♦♥ ♦ t r ♠rt ♣s s ♣♣r♦①♠t ② t ①♣t
t♠ t ts ♦r t ♣r ♦ ♥sr♥ t♦ r♦r r♦♠ ts ♠♥♠♠ ♣ t♦ ts ♠①♠♠
♦♥sst♥t t ♠♣r ♥ r♣♦rt ♥ r ♦r ♥♠r ♥②ss s♦s tt t s♦t
♠rt ♣s sts ♦♥r ♥ ①♣tt♦♥ t♥ t r ♠rt ♣s ♦r♦r ts r♥
tr♥s ♦t t♦ ♠♦r ♣r♦♥♦♥ ♥ t stt② ♦ t ♠♥ ♦r ♥sr♥ s
r ♥sr♥ ♣r ♥①
2000 2002 2004 2006 2008 2010 2012100
110
120
130
140
150
160
170
Years
Insura
nce
price
in
de
x
♦ts ts r r♣♦rts t ♣ttr♥ ♦ t ♥sr♥ ♣r ♥① t r♦♠ ♦♥ ♦ ♥sr♥ ♥ts ♥ r♦rs s s♦♥ ♥ tt t s♦t ♠rt ♣s ♥♥ ♣rs sts ♦♥r t♥ t r ♠rt ♣srs♥ ♣rs
r ♣♣r s rt t♦ t rr trtr tt tt♠♣ts t♦ ♥ t ♠r♥ ♦ ②s t♦
♥s ♥ ♥srrs ♥t ♦rt trt ♥ ♦♦♠ ♦♣ t♦ s♠♣ ♠♦s
♦♥ ts ♥ ② r② ♦♥ ♣rt qr♠ ♣♣r♦ r t sts ♦ ♥sr♥ s♣♣②
rs ①♣♥ t ♥rrt♥ ② ♣♦st♦♥ ♦ t s♣♣② r s t ② ♥srrs ♥t
♦rt ♥ t ♦r♠ ♦ ♥srr ①♣tt♦♥s ①♣♥s t t♠♥ ♥ ♥t ♦ t ♣r ♣s
♥ rr♥t♦♥ ♣r♦♣♦s ♠♦ ♦ ♥sr♥ s♣♣② t ♣t② ♦♥str♥ts ♥
♥♦♥♦s t rs ♥str② ♠♥ s ♥st t rs♣t t♦ ♣r ♥ ♣t t ♣r
♥rs ♦♦♥ ♥t s♦ t♦ ♣t ♦ rr ♥ st ♣r♦r♠ ♥
♠♣r ♦♠♣rs♦♥ ♦ s① tr♥t ♠♦s ♦ ♥sr♥ ♣r♥ s♥ t t♥ ♥
② ♥ tt t ♣t② ♦♥str♥t ♠♦ s ♦♥sst♥t t t ♦t ♥ t ♦♥ ♥
s♦rt tr♠ r♦♥ s♦ ♣r♦s ♠♣r s♣♣♦rt ♦r t ♣r♠r② ♣rt♦♥s ♦ ♣t②
♦♥str♥t t♦rs ♦ ♣r♦♣rt②st② ♥sr♥ ②s
♦sst rt ♣♣r s ❲♥tr ♦ ♦♣s srt t♠ ♠♦ s♠r t♦ ♦rs
♠② ♥ ❲♥tr ♥srrs ♠t qt② ♣t s♦ s t♦ r ♦t t ♥ t♦
t♦ ♠t ♣♦②♦rs ♠s s tr ♠♣s ♥ ♣rs♦♣♥ s♦rtr♥ s♣♣② r ♦s
♣♦st♦♥ s tr♠♥ ② t ♦ ♣t ♥ t♦♣ ♦ ts ♦st r♥t t♥ ♥tr♥ ♥
①tr♥ s♦rs ♦ ♣t ♣r♥ts ♥♥ ♣t r♦♠ q② st♥ ♥sr ♥
rs s t ♦ss ♦ ♦♥ ♦r t ♣r♦t② tt s ♦♠♠♦♥ t♦ ♥srs t s ts
r♥♦♠ s ♥ ♦r ♠♦ t stt r s t t♦t ♥t t ♦ t ♥sr♥ ♦♠♣♥s ♥
t rt♦♥ ①♣tt♦♥ qr♠ s rtr③ ② t ♦♥t ②♥♠s ♦ ♥sr♥ ♣rs ♥
♠rtt♦♦♦ ♦♥s q rt♦ ♦r ♥ ♦♥trst t♦ ♦r ♠♦ t rrs rtr③t♦♥
♦ t qr♠ ②♥♠s ♥ ❲♥tr ♦s ♥♦t ♦ ①♣t s♦t♦♥s ❲♥tr s♦s
tr♦ s♠t♦♥ ♠t♦s tt t qr♠ ♣r s rr t♥ t ①♣t ♦ss ♥ s
♥t② ♦rrt t ♣t②
r ♠♦ ♥ s♥ s ♦♥t♥♦s t♠ rs♦♥ ♦ ❲♥tr ❲ ♥rt♥ t ♠♥
♣rt♦♥s ♦♥sst♥t t t♦s ♦t♥ ♥ srt t♠ t ♦s t ①♣t rtr③t♦♥
♦ qr♠ ②♥♠s ❲ strt t ♥t ♦ s♥ t ♦♥t♥♦st♠ ♣♣r♦ ②
♥②s♥ t ② ♣r♦♣rts ♦ ♣r tt♦♥s
rst ♦ t ♣♣r s ♦r♥③ s ♦♦s t♦♥ ♣rs♥ts t ♠♦ t♦♥ r
tr③s t ♦♠♣tt qr♠ t♦♥ sts t ②♥♠s ♦ ♥sr♥ ♣rs t♦♥
♦♥s
♦
❲ ♦♥sr ♦♠♣tt ♥sr♥ ♠rt tt ♦♥ssts ♦ ♦♥t♥♠ ♦ ♥sr♥ ♦♠♣♥s
♦r♥ ♥sr♥ t♦ ♥s ♦ ♣rt② ♦rrt rss s s tr rss
♥ts s♦♥t t tr t ♦♥st♥t rt r ♠t ♦ss Lt ♥rr ② ♥
♣ t♦ t t s s tt
dLt = ℓdt+ σ0dBt,
r ℓ ♥♦ts ①♣t ♦sss ♣r ♥t ♦ t♠ σ0 s ♥ ①♣♦sr ♣r♠tr ♥ Bt s st♥r
r♦♥♥ ♠♦t♦♥ ♥ ♦♥ t ♣r♦t② s♣(Ω,F,P
)tt ♥rts t trt♦♥ F =
Ft, t ≥
0 ❲ ss♠ tt Bt s t s♠ ♦r ♥s ♣rt ♦rrt♦♥ t♥ rss
♥sr♥ ♦♥trts r s♦rttr♠ ♥ t ♠rt ♣r ♦ ♥sr♥ ♣r ♥t ♦ t♠ s
πt = ℓ+ σ0pt,
r pt s rs ♦♥ t♦r ♦r s♠♣t② tr♦♦t t ♣♣r rr t♦ pt s t♦ t
♣r ♦ ♥sr♥ ♠♥ ♦r ♥sr♥ s ①♦♥♦s② ♥ ② ♥t♦♥ D(p) s tt
D′(.) < 0
♥ ♥sr♥ ♦♥trts ♥ ♥♥ts♠ ♥t ♥srrs ♥♦ ♦♥tr♠ ts
s t ♥ st ♦ ♥sr♥ ♦♠♣♥② s ♦♥② ♦♥ t♠ ♦♥ s qt② et ♦♥ t
t② s ♥ q rsrs ♦♥ t sst s t et = mt r♦r mt Mt ♦♥ rt
♥ s♠t♥♦s② s t ♦♠ ♦ ♥ rs♣t② rt rsrs ♦r s
♦s②♥rt ♣rt ♦ ♥ rs s ♥t s♥ t ♥ ♠♥t ② rst♦♥
t ♦♦ ♦ qt② ♦r ♥ ♥sr♥ ♦♠♣♥② rs♣t② t ♥tr ♥sr♥ ♥str② ♥
♥ ♥♥ts♠ ♣r♦ (t, t + dt) ♥ ♥ ♥srr ♦♣rt♥ t s xt ♥rts r♥♥s
xt(πtdt− dLt) tt ♥ strt s ♥s ♦r ♥ rt♥ ♥ rsrs ♦r♦r ♥sr
♥ ♦♠♣♥s ♥ ♥rs t ♦ rsrs ② rs♥ ♥ qt② ♥ts ♣r♦♣♦rt♦♥
♦st γ
②♥♠s ♦ rsrs mt ♦ ♥ ♥sr♥ ♦♠♣♥② ♦♣rt♥ t s xt t t♠ t s ♥
②
dmt = σ0xt (ptdt− dBt) + dit − dδt,
r dδt ≥ 0 ♥ dit ≥ 0 ♥♦t rs♣t② t ♥s ♥ t ♠t ♥ ♥ r♣
t③t♦♥ ♠♦♥ts ♥ t ♦r ♠♦ ♥ s ♥r qr♠ rs♦♥ ♦ t ss
r♥ t♦r② ♠♦ ♥ ts r♦♥♥ rs♦♥ s ♥ rr r rsrs ②♥♠s s
♥ ②
dmt = µdt− σ0dBt − dδt.
♠♥ r♥ ♦r ♣rt♥s t♦ t t tt ♥ ♦r stt♥ ♣rs ♥ q♥tts r
♥♦♥♦s ♥ ♥sr♥ ♦♠♣♥s t ♣♦sst② t♦ r♣t③
Pr♦ tt t qr♠ t s♣♣② ♦ ♥sr♥ ♦♥trts qs t ♠♥ t ②♥♠s
♦ rt rsrs ♣t② ♦ t ♥tr ♥sr♥ st♦r sts②
dMt = σ0D(pt) (ptdt− dBt) + dIt − d∆t,
r d∆t ≥ 0 ♥ dIt ≥ 0 ♥♦t rs♣t② t ♥s ♥ t rt ♠t ♥
♥ r♣t③t♦♥ ♠♦♥ts
♥ ♦r ♠♦ ♦s ♦♥ r♦♥ stt♦♥r② qr♠ ♥ t ♥sr♥ ♣r s
tr♠♥st ♥t♦♥ ♦ t rt ♦ rsrs ♥ t ♥sr♥ st♦r pt = p(Mt) ♦
♦r♠② ♥ t qr♠ t J = [0, 1] ♥♦t t st ♦ ♥sr♥ ♦♠♣♥s ♦
s ♥① ② j ∈ J
♥t♦♥ stt♦♥r② r♦♥ ♦♠♣tt qr♠ ♦♥ssts ♦ ♥ rt rsrs ♣r♦
ss Mt ♣r ♦ ♥sr♥ p(M) ♥ ♥sr♥ s♣♣② ♥t♦♥s xj(M), j ∈ J tt r ♦♠♣t
t ♥srrs ♣r♦t ♠①♠③t♦♥ ♥ t ♠rt r♥ ♦♥t♦♥∫
Jxj(M)dj = D[p(M)]
♥ t ♦♦♥ st♦♥ sts t ①st♥ ♦ ♥q stt♦♥r② r♦♥ qr♠
♥ st② ts ♠♥ ♣r♦♣rts
♦r♥ t♦ r♦♥ ♥ s ♦r ①♠♣ t rt ♦sts ♦ qt② sss r♥ r♦♠ t♦ % ♦ t ss
❲ ss♠ tt rsrs r♥ ♥♦ ♥trst
qr♠
♥sr♥ ♦♠♣♥② ts t ♣r ♦ ♥sr♥ p(Mt) ♥ t ②♥♠s ♦ Mt s ♥
♥ ♦♦ss t s ♦ ♦♣rt♦♥ xt ≥ 0 ♥ dδt ≥ 0 ♥ r♣t③t♦♥ dit ≥ 0 ♣♦s s♦
s t♦ ♠①♠③ sr♦r
v(m,M) = maxxt≥0,dδt≥0,dit≥0
E
[ ∫ +∞
0e−rt dδt − (1 + γ) dit
]
,
r mt ♦♦s ♥ Mt ♦♦s
♦t tt t ♦t ♥t♦♥ ♥ t stt qt♦♥ r ♦♠♦♥♦s ♥ (m,x, dδ, di)
♠♣②♥ tt t ♥t♦♥ s ♥r ♥ t ♥ ♦ rsrs
v(m,M) = mu(M),
r u(M) s t ♠rtt♦♦♦ ♦ t ♥sr♥ ♦♠♣♥② s t s♠ ♦r t ♥tr
♥sr♥ st♦r
♥ t ♦ ♣r♦♣rt② ♦ t ♥t♦♥ ♦♥ ♥ s② s♦ tt t ♠♥ qt♦♥
♦rrs♣♦♥♥ t♦ sr♦r ♠①♠③t♦♥ ♥ rtt♥ s ♦♦s
ru(M) = maxx≥0,dδ≥0,di≥0
[
xσ0
p(M)u(M) + σ0D[p(M)]u′(M)
+ σ0D(p)p(M)u′(M) +σ20D
2[p(M)]
2u′′(M)
+dδ
m
1− u(M)
− di
m
1 + γ − u(M)]
,
r t tr♠ ♥ t t♥ s s t ①♣t rtr♥ r♦♠ ♦♥ ♦♥ ♥t ♦ qt② ♥ t
♥sr♥ ♦♠♣♥② t rst tr♠ ♥ t rt♥ s ♥ r② rts ♣trs t ♠♣t ♦
♥ rs ①♣♦sr t s♦♥ ♥ t tr tr♠s ♥ t rt♥ s rt t ♠♣t
s ② t ♥s ♥ t rt ♣t② ♦ t ♥sr♥ st♦r ♥ t st t♦ tr♠s
♣tr rs♣t② t ♠♣t ♦ t ♣②♦t ♥ r♣t③t♦♥ ♣♦s
♦♥sr rst t ♦♣t♠ ♥ ♥ r♣t③t♦♥ ♣♦s ①♠③♥ t rs♣t t♦
dδ ≥ 0 ♥ di ≥ 0 ♠♣s str♦♥ ♦♥t♦♥s ♦♥ u(.)
u(M) ≥ 1,
♦t tt u (M) ♥ ♥tr♣rt s t ♠rt ♦ ♦♥ ♦r ♦ ♥t ♦rt ♦♦ ♥ t ♥sr♥st♦r ♥ t♦t ♣t② s M t r♣rs♥ts t ♥sr♥ ♥♦ ♦ ♦♥s q rt♦ s ♥ ❲♥tr
t dδ > 0 ♦♥② ♥ u(M) = 1 ♥
u(M) ≤ 1 + γ,
t di > 0 ♦♥② ♥ u(M) = 1 + γ
s s♦ ♦ t ♠rtt♦♦♦ rt♦ ♦ t ♥sr♥ st♦r s rs♥ ♥t♦♥ ♦
ts rt ♣t② u′(M) < 0 s ♠♣s tt rt ♣t② M rs t♥ t♦
rt♥ ♦♥rs ♥ s ♥ ♠♥② s♦♥ ♠♦s ♥ t t ♦♣t♠ qt② ♠♥
♠♥t t ♦♣t♠ ♥ ♥ r♣t③t♦♥ ♣♦s ♥ ♦r ♠♦ r ♦ s♦ rrr
t②♣ ♥ ♣rtr ♥s r strt ♥ t ♥sr♥ st♦r s s♥t② ♣t③ s♦
tt rt rsrs r rt M s tt t ♠rtt♦♦♦ ♦ t ♥srrs
qt② s t♦ 1
u(M) = 1.
② ♦♥trst r♣t③t♦♥s t ♣ ♥ t ♥sr♥ st♦r s ♥r♣t③ ♥ r
srs t♦ rt M < M s tt t ♠r♥ ♦ ♦♥ ♦♥ sr ♥ t
♥sr♥ ♦♠♣♥② qs t ♠r♥ ♦st ♦ ss♥ ♥ sr
u(M) = 1 + γ.
❲ tr♥ ♥♦ t♦ t ♦♣t♠ ♦ ♦ t s ♦ ♦♣rt♦♥ ♦ ♥ ♥srr ①♠③t♦♥ t
rs♣t t♦ xt ♠♣s tt ♥ ♥tr♦r s♦t♦♥ ①sts ♥ ♦♥② t ♠rtt♦♦♦ u(M)
♥ t ♣r p(M) sts②
p(M) = −u′(M)
u(M)σ0D[p(M)].
❯♥r t ♦ qt② t tr♠ ♥ x ♥ss r♦♠ t ♠♥ qt♦♥ ♥ ♦♥ t ♥tr
(M,M) r rt ♣t② ♦s s♦② t♦ rt♥ r♥♥s t ttr tr♥s♦r♠s
t♦
ru(M) = σ0D(p)p(M)u′(M) +σ20D
2[p(M)]
2u′′(M), M ∈ (M,M).
♥ t ♦ qt♦♥ ② u(M) ♥ s♥ qt♦♥ t♦ r♣ t tr♠ u′(M)u(M) ♦♥
♦t♥s
r + p2(M) =σ20D
2[p(M)]
2
u′′(M)
u(M).
s ♦♥ s t ♣r ♦ ♥sr♥ p(M) stss qt♦♥s t sr♦rs ♦ ♥
♥sr♥ ♦♠♣♥② r ♥r♥t t rs♣t t♦ tr s ♦ ♦♣rt♦♥ s③ ♦ t ♥sr♥
♥♥ ♥ r② ♦t ♥ ❱♥ ♦t♦♥ ♥ ♥ ❲♥ ♥t ♦♥t①t ♦ tr ♠♦s r♦s ♣♣t♦♥s ♦ t ♦♣t♠ qt② ♠♥♠♥t r sss ♥ ♠
st♦r tr♦r ♥tr② tr♠♥ ② t ♠♥ s
♥♠♥t② qt♦♥s st♠ r♦♠ t s♥ ♦ rtr ♦♣♣♦rt♥ts ♥ ♣r
t ♦♠♣tt♦♥ ♥ t ♥sr♥ ♠rt ♦ s ts rs♦rt t♦ t ♥♦♥rtr r♠♥ts
♦♥sr t t♦t ♠rt t♦t ♣t③t♦♥ ♦ t ♥sr♥ st♦r t t♠ t
Wt ≡ u (Mt)Mt.
rtr r ♦♥t♦♥ ♠♣s
E [dWt] = rWtdt,
♦r q♥t②
E [u (Mt + dMt)Mt] + E [u (Mt + dMt) dMt]− u (Mt)Mt = ru (Mt)Mtdt.
♥ t ♦ ♠♦t♦♥ ♦ Mt t s♦♥ tr♠ ♥ t ♦ qt♦♥ ♥ rrtt♥ s
♦♦s
E[u (Mt + dMt) dMt] =
u (Mt) p (Mt) dt− E
[
u (Mt + dMt) dBt
]
σ0D [p(Mt)] ,
r t tr♠ ♥ t rt♥ s s t ①♣t ♠rt ♦ t ♣r♦t ♠r♥ ♥rt
② ♥sr♥ tt② Prt ♦♠♣tt♦♥ ♦♥ t ♥sr♥ ♠rt ♠♣s tt ts ♠rt
s ③r♦
u (Mt) p (Mt) dt = E
[
u (Mt + dMt) dBt
]
.
♦♠♣t♥ t ①♣tt♦♥ ♥ t rt♥ s ♦ s♥ t t tt E [dMtdBt] =
−σ0D[p(M)]dt ②s qt♦♥ ♦r♦r ♦♠♥♥ ♥ s t♦
E [du (Mt)] = ru (Mt) dt,
② s♥ tôs ♠♠ ♥ qt♦♥ ♥ rrtt♥ s
♦ tr♠♥ t qr♠ ♥sr♥ ♣r ♥♦t tt qt♦♥ ♥ rrtt♥ s ♦♦s
u′(M)
u(M)= − p(M)
σ0D[p(M)].
r♥tt♥ t ♦ qt♦♥ ②s
u′′(M)
u(M)−[u′(M)
u(M)
]2= −p′(M)
[ 1
σ0D[p(M)]− p(M)D′[p(M)]
σ0D2[p(M)]
]
.
♥srt♥ u′′(M)u(M) r♦♠ t ♦ qt♦♥ ♥t♦ ②s
2(r + p2(M)) = p2(M)− σ0p′(M)
(
D[p(M)]− pD′[p(M)])
,
t♠t② s t♦ rst♦rr r♥t qt♦♥ tr♠♥♥ t qr♠ ♣r
p′(M) = − 1
σ0
2r + p2(M)
D[p(M)]− p(M)D′[p(M)].
♥ ttD′(.) < 0 t s ♠♠t t♦ s r♦♠ t ♦ qt♦♥ tt t ♣r ♦ ♥sr♥ s
♥rs② rt t♦ t rt ♣t② ♦ t ♥sr♥ st♦r ♦r♦r ♣♣②♥ tôs ♠♠
t♦ pt = p(Mt) ♥ ♦t♥ ♥ ①♣t rtr③t♦♥ ♦ t ♥sr♥ ♣r ♣r♦ss ♥
dpt = σ0D[p(Mt)]
(
pt(Mt)p′(Mt) +
σ0D[p(Mt)]
2p′′(Mt)
)
︸ ︷︷ ︸
µ(pt)
dt−σ0D[p(Mt)]p′(Mt)
︸ ︷︷ ︸
σ(pt)
dBt.
s♠♣ ♦♠♣tt♦♥ ♠♠t② ②s t ①♣rss♦♥ ♦ t ♥sr♥ ♣r ♦tt②
σ(p) =2r + p2
1 + ε1(p),
r ε1(p) s t stt② ♦ t ♠♥ ♦r ♥sr♥
ε1(p) ≡ −pD′(p)
D(p).
r♦r qt♦♥ ♥ rrtt♥ s ♦♦s
p′(M) = − σ[p(M)]
σ0D[p(M)],
♠♦♥strts tt t ♥s ♥ t qr♠ ♥sr♥ ♣r r ♠♥② r♥ ② t
♥♦♥♦s ♦tt②
rtr♠♦r s♠♣②♥ t ①♣rss♦♥ ♦ t rt ♦ t ♥sr♥ ♣r♠♠ ②s
µ(p) = σ(p)
[(2r + p2)ε1(p)ε2(p)
2p(1 + ε1(p))2− pε1(p)
1 + ε1(p)
]
,
r
ε2(p) ≡ −pD′′(p)
D′(p).
♦ ♦♠♣t t rtr③t♦♥ ♦ t ♦♠♣tt qr♠ t V (M) ≡ Mu(M) ♥♦t
t ♠rt ♦ t ♥tr ♥sr♥ ♥str② s♥ ♦ rtr ♦♣♣♦rt♥ts t t
rt♥ ♦♥rs M ♥ M ♠♣s tt
V ′(M) ≡ u(M) +Mu′(M) = 1,
♥
V ′(M) ≡ u(M) +Mu′(M) = 1 + γ.
♦tr t qt♦♥s ♥ ts rs♣t② ♠♣s t♦ ♦♥t♦♥s u′(M) = 0 ♥
M = 0 ♥srt♥ u′(M) = 0 ♥t♦ qt♦♥ t s s② t♦ s tt ♥ t ♥sr♥ ♥str②
♦♣rts t t ♠①♠♠ ♦ rsrs t ♦♥ t♦r ♥ss p ≡ p(M) = 0 ♥
p′(M) < 0 ts ♠♣s tt p(M) > 0 ♦r M > 0 ♥ tr♦r u′(M) < 0 s s ♦♥tr
♦r ♦♦♥ ♣r♦♣♦st♦♥ s♠♠r③s ♦r rsts
Pr♦♣♦st♦♥ r ①sts ♥q stt♦♥r② r♦♥ qr♠ ♥ rt rsrs
♥ t ♥sr♥ st♦r ♦ ♦r♥ t♦
dMt = σ0D[p(Mt)] (p(Mt)dt− dBt) , Mt ∈ (0,M).
♥sr♥ ♣r ♥t♦♥ p(M) stss t r♥t qt♦♥
p′(M) = − σ[p(M)]
σ0D[p(M)],
t t ♦♥r② ♦♥t♦♥ p(0) = p r p s♦s
∫ p
0
p
σ(p)dp = ln(1 + γ).
Pr♦♦ ♦ Pr♦♣♦st♦♥ ❲ r② sts tt ♥② qr♠ ♣r ♥t♦♥ p(M)
♠st sts② t rst♦rr r♥t qt♦♥ ❯♥q♥ss ♦ t qr♠ rst
r♦♠ t ②♣st③ t♦r♠ ♦♥ t ♦♥r② p(0) = p s tr♠♥ ♥ p(M)
s ♥♦♥ ♥t♦♥ u(M) ♥ ♦♠♣t ② s♦♥
u′(M)
u(M)= − p(M)
σ0D[p(M)],
②s
u(M) = u(M)exp(∫ M
M
p(s)
σ0D[p(s)]ds)
.
s ♣r♦♣rt② ♦ ♦r ♠♦ ♠♣s tt ♥sr♥ ♦♠♣♥s ②s ♥rt ♥♦♥♥t ①♣t ♣r♦ts ♥♣rt ♦r ♦♥ t♦rs ♥ ♥t s ♥srrs ♥ tr ♦sss ♦♠♣♠♥tr② ♥♥♠rt tts
♥ u(M) = 1 ♥ u(0) = 1 + γ ts ♠♣s
∫ M
0
p(s)
σ0D[p(s)]ds = ln(1 + γ).
♥② ♥♥ t r ♦ ♥trt♦♥ ♥ t ♦ qt♦♥ t♦ p(s) = p ②s
♦t tt ♥ t ♦♠♣tt qr♠ ♥sr♥ ♦♠♣♥s r♣t③ ♦♥② ♥ r♥♥♥
♦t ♦ rsrs t t s♠ t♠ t trt ♦ rt rsrs ♥ t ♥sr♥ ♥str②
♥ s② ♦♠♣t ♦r♥ t♦
M =
∫ p
0
σ0D(p)
σ(p)dp.
t s ♠♠t t♦ s r♦♠ qt♦♥s ♥ tt ♦t t ♠①♠♠ ♦ ♥sr♥
♣r p ♥ t trt ♦ rsrs M ♥rs t t ♥♥♥ ♦st γ s tr ♦ t
♦♠♣tt qr♠ ♠r♥ ♥ ♦r ♠♦ ssts tt t ♠♥t ♦ ♥rrt♥ ②s
♦sr ♥ ♣rt ♠t ♥tr♥s② rt t♦ t ♠♥t ♦ ♥♥ rt♦♥s ②
♥sr♥ ♦♠♣♥s
②♥♠s ♦ ♥sr♥ ♣rs
♥ ts st♦♥ st② t ②♥♠s ♦ t qr♠ ♣r ♦ ♥sr♥ s ♣rt ② ♦r
♠♦ ♦r ♦♥♥♥ s t ♦♦♥ s♣t♦♥ ♦ t ♠♥ ♦r ♥sr♥
D(p) = (α− p)β ,
r β > 0 ♥ α > 0 Pr♠tr α ♥ ♥tr♣rt s t ♣r ♦ ♠♥ ♦r
♥sr♥ ♥ss ♦t α ♥ β t t stt② ♦ ♠♥ ♦r ♥sr♥ ♦r ♣rs②
ε1(p) =βp
α− p, ε2(p) =
(β − 1)p
α− p.
♥srt♥ ε1(p) ♥ ε2(p) ♥t♦ t ♥r ♦r♠s ♦ σ(p) ♥ µ(p) ♦t♥ t♦ s♠♣
①♣rss♦♥s tt rtr s t♦ strt t ②♥♠s ♦ t qr♠ ♣r
σ(p) =(α− p)
(p2 + 2r
)
p(α+ (β − 1)p),
µ(p) = σ(p)βp
[(β − 1)(2r − p2)− 2αp
2(α+ (β − 1)p)2
]
.
♥ rrs♦♥
t s s② t♦ s r♦♠ ①♣rss♦♥ tt t s♥ ♦ t ♥sr♥ ♣r rt s tr♠♥ ②
t s♥ ♦ t ♣♦②♥♦♠
h(p) ≡ (β − 1)(2r − p2)− 2αp.
t ♥ s♦♥ tt ♦r α >√2r ♥ β > 1 t qt♦♥ h(p) = 0 s ♥q r♦♦t p∗ ♦♥
t ♥tr (0, α) s tt µ(p) > 0 ♦♥ (0, p∗) ♥ µ(p) < 0 ♦♥ (p∗, α) s ♠♣s tt t
♣r ♣r♦ss pt ①ts ♠♥ rrs♦♥ ♦r t ♠♣♦rt♥t t♦ ♥♦t tt ts ♠♥ rrs♦♥
♣r♦♣rt② ♠♥sts ts ♦♥② ♥ t ♥♥♥ ♦st γ s s♥t② ♥ r tt t
♠①♠♠ ♦ ♥sr♥ ♣r p s ♥ ♥rs♥ ♥t♦♥ ♦ γ t p = 0 ♥ γ = 0 s
♥ γ s ♦ p∗ > p ♥ µ(p) > 0 ♦r p ∈ (0, p)
♦r♦r ts ♠♥rrs♦♥ ♣r♦♣rt② s ♥♦t r② str♦♥ ♥ ♦r ♠♦st ♣r♠tr ♦♠
♥t♦♥s |µ(p)| tr♥s ♦t t♦ r② s♠ s ♦♠♣r t♦ σ(p) s t t ♥ t ♥tr ♣♥s ♥
r ♠♥ tr ♦ t ②♥♠s ♦ t ♣r ♣r♦ss pt s ts rt♦♥ t t ♦♥rs
♦ t ♥tr (0, p) ♥s t ♣r rrss
Pr rrss
rt♦♥ ♣r♦♣rt② ♦ t rt rsr ♣r♦ss ♥ ♦r ♠♦ ♥s rrss ♦ t
♣r ♦ ♥sr♥ ♠♣r ♦r s s♦♥ tt t ♥sr♥ ♠rt tr♥ts s♦t ♠rt
♣ss rtr③ ② ♥ ♣r♠♠s t♦tr t ♥ ①♣♥s♦♥ ♦ ♥srrs ♣ts ♥
r ♠rt ♣ss rtr③ ② rs♥ ♣r♠♠s t♦tr t ♦♥trt♦♥ ♦ ♥srrs
♣ts
r ♠♦ ♥ s t♦ ♦♠♣t t ①♣t rt♦♥ ♦ ♣s ♦ t ♥rrt♥ ②
② s♥ t ②♥♠s ♦ t ♣r ♦ ♥sr♥ ♥ ♥ qt♦♥ ♥ ♣rtr t ①♣t
rt♦♥ ♦ t s♦t ♠rt ♣s ♥ ♥sr♥ ♣rs ♥ ♠sr s t ①♣t
t♠ ♥ ♦r t ♣r♦ss pt t♦ r 0 strt♥ r♦♠ t stt p ♥ s♠r s♦♥ t
①♣t rt♦♥ ♦ t r ♠rt ♣s rs♥ ♥sr♥ ♣rs ♥ ♠sr ② t
①♣t t♠ tt t ♣r♦ss pt ♥s t♦ r t stt p strt♥ r♦♠ 0
♦ ♦r♠③ ts t Ts(p) ♥♦t t ①♣t t♠ tt t ♣r ♣r♦ss pt ts ♥ ♦rr t♦
r ♥② stt p ≤ p strt♥ r♦♠ t stt p ♥ t Th(p) t ①♣t t♠ t ts t♦ r
t stt p strt♥ r♦♠ ♥② p ≤ p ❲ rr t♦ T s ≡ Ts(0) s t r rt♦♥ ♦ t s♦t
♠rt ♣s ♥ t♦ T h ≡ Th(0) s t r rt♦♥ ♦ t r ♠rt ♣s
Pr♦♣♦st♦♥ r rt♦♥ ♦ t s♦t ♠rt ♣s ♥ ♦♠♣t s T s ≡ Ts(0)
r ♥t♦♥ Ts(p) stss
1− µ(p)T ′s(p)−
σ2(p)
2T ′′s (p) = 0,
t r ♦ t trtr ♦♥ ♥rrt♥ ②s ♥ ♦♥ ♥ rr♥t♦♥ t
t t ♦♥r② ♦♥t♦♥s Ts(p) = 0 ♥ T ′s(p) = 0
r rt♦♥ ♦ t r ♠rt ♣s ♥ ♦♠♣t s T h ≡ Th(0) r ♥t♦♥
Th(p) stss
1 + µ(p)T ′h(p) +
σ2(p)
2T ′′h (p) = 0,
t t ♦♥r② ♦♥t♦♥s Th(p) = 0 ♥ T ′h(0) = 0
Pr♦♦ ♦ Pr♦♣♦st♦♥ ♦ r t t gp0(p) ♥♦t t ①♣t t♠ tt s
♥ t♦ r s♦♠ stt p0 strt♥ r♦♠ ♥② p ≥ p0 r p0 ≤ p ≤ p ♥ Ts(p0) =
Ts(p) + gp0(p) ♥ ts Ts(p) = Ts(p0) − gp0(p) ② t ②♥♠♥ t♦r♠ ♥t♦♥ gp0(p)
♠st sts② t
1 + µ(p)g′p0(p) +σ2(p)
2g′′p0(p) = 0.
❯s♥ t t tt g′p0(p) = −T ′s(p) ♥ g′′p0(p) = −T ′′
s (p) ♦♥ ♦t♥s ♦♥r②
♦♥t♦♥ Ts(p) = 0 rts t t tt t p t t♠ t♦ r p s ③r♦ rs t ♦♥r②
♦♥t♦♥ T ′s(p) = 0 ♠rs t♦ t rt♦♥ ♦ t ♣r ♦ ♥sr♥ t p s♠ r
♠♥ts ♣♣② t♦ sts t ♦♥r② ♦♥t♦♥s ♦r t ♥t♦♥ Th(p) t♦ t ②♥♠♥
t♦r♠ ♣♣s rt②
t♥ s ♥ t ♥tr ♣♥s ♦ r r♣♦rt rs♣t② t s ♦ T s ♥
T h s ♥t♦♥s ♦ ♣r♠tr α ♦r t♦ r♥t s ♦ β rt♥ s ♣♥ ♦ r
r♣♦rts t ♦rrs♣♦♥♥ r♥ T s − T h
s ♥♠r rsts r② s♦ tt ♦r ♣r♠tr ♦♠♥t♦♥s ♠♣②♥ r s
tt② ♦ ♠♥ ♦r ♥sr♥ r β ♥ ♦r α t ②s ♠r♥ ♥ ♦r ♠♦
r s②♠♠tr ♥ t s♦t ♠rt ♣s t♥s t♦ sst♥t② ♦♥r t♥ t r ♠rt
♣s ♣♦t♥t ①♣♥t♦♥ ♦r ts tr rsts ♦♥ t ♦srt♦♥ tt ♥ t stt♥ t
♥ st ♠♥ ♦r ♥sr♥ t ♥♦♥♦s ♦tt② s ♠♦♥♦t♦♥② rs♥ ♥t♦♥ ♦
p ♥ t ♥♦♥♦s ♦tt② s ♦r ♥ t stts t r ♣r ♦ ♥sr♥ t s②st♠
t② ♥s ♠♦r t♠ t♦ ♠ ♦♥r ♠♦ rtr t♥ t♦ ♠ ♥ ♣r ♠♦ ♦r t
♣r♠tr ♦♠♥t♦♥s ♠♣②♥ ② ♥st ♠♥ ♦r ♥sr♥ t ♥♦♥♦s ♦tt②
♣ttr♥ s U s♣ ♥ t r♥ t♥ t r rt♦♥s ♦ t s♦t ♥ r ♠rt
♣ss s ♠♦st ♥
♦♥r♥ ♦r
♦ st② t ②♥♠s ♦ t ♣r ♦ ♥sr♥ ♥ t ♦♥ r♥ ♦♦ t ts r♦ ♥st②
♥t♦♥ tt rts t ♣r♦♣♦rt♦♥ ♦ t♠ tt t ♣r ♣r♦ss s♣♥s ♥ s stt ♥
t ♦♥ r♥ ♥ t ♣r ②♥♠s ♥ t ttr ♥ ♦♠♣t ② s♦♥ t ♦rr
r ♣tr
r r rt♦♥ ♦ s♦t ♥ r ♠rts
0.3 0.5 0.7 0.9α
5
10
15
Ts
α2.435
2.440
2.445
2.450
2.455
2.460
Th
α
5
10
15
Ts - Th
β = 3
β = 0.5
β = 3
β = 0.5
β = 3
β = 0.5
0.1 0.3 0.5 0.7 0.90.1 0.3 0.5 0.7 0.90.1
♦ts ts r r♣♦rts t r rt♦♥ ♦ t s♦t ♠rt ♣s T s t t ♣♥ t r rt♦♥ ♦ t r♠rt ♣s Th t ♥tr ♣♥ ♥ tr r♥ T s − Th t rt ♣♥ s ♥t♦♥s ♦ ♣r♠tr α ♦ ♥s♦rrs♣♦♥ t♦ β = 0.5 ♥ s ♥s rr t♦ β = 3 Pr♠tr ♦♠♥t♦♥s t r ♦r β ♥ ♦r r α
♦rrs♣♦♥ t♦ t st ♥st ♠♥ ♦r ♥sr♥ tr ♣r♠tr s r st s ♦♦s r = 0.04 σ0 = 0.05γ = 0.1
♦♠♦♦r♦ qt♦♥ t♠t② ②s
f(p) =C0
σ2(p)exp
(∫ p
0
2µ(s)
σ2(s)ds)
,
r t ♦♥st♥t C0 s s tt∫ p
0 f(p)dp = 1
rt ♣♥ ♥ r ♣ts t t②♣ ♣ttr♥ ♦ ts r♦ ♥st② s♦♥ tt
t ttr t♥s t♦ ♦♥♥trt ♥ t stts t t ♦ ♥♦♥♦s ♦tt② s ♣r♦♣rt②
♦ t r♦ ♥st② ♥t♦♥ s♦s tt t ♥t s♦s ♥rr ② ♥srrs ♠② ♥rt
♣rsst♥ s♦ tt t s②st♠ ♥ s♣♥ qt s♦♠ t♠ ♥ t stts t ♥sr♥ ♣rs
♥ ♦ ♥sr♥ ♣t② s ♥ t r♥t qr♠ ♠♦s t ♥♥ rt♦♥s s
r♥♥r♠r ♥ ♥♥♦ ♠♥♦ P ♥ ♦t ♣rsst♥ ♠rs s
♥tr ♦♥sq♥ ♦ t ♣t② st♠♥ts ♠♣♠♥t ② ♥sr♥ ♦♠♣♥s ♦♦♥
♣r♦t ♥ ♦sss ♥ ♣rtr ♥①♣t ♦sss r ♦♦ ② r s ♦ ♦♣rt♦♥s
♠♣②♥ tt t ♠② t ♦♥ t♠ ♦r ♥srrs t♦ rst♦r tr ♣t②
r rt ♦tt② ♥ r♦ ♥st② ♦ t ♥sr♥ ♣r
0.02 0.04 0.06 0.08 0.10p
-0.0008
-0.0006
-0.0004
-0.0002
0.0002
0.0004
μ(p)
0.02 0.04 0.06 0.08 0.10p
0.060
0.065
0.070
0.075
0.080
σ(p)
0.00 0.02 0.04 0.06 0.08 0.10p
7
8
9
10
11
f(p)
♦ts ts r ♣ts t t②♣ ♣ttr♥s ♦ t ♥sr♥ ♣r rt µ(p) t rt ♣♥ ♥sr♥ ♣r ♦tt② σ(p)t ♥tr ♣♥ ♥ t r♦ ♥st② ♦ t ♥sr♥ ♣r f(p) t rt ♣♥ ♦r α >
√2r ♥ β > 1 Pr♠tr
s r = 0.04 σ0 = 0.05 γ = 0.1 α = 0.5 ♥ β = 2
♦s
♦♥s♦♥
s ♣♣r ♣rs♥ts ♥r qr♠ rs♦♥ ♦ t ss r♥ t♦r② ♠♦ ♥ ♦♥t♥♦s
t♠ s st ♦r ①♠♣ ② rr ♥ ❲ ♠♦ ♦♠♣tt ♥sr♥ st♦r
tt ♦rs ♥sr♥ ♦♥trts t♦ ♣♦♣t♦♥ ♦ ♣♦t♥t ♥srs ♦♥r♦♥t t ♦rrt rss
r s ♥q ♦♠♣tt qr♠ ♥ ♣rs ♥ ♣ts r tr♠♥st ♥t♦♥s
♦ t ♦ rt rsrs ♥ t ♥sr♥ st♦r ♣r ②♥♠s ♥ ♦♠♣t
①♣t② tr t♥ r ②s ♥sr♥ ♣rs r rtr③ ② s②♠♠tr rrss
♠rt ①ts tr♥t♥ ♣r♦s r ♣r♠♠ ♥ ♣r♦tt② rs r ♠rts ♥
s♦t ♠rts r rt♦♥ ♦ r ♠rts s s♦rtr t♥ tt ♦ s♦t ♠rts ♣r♦
tt t stt② ♦ t ♠♥ ♦r ♥sr♥ s ♥♦t t♦♦ ♦
❲ s♦ ♥ tt ♥ t♦ t ♥sr♥ ♣r♠♠ s ② ♣rt tr r ♥♦
rtr ♦♣♣♦rt♥ts rs♦♥ s tt ♥st♦rs ♥♥♦t rt② s ♥sr♥ ♦♥trts t
♥ ♦♥② ② ♥ s t st♦s ♦ ♥sr♥ ♦♠♣♥s ♣rs ♦ ts st♦s r s♦♥t
♠rt♥s ♥ t♥ ♥ strt♦♥s ♥ r♣t③t♦♥s t ♥sr♥ ♣r♠♠s r
♥♦t ♠r② ①♣t ♣r♦ts r ♣♦st s♣t t ♣rt ♦♠♣tt♦♥ ♦♥ t ♥sr♥ ♠rt
s s s ♥ t ♣rs♥ ♦ ♥♥ rt♦♥s ♥ ♥ ♥ sr♦rs r rs ♥tr
♥sr♥ ♦♠♣♥s t♦ ♦♠♣♥st ♦r tr ①♣♦sr t♦ rt rss
♥tr ①t♥s♦♥ ♦ ♦r ♠♦ ♦ ♦ ♦r t ♥②ss ♦ t ♠♣t♦♥s ♦ rt♦r②
♠srs s s ♠♥♠♠ rsr rqr♠♥ts ♥♦tr ♣♦t♥t ♥ ♦ rsr ♦
t♦ ♦♥sr ♥ tr♥t ♠♦ s♣t♦♥ t P♦ss♦♥ rs s ttr st ♦r ♠♦♥
t tstr♦♣ ♥sr♥ ♠rt ♥② ♥trst♥ ♥ ♦ ♥qr② ♦ t♦ tst t ♠♦
♣rt♦♥ ♦♥ t s②♠♠tr② ♦ ♥rrt♥ ②s
r♥s
❬❪ ♦♦♠ ②s ♥ ♦t♦♥s sts Pr♦♣rt②st② t♦♥
❬❪ ♦t♦♥ P ♥ ♥ ❲♥ ❯♥ ♦r② ♦ ♦♥s q ♦r♣♦rt ♥st♠♥t
♥♥♥ ♥ s ♥♠♥t ♦r♥ ♦ ♥♥
❬❪ ♦②r qr ♥ ❱♥ ♦r♥ r ❯♥rrt♥ ②s ♥ ♦r
st ♦r♥ ♦ s ♥ ♥sr♥
❬❪ r♥♥r♠r ♥ ❨ ♥♥♦ r♦♦♥♦♠ ♦ t ♥♥ t♦r
♠r♥ ♦♥♦♠
❬❪ ♥ rr♥t♦♥ ♥sr♥ ♣♣② t ♣t② ♦♥str♥ts ♥ ♥♦♥♦s
♥s♦♥② s ♦r♥ ♦ s ♥ ❯♥rt♥t②
❬❪ ♥ ❲♦♥ ♥ ❯♥rrt♥ ②s ♥ s ♦r♥ ♦ s ♥ ♥sr♥
❬❪ ♦ rr ♥ P st Pr♦♣rt②t② ♥sr♥ ②
♦♠♣rs♦♥ ♦ tr♥t ♦s ♦tr♥ ♦♥♦♠ ♦r♥
❬❪ ♠♠♥s ♥ tr ♥ ♥tr♥t♦♥ ♥②ss ♦ ❯♥rrt♥ ②s ♥
Pr♦♣rt②t② ♥sr♥ ♦r♥ ♦ s ♥ ♥sr♥
❬❪ ♦rt② ♥ r♥ ♥sr♥ ②s ♥trst ts ♥ t ♣t②
♦♥str♥t ♦ ♦r♥ ♦ s♥ss
❬❪ r♦♦t ♥ P ♦♥♥ ♥ t Pr♥ ♦ ♥tr♠t ss ♦r② ♥
♣♣t♦♥ t♦ tstr♦♣ ♥sr♥ ❲♦r♥ P♣r
❬❪ ♥ Pttrs♦♥ ♥ ❲tt ❯♥rrt♥ ②s ♥ Pr♦♣rt②
♥ t② ♥sr♥ ♥ ♠♣r ♥②ss ♦ ♥str② ♥ ②♥ t ♦r♥ ♦ s
♥ ♥sr♥
❬❪ rr ❯ ♥ ❲ ♣t♠ ♥s ♥②ss t r♦♥♥ ♦t♦♥
♦rt♠r♥ tr ♦r♥
❬❪ ♦s P r ♥ ♦rr qt♦♥s ♦r s♦♥ Pr♦sss ♥ ❲② ♥②
♦♣ ♦ ♣rt♦♥s sr ♥ ♥♠♥t ♥ t ② ♠s ♦r♥
♦s ♦① P♥r s♥♦ r② P r♦ ♥ ♦ ♠t ♦♦♥ ❲②
❬❪ r♦♥ Pr♦♣rt②st② ♥sr♥ ②s ♣t② ♦♥str♥ts ♥ ♠♣r
sts P ssrtt♦♥ ♣rt♠♥t ♦ ♦♥♦♠s ssstts ♥sttt ♦ ♥♦♦②
❬❪ r♦♥ ♣t② ♦♥str♥ts ♥ ②s ♥ Pr♦♣rt②st② ♥sr♥ rts
♦r♥ ♦ ♦♥♦♠s
❬❪ r♦♥ ♥ s ①tr♥ ♥♥♥ ♥ ♥sr♥ ②s ♥ ♦♥♦♠s ♦
Pr♦♣rt②st② ♥sr♥ t ② r♦r ♦ ❯♥rst② ♦ ♦
Prss
❬❪ rr♥t♦♥ s ♥ ❨ ♦♥ ♥sr♥ Pr ❱♦tt② ♥ ❯♥rrt♥
②s ♥ ♦rs ♦♥♥ ♥♦♦ ♦ ♥sr♥ r ♠ ♥
❬❪ ♥♥ ♥ r② ♣t♠③t♦♥ ♦ t ♦ ♦ ♥s ss♥ t
♠t r②s
❬❪ ♠♥♦ P ♥ ♦t ♣♣s ♦ ❲rt s ♦ t
①tr♠ ♥♥ rt♦♥s ❲♦r♥ P♣r ❯♥rst② ♦ ❩r
❬❪ ❲tt ♥ ♥♥ ♥ P r♦tt rt ♥ ♦t
s♦ rt ①♣tt♦♥s ♥ ♥♦♥♦s ♦♥♦♠ ①♣t♦♥ ♦ ♥sr♥ ②s ♥ t②
rss ♦r♥ ♦ s ♥ ♥sr♥
❬❪ ♠♠♥♥♥t ♥ ❲ss ♥tr♥t♦♥ ♥sr♥ ②s t♦♥ ①♣t
t♦♥s♥sttt♦♥ ♥tr♥t♦♥ ♦r♥ ♦ s ♥ ♥sr♥
❬❪ r ❯ tt♦♥ ❯♥rrt♥ ②s ♥ Pr♦♣rt②t② ♥sr♥ Prt
s♦♠ t♦r② ♥ ♠♣r rsts ♦r♥ ♦ s ♥♥
❬❪ r ❯ ♥ tr s♥ss ②s ♥ ♥sr♥ ♥ ♥sr♥ t
s ♦ r♥ r♠♥② ♥ t③r♥ ♦r♥ ♦ s ♥♥
❬❪ ♦t ♥ ❱♥ qt② ♥♠♥t ♥ ♦r♣♦rt ♠♥ ♦r
♥ ♥ ♥sr♥ ♦r♥ ♦ ♥♥ ♥tr♠t♦♥
❬❪ ♠ t♦st ♦♥tr♦ ♥ ♥sr♥ ♣r♥r ❨♦r
❬❪ r t♦st s ♦r ♥♥ ♣r♥r ❨♦r
❬❪ trt Pr♦t ②s ♥ Pr♦♣rt②t② ♥sr♥ ♥ ♦♥ sss
♥ ♥sr♥ ♠r♥ ♥sttt ♦r P ❯♥rrtrs r♥
❬❪ ❱♥③♥ t♠♥ t♦s ♥ Pr♦t ②s ♥ Pr♦♣rt② ♥ t② ♥sr
♥ ♦r♥ ♦ s ♥ ♥sr♥
❬❪ ❲ss ❯♥rrt♥ ②s ②♥tss ♥ rtr rt♦♥s ♦r♥ ♦
♥sr♥ sss