ZurichOpenRepositoryand Archive Year: 2016

Zurich Open Repository andArchiveUniversity of ZurichMain LibraryStrickhofstrasse 39CH-8057 Zurichwww.zora.uzh.ch

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

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r ♥sr♥ ♣r ♥①

2000 2002 2004 2006 2008 2010 2012100

110

120

130

140

150

160

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price

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♦♥ ts ♥ ② r② ♦♥ ♣rt qr♠ ♣♣r♦ r t sts ♦ ♥sr♥ s♣♣②

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♦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♠ ♥

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♣♦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

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t rt♦♥ ①♣tt♦♥ qr♠ s rtr③ ② t ♦♥t ②♥♠s ♦ ♥sr♥ ♣rs ♥

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♦ 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 ♠♥

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♦ qr♠ ②♥♠s ❲ strt t ♥t ♦ s♥ t ♦♥t♥♦st♠ ♣♣r♦ ②

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rst ♦ t ♣♣r s ♦r♥③ s ♦♦s t♦♥ ♣rs♥ts t ♠♦ t♦♥ r

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♦♥s

♦

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♥ts s♦♥t t tr t ♦♥st♥t rt r ♠t ♦ss Lt ♥rr ② ♥

♣ t♦ t t s s tt

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r♦♥♥ ♠♦t♦♥ ♥ ♦♥ t ♣r♦t② s♣(Ω,F,P

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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

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♥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

❬❪ ❲ss ♥ ♥ ❯ ♥sr♥ Prs ♥♥ t② ♥ ♦

♣t② ♦r♥ ♦ s ♥ ♥sr♥

❬❪ ❲♥tr t② rss ♥ t ②♥♠s ♦ ♦♠♣tt ♥sr♥ rts

❨ ♦r♥ ♦♥ t♦♥

❬❪ ❲♥tr ②♥♠s ♦ ♦♠♣tt ♥sr♥ rts ♦r♥ ♦ ♥♥

♥tr♠t♦♥