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(5)

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uw¢+ar~Jo+¨tq uw}k

H

µ++r~¨¨r~vg @@z+¡£z+}uiqCkav z+v+| r~k'qCz+v+ar~v k¡ q¨qCr~}k{+z+v+q_'qCrf+r~v5nDkv iov++]k!k+z+z+¡}kr~¢0qIr~kv+r# r~qC¨{-uwv+v+r

T

z+v+_/zf+}qIkv3¨r~+r~£¯6r~¨k!+r

τ

^+kr v@z0qC~r~v+¡u 20

¡ kr~¨r~v++r_´8ar +r~ avqC¨a}k}]kwuw¨uw¢+}kr~vD krAp@²!'qCr~¡uiqCk z+v+ AwqLuiqCkavIs|a}kwqD~z+vzef+'q

±°¨rt²ar~Artq uw} * w¤a¤a¥+

kr ar~Cuw¡IqCrSq^}f+}suf+'q

W

r~kv+r~/iov++]k!k+z+z+¡ _¢5r~'qCk¡ ¡IqA¨efy+z+efy+krSq^r~'qCr +r~

a¡ r~vqLuwv+v@z0qC~r~v0s|z+v+qCkav

U

^kv ^3¨r~+r~ ¯6r~¨k!+r

τ

µ°+kr>v+r~uiqC]kkaavQ+r~uf+} +r~ kv

Cuwv+Jf+r~q kar~¨¢+CuefqCr~v|Kuwar

t ^k (τ )

^z+v+ ^{5a¨qC]k!kaav r~kv+r~¡

av+¨z+¡ }o0qCr~¨¢o+v++r~}

X(τ )

uw¢fzwv+wqp

W = Z T

0

e ⁻ ^ρτ U(t ^k (τ), X (τ ))dτ

^*) ⁺

uw av+¨z+¡ }o0qCr~¨¢o+v++r~} ~r~kwqar~¡ z+r~!o+¢+}kef+r~vmg v+v-uf+¡ r {5a¨qC]kar~v z+v+uw¢!

v+rf+¡ r~v++r~v´8¨r~v+~v@z0qC~r~vµ5+r~´8¨r~v+~v@z0qC~r~v>+r~ Cuwv+Jf+r~qC'qLuwark'qAv+r~uiqC]k5µuw¢5r~_k¡

±°rtqCCuw8uw¢+v+rf+¡ r~v+EA¢5r~R+kr ¨r~z+uw¢+}kr~ qCz+v+ar~v

∂ ² U

∂X∂t ^k

z+v+

∂ ² U

∂t ^k ∂X

z+v+-uw¡ q^o+¢5r~R+r~v

wa¡ {+}kr~¡ r~vqzw¨r~v !+r~¨z+¢+'qCqCz0qC]kar~v f-uwCuwqCr~*kaav av+¨z+¡ z+v+W´8r~¨z+v+f+r~q!_k¨

uwvy+kr~¨r~p@qCr~}k}krv+efywr~kv+rEg z+¨CuwarIartqC¨r r~v5uw´8r~¨z+v+f+r~qC¨iuw{+qLuw}Cf-uiq +k¨r~qCr~v

nDkv-z+¨ uwz0sR+krPq^}f+}suf+'qA+r~/iov++]k!k+z+z+¡ ~µ--u +krPkaavyf+¡Sr~¨¢+CuefqCr* r~k'qCz+v++kr

g v+uf+} +r~

Cuwv+Jf+r~qC'qLuwarPkar~¨¨kv+ar~'qp

t ^k (τ) = t ^k (H(τ ))

^* ⁺

¡ q

∂t ^k (H(τ ))

∂H(τ) < 0

^* ^¥+

z+v+

∂ ² t ^k (H(τ ))

∂H(τ) ² > 0

^* ⁺

uwR´8r~¨z+v+f+r~qC¨iuw{+qLuw}_k¨+z+efEiov.kar~'qCqCkav+r~v£uw@@z+¡£z+}kkr~'qµ uwKuwz0s0°r~.q^r~ar~v

r~'s|a}kar~viuwv+v¤p6uwbiov++]k!k+z+z+¡iuwv+v r~q

t ^I

uwz0s°r~v++r~v *|~z+¡ ±°r~k¨{+kr~}<p!{5a'q

+

!+r~

¡ r~+k~kv+kef+rj r~k'qCz+v+ar~v

M

iuwz0s|r~vµ}°a¢5r~+krjiovqCr~¨{+¨rtqLuiqCkav1kaav

M

^°r~^qCr~Y~z ^suw¨¨r~v

k'quw}k v@z+ -uwnDkv+iuwz0s|r~vxkaav zw¨tqC}kef+r~ ±°rf-uwv++}kz+v+^!+r~ {f-uw¨¡uw~r~z0qCkef+r~ ¯D¨i

+z+qCrwK kr~¨r r~k'qCz+v+ar~v av+v+r~v uwzefg z0s°r~v++z+v+ar~v s%o+ ar~¨z+v++r#nD¨vzf+¨z+v+y!+r~

s%o+r~kv+r~v uw}k}kar~¡ r~kv+r~v ar~¨z+v++r~v_ r~¢5r~v+L uwv++r~}Y¢5r~+r~z0qCr~v6´8}kr~ef+~r~qCk,_k¨kv13¨r~+r~

¯6r~¨k!+rAr~kv,g vqCr~k}

δ

kaa¡%´8r~¨z+v+f+r~qC¨iuw{+qLuw}5uw¢+ar~ef+¨kr~¢5r~vµ.°a¢5r~5+kr~¨r~ g vqCr~k}¡ q

+r~¡ g }qCr~ 'qCr~kwqµ+f

∂δ

∂τ > 0

- krPg @@z+¡£z+}uiqCkav+¨a}kr~ef@z+v+ }uwz0qCrtq_uw}k¨!_krs|a}kwqp

Nbsn=b&b_Ne n HRPPb

T

^dNc Q#NoTN]_Ne muQub_cdbTN]_Ne No amuQub_TT[&bsn eNo o0es`

(7)

∂H

∂τ = H ˙ = I(M (τ), t ^I (τ )) − δ(τ)H(τ )

^*¦ ⁺

krA´8¨r~v+~{+¨!+z+qC]k!qziqCr~v +r~Yuwz0s|ar°r~v++rtqCr~v# r~qDz+v+ +r~D¡ r~+k~kv+kef+r~v, r~k'qCz+v!

ar~v¢5r~R+r~Sg @@z+¡£z+}uiqCkav>+r~I´8r~¨z+v+f+r~qC¨iuw{+qLuw}kk'q uwv+v-uf+¡ r~ar~¡ z {5a¨qC]kyz+v+

uw¢+v+rf+¡ r~v+ uwv+r~¢5r~v¡£z+¨-uwPiov++]k!k+z+z+¡:¢5r~K+r~ u 20k¡ kr~¨z+v+ kaav ´8}kr~ef@z+v+>

uw}k 0_r~'qC¨kqCkav+krg @@z+¡£z+} uiqCkav+¨a}kr~ef@z+v+

¢5ruefqCr~vUz+v+W-uw'swr~kv+r>v+r~uiqC]kar~v

Yr~¨¡,aar~v+L°r~'qCr

A

f+kvqCr~¨}uw¨¨r~vRuw£¢5r~'qCrf+r~v++rYr~¨¡,aar~v r~L_k'qCef-uis qCrtq£kv&3¨r~+r~

¯6r~¨k!+r kv+¨r~vkvx/}f+r#kaav

rA

^Y kr ^r~^qµR+kr v-uefxg ¢+~z+^+r~ kv

Cuwv+Jf+r~q1kar~)

¢+CuefqCr~v Kuwarz+v++r~8~z iov.kar~'qCqCkav+¨°reLwr~vuwz0s|ar°r~v++rtqCr~v Kuwar*kar~¨¢+}kr~k¢0qµ<_k¨

~z+NnD¨~kr~}kz+v+r~kv+r~NnDkv+wa¡ ¡ r~v+

Y

kar~L°r~v++rtq0±°r~oeL@¨efqCkwq<¡uwv v@z+v v+ef +kr ¡ q

+r~v ¯D¨r~k¨r~v

P

^¢+

D

¢5r°r~'qCrtqCr~v_nDkv+Wzwz0s|rEkaav av+¨z+¡ }o0qCr~¨vz+v+¡ r~+k~kv+kef+r~v

r~k'qCz+v+ar~vµ-}uwz0qCrtq_+kr0_r~'qC¨kqCkav¤p

∂A

∂τ = ˙ A = rA(τ) + Y (t ^k (τ) + t ^I (τ )) − P (τ)M (τ) − D(τ )X(τ )

^* ⁺

krAx/}f+r_+r~<r~¨~kr~}qCr~v nDkv+wa¡ ¡ r~v+

Y

fzwv+wq<v-uiqo+¨}kef a}kr~ef+r~¨¡uar~vv+r~uiqC]kEkaav

t ^k

^z+v+

t ^I

uw¢+§^of+'q ¡uwv v@z+v v+ef +kr/°r~qCr~¨r~v 0Auwv++¢5r~+kv+az+v+ar~v r~kvµ0-uw¨ ¢+k°~z+¡

nDv++rA+r~<¯D}uwv@z+v+af+a¨k~avqCY+r~°´8r~¨z+v+f+r~qC¨iuw{+qLuw}k¢5r~'qLuwv+ r~kv+r~v ar_k¨¨r~v1q^r~'q

H ⁰

v+efq z+vqCr~¨ef+¨r~qCr~v-uw's)µ z+v+U-uw¨#kaa¡ iov++]k!k+z+z+¡ wr~kv+r pef@z+}k+r~v f+kvqCr~¨} uw¨¨r~v

°r~¨+r~vo+'s|r~v¬µ }vzw¨'q¨ef-uw({0qCk¡ kr~¨z+v+a¨{+¨a¢+}kr~¡ ¡ q xAk}s|r r~kv+r~ ! W H

W

}a¨r~v' z+S6ref+v+k¨r~°uwz0s +r~v_g vf-uwv+y± kar~L_kr~¨r~vKuwS_efqCka'qCr nD¨ar~¢+v+k

+r~)({0qCk¡ kr~¨z+v+ k'qA-uw/g v-uw}kaaav ~zy+r~v7o+¢+}kef+r~v>´8¨r~v+~r~'qCCuwioa}kr~ef!'´8¨r~v+~wa'qCr~v

EA¢5r~¨}kr~az+v+ar~v z+v+} uwz0qCrtq<¡ q

q(τ )

^uw}k ^n9r r~qC]k!{+¨r~k<+r~°´8r~¨z+v+f+r~qC¨kv.kar~'qCqCkav+r~v s%o+

uw}k}krS r~qC{+z+v+qCr

τ

^p

e ⁻ ^ρτ ∂U/∂t ^k (τ)

e ⁻ ^rτ λ(0) + ∂Y (τ)

∂t ^k (τ)

∂t ^k (τ )

∂H(τ ) =

r + δ(τ) − q(τ ˙ ) q(τ )

q(τ )

^* ⁺

xAkr~¨¢5r~R~r~kwq8+kr}kkv+wr p!r~qCr*kaav^´8}kr~ef@z+v+ +r~v^´8¨r~v+~r~'qCCuw+µ+r~8kv+¨r~ a¡

{5av+r~vqCr~vkvqCr~¨{+¨rtqCkr~'qE°r~¨+r~viuwv+vµKz+v++kr¨refqCryp!r~qCr+kr ´8¨r~v+~wa'qCr~v+r~*iov!

kar~'qC qCkav>kv´8r~¨z+v+f+r~qC¨iuw{+qLuw}5 r~d6r~¨¡

∂t ^k (τ)

∂H(τ)

uwz0sY+r~ }kkv+wr~vp!r~qCrIk'qar~¡ z+r~

g v+v-uf+¡ r~vy{5a¨qC]k5iov+v+r~f-uw}k¢>+r~AreL@kar~v }uw¡ ¡ r~ ¢5r~ef+¨r~k¢5r~v>+kr °r~Dp!z+¡ ¡uwv!

+r~v>+krs|a}kar~v++r~vZn9r r~qCrp r~ ¨refqCr*6r~¨¡:r~vqfzw}q +krEnDf}f@z+v+ +r~dnDkv+wa¡ ¡ r~v+

uwz0s|a¨z+v+>¢5r~¨¨r~¨r~I´8r~¨z+v+f+r~q*%iovqCr~¨{+¨rtqLuiqCkav7kaav

H

uw}kiov.kar~'qC qCkav+¨az0q

+ µ +r~}kkv+wr

6r~¨¡cr~vqfzw}q8-uwar~ar~v+r~v uw¢5+k¨wavqCkr~'qCr~vz+v+¡ q +r~¡pef-uiq¨qCr~v+{+¨r~k+r~dYr~¨¡,i

ar~v+

λ(0)

¢5r°r~'qCrtqCr~v +k¨r~qCr~v,g v+'qCkr~ +r~ a¡ r~vqLuwv+v@z0qC~r~v+)*%iovqCr~¨{+¨rtqLuiqCkav1kaav

H

uw}k av+¨z+¡ az0q +

H (T ) ≥ H ⁰

PHQ

A(T ) ≥ 0

`

(8)

uw z+¨¨{+o+v+a}kef+rwµ¬kv8+k¨@¨rtqCr~^ r~qs|a¨¡£z+}kkr~'qCr

!+r~}k}wz+vqCr~¨ef+r~k+rtqK¨efS_krRs|a}kwq

xAkr~R{+¨!+z+~kr~'qY-uwciov++]k!k+z+z+¡ -uw

av+¨z+¡ }o0qCr~¨¢o+v++r~}-¨r~}k¢+'qµ¡ q9g ¨¢5r~qC¨~r~qYz+v+

r~kv+ar~iuwz0sqCr~vW´Po0qCr~¨vuw}k!iov+{+z0q¨suwqCa¨r~v< kr ar~Cuw¡IqCr ~z+!Yr~'s%o+az+v+ 'qCrf+r~v++rZ r~q

¡£z+¨Auw}k¨ uwz0s'k!kr~{g }qCr~¨v-uiqC]kar~v#kar~'qCr~k}q{°r~¨+r~vµ

t = t ⁱ (τ) + t ^w (τ ) + t ^x (τ ) + t ^k (τ ),

^* ⁺

uw}k¨ûwz0s8´8r~¨z+v+f+r~qC's%o+¨äärwµ }f+v-uw¨¢5r~qµ

av+¨z+¡ }o0qCr~¨{+¨!+z+qCkavz+v+

Cuwv+a

f+r~qNioav+a¨kr~'q¡uwv+kr

av+¨z+¡ }o0qCr~¨{+¨!+z+qCkavµ zwv++r~'q ¨ef@z-uw}kqLuiqC]k v+efqC uwv

+r~¡ !+r~}k}

r~I±°k}k+z+v+a¨'qLuwv+µ !+r~+kr x/}f+r +r~IxAz+¡uwv+iuw{+qLuw}k @< ´8r~¨z+v+f+r~qC¨iuw{+qLuw}µ

_k¨qD¨ef{5a¨qC]kIuwz0s5+kr ¯D¨!+z+qCkav£+r~ av+¨z+¡ }o0qCr~¨¢o+v++r~}kDz+v+ +kr iov.kar~'qCqCkav+r~v

kv ´8r~¨z+v+f+r~qYuwz+~µW_k¨ uw¢5r~Rv+efqDr~v++aar~vk¡ !+r~}k}0¢5r~'qCk¡ ¡Iqi r~Y±°k}k+z+v+a¨'qLuwv+

+kr~vqjf+kr~_uw}k{g v-uw}kaakr~z+j6ref+v+a}kaakr8kv +r~A¯D¨!+z+qCkav

lA¡ -uw !+r~}k}~zuwv-uw}²!¨kr~¨r~vUz+v+r~¡ {+k¨kefQ~zUqCr~'qCr~vµ{_k¨Ur~ kvU+r~ 0_r~ar~}

°r~qCr~jkar~¨r~kv0suefqpUiovy+r~´8}kr~ef@z+v+ _k¨ r~vq°r~+r~+r~Awav+¨z+¡ {0qC]kar !+r~Akv.kar~'qC

kar ´8¨r~v+~r~'qCCuw!kaavy´8r~¨z+v+f+r~qC¨kv.kar~'qCqCkav+r~vkar~¨v-uef+}vzw¨¨kwqµ+z+¡%-uwv+v r~kv+r~¡ ¨r~kv+r~v

iov.kar~'qCqCkav+)°!+r~

av+¨z+¡ ¡ !+r~}k} ar~ar~vJo+¢5r~¨~z+'qCrf+r~v

BC4354BC4DPOQv:=JQ>8>8FI\:QWV#I\[5=;];

g z+_+r~4({0qCk¡uw}kqziqC¨¢5r~+kv+az+v+Ikv ´8}kr~ef@z+v+ _k¨ k¡ ¨r~kv+r~v|iov.kar~'qCqCkav+¨¡ !+r~}k}p

∂Y (τ)

∂t ^k (τ)

∂H(τ) =

r + δ(τ ) − q(τ) ˙ q(τ)

q(τ)

^* ⁺

kr!Yr~zwv++r~¨z+v+ +r~bnDkv+wa¡ ¡ r~v+8+z+ef

Cuwv+Jf+r~qC'qLuwark'qr r~v+¨efqC}kefa}kr~ef

+r~¡ r~vqCuwv+ar~v+r~v }f+vµ<f

∂Y

∂t ^k (τ) = −w(τ )

uwCuwz+ }uw¨¨r~vQ¨efQv-uefQr~kv+r~r~kv!

ef+zwv+wr~v++r~v#g v+v-uf+¡ rA+kr uef0s|Cuwar~vv-uef ´8r~¨z+v+f+r~qC¨iuw{+qLuw}-z+v+¡ r~+k~kv+kef+r~v

r~k'qCz+v+ar~v f+r~¨}kr~qCr~v_ krZg v+v-uf+¡ r¢5rtqC¨]r5q +kr g ¢fzwv+akawr~q kv

Cuwv+Jf+r~qkar~)

¢+CuefqCr~SKuwar

t ^k (τ )

^kaa¡ 3¨r°r~k}kkar~v A]karuwz+r~I´8r~¨z+v+f+r~qC¨iuw{+qLuw}Uq^r~v+v+kr ¯D¨i

+z+qCkav+¨r~}uw'qCk~qziq

^kaav

H

@}kr~kv+r~<k'q uw}k<r~kv+~µ!f°r~v+v

H

^¢5r~ ^+r~ ¯D¨!+z+qCkav,kaav

t ^k (τ )

¨kv+wr~v++ryp!iuw}kr~v+r~'qCzwar uwz0s°r~k'qµR}uw¨¨r~v ¨ef+kr ¢5r~k+r~v uef0s|Cuwarts|z+v+qCkav+r~v

_kr kv +r~vxg ¢+¢+k}k+z+v+ar~vQ z+v+ -uw¨'qCr~}k}kr~vD kr~¨rg v+v-uf+¡ r k'qkv +r~vW´8}kr~ef@z+v!

ar~v

¥

z+v+

r~vqf-uw}qCr~vz+v+k'q wa¡ {-uiqCk¢5r~} ¡ q r~kv+r~ °a¢+¢!) az+a} uwp!{5r~~]s-iuiqCkav

kr8¡ q +r~¡ g }qCr~ ~z+v+rf+¡ r~v++rbg ¢+ef+¨r~k¢+z+v+a¨CuiqCrS_k¨ qC¨wqC'qCr~kar~v++r~ uef0s|Cuwar

v-uef¡ r~+k~kv+kef+r~v r~k'qCz+v+ar~vv+efqwa¡ {5r~v+¨kr~'qµ ¨ -uw¨8+r~ ´8r~¨z+v+f+r~qC¨iuw{+qLuw}

¢5r~'qLuwv+>wavqCkv@z+kr~¨}kef uw¢+v+k¡ ¡Iq-uwdnDv++rI+r~¯D}uwv@z+v+af+a¨k~avqC

T

^µ5uw}k¨ +r~/6!µ

_k¨ r~¨¨r~efqµ°r~v+v

H

r~kv+r~v.0_rts|r~¨r~v+°r~'q

H ⁰

r~¨¨r~efq

Nb@jb_]_b_Nje [&b_NJklcmBnn=o;RP U psrrrBZM`

(9)

g ¢+¢+k}k+z+v+ }pUnDvq_eL@}kz+v+ +r~ uef0s|Cuwarv-uef>´8r~¨z+v+f+r~qC¨iuw{+qLuw}

g ¢+¢+k}k+z+v+

pnDvq_eL@}kz+v+ +r~ uef0s|Cuwarv-uefy¡ r~+k~kv+kef+r~v r~k'qCz+v+ar~v

(10)

iovU+kr~¨r~¡ 0Auf+¡ r~vQ}vzw¨'q ¨efuwzef+r~n9r r~q r~kv+r~ }f+v+CuiqCzwv++r~¨z+v+uwz0sI-uw

Yr~f-uw}qCr~vW~r~kar~v kr7 r~k'qCz+v+kaav

H

^k'q +kr7nDf}f@z+v+ +r~Kuwarwµ uwvW+r~v+r~vU-uw

iov++]k!k+z+z+¡ uw¨¢5r~qCr~v iuwv+v;Kr~kvf}f+r~¨r~1 }f+v+CuiqC r~f}fq uw}k¨y+r~v´8¨r~v+~r~'qCCuw>+r~

´8r~¨z+v+f+r~qC¨iuw{+qLuw}k~<u^-uw iov++]k!k+z+z+¡ k¡ ({0qCk¡£z+¡ +r~ ±°r~+kv+az+v+ s|a}kwqµD-uw¨

´8¨r~v+~r~'qCCuwa}kr~ef +r~vy´8¨r~v+~wa'qCr~vkaav

H

^¨r~kv ¡£z+¨~µ+Jav+v+r~v +kr~¨r8v@z+vf}f+r~ ¨r~kvµ z+v+£r~'_k¨£¡ rf+R´8r~¨z+v+f+r~qRv-uef+arts|Cuwwqwu uw¢5r~

H

^uwzef£¡ ^qKxAk}s|rckaavkv.kar~'qCkr~'qCr~

r~q8uw@@z+¡£z+}kkr~'q_k¨µ ak¢0qr~r~kv+r~var~ar~v+}vzwzs-ar~v n9r r~qp\nDkv'qCr~kar~v++r~S }f+v+CuiqC

r~f}fq a}kr~ef+~r~qCk+r~v q^r~'q +r~°kv.kar~'qCkr~'qCr~v r~q

t ^I

^z+v+ ^-uw¡ q<+kr´8¨r~v+~wa'qCr~vkaav

H

@lAvqCr~¨'qCr~}k}q6¡uwvv@z+v£uw}k}kr~¨+kv+a~µt-uw¨

H

v+efqKuwz+¨ef+}kkr a}kef¡ q' r~qµ¨av++r~¨vIuwzef

¡ qD¡ r~+k~kv+kef+r~v, r~k'qCz+v+ar~v {+¨!+z+~kr~'qN_k¨µa¨k'qY+kr~¨r~ n9r r~q°@}kr~kv+r~Yuw}kD+r~Yuwz0s

+r~v>´8¨r~v+~r~'qCCuw +r~ ´8r~¨z+v+f+r~ qµ-¨ -uw¨Akv+¨ar~Cuw¡Iq s%o+ -uwjiov.kar~'qCqCkav+¨¡ !+r~}k}ak}qp

∂H(τ )

∂w(τ) > 0

^*) ^¤+

q^r~v+vv@z+v

t ^I

^¨kv+q z+v++r~, }f+v+CuiqC r r~v+¨efqC}kefwr~kv+r~v nDkv-z+¨ uwz0s +krg ¢!

ef+¨r~k¢+z+v+a¨CuiqCr

δ

f-uiqµ+iuwv+v +kr~¨r~jf}f+r~¨rI´8r~¨z+v+f+r~qC¨iuw{+qLuw}k'qCeLv@z+_+z+ef|f}f+r

¨r uef0s|Cuwar v-uef¡ r~+k~kv+kef+r~vx r~k'qCz+v+ar~v{+¨!+z+~kr~'q!°r~¨+r~vµY¨ -uw¨ r~¢5r~v0suw}k}k

ar~}qCr~v ¡£z+¨p

∂M(τ )

∂w(τ) > 0

^*)a ⁺

u¢5r~8+r~ ¯D¨!+z+qCkav kaavV´8r~¨z+v+f+r~qyuwzefkv.kar~'qCkr~'qCr_ r~q r~kv+r 0_a}k}kr>¨{+kr~}q

*|+kr ¡ q8+r~¡ }f+v+CuiqC ¢5r°r~'qCrtqE_k¨

+

µ ¨kv+ +kr r~v+ar~v+¨ruwqCkav+r~v_kaav

H

^z+v+

M

v+efq a}kr~efa¨a¨ N_k¨W+kr7 r~q

t ^I

^¨r~}^uiqC]k qCr~z+¨r~ uw}k +r~ ¯D¨r~k+r~¡ r~+k~kv+kef+r~v

r~k'qCz+v+ar~vµ-¨,_k¨ +r~Ar~kar~v+rE r~qLuwz0s uwv+ +z+ef

M

¨z+¢+'qCqCz+kr~'q r~/g v+'qCkr~!kaav

H

^uw}k 0_ruwqCkav£uwz0s-r~kv+r~v£{5a¨qC]kar~v* }f+v+ef+eLIk'qRuw}k¨ @}kr~kv+r~Kuw}k6+r~^g v+'qCkr~/kaav

M

uw}k}kr~kvk¡ {+}kk~kr~'q}g v++r~¨rbn9r r~qCrwµ-kv+¨¢5r~¨av++r~¨r8+r~ r~kv+r~_pef+eL@ uwz0s6-uwYr~¨¡,aar~vµ

°r~¨+r~vyk¡ uw{+qCr~} ¦! +k¨@z0qCkr~'q

BC4354tC4DPOQ I\:QW?<V7V#I\[5=;];

g v +kr~¨r~Ap@qCr~}k}krd_k¨ +r~/nDkv-z+¨jkaav>´8r~¨z+v+f+r~qAuwz0sK+kr8¡,aa}kef+rdg ¨¢5r~qC¨~r~q{kar~)

v-uef+}vzw¨¨kwq- kr7({0qCk¡uw}kqziqC¨¢5r~+kv+az+v+£uwz+ ´8}kr~ef@z+v+ kar~¨Jo+¨tq ¨ef -uf+r~_~z

1 λ(0)

∂U

∂t ^k (τ)

∂H(τ ) e ⁻ ^ρτ e ^−rτ = 1

λ(0)

∂U

∂H (τ ) e ⁻ ^ρτ e ^−rτ =

r + δ(τ) − q(τ ˙ ) q(τ )

q(τ )

^*) ⁺

r~´8¨r~v+~r~'qCCuw,kaav7iov.kar~'qCqCkav+r~v>kv´8r~¨z+v+f+r~q¢5r~'qCrfquw}k¨v@z+ v+efuwz+A+r~¡

Az0qC~r~v-uwv+'qCkr~+µ ¢5r°r~'qCrtq8¡ qA+r~¡ pef-uiq¨qCr~v+{+¨r~k+r~Yr~¨¡,aar~v+ ~z+¡ r~qC{+z+v+q+r~

({0qCk¡ kr~¨z+v++uwCuwz+ s|a}kwqµ-uw¨¢5r~K¡ qA+r~¡ g }qCr~A'qCr~kar~v++r~vZg ¢+ef+¨r~k¢+z+v+ar~v +kr

(11)

uef0s|Cuwarv-uef>´8r~¨z+v+f+r~q uwzef kv +kr~¨r~¡ p!z+¢+¡ !+r~}k}¨kv+qAz+v+|g ¢+¢+k}k+z+v+ uwzef

f+kr~K+r~v!Yr~¨}uwz0s-~z0qC¨rr r~v+ ~r~kwqa kr g ¢+¢+k}k+z+v+

k'qD¡ q6+r~¡hYr~¨} uwz0s+r~ uef0s|Cuwar

v-uef ¡ r~+k~kv+kef+r~v, r~k'qCz+v+ar~vz+vqCr~Yr~kv+r~c°r~qCr~¨r~v,g v+v-uf+¡ rAuwv +kr kvqCr~'qCr~¡ {5aCuw}kr

p!z+¢+'qCqCz0qCkav+¨r~}uw'qCk~qziq wa¡ {-uiqCk¢5r~}µ!+kr uw¢5r~ +z+ef-uwz+A¨ruw}kk'qCkef k'q

Ak¡ ¡IqY¡uwv.*_krAk¡ z+¨¨{+o+v+a}kef+r~v !+r~}k}kaav ´8¨a¨¨¡uwv + uwvµ-uw¨ r~q°uwzef¢5r~

+r~ ¯D¨!+z+qCkav +r~ av+¨z+¡ az0qCr~ r~kv+ar~¨rtqCtq{_k¨µ0¨k'q +r~{n9r r~q_r~kv+r~{ }f+vzwv++r

¨z+v+>v+efq£r~kv++r~z0qCk+'nDkv_g v+'qCkr~|kaav

w

_k¨q£¨ef uwz0s °r~ g 'qCr~v uwz+ Kr~Ir~f}fq

¨@°}f+} +kr a'qCr~vy+r~ ¯D¨!+z+qCkavkaav´8r~¨z+v+f+r~qµuw}kAuwzefy+krPkaav av+¨z+¡ q^r~}

ef+r~A´8z0q* av+¨z+¡ !+r~_´8r~¨z+v+f+r~q + v-uef +r~¡ }f+v+ef+eL¡ rf+°v-uef+arts|Cuwwq _k¨µ

fzwv+wq -ukaavWuw¢µ9°r~}ef+ry¯D¨!+z+qCkav+r~v iov+{+z0q r~q kvqCr~v+¨]kar~ v@z0qCtq°p!kr_k¨¨r

}uiqC]k qCr~z+¨r~~µ uw}k¨_k¨-uw8+a'qPf+r~¨ar~'qCr~}k}qCr ´8z0q¨r~} uiqC]k°r~v+kar~8v-uef+arts|Cuwwq kr~¨r

lAv+r~kv++r~z0qCkawr~q k'q r~kvlAvqCr~¨ef+kr~ +r~EnD¨ar~¢+v+k¨¨r+r~Eiov.kar~'qCqCkav+)z+v+^+r~

av!

¨z+¡ ¡ !+r~}k}k~µJ_kr¨krdkaav>´8¨a¨¨¡uwv z+¨¨{+o+v+a}kefkaa¨ar~ef+}uwar~v_z+¨+r~v

r~^_efqCka'qCrlAvqCr~¨ef+kr~}kkr~wq3¨r~+efkvI+r~ 0_ruwqCkavI+r~K¡ !+r~}k}kr~v++aar~v+r~v£´8i

ar~vyuwz0sKr~kv+rdYr~zwv++r~¨z+v+,kaav

A

z+v+_k¨ kv uw{+qCr~} ¦! -uw¨ar~'qCr~}k} q

'(='( 50/>@?0:3&?0:

g zef °r~v+v kv>+r~I´8¨z+v+0s|a¨¡:+r~ ´8¨a¨¨¡uwv! !+r~}k}k +r~SnDkv-z+¨Pkaav^pef+eL@ uwz0s

wa¡ {-uwCuiqC]ko'qLuiqCkef+rq^r~k¨r uwv-uw}²!¨kr~'q!_k¨µR¨ s-v++rtq+r~ r~kar~vqC}kef+rYa¨uwv++r~

({0qCk¡ kr~¨z+v++efWz+vqCr~ p!ef+r~f+r~q'qLuiq¨q ´8r~Cuw+ryk¡ z+Cuw¡ ¡ r~vf-uwv+^¡ q Cuwv+a

f+r~qCr~vz+v++r~1qUk¨@Cuw¡ wr~q1kaav {f+r~Cuw{+kr~vk'q -uw r~kv+r z+v+¨ruw}kk'qCkef+rg v+v-uf+¡ rw

r~f-uw}k¢ k'q -uw !+r~}k}5z+¡ 0_k¨kw£r~L°r~qCr~'q°a¨+r~v;-uw}k}kr~¨+kv+aNf+r~¨¨efqAwr~kv+rPnDkv+ki

wr~qA-uwo+¢5r~~µ-uwv°r~}ef+r~p@qCr~}k}kr+krSlAv+¨ef+r~f+r~qAr~kv+arts%of+'q{°r~¨+r~v>¨a}k}

uw¨-uwv+av+6z+v+q uwa'qLuWr *)@

+

¢+}kr~k¢5r~v w± 5k¡ 0Auf+¡ r~vy+r~diov.kar~'qCqCkav+¨¡ !+r~}k}k

z+v+ s%of+¨r~vlAv+¨ef+r~f+r~qA+a'q_r~kvµ° +r~ s|z+v+qCkav-uw}krS z+Cuw¡ ¡ r~vf-uwv+ _kef+r~v´8r

¨z+v+f+r~qz+v+ nDkv+wa¡ ¡ r~v¢5r~'qCk¡ ¡Iq!_k¨YuwnDkv+wa¡ ¡ r~vz+vqCr~ p!ef+r~f+r~q,_k¨

uw}k

Y [t ^k (τ ), t ^I (τ)]

+rs-v+kr~'qµ°a¢5r~ -uw ´8r~¨z+v+f+r~qC¨v+]karuwzo+¢5r~ +kr/Yr~¨¨kv+ar~¨z+v+£+r~

Cuwv+Jf+r~qC'qLuwarD-uw'nDkv+wa¡ ¡ r~v+rtqCr~¨¡ kv+k'qCkefr~f}fqYr~¨r~kv0suefq6¡uwv-uw !+r~}k}µ

}vzw¨'q_¨ef +r~ z+Cuw¡ ¡ r~vf-uwv+ uwzef>uw}k

Y [t ^k (H(τ ))] = Y [H(τ )]

^*) ^¥+

ef+¨r~k¢5r~vRuw¨-uwv+av+<z+v+q uwa'qLuWr *)@ + ef+}uwar~v v@z+v_kaa~µ'f+kr~ r~kv 'qCef-uw'qC

ef+r~nD}kr~¡ r~vqr~kv+~z0s%of+¨r~vµ°kv+r~¡ ´8}kr~ef@z+v+ ¥ {-uwCuw¡ rtqC¨k¨kr~'q1_k¨µYz+v+Q+kr~¨r~

¯KuwCuw¡ rtqCr~<uw}k z0suw}k}kLkwuw¨uw¢+}kr<¡ qYar~ar~¢5r~v+r~Nq uf+¨ef+r~kv+}ef+wr~qCLkar~'qCr~k}kz+v+Ikv+v+r~f-uw}k¢

r~kv+r~jiovqCr~Lkwuw}k}k +rs-v+kr~'q{_k¨¤p

Nb;NPBeb_c=eb_o&mc RTb

[Hn eNe

eNmPHn=b_TRneN]_Ne emP

H

jN[ue0RPO^dNb&b_cbBeQRn@klbsn

PHQuSb_Ne n R Ne RT

]_^dNn Sb_P9Qub_PJb_cNmuQub_P#jb*e R

n SBed^Db_c Qub_P RPP

PHQ#o

nnTb_NPb_c@p@n=b_NPO`

(12)

Y [H(τ)] = kH (τ )

^*) ⁺

¡ q

k ∈ [k, k]

^*)]¦ ⁺

z+v+

E[k] = ˆ k

^*) ⁺

nDkv ¨k¨kwav+r~z0qCCuw}kr~{iov++]k!k+z+z+¡ jo+¨+rv@z+vyz+vqCr~_+r~ ±°r~v@z0qC~z+v++r~/nDL uw'qCz+v+a)

°r~'qCr~

ˆ k

¨r~kv+r~v Az0qC~r~v¡u 20k¡ kr~¨r~v- krAz0qC~r~v0s|z+qCkav_k¨>uw}k¨ ¨ +rs-v+kr~'qµ-uw¨

¨ef -uwjiov++]k!k+z+z+¡S¨k¨kwukar~¨{kar~fzw}qµ°a¢5r~ +r~ ´8Cuw +r~#0_k¨kwukar~¨¨kav ¡ q +r~

x/}f+r +r~dnDkv+wa¡ ¡ r~v+A¢+UYr~¨¡,aar~v+A¨kv+qdlA¡ +kr

a¡ {+}kr-20qziq_~zy¨r~+z+~kr~¨r~vµ

_k¨£-uw !+r~}k}0uwz0s°r~+¯6r~¨k!+r~v¢5r~ef+zwv+q@p!a¡ qK¡£z+¨R+r~Dx uwz+f-uw}qR¨r~kv+r~v Az0q'

~r~v z+vqCr~£r~kv+r~£z+v+¨ef+r~¨r~v 0_r~'qC¨kqCkav^¡u 20k¡ kr~¨r~v krg z0qCa¨r~v ~r~kar~v v@z+vµK-uw¨

z+vqCr~Y+kr~¨r~v,g v+v-uf+¡ r~v s|a}kar~v++r~NlAvqCr~¨ef+kr~_kef+r~v p!ef+r~f+r~ q<z+v+ lAv+¨ef+r~f+r~q

¢5r~'qCrfqp kr£§-CuwarwµU_kr£-uwdiov++]k!k+z+z+¡ kv av+¨z+¡ }o0qCr~z+v+kv>¡ r~+k~kv+kef+r* r~k)

qCz+v+ar~vIkv.kar~'qCkr~'qµ]iuwv+v v+efqK¡ rf+\kaav+r~kv-uwv++r~6artqC¨r~v+vq'°r~¨+r~vw kr uef0s|CuwarYv-uef

¡ r~+k~kv+kef+r~v* r~k'qCz+v+ar~v'qCr~kwqK¨@°}f+}+¡ q6+r~¡0_k¨kw+µiuw}kKuwzef¡ qK+r~¡ Yr~¨¡,aar~v;

_kr8´8¨a¨¨¡uwv *)@}} + 3¨r~+ef uwv++r~z0qCrtqµ+}kkr~wq<-uw r~¨¨r~vqCkr~}k} uwv +r~°s|z+v+qCkav-uw}kr~v §-a¨¡

kaav>´8}kr~ef@z+v+

nDkv+rSzf+v+}kef+rSlAvqCr~¨¨zef@z+v+s%of+¨r~v uw¨-uwv+av+ z+v+#q uwa'qLuWr *)@}

¤+

s%o+ -uw

av!

¨z+¡ ¡ !+r~}k}+z+efif+ !+r~}k}@k'q 3¨r~+ef£¨r~kv£'qLuiqCkefµwfir~^_k¨uwv-uw}²!¨kr~'qµ@°r~}ef+r~v

n9r r~q9lAv+ëf+r~f+r~qRuwz0s+-uw ({0qCk¡ kr~¨z+v+a¨{+ä¢+}kr~¡kv v@z+6r~kv+r~6¯6r~¨k!+rNf-uiqWlAv+ëf+r~)

f+r~q{_k¨|f+kr~_uwv °r~6uwv++r~¨r~vp@qCr~}k}kr~v r~kv+arts%of+'qµ+vzw¡ }kef ¢5r~o+a}kef +r~nDkvqC¨rtqCr~v+

kaav Cuwv+Jf+r~qCr~vAz+v+ ¢5r~o+a}kef +r~qUk¨@Cuw¡ wr~qkaav ¡ r~+k~kv+kef+r~ ±°rf-uwv++}kz+v++

±°r~k+r~}vzw¨'q¨ef+z+efQr~kv+r 'qCef-uw'qCkef+r´8ar

µ

r~¨~r~z+ar~vµ<+krnDkv-z+¨ uwz0s8+kr g @@z+¡£z+}uiqCkavkaav>´8r~¨z+v+f+r~qC¨iuw{+qLuw} v+k¡ ¡Iqµ!¨ -uw¨_ak}qp

H = g(M, µ)

^*) ⁺

u +krbq uf+¨ef+r~kv+}kef+wr~q8+r~SnDkvqC¨rtqCr~v+8r~kv+r~ Cuwv+Jf+r~qkaav+r~x/}f+r£+r~ ´8r

¨z+v+f+r~qC¨iuw{+qLuw}k'qCeL@ uw¢fzwv+wqµY}vzw¨'q¨ef 0_k¨kw^r~kv0suef+z+efWr~kv+r|Yr~¨@vJo+{0s|z+v+

+r~°§-a¨¡

H = g ⁰ (M + µ)

-uw¨'qCr~}k}kr~vµkv+r~°uw}k¨ }krtqCtqC}kef lAv+¨ef+r~f+r~qo+¢5r~<+kr x/}f+r kaav

H

f+r~¨¨efq iovW+kr~¨r~§-a¨¡ ¨kv+W+kr

¨r~z+uw¢-uw¢+}kr~qCz+v+ar~vv@z+}k}µc uw¢5r~+r~z0qCrtqµ

-uw¨A+krSqUk¨@Cuw¡ wr~qA+r~A¡ r~+k~kv+kef+r~v r~k'qCz+v+ar~vyv+efqjkaa¡A]karuwzy+r~´8r~¨z+v+!

f+r~q uw¢fzwv+wq efqCr£¡uwv +kr°r~qCr aa}kef+wr~qjkaavZlAv+¨ef+r~f+r~q¡ !+r~}k}kkr~¨r~vµ0¨

_k¨ r~kv+r §-a¨¡

H = g ⁰⁰ (µM )

kaa¨ar~ef+}uwar~vY krg z0qCa¨r~v arf+r~vv+efm°r~qCr~z+v+

kar~¨~efqCr~v>uwz0sRwa¡ }kr-20r~¨r8s|z+v+qCkav-uw}kr8§-a¨¡ r~vµ0¨ -uw¨Aar~}qCr~v ¨a}k}p

b *cbsRn=NPj [Hn=mT

eb dNn b_c n=NmPO`

Rc QRPmPN

PHQ RjBn e R UpsrrYBZDn cb Sb_P SNb_c mP PHn=N Sb_cSb_Nes`

K_m O` *Nes` PHn=N Sb_cSb_Nes`

(13)

g ⁰ (M, µ) = M + µ

^*) ⁺

g ⁰⁰ (M, µ) = µM

^*)@ ⁺

krEqUk¨@z+v+kaav lAv+ëf+r~f+r~qP_k¨v@z+v uwvf-uwv+7kar~ëf+kr~+r~v+r~SYr~ä}kr~ef+r£z+vqCr~)

¨zefqµ +kr¨ef^uwz0s°+kr*Yr~zwv++r~¨z+v+ +r~PYr~'qCr~k}kz+v+a's|z+v+qCkav7kaav

µ

¢5r~~krf+r~v'Czw¨'q

¨ef +kr~¨r/Yr~'qCr~k}kz+v+a's|z+v+qCkav,kaa}k}k'qzwv++kI+z+ef f+¨r~v#nDL uw'qCz+v+aL°r~'q_z+v+ f+¨r/<u]

¨uwv+ ¢5r~ef+¨r~k¢5r~vµ@¨ r~vqC'qCrf+r~v +¨r~-¡,aa}kef+r §9zw}k}krp}nDkv+r/nDf}f@z+v+I+r~ nDL uw'qCz+v+a)

°r~'qCr~

E[µ]

^µ r~kv+r!nDf}f@z+v+ +r~<uw¨uwv+

V [µ]

z+vqCr~±°r~k¢5rf-uw}qCz+v++r~uw}qCr~vZnDL uw)

qCz+v+aL°r~'qCr~ z+v+Wr~kv+r a¡£¢+kv-uiqCkavuwz+¢5r~k+r~¡ < r~ EA¢5r~¨uwv+_kaavUp!ef+r~f+r~q ~z

lAv+¨ef+r~f+r~qAk'qAuw}kAp!av++r~'suw}k}kv r~kv+r~/nDf}f@z+v+ +r~{<uw¨uwv+8r~vqf-uw}qCr~v

§ô+°+krlAv+ëf+r~f+r~q +r~°r~¨'qCr~v#Y²!{+ ¢5r~o+a}kef+r~ x/}f+r +r~ ´8r~¨z+v+f+r~qCïuw{+qLuw}k

_k¨q ¨ef>r~kv+r 0_refqCLkar~¨ef+kr~¢+z+v+ +r~/Yr~'qCr~k}kz+v+1kaav

µ

^_kr r~kv+rEnDf}f@z+v+kaav

H

¨r~}k¢+'q uwz+ <+kr uef0s|Cuwar v-uefW¡ r~+k~kv+kef+r~v r~k'qCz+v+ar~vW¨kv+qkv+r~§-a}kar M µY-u

-uwdiov++]k!k+z+z+¡3¨r~+r~varjo+v+efqCr~v ´8r~¨z+v+f+r~qC¨~z+'qLuwv+v@z+v¡ qj°r~v+kar~dg z0s uwv+

r~¨¨r~ef+r~viuwv+v{nDf}fq ¡uwv 3¨r~+ef+kr<uw¨uwv+>z+v+-uw¡ q -uw&0_k¨kw+µ°¨'qCr~kwq

+kr uef0s|Cuwar v-uef

M

kv^§-a}karwµ<°r~v+v ¡uwv ¢5rtqCCuwa¨¡ zak uw¢+v+rf+¡ r~v++r0_k¨kwukar~)

¨kav¢5r~o+a}kef

H

kaaCuwz+¨¨rtqCtqNYr~zwv++r~'q¡uwv¢5r~k+ry´8ar~vµR+r~v nDL uw'qCz+v+aL°r~'q z+v+ +kr/<uw¨ uwv+{kaav

µ

a}kr~ef+~r~qCk+µ¨£k'q°+r~n9r r~q_uwz0s +kr uef0s|Cuwar v-uef

M

^-uwv+v

r~kv++r~z0qCk+µW°r~v+v ¢5r~k+rYr~¨ef+kr~¢+z+v+ar~v kv+kr kaa¡~iov++]k!k+z+z+¡ r~Ljo+v+efqCr !+r~Rz+v+r~)

jo+v+efqCr)0_efqCz+v+r~'s|a}kar~v;wfµr~f}fq<¡uwv+kr<uw¨uwv+ ¢5r~-a}kr~ef+~r~qCkar~Dp!r~v+@z+v+

+r~dnDL uw'qCz+v+aL°r~'qCr~1*|¢5r~k+r~ z+v+r~Ljo+v+efq

+

µ¨f-uiq +kr~Ar~kv+r ar~'qCr~kar~'qCr uef0s|Cuwar

v-uef ¡ r~+k~kv+kef+r~v| r~k'qCz+v+ar~v ~z+A§-a}karw

io¡N°r~qCr~v§uw}k}a¢5r~+r~z0qCrtqNlAv+¨ef+r~f+r~q 0_k¨kw/o+¢5r~K+kr<¯D¨!+z+qC]k!qziq6¡ r~+k~kv+kef+r~

r~k'qCz+v+ar~v ¢5r~°+r~´8r~¨z+v+f+r~qC¨{+¨!+z+qCkavR krn9r r~qCr r~kv+r~I¨r~{-uwCuiqCr~v v++r~¨z+v+

kaav,nDL uw'qCz+v+aL°r~'q<z+v+*<uw¨uwv+ uwz0s+kr uef0s|Cuwar v-uef

M

¨kv+v+efqDr~kv++r~z0qCkw+kr g z0qCa¨r~v~r~kar~v13¨r~+efµ-uw¨r~kv 0joeL@uwv+ +r~ uef0s|Cuwar£v-uefr~kv+r~PnDf}f@z+v+ +r~

nDL uw'qCz+v+aL°r~'qCr~v+wq°r~v++kyk'qs%o+ r~kv+r~v_g v+'qCkr~ +r~ uef0s|Cuwaruw}k70_ruwqCkav^uwz0s

r~kv+rjnDf}f@z+v++r~<0_k¨kwa~Wq^r~qCr~¨rYaCuwz+¨¨rtqC~z+v+ -uis%o+Rk'qµw-uw¨Y+kr ¯D¨r~k¨r~}uw'qCk~qziq

+r~ uef0s|Cuwar v-uef ¡ r~+k~kv+kef+r~v ´Po0qCr~¨v @}kr~kv+r~ uw}k°r~kv+°k'q z+v+ +kr)0_k¨kwukar~¨¨kav

¢5r~o+a}kef

M

^v+efq ^~z+v+k¡ ¡IqJlAvqCr~ +kr~¨r~v#g v+v-uf+¡ r~v _k¨q°r~kv+r a}kr~ef+~r~qCkar v++r

¨z+v+bkaavnDL uw'qCz+v+aL°r~'q_z+v+,<uw¨ uwv+_r~kv++r~z0qCk kv +krAr~L uw'qCrtqCr)0_efqCz+v++µ@vzw¡ }kef

_kr8¢5r~\Y²!{!i{lAv+¨ef+r~f+r~q

nDs zw}k}q uwv+kr~¨r~p@qCr~}k}kr ef+av uwz0s)µ5-uw¨8r~~z+¡ kv++r~'q8¢+k ~z+¡ r~qC{+z+v+q8+r~S z!

Cuw¡ ¡ r~v0suw¨¨z+v+kaav ´8¨a¨¨¡uwv *)@}} + wr~kv+r r~kvf+r~qC}kef+r£qf+r~a¨rtqCkef+r§uw¨¨z+v+|o+¢5r~

lAv+¨ef+r~f+r~q°k¡ iov.kar~'qCqCkav+)Rz+v+ av+¨z+¡ ¡ !+r~}k}+ak¢0q@ kr 3¨r°r~k}kkar~v §-a¨ef@z+v+aCuwv!

ziqC~r°z+vqCr~¨ef+r~k+r~v¨efIa¨z+v++}kr~ar~v+Ikv +r~v*g v+v-uf+¡ r~vIz+v+I+r~ !+r~}k}s|a¨¡£z+}kkr~¨z+v++µ

vzw¡ }kef^kv +r~ §-CuwarwµRuwv°r~}ef+r~ p@qCr~}k}kr#lAv+¨ef+r~f+r~q r~kv+arts%of+'q*_k¨µKz+v+a¢ -uw

KK PBeb_cd^Db_Neb_cb_P PPHRSo b_P^dNb0RP9QuNb mco;RTNe e Qubsn k

ebsn `

(14)

iov++]k!k+z+z+¡ 'qLuiqCkef¢+RkvqCr~'qCr~¡ {5aCuw}°a{0qCk¡ kr~'q6u¨efuw¢5r~!iov.kar~'qCqCkav+) z+v+

av+¨z+¡ ¡ !+r~}k}5¢5r~o+a}kef f+¨r~ 0_ruwqCkav uwz0s 0_k¨kw£z+vqCr~¨ef+r~k+r~vµiuwv+v -uw v@z+ r~kv

g z0s|¨z0sR~z r~kv+r~_r~kvf+r~qC}kef+r~v z+v+y¡,aa}kef+'q uw}k}kar~¡ r~kv+r~v §-a¨¡£z+}kkr~¨z+v+£¨r~kvµ-+krdkaav

±°r~ef+zwv+@z+v+ uwz0sR'qLuiqCkef+rbg v-uw}²!¨r !+r~ ±°r~ef+zwv+@z+v+ ¢5r~o+a}kef +r~_s|z+v+qCkav-uw}kr~v

§-a¨¡%¨1°r~q{_kr¡,aa}kef uw¢+¨krfq

nDkv+r9°r~qCr~¨r aa}kef+wr~qµ~'qCef-uw'qCkef+r nDkvo+¨¨rY~z ¡ !+r~}k}kkr~¨r~vµ r~vq¨s|r~¨vq6+kr9{f+r~i

¨kr°'qzw¨wr~9kaa¡ z+¨¨{+o+v+a}kef+r~v ´8¨a¨¨¡uwv! !+r~}k}0uw}kK+kr°¢5r~k+r~v*kaaCuwv+ar~'qCr~}k}qCr~vWqUkr

kv£+r~<´8¨z+v+0s|a¨¡ µik'qD+kr x/}f+rkaav

H

r~vqCef+r~k+r~v+ s%o+R+kr §-Cuwarwµia¢r~kv!iov++]k!k+z+z+¡

@Cuwv+S_k¨!+r~Rv+efq'w-uwcnDkvqC¨rtqCr~vr~kv+r~ Cuwv+Jf+r~qRz+vqCr~¨}kkr~wq 3¨r~+efa¨z+v++ziqC~}kef

+r~¡ z0suw}k}µz+v+ -uw ´8r~¨z+v+f+r~qC¨v+]karuwz ¢5r~r~kv-z+¨'q_v@z+ +kr/q uf+¨ef+r~kv+}kef+wr~qµ0_

ef+r~v +r~v z+'qzwv++r~v

W z+v+

J

~z °ref+¨r~}kv !+r~°v+efq0±°¨rt²ar~ rtq uw} *

w¤a¤a¥+

uw¢+'qCCuf+kr~¨r~v-ukaav v+efZ°r~qCr~~µz+v+ kar~¨~efqCr~v k}a}k}kk uwz0s<+r~v^±°r~a¨]r +r~I´8r~¨z+v+!

f+r~qCïuw{+qLuw}k~µwkv++r~¡[+krEA¢5r~üwv+aL uf+ëf+r~kv+}kef+wr~qCr~v v@z+<v+ef1kaav~r~qC}kef+r~v z+v+

s-v-uwv+~kr~}k}kr~vZiov.kar~'qC qCkav+r~v^uw¢fzwv+ar~vµ¤°a¢5r~Y¢5r~k+r q¨qCr~}Rv+efq kv¢5r~k+r~v_ z+'qzwv++r~v

a}kr~ef ~z+{Yr~'s%o+az+v+ 'qCrf+r~v

§-a¨¡uw}D}vzw¨'q£¨ef r~kv^¨a}ef+r~ !+r~}k}Y+z+ef r~kv+r uw¨w@k rtq¨qCrsuw¨¨r~vµ'°a¢5r~°+kr

~r~qC}kef+r~v!+r~ds-v-uwv+~kr~}k}kr~v7iov.kar~'qCqCkav+r~v_nDkv-z+¨uwz0s°+kr!nD}kr~¡ r~vqCr+r~EA¢5r~¨uwv+a)

¡uiqC¨2 G v+rf+¡ r~vUnDkv+r8wa¡ {+}kr-20r~¨r §-a¨¡ _k¨ k¡ uw{+qCr~} ¥ ¥ kaa¨ar~'qCr~}k}q

'(( 5#<X:>@??0:

iov +r~vkaaCuwv+ar~uwv+ar~v uw{+qCr~}kv_z+¨+r~v qf+r~a¨rtqCkef+r ±°r~ao+v++z+v+ar~v s%o+_+r~v>´8}kr~ef!

}uwz0s9kaavq^}f+}k'qLuwv+>z+v+ZnDkv+wa¡ ¡ r~vkaa¨ar~'qCr~}k}q5nD¡uw -uwv+r~¢5r~vuwzefkvqCr~¨r~¨Cuwvq

¨r~kvµ°a¢Uv+efq v@z+ -uw uw¢+¨a}kz0qCr A]karuwzU+r~nDkv+wa¡ ¡ r~v+~µ°¨av++r~¨vuwzefU+kr>¨r~}u]

qC]karnDkv+wa¡ ¡ r~v+¨{5a¨qCkav kvr~kv+r~ ´8r~¨r~}k}kef-uis q !+r~+r~´8Cuw+r~,lAv+a}kr~eff+r~qkv

r~kv+r~ ´8r~¨r~}k}kef-uisqA+k¨r~qCr~vy!+r~_kv++k¨r~qCr~v7nDkv-z+¨ uwz0sR+krkv++]k!k+z+r~}k}kr´8r~¨z+v+f+r~q

v+k¡ ¡Iq r~ r~kv0suef+'qCr ref-uwv+k¨¡£z+ v+k¡ ¡Iqw± 6uwvµ -uw¨ElAv+a}kr~eff+r~q¨r~}k¢+'qI!+r~

+r~ Ar~kµ+r~ -uwCuwz+_s|a}kwqµ-¢5r~Ciov++]k!k+z+r~vyz+v+ar~¨z+v++r~v^p@qC¨r~¨/kar~¨z+¨Cuef+r~v IIuw¡ q

+kr~¨r~ g ¨az+¡ r~vq 3¨r~+ef uwzefs%o+°+krA¨r~} uiqC]k0_r~ef+r~v r~kv+r~ ´8r~¨r~}k}kef-uisq°{+} uwz+¨k¢5r~}k'qµ

¡£z+¨ ¡uwv }krtqCtqC}kef uwv+v+rf+¡ r~vµ0-uw¨ r~kviov++]k!k+z+z+¡ r~kv+ruwv+ar~¢5a¨r~v+rdgCkar~¨¨kav ar~ar~v

lAv+a}kr~eff+r~q{f-uiq

ruiqCav *

w¤a¤a¥+

suw¨'qR+krN_efqCka'qCr~v£qf+r~a¨rtqCkef+r~v!g v+ziqC~r z+v+£r~¡ {+k¨kef+r~v!nD¨ar~¢!

v+k¨¨rA~z+Cuw¡ ¡ r~vµ@z+¡[~z+r~¡ pef+}kz+¨°~zwa¡ ¡ r~vµ@-uw¨<wr~kv+k¨r~qCr~ z+Cuw¡ ¡ r~vf-uwv+

K h NoHRTTmP ]_^Db_N n ePHQub_P

T =

π(g → g) π(g → k)

π(k → g) π(k → k)

,

o Ne

π(g → g)

RTn RSc n Sb_NPTN S b_Ne mo,jbsn

PHQub_P n e RPHQ R

n ^dNbsQub_c NP b_NPb_P jbsn

PHQub_P

n e RPHQ ]

jb_TRPjb_PO`

Nb RSc n Sb_NPTN S b_Neb_P NP bsQub_c &b_NTb RQQuNb_cb_P n=N S ]

b_NPHn

PHQ n=NPHQ

P BeNmPb_PmP

M

PHQ

t ^I

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K bsR emP UXWYYgBZVPb_PPBe bsn ! "! #$ !%& !' ( *) H`

(15)

_kef+r~v nDkv+wa¡ ¡ r~v+¨z+v+a}kr~eff+r~qz+v+

Cuwv+Jf+r~q8¢5r~'qCrfq¤g zef

r~}k}ka z+v+

k}²a

*

w¤a¤+

wa¡ ¡ r~v kv£r~kv+r~Dv+r~z+r~¨r~v lAvqCr~¨¨zef@z+v+I~z +kr~¨r~¡ nD¨ar~¢+v+k~@p!a}k}qCr 3¨r~+ef r~kv

¨a}ef+r~c z+Cuw¡ ¡ r~vf-uwv+¨r~}k¢+'qRv+efqD+k¨r~qR¢5r~'qCrf+r~vµzf+¨r~v+¨ef63¨r~+ef+kr Yr~¨¢+kv!

+z+v+_kef+r~v#q^}f+}k'qLuwv+ z+v+ ´8r~¨z+v+f+r~q ¢5r~'qziqCkwqµ!¨}vzw¨'q ¨ef v-uef ruiqCav -uw)

uwz+ qC¨wqC~+r~¡Vr~kvPg ¨az+¡ r~vq s%o+CnDkv+wa¡ ¡ r~v+¨z+¡Ekar~'qCr~k}kz+v+_uw¢+}kr~ qCr~v¤pq^r~v+v ¡uwv8-ukaav

uwz+¨arfqµ5-uw¨/q^}f+}k'qLuwv+y¢5r~6+r~dnD¨~r~z+az+v+ kaav´8r~¨z+v+f+r~q8uw¢+v+rf+¡ r~v++r´8¨r~v+~r~)

qCzwar8uwz0s°r~k'q ¬µ+¨wa¡ ¡Iq r~kv+rSlA¡Ekar~'qCr~k}kz+v+Ekaavyuw¨¡ ~z ¨r~ef +r~A´8r~¨z+v+f+r~q_+r~

g ¨¡ r~v>¡ rf+A~z ´8z0qCrwµ uw}kA¨krI+r~v 0_r~ef+r~v>ef-uw+rtqµ< uw -uw+z+ef+v+ef+v+q¨qC}kef+r ´8r

¨z+v+f+r~qC¨v+]karuwz r~kv+r~ ´8r~¨r~}k}kef-uis qSf+r~¢5r~v iuwv+v\g }k}kr~¨+kv+a }kr~k+rtq8-uwSg ¨az+¡ r~vq uwv

¡ kv++r~'qCr~v+I°r~ g v+v-uf+¡ r~vµCkaav +r~v+r~v ruiqCav^r~kv+r¨r~}k¢+'quwv+¨{+¨efqK kr,YaCuwz+)

¨rtqC~z+v++µ5-uw¨_qLuiqCzef+}kef r~kv iuwz+Cuw}kr~ z+Cuw¡ ¡ r~vf-uwv+!kaav7nDkv+wa¡ ¡ r~v kv 0_efqCz+v+

´8r~¨z+v+f+r~q6¢5r~'qCrfqµik'q ef+}kkr a}kefz+¡ 'qC¨q¨qCr~vz+v+b_k¨ ar~Cuw+rYkv }krtqCtqCr~' r~q6kv §-Cuwar

ar~'qCr~}k}q'0-uwv+r~¢5r~v#_k¨ wr~kvg äz+¡ r~vq{kaaär~¢+Cuefqµ°r~}ef+r~ uwz+A´8r~¨refqCkawr~qCäo+v!

+r~v^r~kv+r¨a}ef+r lA¡Ekar~'qCr~k}kz+v+*|+kr43Cus-v-uwv+~kr~}k} ar~¨z+v+f+r~qC}kef~z uw'qCr~v r~kv+r~

´8¨z+{+{5r8arf+r~v|jo+¨+r + ¨refq¨s|r~'qCkwq

uwv iuwv+v ¨efyv@z+v s|Cuwar~vµ°r~}ef+rEnDkv+wa¡ ¡ r~v+¨a¨z+{+{5r uw¡%rf+r~'qCr~vkaavlAv+a}kr~ef!

f+r~q ¢5rtqC¨r r~vk'qg v+ar~v+a¡ ¡ r~vµ@nDkv+wa¡ ¡ r~vdf-uiq qLuiqCzef+}kef8r~kv+rYuw¢+v+rf+¡ r~v++r°´8¨r~v+

{+¨!+z+qC]k!qziq ¢5r~K+r~8¯D¨!+z+qCkavkaav^´8r~¨z+v+f+r~q' -uwv+v}vzw¨'q¨ef *_k}k}kJo+¨}kef + r~kv

¯Dz+v+q+rs-v+kr~¨r~vµ<uwv+r~¡ +r~ nDkv-z+¨ r~kv+r~ ~z+ziqC~}kef+r~v nDkvf+r~q,nDkv+wa¡ ¡ r~v+kr

´8r~¨z+v+f+r~qv+efq¡ rf+¡uaar~¢+}kef>r~f}fq<q^r~v+v v@z+vr~kv uwv+¡ q¨r~kv+r~¡ +z+ef!

ef+v+q¨qC}kef+r~v{+¨i

a{0s nDkv+wa¡ ¡ r~vmo+¢5r~ +r~¡ r~kv+r~ uwv++r~¨r~v}kkr~wqµR}vzw¨'q ¨efz+vqCr~

ar_k¨¨r~vZlA¡ 'qzwv++r~vys|a}kar~v++r~ uw¢+}kr~qCr~vUio¡ r~¨'qCr~vpef+¨q¨q ¨a}k} +krEnDkv+wa¡ ¡ r~v+Lkar~)

qCr~k}kz+v+k¡ ¨r~ef+r~v!_kr k¡ uw¨¡ r~v, uwv+ z+¡ -uw< z+ef+ef+v+q¨qC¨r~kv+wa¡ ¡ r~v +kr a}kr~ef+r~v

uw¢+¨a}kz0qCr~v7g ¢.°r~ef@z+v+ar~v v-uefz+vqCr~v7_krIv-uefa¢5r~vuwz0s°r~k¨r~vuwv+v}kkr~ar~vyk¡ ¨r~

ef+r~v uwv+¡ rf+iov++]k!k+z+r~va¢5r~f-uw}k¢>+r~_k}k}kJo+¨}kefyar~¨rtqCtqCr~v ´8¨r~v+~ra}kr~ef+~r~qCk

¢5r~+r~z0qCrtq -uw~µ!-uw¨°k¡ uw¨¡ r~v uwv+-uw uw¢+¨a}kz0qCrAkv++]k!k+z+r~}k}krjnDkv+wa¡ ¡ r~vr~kv+r~v 'qzw)

wr~¨r~vZnDkv-z+¨uwz0sD+kr kv++]k!k+z+r~}k}kr´8r~¨z+v+f+r~ qdf-uiqUCzw¨'q¡uwv v@z+v>z+vqCr~¨ef+kr~+}kef+r

Yr~'qCr~k}kz+v+ar~v z+¡ -uw#3¨r°r~k}kkar z+ef+ef+v+q¨qC¨r~kv+wa¡ ¡ r~v~zz+v++rs-v+kr~'q!o+¢5r~+kr

p!{-uwv+v.°r~qCr !+r~+kr!<uw¨ uwv++kr~¨r~PYr~'qCr~k}kz+v+ar~v r~kv lAv+a}kr~eff+r~qC¨¡u+µ¨|f-uiq +r~

´8Cuw>+r~SlAv+a}kr~eff+r~qrf+r~8r~kv+r~v nDkv-z+¨uwz0s<r~kvZiov++]k!k+z+z+¡:k¡:¨r~ef+r~v uwv+uw}k

k¡Suw¨¡ r~v- r~v+vyk¡Suw¨¡ r~v| uwv+ }kkr~ar~v +kr8¡ r~k'qCr~v z+vqCr~f-uw}k¢ +r~ ´8¨r~v+~rwµr~uw}µ_kr

z+v+a}kr~ef +kr´8r~¨r~}k}kef-uis qIk'qµ¤zf+¨r~v+ +r~£´8Cuw^+r~ElAv+a}kr~eff+r~q k¡ ¨r~ef+r~v uwv+

¢5r~'qCk¡ ¡Iqµ@_kr k!kr~}kr ´8r~¨r~}k}kef-uis qC¨¡ qCa}kkr~+r~ z+vqCr~f-uw}k¢+r~cnDkv+wa¡ ¡ r~v+¨a¨r~v+~r }kkr~ar~v

K \ndcb_N SBe R

nu^Db_PPQuNb o NelQub_o mSTne RPHQ b_c^Dmc[&b_Pb_P o bsQuN]_NPNn Sb_PLOb_Nn e

Pjb_P9R[Pb_So b_PHQub

klcb_P] cmuQ

BeNNe e R ^Db_Nn=b_PO`

(16)

g ¢+¢+k}k+z+v+

¥

plAvqCr~¨ef+kr~+}kef+rSqUk¨@z+v+!kaav7lAv+a}kr~eff+r~q

qUkr ¡uwvkv g ¢+¢+k}k+z+v+ ¥ ¨rf+r~vQiuwv+v µDs%of+'qr~kv+r.0_r~+z+qCkav+r~ lAv+a}kr~eff+r~q

k¡ uw¨¡ r~v_ uwv+iuwz+¡~z r~kv+r~70_r~+z+qCkav +r~ 0_k¨kwa~µ@Cuwv+~z °r~¨+r~vµC°r~k}D¡uwv

z+vqCr~f-uw}k¢^+r~*nDkv+wa¡ ¡ r~v+¨a¨r~v+~r }kkr~wq\io¡ ¨r~ef+r~v uwv+ -uwar~ar~v^¨kv+q£+r~bg vqCr~k}

+r~8uwz0s|a¨z+v+f+¨r~PnDkv+wa¡ ¡ r~v+ @Cuwv+wr~v ´8r~¨r~}k}kef-uis qC¨¡ qCa}kkr~+r~ *|+r~+z+v+@}kr~¨r±°r

¨r~ef

+

<lAvqCr~YaCuwz+¨¨rtqC~z+v+ar~v7_krIk¡ f+kr~ +rs-v+kr~'qCr~v>¨r~ef+r~v uwv+>¨r~v+quw}k¨r~kv+r

0_r~+z+qCkav +r~{lAv+a}kr~eff+r~q -uw kv++]k!k+z+r~}k}kr Cuwv+Jf+r~qC¨¨k¨kw+0uwg ¨az+¡ r~vqfzwv+wq

uw}k}kr~¨+kv+a ¨rf+A@¨qCkefkaavy+r~/ uwar+r~A¢5r~k+r~v +z+ef+ef+v+q¨qC}kef+r~v7nDkv+wa¡ ¡ r~v uw¢;

¡uwviuwv+v}f+v+r#°r~qCr~¨r~ +krYr~'qCr~k}kz+v+ar~v+r~uw¨¡ r~vx uwv++r~ v-uefa¢5r~vef+kr~¢5r~vµ

z+¡ -uwv+v}krtqCtqC}kefU+r~va}kr~ef+r~v n9r r~quwz0s£-uw Cuwv+Jf+r~qC¨¨k¨kw~zr~¨~r~z+ar~v _kr

k¡ ¨r~ef+r~v| uwv+;k!kr~}k}kr~efq ¨{+¨efq_ ruiqCav*w¤a¤a¥+ +r~f-uw}k¢ ¢5r_z+¨'q{kaav +r~v -E

Czwv++r~¨vµa+r~¨r~v nDkv+wa¡ ¡ r~v.kar~'qCr~k}kz+v+Pf+kv+¨r~ef+r~v+ v+kr~+¨k8uwv+ar~¨kr~+r~}qYk'qg zar~¨+r~¡

k'q8+krEq uf+}K+r~ ´8¨r~v+~r!f}ef+'qP_k}k}kJo+¨}kef;5uw}k}kr~¨+kv+aAk'q8-uwdg ¨az+¡ r~vq uwzef}f+v+r

r~kv+ruw¢+¨a}kz0qCr´8¨r~v+~r~z^¨rtqCtqCr~v v+ef v-uef.kaa}k}k~krf@¢-uw~µ¤°r~v+v ¡uwv +kr uw¢+v+rf+¡ r~v++r

´8¨r~v+~{+¨!+z+qC]k!qziqD+r~ nDkv+wa¡ ¡ r~v+Duw@~r~{0qCkr~'q.g }k}kr~¨+kv+a's-v++r~v r~}k}kaRz+v+ k}²a

* w¤a¤+ wr~kv+r£±°r~'qziqCkaz+v+ -uis%o+~µ-uw¨ kvy+r~v>¨r~ef+r~¨r~vZCzwv++r~¨v7lAv+¨ef+r~f+r~q¢5r~¨av!

+r~¨jg z+L_k¨@z+v+ uwz0sK+kr8z+vqCr~¨'qCr~v7nDkv+wa¡ ¡ r~v+¨a¨z+{+{5r~v|f-uiq

Czw¨'q£¡uwv !+r~}k}krwµkv +r~v+r~vmlAv+a}kr~eff+r~q @ uw}kSg ¨az+¡ r~vqIkv+kr ´8r~¨z+v+!

f+r~qC¨{+¨!+z+qCkav+'s|z+v+qCkav*|!+r~ Az0qC~r~v0s|z+v+qCkav + r~kv+arts%of+'q{_k¨µ-uwzar~Auefqµ¨ r~)

ak¢0q ¨ef +r~v+v+ef r~kv °r~qCr~¨r~ qf+r~a¨rtqCkef+r~g v+CuiqCw Ak¡ ¡Iq°¡uwv uwvµ0-uw¨{lAv+a}kr~ef!

f+r~q k¡ nDkv+wa¡ ¡ r~v¨r~kv+r~v lA¨¨{+¨z+v+ uwzefkv+r~xArtqCr~¨aar~v+qziq +r~ kv++]k!k+z+r~}k}kr~v

¯Dzis|r~¨r~v+~r~v f-uiqµa¨P_k¨qY¨ef +kr~¨r_xArtqCr~¨aar~v+qziq<uwzefuwz0s +kr ar~¨r~}k}kef-uisqC}kef+r qUk}

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Wohlstand und Gesundheit : neuere empirische Resultate und theoretische Erklärungsversuche

H

T

τ

W

U

τ

t k (τ )

X(τ )

W = Z T

0

e − ρτ U(t k (τ), X (τ ))dτ

∂ 2 U

∂X∂t k

∂ 2 U

∂t k ∂X

t k (τ) = t k (H(τ ))

∂t k (H(τ ))

∂H(τ) < 0

∂ 2 t k (H(τ ))

∂H(τ) 2 > 0

t I

M

M

δ

∂δ

∂τ > 0

T

∂H

∂τ = H ˙ = I(M (τ), t I (τ )) − δ(τ)H(τ )

A

rA

Y

P

D

∂A

∂τ = ˙ A = rA(τ) + Y (t k (τ) + t I (τ )) − P (τ)M (τ) − D(τ )X(τ )

Y

t k

t I

H 0

q(τ )

τ

e − ρτ ∂U/∂t k (τ)

e − rτ λ(0) + ∂Y (τ)

∂t k (τ)

∂t k (τ )

∂H(τ ) =

r + δ(τ) − q(τ ˙ ) q(τ )

q(τ )

∂t k (τ)

∂H(τ)

H

λ(0)

H

H (T ) ≥ H 0

A(T ) ≥ 0

t = t i (τ) + t w (τ ) + t x (τ ) + t k (τ ),

∂Y (τ)

∂t k (τ)

∂t k (τ)

∂H(τ) =

r + δ(τ ) − q(τ) ˙ q(τ)

q(τ)

∂Y

∂t k (τ) = −w(τ )

t k (τ )

H

H

t k (τ )

T

H

H 0

H

H

H

t I

H

H

∂H(τ )

t ^k (τ )

e ⁻ ^ρτ U(t ^k (τ), X (τ ))dτ

∂ ² U

∂X∂t ^k

∂ ² U

∂t ^k ∂X

t ^k (τ) = t ^k (H(τ ))

∂t ^k (H(τ ))

∂ ² t ^k (H(τ ))

∂H(τ) ² > 0

t ^I

∂τ = H ˙ = I(M (τ), t ^I (τ )) − δ(τ)H(τ )

∂τ = ˙ A = rA(τ) + Y (t ^k (τ) + t ^I (τ )) − P (τ)M (τ) − D(τ )X(τ )

t ^k

t ^I

H ⁰

e ⁻ ^ρτ ∂U/∂t ^k (τ)

e ⁻ ^rτ λ(0) + ∂Y (τ)

∂t ^k (τ)

∂t ^k (τ )

∂t ^k (τ)

H (T ) ≥ H ⁰

t = t ⁱ (τ) + t ^w (τ ) + t ^x (τ ) + t ^k (τ ),

∂t ^k (τ)

∂t ^k (τ)

∂t ^k (τ) = −w(τ )

t ^k (τ )

t ^k (τ )

H ⁰

t ^I

t ^I

t ^I

∂t ^k (τ)

∂t ^k (τ)

∂H(τ ) e ⁻ ^ρτ e ^−rτ = 1

∂H (τ ) e ⁻ ^ρτ e ^−rτ =

Y [t ^k (τ ), t ^I (τ)]

Y [t ^k (H(τ ))] = Y [H(τ )]

H = g ⁰ (M + µ)

H = g ⁰⁰ (µM )

g ⁰ (M, µ) = M + µ

g ⁰⁰ (M, µ) = µM

t ^I