35=(à&=$1,( (/(0(17Ï:3Ïà35=(:2'1,.2
:<&+,8.à$'<35=(à&=1,.2:(

:352:$'=(1,(

3RGVWDZ UHDOL]DFML XNáDGyZ LPSXOVRZ\FK RUD] QLH]Z\NOH

ZD*Q\FK ZH ZVSyáF]HVQHM HOHNWURQLFH XNáDGyZ F\IURZ\FK MHVW

Z\NRU]\VWDQLH ZLHONRV\JQDáRZHM SUDF\ HOHPHQWyZ DNW\ZQ\FK

SU]HáF]DQ\FK

PLG]\

VWDQDPL

RGFLFLD

L

SU]HZRG]HQLD

RGSRZLDGDMF\FKVWDQRP]HUDORJLF]QHJRLMHG\QNLORJLF]QHM3U]HMFLD

PLG]\ W\PL VWDQDPL SRZLQQ\ ]DFKRG]Lü Z PR*OLZLH MDN QDMNUyWV]\P

F]DVLH D SRQDGWR XNáDG SRZLQLHQ FHFKRZDü VL PR*OLZLH GRNáDGQ\P

RGWZRU]HQLHPSR]LRPyZQDSLüRGSRZLDGDMF\FKVWDQRP]HUDLMHG\QNL

ORJLF]QHM

:áDFLZRFL G\QDPLF]QH SU]HáF]QLND RSLVXMH VL SU]HELHJLHP L

F]DVHPWUZDQLDSRV]F]HJyOQ\FKID]SURFHVyZSU]HáF]DQLD=DOH*QLHRG

X*\W\FK HOHPHQWyZ VWRVXMH VL FKDUDNWHU\VW\F]QH GOD QLFK Z]RUFRZH

XNáDG\SU]HáF]DMFHRUD]W\SRZHV\JQDá\VWHUXMFH

2JyOQD WHQGHQFMD UHDOL]DFML FRUD] V]\EV]\FK XNáDGyZ

LPSXOVRZ\FKLF\IURZ\FKZ\QLND]NRQLHF]QRFLSU]HWZDU]DQLDFRUD]WR

ZLNV]HMLORFLLQIRUPDFMLZMHGQRVWFHF]DVX:VSyáF]HVQHWHFKQRORJLH

PRQROLW\F]QHLWHFKQLNLUHDOL]DFMLXNáDGyZSU]HáF]QLNRZ\FKSR]ZDODM

X]\VNDü F]DV\ SU]HáF]DQLD SRZV]HFKQLH VWRVRZDQ\FK HOHPHQWyZ

SyáSU]HZRGQLNRZ\FK Z ]DNUHVLH RG QDQRVHNXQG GR NLONXG]LHVLFLX

SLNRVHNXQG

6WDQ\]DáF]HQLDLZ\áF]HQLDXNáDGXSU]HáF]QLNRZHJRNOXF]D

PR*QD VFKDUDNWHU\]RZDü SU]\ SRPRF\ SDUDPHWUyZ RGQLHVLRQ\FK GR

LGHDOQHJR NOXF]D R ]HURZHM UH]\VWDQFML Z VWDQLH ZáF]HQLD

LQLHVNRF]RQHMUH]\VWDQFMLZVWDQLHZ\áF]HQLDU\V

5]HF]\ZLVW\XNáDGSU]HáF]QLNRZ\SRVLDGDQLH]HURZUH]\VWDQFM

r

f

Z VWDQLH SU]HZRG]HQLD ZáF]HQLD RUD] VNRF]RQ ZDUWRü

UH]\VWDQFML

r

r

Z VWDQLH RGFLFLD Z\áF]HQLD =DWHP FKDUDNWHU\VW\NL

SUGRZR QDSLFLRZH U]HF]\ZLVWHJR XNáDGX SU]HáF]QLNRZHJR V QD

RJyá OLQLDPL QDFK\ORQ\PL D SRQDGWR MDN Z SU]\SDGNX HOHPHQWyZ

ELSRODUQ\FKSU]HVXQLW\PLZ]JOGHPSRF]WNXXNáDGXZVSyáU]GQ\FK

1DV]H UR]ZD*DQLD VNRQFHQWUXMHP\ JáyZQLH QD RNUHOHQLX

VWDW\F]Q\FK L G\QDPLF]Q\FK ZáDFLZRFL SRGVWDZRZ\FK XNáDGyZ

SU]HáF]QLNRZ\FKZRSDUFLXRNWyUHUHDOL]RZDQHVXNáDG\LPSXOVRZHL

F\IURZH

0

U

U

U

I

X

.

L

.

8

)

b)

a)

X

.

+

-

X

L

L

.

6

8

)

U

U

U

I

>>

U

I

.OXF]

LGHDOQ\

.OXF]

U]HF]\ZLVW\

$

%

$

%

VWDQ SU]HZRG]HQLD

ZáF]HQLD

VWDQ RGFLFLD

Z\áF]HQLD

5\V 0RGHO ]DVWSF]\ XNáDGX SU]HáF]QLNRZHJR GOD VWDQyZ SU]HZRG]HQLD

L RGFLFLD D RUD] FKDUDNWHU\VW\ND SUGRZR QDSLFLRZD GOD W\FK

VWDQyZE

3HZQ\P X]XSHáQLHQLHP EG]LH SU]HGVWDZLHQLH Z\EUDQ\FK

XNáDGyZ SU]HáF]QLNyZ VWHURZDQ\FK QD]\ZDQ\FK NOXF]DPL

DQDORJRZ\PL XPR*OLZLDMF\FK SU]HV\áDQLH V\JQDáyZ Z WRU]H W\ONR Z

RNUHORQ\P SU]HG]LDOH F]DVRZ\P Z\]QDF]RQ\P F]DVHP WUZDQLD

LPSXOVX VWHUXMFHJR 3U]\ ]DáF]HQLX NOXF]D DQDORJRZHJR QDSLFLH ]

MHJRZHMFLDSRZLQQRE\üSU]HQLHVLRQHQDMHJRZ\MFLH]PR*OLZLHGX*

GRNáDGQRFL]DSU]\Z\áF]HQLXNOXF]DQDSLFLHZ\MFLRZHSRZLQQR

E\üQLH]DOH*QHRGQDSLFLDZHMFLRZHJR

35=(à&=$1,((/(0(172:3Ïà35=(:2'1,.2:<&+

3U]HáF]DQLHGLRG\

3URFHV SU]HáF]DQLD GLRG\ SyáSU]HZRGQLNRZHM UR]SDWU]\P\ Z

W\SRZ\P XNáDG]LH SU]HáF]DQLD U\VD Z NWyU\P GLRGD ' MHVW

SRáF]RQD SU]H] UH]\VWRU

R

]H (UyGáHP V\JQDáX LPSXOVRZHJR

e

G

]DSHZQLDMF\P]DUyZQRSRODU\]DFMZNLHUXQNXSU]HZRG]HQLDMDNLZ

NLHUXQNX]DSRURZ\P=Z\NOHUH]\VWDQFMD

R

REZRGX]HZQWU]QHJRMHVW

GX*R ZLNV]D QL* UH]\VWDQFMD

r

d

GLRG\ Z NLHUXQNX SU]HZRG]HQLD

(

)

R

r

d

>>

WDN *H Z VWDQLH SU]HZRG]HQLD VSHáQLRQH V ZDUXQNL

VWHURZDQLDSUGRZHJRQDWRPLDVWGX*RPQLHMV]DQL*UH]\VWDQFMD

r

r

GLRG\

GOD SRODU\]DFML ]DSRURZHM

R

r

r

<<

WDN *H Z VWDQLH ]DSRURZ\P

VSHáQLRQHVZDUXQNLVWHURZDQLDQDSLFLRZHJRGLRG\

0

U

U

U

G

L

'

8

)

b)

a)

U

5

U

G

U

<< <<

( )

,

'

0

4

)

4

)

4

5

4

5

+

(

)

−

(

5

X

'

5

,

)

−

,

5

H

*

W

'

5

X

'

H

*

L

'

H

*

−

(

5

+

(

)

W

0

5\V 7\SRZ\ XNáDG SU]HáF]DQLD GLRG\ D VFKHPDW XNáDGX E Z\]QDF]HQLH

SXQNWyZSUDF\QDFKDUDNWHU\VW\FHGLRG\

1DU\VESU]HGVWDZLRQRJUDILF]QHZ\]QDF]HQLHSX QNWyZSUDF\

GLRG\ Z VWDQLH ]DSRURZ\P

Q

R

L Z VWDQLH SU]HZRG]HQLD

Q

F

NWyUH

Z\QLNDM]RGG]LDá\ZDQLDQDSLü

−

(

5

)

L(

>@3RPLMDMFEDUG]RPDá\

SUGQDV\FHQLD]áF]DSU]\SRODU\]DFML]DSRURZHMQDSLFLHQDGLRG]LHZ

VWDQLHZ\áF]HQLDZ\QRVL

−

E

R

.

:VWDQLHSU]HZRG]HQLDSU]\W\SRZ\FK

ZDUXQNDFK

SU]HáF]DQLD

E

U

R

r

F

F

d

>>

>>

L

QDSLFLH

QD

SU]HZRG]FHMGLRG]LHMHVWEOLVNLH

U

F

DSUGGLRG\ZSXQNFLHSUDF\

Q

F

RNUHORQ\MHVW]DOH*QRFL

( )

I

E

U

Q

R

r

E

U

R

F

F

D

F

d

F

F

=

−

+

≈

−

3URFHV\ SU]HáF]DQLD GLRG\ RG VWDQX RGFLFLD GR VWDQX

SU]HZRG]HQLDLRGZURWQLHPRJE\üVFKDUDNWHU\]RZDQHRGSRZLHGQLPL

F]DVDPL WUZDQLD SRV]F]HJyOQ\FK ID] W\FK SURFHVyZ Z\QLNDM F\FK

EH]SRUHGQLR]NRQLHF]QRFL]JURPDG]HQLDOXERGSURZDG]HQLDáDGXQNX

HOHNWU\F]QHJRZED]LHGLRG\LZMHMSRMHPQRFL]áF]RZHM

,GHDOL]RZDQHSU]HELHJLF]DVRZHSUGXLQDSLFLDZ\VWSXMFHZ

SURFHVDFKSU]HáF]DQLDGLRG\V\JQDáHPSURVWRNWQ\PSU]HGVWDZLRQRQD

U\V *G\ QDSLFLH (UyGáD V\JQDáX MHVW XMHPQH GLRGD MHVW RGFLWD L

SRMHPQRü ]áF]RZD GLRG\ C

j

MHVW QDáDGRZDQD GR QDSLFLD

−

E

R

6NRNRZD ]PLDQD QDSLFLD

e

G

RG

−

E

R

GR

+

E

F

UR]SRF]\QD SURFHV

áDGRZDQLD SRMHPQRFL ]áF]RZHM SU]\ F]\P Z SLHUZV]HM FKZLOL SUG

áDGRZDQLDZ\QRVL

( )

I

E

E

R

D

F

R

0

=

+

W

V

W

I

−

0 1

,

5

−

,

5

L

'

H

*

0

0

W

L

(

)

−

(

5

W

W

0

0

( )

,

'

0

,

)

0

0

−

(

5

+

(

)

5&

M

W

G

8

, U

)

) G

+

5&

M

5&

M

5&

M

(

)

,

, U

)

5 G

+

−

0 9

(

5

W

3U]\ GX*HM

UH]\VWDQFML GRSURZDG]H GLRG\ U

6

5\V ,GHDOL]RZDQH SU]HELHJL

F]DVRZH SUGX L QDSLFLD

Z SURFHVDFK SU]HáF]DQLD

GLRG\

1DSLFLH QD GLRG]LH QDUDVWD ]H VWDá F]DVRZ R C

j

G*F GR

DV\PSWRW\F]QHMZDUWRFL

E

F

( )

( )

[

]

( )

u

U

U

t

R C

U

D

D

D

D

=

−

∞

−









 +

∞

0

exp

SU]\F]\P

( )

( )

U

E

U

E

D

R

D

F

0

= −

∞ = +

;

*G\QDSLFLHQDGLRG]LHRVLJQLHZDUWRü

U

F

UyZQRN9GLRGD

ZFKRG]LZVWDQSU]HZRG]HQLDRJUDQLF]DMFGDOV]HQDUDVWDQLHQDSLFLD

=DWHP]ZDUXQNX

( )

U

t

U

D

d

F

=

PR*HP\Z\]QDF]\ü F]DVRSy(QLHQLDZáF]HQLDGLRG\

t

d

Z\QLNDMF\]

áDGRZDQLDSRMHPQRFL]áF]RZHM

3RGVWDZLDMFZDUXQHNGRUyZQRWU]\PXMHP\

( )

( )

( )

( )

t

R C

U

U

U

U

t

R C

E

E

E

U

d

j

D

D

D

D

d

j

F

R

F

F

=

∞ −

∞ −

=

+

−

ln

ln

0

: V]F]HJyOQ\P SU]\SDGNX JG\

E

E

U

F

R

F

=

>>

Z\UD*HQLH

XSUDV]F]DVLGRSRVWDFL

t

R C

R C

d

j

j

≈

=

ln

,

2

0 69

3U]\ Z\SURZDG]DQLX ]DOH*QRFL QLH XZ]JOGQLRQR

]DFKRG]FHM ]PLDQ\ SRMHPQRFL ]áF]RZHM

( )

C u

j

D

ZUD] ]H

]PLHQLDMF\P VL QDSLFLHP QD WHM SRMHPQRFL SRQLHZD* DQDOLW\F]QH

UR]ZL]DQLH WHJR SUREOHPX QLHOLQLRZHJR QLH MHVW áDWZH 3UDNW\F]QLH Z

PLHMVFH SRMHPQRFL C

j

QDOH*\ ZSURZDG]Lü SHZQ XUHGQLRQ

SRMHPQRü

(

)

&

N &

(

&

U

&

M

5

R

=

−

+

SU]\ F]\P

C

o

MHVW SRMHPQRFL PRQWD*X ]áF]D D

k

C

MHVW

ZVSyáF]\QQLNLHPXUHGQLDMF\PUyZQ\PRNGOD]áF]\OLQLRZ\FKL

RNGOD]áF]\VNRNRZ\FK>@

3RF]ZV]\ RG FKZLOL

t

d

SU]H] GLRG Sá\QLH SUG R ZDUWRFL

FKZLORZHM RNUHORQHM ]DOH*QRFL &KRFLD* RVLJQLFLH QDSLFLD

U

F

MHVW ]HZQWU]Q R]QDN ]DNRF]HQLD SURFHVX ZáF]DQLD GLRG\ WR

MHGQDNZED]LHGLRG\WUZDQDGDOSURFHVXVWDODQLDVLáDGXQNXQRQLNyZ

QDGPLDURZ\FK NWyU\ GOD GLRG\ R NUyWNLHM ED]LH RNUHORQ\ MHVW

LORF]\QHP SUGX SU]HZRG]HQLD

I

F

L VWDáHM F]DVRZHM

α

0

t

t

t

t

≈

t

t

MHVW

F]DVHPSU]HORWXQRQLNyZSUGX

Q

I t

F

t

=

'OD GLRG\ R GáXJLHM ED]LH ]DPLDVW F]DVX SU]HORWX

t

t

QDOH*\ VWRVRZDü

F]DV *\FLD QRQLNyZ

τ

3URFHV JURPDG]HQLD áDGXQNX QDGPLDURZ\FK

QRQLNyZPQLHMV]RFLRZ\FKZ GLRG]LH R NUyWNLHM ED]LH SU]HGVWDZLRQR

V]NLFRZR QD U\VD 6WDáD ZDUWRü SUGX

I

F

R]QDF]D *H UyZQLH*

JUDGLHQWNRQFHQWUDFMLQRQLNyZZED]LHGLRG\MHVWVWDá\

∆

S

[

0

L

,

'

)

=

W

= ∞

W

≈

0

:

∆

S

[

0

L

,

FRQVW

'

5

= − =

=

W

= ∞

:

L

FRQVW

'

≠

W W

V

=

W

W

≈

0

W

=

0

5R]NáDG VWDFMRQDUQ\

5R]NáDG SRF]WNRZ\

)D]D

SLHUZV]D

)D]D GUXJD

5\V &KZLORZH UR]NáDG\ QDGPLDURZ\FK QRQLNyZ áDGXQNX G]LXURZ\FK

∆

S

ZED]LHGLRG\W\SXQDSU]\ZáF]DQLXESU]\Z\áF]DQLX

: SRF]WNRZHM ID]LH JURPDG]HQLD QDGPLDURZ\FK QRQLNyZ

áDGXQNXZED]LHGLRG\PR*QD]DREVHUZRZDüPDáR]QDF]F\FKZLORZ\

Z]URVW QDSLFLD QD GLRG]LH V]F]HJyOQLH GOD GLRG R GX*HM UH]\VWDQFML

REV]DUyZGRSURZDG]HOXEGX*HMLQGXNF\MQRFLGRSURZDG]HGLRG\

=QDF]QLH GáX*V]H L EDUG]LHM ]áR*RQH V SURFHV\ SU]\ Z\áF]DQLX

GLRG\UR]SRF]\QDMFHVL]FKZLOVNRNRZHM]PLDQ\V\JQDáX]ZDUWRFL

E

F

GRZDUWRFL

−

E

R

3RQLHZD* Z ED]LH GLRG\ QDJURPDG]RQ\ MHVW áDGXQHN QDGPLDURZ\FK

QRQLNyZ PQLHMV]RFLRZ\FK WR EH]SRUHGQLR SR VNRNX QDSLFLD

Z\áF]DMFHJRQDVWSXMH]PLDQDSUGXGLRG\]

I

F

QD

−

I

R

DQDGLRG]LH

XWU]\PXMH VL MHV]F]H SU]H] SHZLHQ F]DV QDSLFLH GRGDWQLH 3RQLHZD*

]PLDQ\QDSLFLD

u

D

VQLHZLHONLHGODWHJRSUG

I

R

MHVWSUDNW\F]QLHVWDá\

LZ\QRVL

I

E

U

R

R

R

F

≈

+

1LHZLHONL XMHPQ\ VNRN QDSLFLD QD ]DFLVNDFK GLRG\ MHVW

Z\QLNLHP ]PLDQ\ NLHUXQNX VSDGNX QDSLFLD QD UH]\VWDQFML V]HUHJRZHM

r

S

GLRG\

(

)

∆

U

r

I

I

S

F

R

=

+

=H Z]JOGX QD ]PLDQ NLHUXQNX SUGX Sá\QFHJR SU]H] GLRG

]PLHQLD VL UyZQLH* NLHUXQHN JUDGLHQWX NRQFHQWUDFML QRQLNyZ

PQLHMV]RFLRZ\FKZED]LHSU]\]áF]XFRV]NLFRZRSU]HGVWDZLRQRQD

U\VE 'LRGD SU]HFKRG]L ]H VWDQX SU]HZRG]HQLD Z VWDQ RGFLFLD

GRSLHUR ZWHG\ JG\ FDá\ áDGXQHN QDGPLDURZ\FK QRQLNyZ

PQLHMV]RFLRZ\FK Z ED]LH ]RVWDQLH ] QLHM XVXQLW\ : SLHUZV]HM ID]LH

Z\áF]DQLD QD]\ZDQHM UyZQLH* ID] PDJD]\QRZDQLD XWU]\PXMH VL

VWDá\ JUDGLHQW NRQFHQWUDFML QRQLNyZ PQLHMV]RFLRZ\FK SU]\ ]áF]X D

W\PVDP\PSá\QLHVWDá\SUGGLRG\

−

I

R

6WRVXMF TXDVL VWDFMRQDUQ\ PRGHO GLRG\ VNáDGDMF\ VL ]

LGHDOQHM GLRG\ L SRMHPQRFL G\IX]\MQHM SURFHV XVXZDQLD áDGXQNX Z

SLHUZV]HMID]LHZ\áF]DQLDPR*HE\üRSLVDQ\UyZQDQLHPUy*QLF]NRZ\P

ZSRVWDFL

− = +

= +

I

i

dq

dt

i

t

di

dt

R

D

D

D

t

D

5R]ZL]XMF UyZQDQLH Uy*QLF]NRZH SU]\ ZDUXQNX

SRF]WNRZ\P

i

I

D

F

=

RWU]\PXMHP\

(

)

(

)

i

I

I

t t

I

D

F

R

t

R

=

+

−

−

exp

&]DV WUZDQLD WHM ID]\

t

s

DQJ VWRUDJH WLPH Z\]QDF]\P\ ]

ZDUXQNX

( )

i t

D

s

=

0

3RGVWDZLDMF WHQ ZDUXQHN GR UyZQDQLD

RWU]\PXMHP\

t

t

I

I

s

t

F

R

=

+











ln 1

'UXJDID]DZ\áF]DQLDUR]SRF]\QDVL]FKZLOJG\QDSLFLHQD

]áF]XRVLJDZDUWRü]HURZLUR]SRF]\QDVLáDGRZDQLHPXUHGQLRQHM

SRMHPQRFL ]áF]RZHM UyZQ GR QDSLFLD

−

E

R

&]DV áDGRZDQLD

EGF\ MHGQRF]HQLH F]DVHP t

f

RSDGDQLD SUGX GLRG\ DQJ IDOOLQJ

WLPHZ\QRVL

W

5 &

I

U

=

&DáNRZLW\F]DVZ\áF]DQLDGLRG\Z\QRVL

t

t

t

OFF

s

f

= +

3U]HSURZDG]RQD DQDOL]D F]DVyZ WUZDQLD SRV]F]HJyOQ\FK ID] SURFHVX

SU]HáF]DQLD GLRG\ SRPLPR *H GRW\F]\ SU]\SDGNX QDMSURVWV]HJR

V\JQDáXSU]HáF]DMFHJRRNV]WDáFLHLGHDOQHJRSU]HELHJXSURVWRNWQHJR

MHVWQLH]E\WGRNáDGQD]XZDJLQDZ\NRU]\VWDQLHXSURV]F]RQ\FKPRGHOLL

QLHXZ]JOGQLHQLH ZLHOX ]MDZLVN Z\VWSXMF\FK Z W\FK SURFHVDFK

3RPLPRWHJRSR]ZDODRQDQDRJyOQH]UR]XPLHQLH]MDZLVN]DFKRG]F\FK

ZW\FKSURFHVDFKLRV]DFRZDQLHZSá\ZXEH]ZáDGQRFLGLRG\ZUy*Q\FK

XNáDGDFKVWRVXMF\FKGLRG\

$QDOL]D XNáDGyZ ]DZLHUDMF\FK GLRG\ PR*H E\ü RF]\ZLFLH

SU]HSURZDG]RQD ]QDF]QLH GRNáDGQLHM PHWRGDPL QXPHU\F]Q\PL ]

]DVWRVRZDQLHPW\SRZ\FKPRGHOLGLRG]HOHPHQWDPLQLHOLQLRZ\PL

3U]HáF]DQLHWUDQ]\VWRUDELSRODUQHJR

-HGQ\P]QDMZD*QLHMV]\FKXNáDGyZ SU]HáF]QLNRZ\FK MHVW NOXF]

WUDQ]\VWRURZ\ QDV\FRQ\ : XNáDG]LH W\P WUDQ]\VWRU SUDFXMH Z

NRQILJXUDFML2(LMHVWVWHURZDQ\RGVWDQXRGFLFLDGRVWDQXQDV\FHQLD

7\SRZ\ XNáDG SU]HáF]DQLD WUDQ]\VWRUD ELSRODUQHJR RUD] JUDILF]QH

Z\]QDF]HQLH SXQNWyZ SUDF\ QD FKDUDNWHU\VW\NDFK ED]RZ\FK L

NROHNWRURZ\FK RGSRZLDGDMF\FK VWDQRP RGFLFLD L QDV\FHQLD

SU]HGVWDZLRQRQDU\V>@

6WHURZDQLH REZRGX ZHMFLRZHJR WUDQ]\VWRUD MHVW EDUG]R SRGREQH GR

VWHURZDQLDSU]\SU]HáF]DQLXGLRG\

L

%

b)

+

(

)

−

(

5

X

%(

W

0

X

&(

L

&

8

5

&&

&

8

&&

c)

a)

5

&

H

*

−

(

5

+

(

)

W

5

%

X

%(

8

&&

6

4

8

%(3

0

4

8

&(6

6

X

&(

L

&

H

*

H

*

U

EH

,

%)

−

,

%5

8

%()

U

EU

≈ ∞

( )

,

%

1DFK\OHQLH

−

5

%

U

5

U

EH

%

EU

<<

<<

6

,

&0

,

%*

1DFK\OHQLH

−

5

&

,

%)

L

%

7

0

5\V 3U]HáF]DQLH WUDQ]\VWRUD ELSRODUQHJR D W\SRZ\ XNáDG SU]HáF]DQLD

JUDILF]QH Z\]QDF]HQLH SXQNWyZ SUDF\ QD FKDUDNWHU\VW\NDFK

EED]RZ\FKFNROHNWRURZ\FK

3UGED]\

I

BF

ZVWDQLHZáF]HQLDWUDQ]\VWRUDZ\QRVL

I

E

U

R

BF

F

BEP

B

=

−

SU]\ F]\P QDSLFLH ED]DHPLWHU MHVW UyZQH Z SU]\EOL*HQLX

8

%(3

≈

−

9

3UG ED]\ MHVW Z\VWDUF]DMFR GX*\ DE\ WUDQ]\VWRU ZSURZDG]Lü Z VWDQ

QDV\FHQLDNLHG\WRSUGNROHNWRUDRJUDQLF]RQ\MHVWSU]H]UH]\VWDQFMZ

REZRG]LHNROHNWRUD

I

U

U

R

CM

CC

CES

C

=

−

1DSLFLH NROHNWRU HPLWHU WUDQ]\VWRUD Z VWDQLH QDV\FHQLD MHVW

PDáH L Z\QRVL Z SU]\EOL*HQLX

8

&(6

≈

−

9

SRQLHZD* MHVW RQR

Uy*QLFQDSLüQDREX]áF]DFKWUDQ]\VWRUDVSRODU\]RZDQ\FKZNLHUXQNX

SU]HZRG]HQLD

U

U

U

CES

BEP

BCP

=

−

1DSLFLH QDV\FHQLD

U

CES

MHVW W\P

PQLHMV]H LP ZLNV]\ MHVW SUG ED]\ WUDQ]\VWRUD SRQDG ZDUWRü

JUDQLF]Q

I

I

BG

CM

=

β

0

SU]\NWyUHM WUDQ]\VWRU ]QDMGXMH VL QD JUDQLF\

VWDQXQDV\FHQLD,GHDOL]RZDQHSU]HELHJLF]DVRZHQDSLüLSUGyZSU]\

SUGRZ\PSU]HáF]DQLXWUDQ]\VWRUDW]QJG\VSHáQLRQ\MHVWZDUXQHN*H

UH]\VWDQFMD ZHMFLRZD WUDQ]\VWRUD SU]\ SRODU\]DFML Z NLHUXQNX

SU]HZRG]HQLD MHVW PDáD Z SRUyZQDQLX GR

R

B

F]\OL

r

R

be

B

<<

SU]HGVWDZLRQRQDU\V

−

,

%5

L

%

H

*

0

0

W

L

+

(

)

−

(

5

W

W

0

0

0

0

−

(

5

+

(

)

(

)

5 &

&

%

MH

MF

+

8

%(3

(

)

,

, U

%)

%5 EH

+

W

−

(

5

−

(

5

(

)

5 &

&

%

MH

MF

+

(

)

5 &

&

%

MH

MF

+

(

)

5 &

&

%

MH

MF

+

X

%(

β

0

,

%)

−

β

0

,

%5

W

W

0

0

0

0

8

&&

8

&&

8

&(6

X

&(

L

&

,

&0

τ

τ

W

U

W

G

W

V

W

I

W

L

( ) (

)

,

(

(

5

%

)

5

%

0

=

+

,

%)

5\V,GHDOL]RZDQHSU]HELHJL

F]DVRZHSUGyZLQDSLüSU]\

SU]HáF]DQLXWUDQ]\VWRUDELSRODUQH

JR

'R DQDOL]\ SURFHVX SU]HáF]DQLD PR*HP\ Z\NRU]\VWDü PRGHO

WUDQ]\VWRUD (EHUVD 0ROOD SU]\ SRODU\]DFML QRUPDOQHM WUDQ]\VWRUD

$QDOL]SURFHVXSU]HáF]DQLDUR]SRF]QLHP\RGPRPHQWXJG\QDSLFLH

(UyGáDV\JQDáX

e

G

]PLHQLDVLRGZDUWRFL

−

E

R

GR

+

E

F .

3RGREQLHMDN

ZSU]\SDGNXSU]HáF]DQLDGLRG\WUDQ]\VWRUSR]RVWDMHZVWDQLHRGFL FLD

D*GRFKZLOLJG\ SRMHPQRFL ]áF]RZH C

C

je

jc

L

]RVWDQ QDáDGRZDQH

WDN *H QDSLFLH QD ]áF]X ED]DHPLWHU RVLJQLH ZDUWRü

U

BEP

3RF]WNRZD ZDUWRü SUGX áDGRZDQLD W\FK SRMHPQRFL Z\QRVL

(

)

(

(

5

5

)

%

+

D SURFHV QDUDVWDQLD QDSLFLD QD ED]LH WUDQ]\VWRUD

ZL*F\VL]áDGRZDQLHPSRMHPQRFL]áF]RZ\FKMHVWWDNLVDPMDNZ

SU]\SDGNX SU]HáF]DQLD GLRG\ 2VLJQLFLH QDSLFLD

U

BEP

QD ED]LH

WUDQ]\VWRUD QDVWSXMH SR F]DVLH RSy(QLHQLD ZáF]HQLD WUDQ]\VWRUD

t

d

DQJGHOD\WLPH

(

)

t

R

C

C

E

E

E

U

d

B

je

jc

F

R

F

BEP

=

+

+

−

ln

: UyZQDQLX MDN UyZQLH* Z GDOV]\FK UR]ZD*DQLDFK

SURFHVyZ SU]HáF]DQLD WUDQ]\VWRUD ELSRODUQHJR QDOH*\ XZ]JOGQLü

XUHGQLRQH SRMHPQRFL ]áF]RZH C

je

L C

jc

Z\]QDF]RQH WDN MDN Z

SU]\SDGNXXUHGQLRQHMSRMHPQRFL]áF]RZHMGLRG\UyZQ

3R F]DVLH

t

d

WUDQ]\VWRU RVLJD VWDQ DNW\ZQHJR SU]HZRG]HQLD

QRUPDOQHJRSU]\F]\PSUGED]\RVLJDZDUWRü

I

BF

UyZQ

1DUDVWDQLH SUGX NROHNWRUD Z\]QDF]\P\ QD SRGVWDZLH UyZQD

áDGXQNRZ\FK SU]\ F]\P ]RVWDQLH SRPLQLW\ áDGXQHN Z SRMHPQRFL

]áF]RZHMED]DHPLWHU q

q

je

jc

<<

SRQLHZD*

u

u

BE

CB

<<

i

q

dq

dt

dq

dt

B

B

B

jc

=

+

+

τ

i

q

t

C

B

t

=

0LG]\F]DVHP*\FLDQRQLNyZ

τ

LF]DVHPSU]HORWXQRQLNyZ

t

t

]DFKRG]QDVWSXMFHUHODFMH

τ β

ω

ω

β

=

=

=

0

1

1

t

t

t

t

T

SU]\ F]\P

ω

α ω

β ω

π

α

β

T

T

f

=

=

=

0

0

2

MHVW SROHP Z]PRFQLHQLD

WUDQ]\VWRUD

3RGVWDZLDMF]DOH*QRFLLGRUyZQDQLDRWU]\PXMHP\

i

i

t

di

dt

dq

dt

B

C

t

C

jc

=

+

+

β

0

3U]\EOL*RQZDUWRüRVWDWQLHJRVNáDGQLNDUyZQZ\]QDF]\P\QD

SRGVWDZLH GHILQLFML SRMHPQRFL C

jc

L VFKHPDWX LGHRZHJR XNáDGX ]

U\VD

(

)

dq

dt

C

du

dt

C

du

dt

C

d U

i R

dt

C R

di

dt

jc

jc

CB

jc

CE

jc

CC

C

C

jc

C

C

= −

≈ −

=

= −

−

=

3RGVWDZLDMF]DOH*QRüGRUyZQDQLDRWU]\PXMHP\

(

)

i

I

i

t

C R

di

dt

B

BF

C

t

jc

C

C

=

=

+

+

β

0

SRQLHZD* Z VWDQLH DNW\ZQ\P SUG ED]\ MHVW VWDá\ L UyZQ\

I

BF

UyZQ

5R]ZL]DQLHP UyZQDQLD Uy*QLF]NRZHJR SU]\

( )

i

C

0

0

=

MHVW

IXQNFMD

( )

(

)

i t

I

t

t

R C

C

BF

t

C

jc

=

−

−

+

















β

β

0

0

1 exp

3UGNROHNWRUDQDUDVWD]HVWDáF]DVRZ

(

)

τ β

β

τ

ω

'

=

+

=

+

0

0

1

t

R C

R

C

t

C

jc

C

T

jc

D

NWyUDMHVWZLNV]DRG

τ

RZVSyáF]\QQLNZ\QLNDMF\]HIHNWX0LOOHUD

3UG NROHNWRUD RSLVDQ\ UyZQ G*\ GR DV\PSWRW\F]QHM

ZDUWRFL

β

0

I

BF

-H*HOL ]DFKRG]L QDV\FHQLH WUDQ]\VWRUD W]Q JG\

β

β

0

0

I

I

I

BF

CM

BG

>

=

UyZQWRF]DVQDUDVWDQLDSUGXNROHNWRUD

t

r

DQJULVHWLPHRGGR

I

CM

MHVWUyZQ\

(

)

(

)

t

t

R C

I

I

I

t

R C

k

k

r

t

C

jc

BF

BF

CM

t

C

jc

F

F

=

+

−

=

+

−

β

β

β

β

0

0

0

0

1

ln

ln

SU]\F]\PZLHONRü

k

I

I

F

N

BF

CM

=

>

β

1

D

MHVWQD]\ZDQDZVSyáF]\QQLNLHPSU]HVWHURZDQLDDOERZVSyáF]\QQLNLHP

JáHERNRFLQDV\FHQLD

3U]HVWHURZDQLH WUDQ]\VWRUD VNUDFD F]DV QDUDVWDQLD SU GX

NROHNWRUD

3U]\

(

)

β

0

1

I

I

k

BF

CM

F

>>

>>

QDUDVWDQLH SUGX NROHNWRUD GR SR]LRPX

I

CM

MHVW Z SU]\EOL*HQLX OLQLRZH L Z W\P SU]\SDGNX ]DOH*QRü

XSUDV]F]DVLGRSRVWDFL

(

)

t

t

R C

I

I

R C

k

r

t

C

jc

CM

BF

T

C

jc

F

≈

+

=

+

β

β

ω

ω

β

0

0

1

1

1DV\FHQLHWUDQ]\VWRUDNRF]\]HZQWU]QHVWDQ\QLHXVWDORQHMHGQDNSR

F]DVLHZáF]DQLD

t

t

t

ON

d

r

= +

WUZD MHV]F]H SURFHV JURPDG]HQLD QDGPLDURZHJR áDGXQNX QR QLNyZ

PQLHMV]RFLRZ\FKZED]LH

: VWDQLH QDV\FHQLD RED ]áF]D WUDQ]\VWRUD V VSRODU\]RZDQH Z

NLHUXQNX SU]HZRG]HQLD ]DWHP NRQFHQWUDFMH EU]HJRZH QR QLNyZ

PQLHMV]RFLRZ\FKZED]LHSU]\REX]áF]DFKVUy*QHRG]HUDàDGXQHN

Q

B

QDJURPDG]RQ\ZED]LHPR*QDUR]ZD*DüMDNRVXSHUSR]\FMáDGXQNX

JUDQLF]QHJR

Q

BN

RGSRZLDGDMFHJRUR]NáDGRZLJUDQLF]QHPXQRQLNyZ

RVLJQLWHPX Z FKZLOL ZHMFLD Z VWDQ QDV\FHQLD RUD] QDGPLDURZ\

áDGXQHN QDV\FHQLD

Q

BS

Z\QLNDMF\ ] SU]HVWHURZDQLD WUDQ]\VWRUD L

JáERNLHJR ZHMFLD Z VWDQ QDV\FHQLD 5R]NáDG áDGXQNX QRQLNyZ

QDGPLDURZ\FKZED]LHWUDQ]\VWRUDSU]HGVWDZLRQRV]NLFRZRQDU\V

4

%6

( )

Q

SH

0

[

0

Q

SH

( )

Q :

SH

:

4

4

4

%

%6

%6

=

+

4

%1

5\V 5R]NáDG áDGXQNX QRQLNyZ

QDGPLDURZ\FKZED]LHWUDQ]\VWRUD

GOD]DNUHVXQDV\FHQLD

: VWDQLH QDV\FHQLD WUDQ]\VWRU MHVW HOHPHQWHP SDV\ZQ\P L QLH

Z\VWSXMHZQLPHIHNW0LOOHUD &]DV*\FLDQRQLNyZPQLHMV]RFLRZ\FK

τ

s

ZVWDQLHQDV\FHQLDRSLVDQ\MHVW]DOH*QRFL-/0ROOD>@

(

)

τ

ω

ω

α α ω

ω

α

α

α

α

s

N

I

N

I

N

I

=

+

−

1

SU]\ F]\P

ω

ω

α

α

N

I

,

R]QDF]DM SXOVDFMH JUDQLF]QH ZVSyáF]\QQLNyZ

Z]PRFQLHQLD SUGRZHJR

( )

( )

α

ω

α

ω

N

I

j

j

,

RGSRZLHGQLR GOD

WUDQVPLVML QRUPDOQHM L LQZHUV\MQHM 1DGPLDU áDGXQNX

Q

BS

PR*QD

Z\UD]Lü SU]\ SRPRF\ QDGPLDUX SUGX ED]\

I

BF

SRQDG ZDUWRü

JUDQLF]Q

I

BG

SU]\ NWyUHM WUDQ]\VWRU RVLJD VWDQ QDV\FHQLD

β

0

I

I

BG

CM

=

=DWHP

(

)

Q

I

I

I

I

BS

s

BF

BG

s

BF

CM

=

−

=

−











τ

τ

β

0

:PRPHQFLH

t

i

QDSLFLH(UyGáDV\JQDáX

( )

e t

G

]PLHQLDVLRGZDUWRFL

+

E

F

GR

E

R

LUR]SRF]\QDVLSURFHVZ\áF]DQLDWUDQ]\VWRUD

6WDQQDV\FHQLDWUDQ]\VWRUDXWU]\PXMHVLGDOHMSU]H]F]DV

t

s

D*

GRFKZLOLXVXQLFLD]ED]\WUDQ]\VWRUDQDGPLDURZHJRáDGXQNXQDV\FHQLD

Q

BS

&]DV

t

s

QD]\ZD VL F]DVHP PDJD]\QRZDQLD OXE F]DVHP

SU]HFLJDQLDSUGXNROHNWRUDDQJVWRUDJHWLPH

3UGED]\]PLHQLDNLHUXQHNSU]HSá\ZXLSU]\MPXMHZDUWR ü

−

I

BR

SU]\F]\P

I

E

U

R

BR

R

BEP

B

≈

+

: SLHUZV]\P RNUHVLH Z\áF]DQLD XVXZDQ\ MHVW QDGPLDURZ\

áDGXQHN

Q

BS

: W\P RNUHVLH QDSLFLD QD ]áF]DFK V VWDáH ]DWHP QLH

Z\VWSXM ]PLDQ\ áDGXQNyZ Z SRMHPQRFLDFK ]áF]RZ\FK 3URFHV

XVXZDQLD áDGXQNX

Q

BS

PR*H E\ü RSLVDQ\ áDGXQNRZ\P UyZQDQLHP

Uy*QLF]NRZ\P

i

q

dq

dt

I

I

BS

BS

s

BS

BR

CM

=

+

= −

−

τ

β

0

5R]ZL]XMFUyZQDQLHUy*QLF]NRZHSU]\ZDUXQNXSRF]WNRZ\P

( )

(

)

q

Q

I

I

BS

BS

s

BF

CM

0

0

=

=

−

τ

β

UyZQRWU]\PXMHP\

( )

T W

,

,

W

,

,

W

%6

V

%5

&0

V

V

%)

&0

V

= −

+











−









 +

+

−











−











τ

β

τ

τ

β

τ

H[S

H[S

7UDQ]\VWRU SU]HFKRG]L GR VWDQX DNW\ZQHJR SR F]DVLH

W

V

JG\

( )

q

t

BS

s

=

0

3RGVWDZLDMFWHQZDUXQHNGRUyZQRWU]\PXMHP\

t

I

I

I

I

k

k

k

s

s

BF

BR

BR

CM

s

F

R

R

=

+

+

=

+

+

τ

β

τ

ln

ln

0

1

JG]LH

k

I

I

R

BR

CM

=

≥

β

0

0

D

MHVWZVSyáF]\QQLNLHPSU]HVWHURZDQLDWUDQ]\VWRUDSU]\Z\áF]DQLX

&]DV

t

s

PDOHMH JG\ ZVSyáF]\QQLN SU]HVWHURZDQLD

k

R

MHVW GX*\

W]Q JG\ G]LDáDQLH Z\áF]DMFH SROHJDMFH QD XVXZDQLX áDGXQNyZ ]

ED]\XMHPQ\PSUGHPED]\MHVWVLOQH1DOH*\MHGQDNSDPLWDü*HF]DV

t

s

SRZLNV]D VL JG\ WUDQ]\VWRU MHVW ZSURZDG]DQ\ Z VWDQ Já ERNLHJR

QDV\FHQLD SRSU]H] VLOQH SU]HVWHURZDQLH WUDQ]\VWRUD SU]\ Zá F]DQLX

GX*H

k

F

:\VWSXM ]DWHP SU]HFLZQH Z\PDJDQLD GOD ]DSHZQLHQLD

NUyWNLFKF]DVyZ

t

t

r

s

L

3R XSá\ZLH F]DVX

t

s

WUDQ]\VWRU SRQRZQLH ]QDMGXMH VL Z VWDQLH

DNW\ZQ\P5R]SRF]\QDVLRSDGDQLHSUGXNROHNWRUDRGZDUWRFL

I

CM

GRDV\PSWRW\F]QHMZDUWRFL

−

β

0

I

BR

]HVWDáF]DVRZ

τ

'

WMWDNVDP

MDNSU]\]DáF]DQLXUyZQD

( )

(

)

(

)

i t

I

I

t

t

R C

I

C

CM

BR

t

C

jcc

BR

=

+

−

+

















−

β

β

β

0

0

0

exp

-HGQDN Z FKZLOL RVLJQLFLD ZDUWRFL

( )

i t

C

=

0

WUDQ]\VWRU ZFKRG]L Z

VWDQ RGFLFLD L NRF]\ VL SURFHV MHJR Z\áF]DQLD 1DVWSXMH WR SR

F]DVLHRSDGDQLDSUGXNROHNWRUDt

f

DQJIDOOWLPH

(

)

(

)

t

t

R C

I

I

I

t

R C

k

k

f

t

C

jc

CM

BR

BR

t

C

jc

R

R

=

+

+

=

+

+

β

β

β

β

0

0

0

0

1

ln

ln

3U]\ VLOQ\P SU]HVWHURZDQLX

(

)

β

0

1

I

I

k

BR

CM

R

>>

>>

;

RSDGDQLH SUGX

NROHNWRUD MHVW Z SU]\EOL*HQLX OLQLRZH L ]DOH*QRü PR*QD

SU]HGVWDZLüZSU]\EOL*RQHMSRVWDFL

(

)

t

t

R C

I

I

R C

k

f

t

C

jc

CM

BR

T

C

jc

T

R

≈

+

=

+

β

β

ω

ω

0

0

1

1

3RZHMFLXZVWDQRGFLFLDWUDQ]\VWRUMHVWHOHPHQWHPSDV\ZQ\PL

UR]SRF]\QD VL áDGRZDQLH SRMHPQRFL ]áF]RZ\FK C

C

je

jc

+

]H VWDá

F]DVRZ

R

B

C

C

je

jc

+

&]DVáDGRZDQLDW\FKSRMHPQRFLMHVWUyZQ\

t

R

pas

B

=

2 2

,

C

C

je

jc

+

àF]Q\F]DVZ\áF]DQLDWUDQ]\VWRUDZ\QRVL

t

t

t

t

OFF

s

f

pas

= + +

$E\ X]\VNDü NUyWNLH F]DV\ SU]HáF]DQLD WUDQ]\VWRUD QDOH*\

VWRVRZDü QLHZLHONLH ZVSyáF]\QQLNL SU]HVWHURZDQLD WUDQ]\VWRUD SU]\

]DáF]DQLX L SU]\ Z\áF]DQLX Sá\WNLH QDV\FHQLH L Sá\WNLH ]DWNDQLH

WUDQ]\VWRUD : SUDNW\F]Q\FK UR]ZL]DQLDFK ]DFKRG]L MHGQDN EDUG]R

F]VWRSRWU]HEDJáERNLHJRQDV\FHQLDLJáERNLHJR]DWNDQLDWUDQ]\VWRUD

FRZL*HVL]NRQLHF]QRFLXVXZDQLDGX*\FKáDGXQNyZ]SRMHPQRFL

]áF]RZ\FKL]ED]\WUDQ]\VWRUD:WDNLFKSU]\SDGNDFKPR *QDVNUyFLü

F]DV\ SU]HáF]DQLD ]D SRPRF W]Z SRMHPQRFL SU]\VSLHV]DMFHM

ERF]QLNXMFHM F]ü OXE FDá UH]\VWDQFM

R

B

QDMF]FLHM UH]\VWDQFMD

(UyGáD VWHUXMFHJR VWDQRZL QLHGRVWSQ F]ü UH]\VWDQFML Z REZRG]LH

ED]\

&

S

5

%2

5

%1

L

%

L

&

X

&(

X

%(

( )

X W

&

L

S

+

8

&&

5

&

( )

H W

*

+

(

)

−

(

5

0

W

L

5\V3U]HáF]DQLHWUDQ]\VWRUD

]SRMHPQRFLSU]\VSLHV]DMF

6WDáD F]DVRZD

τ

r

REZRGX Uy*QLF]NXMFHJR Z REZRG]LH ED]\

U\VSRZLQQDE\üPQLHMV]DRGF]DVXWUZDQLDLPSXOVXZHM FLRZHJR

t

i

F]\OL

τ

r

p

B

B

B

B

i

C

R

R

R

R

t

=

+

<

1

2

1

2

-HGQRF]HQLH VXPDU\F]QD UH]\VWDQFMD

R

R

B

B

1

2

+

PXVL ]DSHZQLü

X]\VNDQLH SUGX ED]\ Z\VWDUF]DMFHJR GR ZHMFLD WUDQ]\VWRUD Z

QDV\FHQLH

β

β

0

0

2

I

E

U

R

R

I

U

U

R

B

F

BEP

B

B

CM

CC

CES

C

=

−

+

>

=

−

!

3RMHPQRü SU]\VSLHV]DMFD C

p

NLOND GR NLONDQDFLH S)

ERF]QLNXMH JáyZQ F]ü UH]\VWDQFML Z REZRG]LH ED]\

R

B 2

3U]\

VNRNRZ\FK ]PLDQDFK V\JQDáX VWHUXMFHJR

( )

e t

G

SRMHPQRü C

p

XWU]\PXMH SUDZLH QLH]PLHQLRQH QDSLFLH QD UH]\VWDQFML

R

B 2

ZVNXWHN

F]HJRSUDZLHFDáD]PLDQDSU]HáF]DMFHJRQDSLFLDRGNáDGDVLQDPDáHM

UH]\VWDQFML

R

B1

']LNLWHPXPR*OLZHVEDUG]RGX*H]PLDQ\SUGXZ

REZRG]LH ED]\ ]DUyZQR GRGDWQLH MDN L XMHPQH =DUD] SR VNRNRZHM

]PLDQLH QDSLFLD VWHUXMFHJR SUG ED]\ R GX*HM FKZLORZHM ZDUWRFL

Sá\QF\ ]H (UyGáD VWHUXMFHJR JáyZQLH SU]H] SRMHPQRü

SU]\VSLHV]DMF SRZRGXMH VLOQH SU]HVWHURZDQLH WUDQ]\VWRUD FR Z

HIHNFLHGDMH]QDF]QHVNUyFHQLHF]DVyZSU]HáF]DQLD

1DOH*\]DXZD*\ü*HQDZHWZSU]\SDGNXJG\QDSLFLHVWHUXMFH

( )

e t

G

]PLHQLDVLVNRNRZRRG

+

E

F

GR]HUDW]Q

E

R

=

0

WRZXNáDG]LH

UyZQLH*Z\VWSXMHSU]HVWHURZDQLHWUDQ]\VWRUDSU]\Z\áF]DQLXERZLHP

XMHPQ\SUGED]\MHVWZ\PXV]DQ\SUGHPUR]áDGRZ\ZDQLDSRMHPQRFL

SU]\VSLHV]DMFHM:W\PSU]\SDGNX

( )

−

≈ −

= −

−

+

I

U

t

R

E

U

R

R

R

R

BR

C

i

B

F

BEP

B

B

B

B

p

2

1

2

1

2

=QDF]QH VNUyFHQLH F]DVX SU]HFLJDQLD SUGX NROHNWRUD PR*QD

X]\VNDü SU]H] GRáF]HQLH GLRG\ 6FKRWWN\¶HJR UyZQROHJOH GR ]áF]D

NROHNWRUED]DU\V

L

%

L

*

'6

&

(

%

&

(

%

5\V7UDQ]\VWRU]GLRG

6FKRWWN\¶HJRSU]HFLZG]LDáDMFD

QDV\FHQLXLMHJRV\PEROJUDILF]Q\

8MHPQH QDSLFLRZH VSU]*HQLH ]ZURWQH SRZRGXMH RJUDQLF]HQLH

QDV\FHQLDVLWUDQ]\VWRUDSRQLHZD*XQLHPR*OLZLDRQRVSDGHNQDSLFLD

NROHNWRUHPLWHUSRQL*HMZDUWRFLRN93U]\SU]HZRG]FHMGLRG]LH

F]üSUGX]H(UyGáDVWHUXMFHJRVNLHURZDQDMHVWGRREZRGXNROHNWRUD

]DSRELHJDMFJáERNLHPXQDV\FHQLXWUDQ]\VWRUD

3U]HáF]DQLHWUDQ]\VWRUD026)(7

8NáDG\SU]HáF]QLNRZH]WUDQ]\VWRUDPL026JáyZQLH]NDQDáHP

Z]ERJDFDQ\P V SRGVWDZRZ\PL XNáDGDPL Z\NRU]\VW\ZDQ\PL GR

UHDOL]DFML VFDORQ\FK XNáDGyZ F\IURZ\FK DQDORJRZ\FK XNáDGyZ ]

SU]HáF]DQ\PL SRMHPQRFLDPL 6& F]\ WH* XNáDGyZ ] SU]HáF]DQLHP

SUGyZ 6, 1D U\V SU]HGVWDZLRQR XNáDG SU]HáF]QLNRZ\ ]

WUDQ]\VWRUHP Q026 ] NDQDáHP Z]ERJDFDQ\P ] OLQLRZ\P UH]\VWRUHP

R

D

ZREZRG]LHGUHQXLREFL*RQ\SRMHPQRFL

C

L

NWyU Z SUDNW\FH

VWDQRZL SRMHPQRFL ZHMFLRZH WUDQ]\VWRUyZ 026 GRáF]RQ\FK GR

Z\MFLD DQDOL]RZDQHJR XNáDGX 7UDQ]\VWRU MHVW VWHURZDQ\ ]D (UyGáD

V\JQDáX

( )

e t

G

R UH]\VWDQFML ZHZQWU]QHM

R

G

L ]PLHQLDMFHJR VL

VNRNRZR RG XMHPQHM ZDUWRFL

−

E

R

GR GRGDWQLHM

E

F

L SR F]DVLH

t

i

SRQRZQLHGRZDUWRFL

−

E

R

L

'

b)

+

(

)

−

(

5

X

*6

W

0

X

'6

L

'

8

5

''

'

8

''

c)

a)

5

'

( )

H W

*

−

(

5

+

(

)

W

5

*

X

*6

+

8

''

6

8

8

8

'6VDW

*6

7

=

−

4

8

7

0

4

8

')

8

'6VDW

,

')

6

W

3

W

2

W

SRVV

W

U

W

I

8

(

*6

)

= +

X

'6

&

/

L

'

( )

H W

*

0

5\V 3U]HáF]DQLH WUDQ]\VWRUD Q026 D XNáDG SU]HáF]DQLD E JUDILF]QH

Z\]QDF]HQLH SXQNWyZ SUDF\ QD FKDUDNWHU\VW\NDFK EUDPNRZ\FK

LGUHQRZ\FK

: SUDNW\F]Q\FK UHDOL]DFMDFK VFDORQ\FK ]DPLDVW OLQLRZHM

UH]\VWDQFMLG\IX]\MQHM

R

D

VWRVXMHVLREFL*HQLHG\QDPLF]QHZSRVWDFL

LQQHJRWUDQ]\VWRUD8NáDG\SU]HáF]QLNRZH]G\VNUHWQ\PLWUDQ]\VWRUDPL

SRORZ\PL 026 L REFL*HQLHP

R

D

V VWRVRZDQH Z Uy*QHJR URG]DMX

XNáDGDFKLPSXOVRZ\FKUHDOL]RZDQ\FKQDWUDQ]\VWRUDFK9026

,GHDOL]RZDQH SU]HELHJL F]DVRZH QDSLü L SUGyZ Z SURFHVDFK

SU]HáF]DQLDWUDQ]\VWRUDQ026SU]HGVWDZLRQRQDU\V>@

2Sy(QLHQLD SU]\ SU]HáF]DQLX WUDQ]\VWRUD 026 Z\QLNDM ]

SU]HáDGRZ\ZDQLD SRMHPQRFL PLG]\HOHNWURGRZ\FK L ]H VNRF]RQHJR

F]DVXSU]HORWXQRQLNyZSU]H]NDQDá&]DVSU]HORWXQRQLNyZ

t

t

RNUHOD

]QDQD]DOH*QRü

(

)

t

L

U

U

t

GS

T

=

−

4

3

2

µ

L

'

H

*

0

0

W

L

+

(

)

−

(

5

W

0

0

X

*6

0

0

−

(

5

W

8

7

τ

τ

W

G

W

I

W

U

0

0

8

'6VDW

8

DV\PSW

∆

8

'

W

8

')

,

')

W

W

X

'6

W

8

''

W

SDVV

,

'6

5\V ,GHDOL]RZDQH SU]HELHJL

QDSLüLSUGyZSU]\SU]HáF]DQLX

WUDQ]\VWRUD026

&]DV

t

t

MHVWSURSRUFMRQDOQ\GRNZDGUDWXGáXJRFLNDQDáX

L

RUD]

RGZURWQLHSURSRUFMRQDOQ\GRUXFKOLZRFLQRQLNyZLQDGZ\*NLQDSLFLD

U

GS

QDG QDSLFLHP SURJRZ\P

U

T

'OD ZVSyáF]HVQ\FK WUDQ]\VWRUyZ

F]DV SU]HORWX MHVW EDUG]R PDá\ Z SRUyZQDQLX ] F]DVDPL WUZDQLD

SRV]F]HJyOQ\FKID]SURFHVyZSU]HáDGRZDQLDGHWHUPLQRZDQ\FKJáyZQLH

SU]H]SRMHPQRFLPLG]\HOHNWURGRZH

:FKZLOLSRMDZLHQLDVLVNRNRZHM]PLDQ\QDSLFLD(UyGáDV\JQDáXRG

ZDUWRFL

−

E

R

GR

+

E

F

QDSLFLHQDEUDPFHQDUDVWD]DVWDáF]DVRZ

(

)

τ

1

=

+

+

R

C

C

C

G

gs

gd

gb

3RF]DVLHRSy(QLHQLD

t

d

UyZQ\P

t

E

E

E

U

d

F

R

F

T

=

+
−

τ

1

ln

QDSLFLHQDEUDPFHRVLJDZDUWRüSURJRZ

U

T

L]DF]\QDSá\QüSUG

GUHQX NWyU\ QDUDVWD GR ZDUWRFL

I

DS

RGSRZLDGDMFHM QDSLFLX

E

F

QD

EUDPFH &]DV QDUDVWDQLD SUGX GUHQX

t

r

MHVW Uy*QLF SRPLG]\ F]DVHP

XVWDODQLHVLQDSLFLDQDEUDPFHLF]DVHPRSy(QLHQLD

t

d

t

t

r

d

=

−

2 2

1

,

τ

1DUDVWDMFH QDSLFLH QD EUDPFH SU]HQRVL VL QD Z\MFLH SU]H]

G]LHOQLN SRMHPQRFLRZ\

(

)

C

C

C

gd

L

db

−

+

Z\ZRáXMF Z\VNRN QDSLFLD

u

DS

SRQDGQDSLFLH]DVLODQLD

U

CC

RQLHZLHONZDUWRü

∆

U

D

(

)

∆

U

E

E

C

C

C

C

D

F

R

gd

gd

db

L

=

+

+

+

: F]DVLH

t

2

QDSLFLH

u

DS

PDOHMH ZVNXWHN UR]áDGRZ\ZDQLD VL

SRMHPQRFL C

C

C

gd

db

L

+

+

]HVWDáF]DVRZ

(

)

τ

2

=

+

+

R

C

C

C

D

gd

db

L

RG

ZDUWRFL SRF]WNRZHM

U

U

DD

D

+ ∆

G*F GR DV\PSWRW\F]QHM ZDUWRFL

QDMF]FLHMXMHPQHMZ\QLNDMFHM]SU]HSá\ZXSUGXV]F]\WRZHJR

I

DS

U

U

I

R

asympt

DD

DS

D

=

−

3URFHV WHQ NRF]\ VL SR F]DVLH

t

2

JG\ FKZLORZ\ SXQNW SUDF\

WUDQ]\VWRUDSU]HMG]LH]REV]DUXQDV\FHQLDQDJUDQLF REV]DUXOLQLRZHJR

WULRGRZHJRW]QJG\QDSLFLHGUHQ(UyGáRRVLJQLHZDUWRü

U

U

U

E

U

DSsat

GS

T

F

T

=

−

=

−

6WG

t

U

U

U

U

U

DD

D

asympt

DSsat

asympt

2

2

=

+

−

−

τ

ln

∆

'DOV]H UR]áDGRZ\ZDQLH SRMHPQRFL C

C

C

gd

db

L

+

+

GR ZDUWRFL

u

U

DS

DF

=

Z\QLNDMFHM]SU]HFLFLDVLSURVWHMURERF]HMRQDFK\OHQLX

1 / R

D

] FKDUDNWHU\VW\N GUHQRZ

( )

I

f U

U

E

D

DS

GS

F

=

= +

GOD

Z

REV]DU]HOLQLRZ\PU\VERGE\ZDVLZNUyWNLPF]DVLH

t

3

3

2 2

=

,

τ

]HVWDáF]DVRZ

(

)

(

)

τ

3

1

1

=

+

+









 =

+

+

+

C

C

C

R

g

C

C

C

R

R g

gd

db

L

D

m

gd

db

L

D

D

m

&DáNRZLW\F]DVZáF]DQLDWUDQ]\VWRUDZ\QRVL

t

t

t

t

t

ON

d

r

= + + +

2

3

3URFHVZ\áF]DQLDWUDQ]\VWRUDUR]SRF]\QDVL]FKZLOSRMDZLHQLD

VL XMHPQHJR VNRNX QDSLFLD (UyGáD VWHUXMFHJR RG

+

E

F

GR

(

5

1DSLFLH QD EUDPFH ]PLHQLD VL ]H VWDá F]DVRZ

τ

1

RNUHORQ

]DOH*QRFL 3UG GUHQX ]PDOHMH GR ]HUD JG\

u

GS

RVLJQLH

ZDUWRüSURJRZ

U

T

SRF]DVLH

t

E

E

E

U

f

F

R

R

T

=

+
+

τ

1

ln

1DSLFLH

u

GS

XVWDODVLQDSR]LRPLH

E

R

SRF]DVLH

τ

1

SU]\ F]\P

SU]H]NUyWNLF]DV t

f

QDSLFLH

u

U

DS

DF

=

3RGREQLHMDNWRPLDáRPLHMVFH

SU]\ QDUDVWDQLX QDSLFLD QD EUDPFH WDN L W\P UD]HP SR RGFLFLX

WUDQ]\VWRUDQDGUHQLHSRMDZLDVLQDSLFLH

∆

U

D

RZDUWRFLRNUHORQHM

UyZQSRG*DMFH]DXMHPQ]PLHQQQDSLFLD

u

GS

:VWDQLHRGFLFLDWUDQ]\VWRUMHVWHOHPHQWHPSDV\ZQ\PLUR]SRF]\QDVL

áDGRZDQLH SRMHPQRFL C

C

C

gd

db

L

+

+

]H VWDá F]DVRZ

τ

2

1DSLFLH

GUHQXXVWDODVLSRF]DVLH

t

pass

=

2 2

2

,

τ

&DáNRZLW\F]DVZ\áF]DQLDMHVWUyZQ\

t

t

OFF

pass

=

+

2 2

1

,

τ

35=<.à$'<8.à$'Ï:35=(à&=1,.2:<&+

,QZHUWHU]WUDQ]\VWRUHPELSRODUQ\P

,QZHUWHU\ F]\OL XNáDG\ SU]HáF]QLNRZH Z NWyU\FK NOXF]

WUDQ]\VWRURZ\ ]DSHZQLD LQZHUVM V\JQDáX ZHMFLRZHJR V

QDMSURVWV]\PLXNáDGDPLORJLF]Q\PLVWDQRZLF\PLUyZQLH*SRGVWDZGOD

UHDOL]DFMLEDUG]LHM]áR*RQ\FKIXQNWRUyZORJLF]Q\FK

1D U\VD SU]HGVWDZLRQR VFKHPDW LGHRZ\ LQZHUWHUD

ELSRODUQHJR QDV\FRQHJR Z MHGQHM ] QDMVWDUV]\FK WHFKQLN UHDOL]DFML

XNáDGyZ ORJLF]Q\FK W]Z WHFKQLFH UH]\VWRURZRWUDQ]\VWRURZHM 57/

DQJUHVLVWRUWUDQVLVWRUORJLF

b)

c)

a)

5

&

X

,

5

%

L

%

L

&

X

2

+

8

&&

7

5

/

−

8

%%

8

,+

8

,/

0

X

L

X

R

X

L

X

R

8

&&

8

2+

8

2/

8

2/ PD[

8

2+ PLQ

8

,/

8

,/ PD[

8

,+ PLQ

8

,+

8

8

2/

&(6

=

5\V,QZHUWHU]WUDQ]\VWRUHPQDV\FRQ\PZWHFKQLFH57/DVFKHPDWXNáDGX

EV\PEROFFKDUDNWHU\VW\NDSU]HMFLRZD

'OD ]DSHZQLHQLD VNXWHF]QHJR RGFLFLD WUDQ]\VWRUD SU]\ QLVNLP

SR]LRPLH QDSLFLD ZHMFLRZHJR

U

IL

MDN UyZQLH* GOD VNUyFHQLD F]DVX

Z\áF]DQLDVWRVXMHVLF]DVDPLXMHPQHQDSLFLH

U

BB

ZREZRG]LHED]\

FKRFLD*QDMF]FLHMSU]\MPXMHVL

U

BB

=

0

: VWDQLH JG\ QD ZHMFLX MHVW QLVNL SR]LRP QDSLFLD QD ED]LH

WUDQ]\VWRUDZ\VWSXMHXMHPQHQDSLFLH

U

BEZ

RZDUWRFL

−

=

+

−

+

U

U

R

R

R

U

R

R

R

BEZ

IL

BB

2

1

2

1

1

2

:WHG\WUDQ]\VWRU]QDMGXMHVLZVWDQLHRGFLFLDDQDSLFLHQDZ\MFLX

LQZHUWHUD Z VWDQLH Z\VRNLP Z\QRVL SU]\ SRPLQL FLX ]HURZHJR SUGX

WUDQ]\VWRUD

U

U

R I

OH

CC

C

OH

=

−

JG]LH

I

U

R

OH

OH

L

=

MHVWSUGHPZ\Sá\ZDMF\P]Z\MFLDGRUH]\VWDQFML

REFL*HQLD

$E\ QDSLFLH

U

OH

QLH REQL*\áR VL Z W\P VWDQLH SRQL*HM SHZQHM

GRSXV]F]DOQHMZDUWRFL

U

OH min

U\VFQDOH*\VSHáQLüZDUXQHN

U

U

R

R

R

OH

CC

L

C

L

min

min

min

≥

+

3U]\ Z\VRNLP SR]LRPLH QDSLFLD Z\MFLRZHJR

U

IH

WUDQ]\VWRU

]QDMGXMH VL Z VWDQLH QDV\FHQLD ]DSHZQLDMF QLVNL SR]LRP QDSLFLD

Z\MFLRZHJR

U

U

OL

CES

=

3DUDPHWU\ XNáDGX PXV] E\ü WDN GREUDQH DE\ VSHáQLü ZDUXQHN

QDV\FHQLD

β

0

I

I

BF

C

>

max

: RJyOQ\P SU]\SDGNX JG\ GR Z\MFLD LQZHUWHUD Z VWDQLH

QDV\FHQLD WUDQ]\VWRUD ZSá\ZD SUG

I

OL max

PDNV\PDOQD ZDUWRü SUGX

NROHNWRUDMHVWUyZQD

I

U

U

R

I

C

CC

CES

C

OL

max

max

=

−

+

QDWRPLDVWSUGED]\

I

BF

ZW\PVWDQLHZ\QRVL

I

U

U

R

U

U

R

BF

IH

BEP

BB

BEP

=

−

−

+

1

2

3RGVWDZLDMF ]DOH*QRFL GR QLHUyZQRFL

RWU]\PXMHP\ ZDUXQHN QDV\FHQLD WUDQ]\VWRUD ] NWyUHJR PR *HP\

Z\]QDF]\üSDUDPHWU\XNáDGX

1D U\V SU]HGVWDZLRQR VFKHPDW LGHRZ\ QDMSURVWV]HJR

LQZHUWHUD ] GLRG 6FKRWWN\¶HJR ]DSRELHJDMF JáERNLHPX QDV\FHQLX

WUDQ]\VWRUD

L

%

'6

&

(

%

5

&

L

&

+

8

&&

&

(

%

L

L

,1

%

−

5

%

L

,1

X

2

X

,

5\V ,QZHUWHU ] GLRG

6FKRWN\¶HJR

-DN MX* Z\MDQLRQR Z UR]G]LDOH QLHOLQLRZH VSU]*HQLH

NROHNWRURZH ]D SRPRF QLVNRSURJRZHM GLRG\ 6FKRWWN\¶HJR ]DSHZQLD

RGSURZDG]HQLHQDGPLDUXSUGXZHMFLRZHJR]REZRGXED]\GRREZRGX

NROHNWRUD:WHQVSRVyEQDVWSXMHRJUDQLF]HQLHáDGXQNXJURPDG]RQHJR

Z ED]LH WUDQ]\VWRUD FR Z NRQVHNZHQFML SURZDG]L GR ]QDF]QHJR

VNUyFHQLD F]DVX SU]HFLJDQLD SUGX NROHNWRUD Z SURFHVLH Z\áF]DQLD

WUDQ]\VWRUD:XNáDGDFKVFDORQ\FKGLRG\6FKRWWN\¶HJRZ\NRQ\ZDQHV

SRSU]H]F]FLRZHQDáR*HQLHPHWDOL]DFMLED]\QDREV]DUNROHNWRUD

,QZHUWHU\]WUDQ]\VWRUDPL026RMHGQDNRZ\PW\SLHNDQDáX

-DN MX* ZVSRPQLDQR Z UR]G]LDOH Z VFDORQ\FK XNáDGDFK

SU]HáF]QLNRZ\FK ] WUDQ]\VWRUDPL 026 ]DPLDVW UH]\VWDQFML

R

D

U\V VWRVXMH VL REFL*HQLH G\QDPLF]QH Z SRVWDFL LQQHJR

WUDQ]\VWRUD

1DU\VZ\V]F]HJyOQLRQRWU]\URG]DMHREFL*HG\QDPLF]Q\FK

SU]\ ]DVWRVRZDQLX WUDQ]\VWRUyZ R W\P VDP\P W\SLH NDQDáX FR

WUDQ]\VWRU SHáQLF\ URO NOXF]D 'OD VNXSLHQLD XZDJL ]DMPLHP\ VL

LQZHUWHUDPL Z WHFKQRORJLL Q026 NWyUH ]DSHZQLDM ZLNV] V]\ENRü

G]LDáDQLD L ]DMPXM PQLHMV] SRZLHU]FKQL Z XNáDG]LH VFDORQ\P QL*

LQZHUWHU\ZWHFKQRORJLLS026

&KDUDNWHU\VW\NL SUGRZR QDSLFLRZH SU]HGVWDZLRQ\FK QD

U\VREFL*HG\QDPLF]Q\FKMDNR QLHOLQLRZ\FK GZyMQLNyZ PR*QD

Z\]QDF]\üJUDILF]QLHZRSDUFLXRFKDUDNWHU\VW\NLGUHQRZHWUDQ]\VWRUyZ

REFL*DMF\FK

0

/

U\V>@

a)

X

L

X

R

L

'/

+

8

''

b)

L

'/

+

8

''

+

8

**

X

'6/

X

'6/

c)

L

'/

+

8

''

X

'6/

d)

+

8

''

X

'6/

L

'/

L

'

X

2

M

&

/

/

X

,

M

L

M

L

M

L

5\V ,QZHUWHU Q026 ] REFL*HQLHP G\QDPLF]Q\P Z SRVWDFL D WUDQ]\VWRUD

]NDQDáHPZ]ERJDFDQ\PZSRáF]HQLXGLRGRZ\PEWUDQ]\VWRUD]NDQDáHP

Z]ERJDFDQ\P SUDFXMF\P Z REV]DU]H OLQLRZ\P SU]\

8

8

8

**

7

''

−

>

FWUDQ]\VWRUD]NDQDáHP]XER*DQ\PSUDFXMF\PMDNR(UyGáRSUGRZHSU]\

8

*6

=

GV\PEROLQZHUWHUD

2EV]DU QLHQDV\FHQLD

8

8

8

**

7/

''

−

>

2EV]DU QDV\FHQLD

8

8

8

**

7/

''

−

<

8

8

8

**

7/

''

−

=

&KDUDNWHU\VW\NL

REFL*HQLD

E1

E2

,

'

8

'6

8

7/

8

7/

F

D

&KDUDNWHU\VW\ND

REFL*HQLD SU]\ 8

8

**

''

=

8

*6

&KDUDNWHU\VW\ND

REFL*HQLD

F

(

)

8

*6

=

0

5\V

&KDUDNWHU\VW\NL

GUHQRZH WUDQ]\VWRUyZ REFL

*DMF\FK L FKDUDNWHU\VW\NL

SUGRZR QDSLFLRZH REFL

*H MDNR QLHOLQLRZ\FK GZyM

QLNyZ

&KDUDNWHU\VW\ND D REFL*HQLD G\QDPLF]QHJR Z SRVWDFL

WUDQ]\VWRUD

M

L

ZSRáF]HQLXGLRGRZ\P

U

U

GG

DD

=

Z\]QDF]RQDMHVW

SU]H]SXQNW\ZNWyU\FK

U

U

GS

DS

=

&KDUDNWHU\VW\NL

( )

b

1

L

( )

b

2

GRW\F]

REFL*H G\QDPLF]Q\FK Z SRVWDFL WUDQ]\VWRUyZ Z]ERJDFDQ\FK

SUDFXMF\FK Z REV]DU]H OLQLRZ\P G]LNL ]DVWRVRZDQLX GRGDWNRZHJR

QDSLFLD

U

GG

R ZDUWRFL

U

U

U

GG

TL

DD

−

>

U

TL

MHVW QDSLFLHP

SURJRZ\PWUDQ]\VWRUDREFL*DMFHJR

&KDUDNWHU\VW\NL

( )

b

1

L

( )

b

2

RGQRV] VL GR GZyFK Uy*Q\FK QDSLü

(

)

U

U

U

GG

GG

GG

1

2

<

L Z\]QDF]RQH V SU]H] SXQNW\ Z NWyU\FK

U

U

k U

GS

TL

DS

−

=

JG]LH

k

>

1

/LQLD F MHVW FKDUDNWHU\VW\N GUHQRZ WUDQ]\VWRUD ] NDQDáHP

]XER*DQ\PZ\]QDF]RQSU]\

U

GS

=

0

&KDUDNWHU\VW\NL SU]HMFLRZH LQZHUWHUD PR*QD Z\]QDF]\ü

JUDILF]QLH ZU\VRZXMF Z SROH FKDUDNWHU\VW\N GUHQRZ\FK WUDQ]\VWRUD

ZHMFLRZHJR 0 XSU]HGQLR Z\]QDF]RQH OLQLH REFL*H Z SRGREQ\

VSRVyE MDN Z\]QDF]RQR SURVW SUDF\ GOD OLQLRZHM UH]\VWDQFML

R

D

U\V

L

'

X

'6

( )

8 E

2/

1

X

,

( )

8 D

2/

( )

8 F

2/

( )

( )

8

F

8

E

2+

2+

1

8

''

8

8

''

7/

−

( )

8

D

2+

$

1

$

2

$

3

( )

E2

( )

E1

( )

D

( )

F

8

,+

5\V &KDUDNWHU\VW\NL GUH

QRZH WUDQ]\VWRUD ZHMFLRZHJR

LNU]\ZHREFL*H]U\V

5R]ZD*P\ SU]\NáDGRZR SURFHV SU]HáF]DQLD LQZHUWHUD ]

REFL*HQLHP G\QDPLF]Q\P R FKDUDNWHU\VW\FH

( )

b

1

3U]\MPXMF *H

SRF]WNRZRQDSLFLHZHMFLRZHMHVWUyZQH

U

U

IL

T

<

U

T

MHVWQDSLFLHP

SURJRZ\PWUDQ]\VWRUDZHMFLRZHJR0QDSLFLHQDZ\MFLXMHVWUyZQH

U

U

OH

DD

=

3U]\ VNRNRZHM ]PLDQLH QDSLFLD ZHMFLRZHJR GR ZDUWRFL

U

IH

SXQNWSUDF\SU]HPLHV]F]DVLVNRNRZR]SXQNWX

A

1

GRSXQNWX

A

2

SR F]\P QDVWSXMH SURFHV UR]áDGRZDQLD SRMHPQRFL

C

L

RG QDSLFLD

U

OH

GR

U

OL

NRF]F\ VL Z SXQNFLH

A

3

3U]\ QDVWSQHM VNRNRZHM

]PLDQLHQDSLFLDZHMFLRZHJR]

U

IH

GR

U

IL

SXQNWSUDF\SU]HPLHV]F]D

VL]

A

3

GR

A

4

SU]\F]\PSRMHPQRü

C

L

áDGXMHVLRGQDSLFLD

U

OL

GR

QDSLFLD

U

OH

L

'

X

'6

( )

E2

( )

E1

( )

D

( )

F

8

,+

8

7

8

2/

8

8

''

7/

−

8

''

5\V

&KDUDNWHU\VW\NL

SU]HMFLRZH

LQZHUWHUD

GOD

Uy*Q\FKREFL*H G\QDPLF]Q\FK

1DU\VSU]HGVWDZLRQRFKDUDNWHU\VW\NLSU]HM FLRZHLQZHUWHUD

Z\]QDF]RQHGODZV]\VWNLFKUR]ZD*DQ\FKREFL*HG\QDPLF]Q\FK

= SRUyZQDQLD FKDUDNWHU\VW\N SU]HMFLRZ\FK QD U\V Z\QLND

*HREFL*HQLHDGDMHQDMPQLHMV]HQDSLFLH

U

OH

UyZQH

U

U

DD

TL

−

FR

MHVWZDGWHMNRQILJXUDFML

'OD SR]RVWDá\FK REFL*H QDSLFLH

U

U

OH

DD

=

1DSLFLH Z\MFLRZH Z

VWDQLH QLVNLP

U

OL

]DOH*\ Z GHF\GXMF\P VWRSQLX RG VWRVXQNX

UH]\VWDQFML REFL*HQLD GR UH]\VWDQFML SU]HZRG]FHJR WUDQ]\VWRUD

ZHMFLRZHJR : FHOX X]\VNDQLD PDáHM ZDUWRFL

U

OL

QDOH*\ ]DSHZQLü

RGSRZLHGQLH UHODFMH SRPLG]\ Z\PLDUDPL WUDQ]\VWRUyZ

0

/

L

(

)

(

)

0 : /

: /

0

0

/

<<

$QDOL]D ZáDFLZRFL G\QDPLF]Q\FK LQZHUWHUD ] Uy*Q\PL

URG]DMDPLREFL*HZ\ND]XMH*HF]DVZáF]DQLDLQZHUWHUDJG\QDSLFLH

Z\MFLRZH]PLHQLDVL]

U

OH

GR

U

OL

ZQLHZLHONLPVWRSQLX]DOH*\RG

REFL*HQLD 1DWRPLDVW F]DV Z\áF]DQLD LQZHUWHUD ]DOH*\ EH]SRUHGQLR

RG GRVWDUF]RQHJR SU]H] REFL*HQLH SUGX áDGRZDQLD SRMHPQRFL

C

L

1DMNUyWV]\ F]DV Z\áF]DQLD ]DSHZQLD REFL*HQLH F NWyUH UyZQLH*

]DSHZQLDNRU]\VWQHSDUDPHWU\FKDUDNWHU\VW\NLSU]HM FLRZHM

U

U

OH

DD

=

PDáD ZDUWRü

U

OL

1DOH*\ MHGQDN SRGNUHOLü *H UHDOL]DFMD

WHFKQRORJLF]QD LQZHUWHUD ] REFL*HQLHP F MHVW QDMWUXGQLHMV]D

SRQLHZD* ZL*H VL ] NRQLHF]QRFL VHOHNW\ZQHJR GRPLHV]NRZDQLD

NDQDáyZ WUDQ]\VWRUyZ REFL*DMF\FK SU]\ X*\FLX WHFKQLNL LPSODQWDFML

MRQyZ

,QZHUWHU&026

,QZHUWHU &026 DQJ FRPSOHPHQWDU\ 026 MHVW XNáDGHP

SU]HáF]QLNRZ\P ] REFL*HQLHP G\QDPLF]Q\P Z SRVWDFL

NRPSOHPHQWDUQHJR WUDQ]\VWRUD ] NDQDáHP W\SX S R ]EOL *RQ\FK

FKDUDNWHU\VW\NDFK VWDW\F]Q\FK SU]\ F]\P EUDPNL REX WUDQ]\VWRUyZ V

SRáF]RQHU\V

']LNL SU]HFLZQ\P W\SRP SU]HZRGQLFWZD WUDQ]\VWRU\ LQZHUWHUD

Z VWDQDFK VWDW\F]Q\FK SU]HZRG] QD SU]HPLDQ ]DOH*QLH RG QDSLFLD

VWHUXMFHJR*G\ QD ZHMFLX MHVW QLVNL SR]LRP QDSLFLD

U

IL

WUDQ]\VWRU

M

n

MHVW RGFLW\ D SU]HZRG]L WUDQ]\VWRU M

p

QDSLFLH Z\MFLRZH MHVW

ZWHG\ UyZQH

U

U

OH

DD

=

1DWRPLDVW SU]\ Z\VRNLP SR]LRPLH QDSLFLD

Z\MFLRZHJR

U

IH

SU]HZRG]L WUDQ]\VWRU

M

n

D WUDQ]\VWRU M

p

MHVW

RGFLW\QDSLFLHZ\MFLRZHMHVWZWHG\UyZQH

U

OL

=

0

+

8

''

L

'S

L

'Q

X

2

X

,

M

n

M

p

5\V,QZHUWHU&026

:SHZQ\PSU]HG]LDOHQDSLüZHMFLRZ\FKRSRUHGQLHMZDUWRFLPRJ

SU]HZRG]Lü RED WUDQ]\VWRU\ FR QLH MHVW NRU]\VWQH ]H Z]JO GX QD

]ZLNV]RQ\ Z W\P F]DVLH SREyU PRF\ ] ]DVLODF]D L VWUDW\ PRF\ Z

WUDQ]\VWRUDFK 6WDW\F]Q FKDUDNWHU\VW\N SU]HMFLRZ LQZHUWHUD PR*QD

Z\]QDF]\ü JUDILF]QLH MH*HOL Z SROX FKDUDNWHU\VW\N GUHQRZ\FK

WUDQ]\VWRUD Q026 ZU\VXMHP\ FKDUDNWHU\VW\NL GUHQRZH WUDQ]\VWRUD

S026 U\V L Z\]QDF]\P\ SXQNW\ SU]HFL FLD VL W\FK

FKDUDNWHU\VW\NGODZVSyOQ\FKQDSLü

X

,

XZ]JOGQLDMF

u

u

u

u

u

u

u

U

i

i

In

Ip

I

GSn

I

GSp

I

DD

Dn

Dp

=

=

=
= −

= −















L

'Q

X

'6Q

X

,Q

$

1

X

'6S

8

''

0

L

'S

3

3

3

3

X

,S

5\V *UDILF]QH Z\]QDF]H

QLH VWDW\F]Q\FK FKDUDNWHU\VW\N

SU]HMFLRZ\FK LQZHUWHUD &026

1DU\VSU]HGVWDZLRQRSU]\NáDGRZVWDW\F]QFKDUDNWHU\VW\N

SU]HMFLRZ LQZHUWHUD

( )

u

f u

O

I

=

RUD] VWDW\F]Q FKDUDNWHU\VW\N

RNUHODMF ZLHONRü SUGX VNURQHJR Sá\QFHJR SU]H] WUDQ]\VWRU\ ]

]DVLODF]DZIXQNFMLQDSLFLDZHMFLRZHJR

( )

i

f u

DD

I

=

6WDW\F]Q FKDUDNWHU\VW\N SU]HMFLRZ PR*QD SRG]LHOLü QD SLü

]DNUHVyZ VWRVRZQLH GR Uy*Q\FK WU\EyZ SUDF\ WUDQ]\VWRUyZ NWyUH

NROHMQRSU]HDQDOL]XMHP\

X

2

8

''

0

,

,, ,,, ,9

9

8

''

X

,

1

2

8

''

1

2

8

''

8

7S

8

7Q

X

2

L

''

5\V 6WDW\F]QD FKDUDNWHU\VW\ND

SU]HMFLRZDLQZHUWHUD&026

=DNUHV,

: W\P ]DNUHVLH QDSLFLH ZHMFLRZH

0

≤ ≤

u

U

I

Tn

GODWHJR

WUDQ]\VWRU

M

n

MHVW Z VWDQLH RGFLFLD D WUDQ]\VWRU M

p

]QDMGXMH VL Z

REV]DU]H OLQLRZ\P SU]\ ]HURZ\P SUG]LH GUHQX i

i

Dp

Dn

= −

1DSLFLH

u

DSp

=

0

]DWHP

u

U

U

OH

DD

0

=

=

=DNUHV,,

: W\P ]DNUHVLH QDSLFLH ZHMFLRZH ]DZLHUD VL Z SU]HG]LDOH

U

u

U

Tn

I

DD

<

<

2

WUDQ]\VWRU

M

n

SUDFXMH Z REV]DU]H QDV\FHQLD ]D

WUDQ]\VWRU M

p

Z REV]DU]H OLQLRZ\P 3RQLHZD* Z XNáDG]LH ]DFKRG]

]ZL]NL

u

u

u

u

u

u

U

u

u

U

GSn

I

DSn

O

GSp

I

DD

DSp

O

DD

=

=

= −

=

−







,

,

GODWHJRSUG\WUDQ]\VWRUyZRSLVXM]DOH*QRFL

(

)

i

u

U

Dn

n

I

Tn

=

−

β

2

2

SU]\F]\P

β

n

n

n

n

K

W

L

=











'

D

(

)

(

) (

)

i

u

U

U

u

U

u

U

Dp

p

I

DD

Tp

O

DD

O

DD

= −

−

−

−

−

−







β

1

2

2

SU]\F]\P

β

p

p

p

p

K

W

L

=











'

D

3U]\UyZQXMF

i

i

Dp

Dn

= −

QDSRGVWDZLH]DOH*QRFL

÷

RWU]\PXMHP\

(

)

(

)

(

)

u

u

U

u

U

u

U

U

U

u

U

O

I

Tp

I

Tp

I

DD

Tp

DD

n

p

I

Tn

=

−

+

+

−

−

−

−









−

−

2

2

2

2

β

β

=DNUHV,,,

: ]DNUHVLH ,,, RED WUDQ]\VWRU\ SUDFXM Z REV]DU]H QDV\FHQLD

]DWHP SUG\ GUHQyZ WUDQ]\VWRUyZ SU]\ Z\NRU]\VWDQLX ]DOH *QRFL

PR*HP\SU]HGVWDZLüZSRVWDFL

(

)

i

u

U

U

Dp

p

I

DD

Tp

= −

−

−

1

2

2

β

(

)

i

u

U

Dn

n

I

Tn

=

−

1

2

2

β

3RGVWDZLDMFLGRRWU]\PXMHP\

u

U

U

U

I

DD

Tp

Tn

n

p

n

p

=

+

+

+

β β

β β

1

=DNáDGDMF

β

β

n

p

Tn

Tp

U

U

=

= −

,

]DOH*QRü XSUDV]F]D VL GR

SRVWDFL

u

U

I

DD

=

2

0R*OLZH ZDUWRFL QDSLFLD Z\MFLRZHJR

u

O

Z W\P ]DNUHVLH

Z\QLNDM ] ZDUXQNyZ DE\ RED WUDQ]\VWRU\ SUDFRZDá\ Z REV]DU]H

QDV\FHQLD

(

)

U

U

U

DS

GS

T

>

−

]DWHPGODWUDQ]\VWRUD

M

n

u

u

U

O

I

Tn

≥ −

GODWUDQ]\VWRUD M

p

u

U

U

u

U

I

DD

Tp

O

DD

−

−

≤

−

1DSRGVWDZLHQLHUyZQRFLLRWU]\PXMHP\

u

U

u

u

U

I

Tn

O

I

Tp

−

≤

≤ −

: ]DNUHVLH ,,, FKDUDNWHU\VW\ND SU]HMFLRZD MHVW EDUG]R VWURPD

SRQLHZD*WUDQ]\VWRU\SUDFXMFHZREV]DU]HQDV\FHQLDWZRU]QLHVWDELOQ\

XNáDGV]HUHJRZHMSUDF\GZyFK(UyGHáSUGRZ\FK
=DNUHV,9

=DNUHV ,9 Z\VWSXMH GOD QDSLFLD ZHMFLRZHJR ] SU]HG]LDáX

U

u

U

U

DD

I

DD

Tp

2

< ≤

+

WUDQ]\VWRU M

p

SUDFXMH Z REV]DU]H QDV\FHQLD

]D WUDQ]\VWRU

M

n

Z REV]DU]H OLQLRZ\P -HVW WR ]DWHP RGZUyFRQD

V\WXDFMD Z SRUyZQDQLX GR ]DNUHVX ,, 3UG\ WUDQ]\VWRUyZ RSLVXM

UyZQDQLD

(

)

i

u

U

U

Dp

p

I

DD

Tp

= −

−

−

1

2

2

β

(

)

i

u

U

u

u

Dn

n

I

Tn

O

O

=

−

−







β

1

2

2

'ODi

i

Dp

Dn

= −

RWU]\PXMHP\

(

) (

)

(

)

u

u

U

u

U

u

U

U

O

I

Tn

I

Tn

p

n

I

DD

Tp

=

−

−

−

−

−

−

2

2

β

β

=DNUHV9

=DNUHV 9 Z\]QDF]D QDSLFLH ZHMFLRZH u U

U

I

DD

Tp

≥

+

L

Z\VWSXMH Z QLP RGZURWQD V\WXDFMD GR ]DNUHVX , 7UDQ]\VWRU M

p

]QDMGXMH VL Z REV]DU]H RGFLFLD ]D WUDQ]\VWRU

M

n

Z REV]DU]H

OLQLRZ\PSU]\]HURZ\PSUG]LHGUHQXGODWHJR

U

DSn

=

0

F]\OL

u

O

=

0

3HZQ ZDG LQZHUWHUD &026 MHVW NRQLHF]QRü MHGQRF]HVQHJR

VWHURZDQLDQDZHMFLXGZyFKWUDQ]\VWRUyZNWyU\FKEUDPNLV]áF]RQH

UD]HP 3RZRGXMH WR GZXNURWQH ]ZLNV]HQLH SRMHPQRFL ZHMFLRZHM

VWDQRZLFHMREFL*HQLHGODSRSU]HGQLHJRVWRSQLD

6WDQ\SU]HMFLRZHSU]\SU]HáF]DQLXLQZHUWHUD&026UR]ZD*\P\

SU]\ SRPLQLFLX SRMHPQRFL PLG]\HOHNWURGRZ\FK L XZ]JOGQLHQLX

W\ONRSRMHG\QF]HMVNXSLRQHMSRMHPQRFL

C

L

3U]\ VNRNRZHM ]PLDQLH QDSLFLD ZHMFLRZHJR

( )

u t

I

RG GR

U

DD

WUDQ]\VWRU M

p

SU]HFKRG]L Z VWDQ RGFLFLD D WUDQ]\VWRU

M

n

]DF]\QD

SU]HZRG]LüSUGUR]áDGRZDQLDSRMHPQRFL

C

L

3RQLHZD*WUDQ]\VWRU

M

n

SUDFXMH SRF]WNRZR Z REV]DU]H QDV\FHQLD GODWHJR NRQGHQVDWRU

C

L

UR]áDGRZ\ZDQ\MHVWVWDá\PSUGHPDQDSLFLHZ\MFLRZH

( )

u t

O

PDOHMH

OLQLRZRU\V

( )

(

)

u t

U

C

U

U

t

O

DD

n

L

DD

Tn

=

−

−

1

2

2

β

X

2

0

8

''

1

2

8

''

8

8

''

7Q

−

τ

W

S1

W

S

W

5\V3U]HELHJF]DVRZ\QDSLFLD

Z\MFLRZHJR LQZHUWHUD &026 SU]\

ZáF]DQLX

3URFHVWHQWUZDGRFKZLOLJG\WUDQ]\VWRU

M

n

SUDFXMHZREV]DU]H

QDV\FHQLDW]QJG\QDSLFLHZ\MFLRZHRVLJQLHZDUWRü

( )

u

u t

U

U

DSn

O

p

DD

Tn

=

=

−

1

3RGVWDZLDMFZDUXQHNGRRWU]\PXMHP\

(

)

t

C U

U

U

p

L

Tn

n

DD

Tn

1

2

2

=

−

β

'DOV]H UR]áDGRZDQLH NRQGHQVDWRUD

C

L

QDVWSXMH SUGHP GUHQX

i

Dn

WUDQ]\VWRUD

M

n

SUDFXMFHJRZREV]DU]HOLQLRZ\P3URFHVWHQPR*QD

Z SU]\EOL*HQLX ]DPRGHORZDü SU]H] ]DVWSLHQLH WUDQ]\VWRUD

M

n

SRVLDGDMFHJRQDSLFLHEUDPND(UyGáR

U

U

GS

DD

=

UH]\VWDQFMOLQLRZ

(

)

r

u

i

U

U

DSn

DSn

Dn

n

DD

Tn

=

≈

−

1

β

5R]áDGRZDQLH SRMHPQRFL

C

L

PR*QD RSLVDü Z]RUHP MDN GOD

REZRGX5&

( )

(

)

( )

(

)

[

]

u t

U

U

t

t

r

C

O

DD

Tn

p

DSn

L

=

−

− −

exp

&]DV SURSDJDFML t

pHL

RNUHORQ\ UyZQLH* F]DVHP RSy(QLHQLD

GHILQLXMH VL MDNR RGVWS F]DVRZ\ PLG]\ ]ERF]HP LPSXOVX

ZHMFLRZHJR L Z\ZRáDQ\P SU]H]H ]ERF]HP LPSXOVX Z\MFLRZHJR

SU]\ SR]LRPLH QDSLFLD QD W\FK ]ERF]DFK =DWHP ] ZDUXQNX

( )

u t

U

O

pHL

DD

=

2

L]]DOH*QRFL±PR*HP\Z\]QDF]\ü

t

t

r

C

U

U

U

U

U

U

U

pHL

p

DSn

L

Tn

DD

Tn

DD

Tn

Tn

DD

=

+

−



















 =

=

−

+

−



































1

2 1

2

2 1

ln

ln

τ

JG]LH

(

)

τ

β

=

−

C

U

U

L

n

DD

Tn

D

:SRGREQ\ VSRVyE PR*QD Z\]QDF]\ü F]DV t

pLH

JG\ QDVWSXMH

áDGRZDQLH SRMHPQRFL

C

L

SU]H] SU]HZRG]F\ WUDQ]\VWRU M

p

SU]\

RGFLW\PWUDQ]\VWRU]H

M

n

./8&=($1$/2*2:(

'RUHDOL]DFMLNOXF]\DQDORJRZ\FKVWRVXMHVLWUDQ]\VWRU\SRORZH

GLRG\LWUDQ]\VWRU\ELSRODUQH1DU\VSU]HGVWDZLRQRLGHDOQ\NOXF]

DQDORJRZ\ RUD] MHJR FKDUDNWHU\VW\N SU]HMFLRZ : VWDQLH ZáF]HQLD

NOXF] SU]HQRVL QDSLFLH ZHMFLRZH QD ]DFLVNL Z\MFLRZH EH] VWUDW L

]QLHNV]WDáFH QDWRPLDVW Z VWDQLH Z\áF]HQLD QDSLFLH Z\MFLRZH QLH

]DOH*\RGZHMFLRZHJR

X

2

X

,

b)

X

2

a)

~

X

,

5

J

H

*

5

/

.

6WDQ

Z\áF]HQLD

6WDQ

ZáF]HQLD

5\V .OXF] DQDORJRZ\ D VFKHPDW EORNRZ\ E FKDUDNWHU\VW\NL SU]HMFLRZH

ZVWDQDFKZáF]HQLDLZ\áF]HQLD

&KDUDNWHU\VW\NLSU]HMFLRZHU]HF]\ZLVW\FKXNáDGyZRGELHJDMRG

LGHDOQ\FKSU]\F]\PJáyZQHUy*QLFHSROHJDMQDW\P*H]DNUHVQDSLü

ZHMFLRZ\FKU]HF]\ZLVW\FKNOXF]\DQDORJRZ\FKMHVWRJUDQLF]RQ\ RUD]

QDFK\OHQLHFKDUDNWHU\VW\NLZVWDQLHZáF]HQLDMHVWPQLHMV]HRGMHGQRFL

D Z VWDQLH Z\áF]HQLD QDSLFLH QD Z\MFLX NOXF]D MHVW Uy*QH RG ]HUD

:áDFLZRFL U]HF]\ZLVWHJR NOXF]D DQDORJRZHJR GHILQLRZDQH V ]D

SRPRFQDVWSXMF\FKQDMZD*QLHMV]\FKSDUDPHWUyZ

UH]\VWDQFMNOXF]DZVWDQLHZáF]HQLD

UH]\VWDQFMNOXF]DZVWDQLHZ\áF]HQLD

]DNUHVQDSLüZHMFLRZ\FK

F]DV\SU]HáF]H

SU]HQLNDQLHV\JQDáXVWHUXMFHJRQDZ\MFLH

ZSURZDG]RQH]QLHNV]WDáFHQLD

V]F]WNRZHQDSLFLHNOXF]D

PDNV\PDOQ\SUGSU]HZRG]HQLD

PDNV\PDOQDPRFVWUDW

1D U\VD SU]HGVWDZLRQR QDMSURVWV]\ NOXF] DQDORJRZ\

]UHDOL]RZDQ\QDWUDQ]\VWRU]H026]NDQDáHPZ]ERJDFDQ\PW\SXQ

+

8

''

X

2

X

,

*1'

M

n

M

p

X

67

5

/

X

2

*1'

M

n

X

67

5

/

b)

a)

X

,

5\V.OXF]HV]HUHJRZH]WUDQ]\VWRUDPL026D]SRMHG\QF]\PWUDQ]\VWRUHP

Q026E&026

7UDQ]\VWRU 026 ] NDQDáHP W\SX Q PR*QD ZSURZDG]Lü Z VWDQ

SU]HZRG]HQLD SU]H] SRGDQLH QDSLFLD VWHUXMFHJR

U

ST

ZLNV]HJR RG

QDMZLNV]HJR GRGDWQLHJR QDSLFLD ZHMFLRZHJR

u

I

3U]\ PDá\FK

QDSLFLDFK GUHQ (UyGáR WUDQ]\VWRU SUDFXMH Z REV]DU]H OLQLRZ\P L

]DFKRZXMH VL Z SU]\EOL*HQLX MDN UH]\VWRU =H Z]URVWHP QDSLFLD

ZHMFLRZHJR PDOHM QDSLFLD EUDPND (UyGáR L EUDPND GUHQ

WUDQ]\VWRUD SRZRGXMF ]ZLNV]DQLH VL UH]\VWDQFML WUDQ]\VWRUD D W\P

VDP\PNOXF]D*G\QDSLFLH

u

I

RVLJQLHZDUWRü

U

U

ST

Tn

−

WUDQ]\VWRU

ZRJyOHSU]HVWDMHSU]HZRG]Lü

$E\ ]ZLNV]\ü ]DNUHV GRSXV]F]DOQ\FK QDSLü ZHMFLRZ\FK

]DPLDVW SRMHG\QF]HJR WUDQ]\VWRUD 026 QDOH*\ ]DVWRVRZDü NOXF]

&026 VNáDGDMF\ VL ] GZX NRPSOHPHQWDUQ\FK WUDQ]\VWRUyZ 026

SRáF]RQ\FKUyZQROHJOHMDNQDU\VE=DáF]HQLHNOXF]DZ\VWSXMH

SU]\ Z\VRNLP SR]LRPLH V\JQDáX VWHUXMFHJR

U

STH

EH]SRUHGQLR

SRGDZDQHJRQDEUDPNWUDQ]\VWRUDQ026LZSRVWDFL]DQHJRZDQHMQD

EUDPN WUDQ]\VWRUD S026 1DSLFLH ZHMFLRZH PR*H SU]\MPRZDü

GRZROQZDUWRü]SU]HG]LDáX

0

≤ ≤

u

U

I

DD

=H Z]URVWHP QDSLFLD ZHMFLRZHJR ]PQLHMV]D VL QDSLFLH

u

GSn

ZVNXWHNF]HJRUH]\VWDQFMDWUDQ]\VWRUDQ026Z]UDVWD1LHSRZRGXMHWR

MHGQDN LVWRWQHJR SRJRUV]HQLD ZáDFLZRFL NOXF]D SRQLHZD*

UyZQRF]HQLHURQLHPRGXá|

|

u

GSp

ZVNXWHNF]HJRUH]\VWDQFMDWUDQ]\VWRUD

S026PDOHMH

-DN SRND]DQR QD U\V Z\SDGNRZD UH]\VWDQFMD NOXF]D XOHJD

W\ONR QLHZLHONLP ]PLDQRP Z FDá\P GRSXV]F]DOQ\P SU]HG]LDOH

]PLHQQRFLQDSLFLDZHMFLRZHJR

0

8

''

X

,

U

N

U

N

U

'6S

U

'6Q

8

''

2

5\V =DOH*QRü UH]\VWDQFML

Z\SDGNRZHM NOXF]D &026 RG

QDSLFLDZHMFLRZHJR

1D U\V SU]HGVWDZLRQR VFKHPDW LGHRZ\ GLRGRZHJR NOXF]D

DQDORJRZHJR

X

67

,

X

,

,

−

'

1

'

2

'

3

'

4

'

5

'

6

X

2

5

/

&

/

−

8

+

8

5\V

'LRGRZ\

NOXF]

V]HUHJRZ\

-H*HOLQDZHMFLHXNáDGXVWHUXMFHJRSRGDP\GRGDWNRZHQDSLFLH

VWHUXMFHWRGLRG\

D

5

L

D

6

]RVWDQVSRODU\]RZDQH]DSRURZRLZyZF]DV

GLRG\

D D

1

4

,

L

D D

2

3

,

EGSU]HZRG]LüSUG\

I

Z\PXV]RQH]H (UyGHá

SUGRZ\FK :VNXWHN WHJR QDSLFLH ZHMFLRZH SU]HQRVL VL QD Z\MFLH

SU]H] Z\SDGNRZ UH]\VWDQFM SU]HZRG]F\FK GLRG

D

D

1

4

−

PRVWND

Z\QRV]F r

r

r

r

k

D

D

D

=

=

2

2

-H*HOL QDSLFLD QD SU]HZRG]F\FK

GLRGDFK

D

D

D

D

1

2

4

3

−

−

,

QLH V UyZQH WR QD Z\MFLX SRMDZLD VL

QDSLFLH UHV]WNRZH 3U]\ XMHPQ\P QDSLFLX VWHUXMF\P ]DF]\QDM

SU]HZRG]Lü GLRG\

D

5

L

D

6

D XNáDG PRVWNRZ\ ]RVWDMH ]DEORNRZDQ\

:\MFLH ]RVWDMH RGG]LHORQH RG ZHMFLD D URGHN XNáDGX SU]\ SHáQHM

V\PHWULL XNáDGX PD VWDá\ SRWHQFMDá UyZQ\ ]HUX 3U]\ ]DVWRVRZDQLX

V]\ENLFK GLRG 6FKRWWN\¶HJR SRZ\*V]D ]DVDGD XPR*OLZLD RVLJQLFLH

F]DVyZSU]HáF]HSRQL*HM

µ

s

1D U\V SU]HGVWDZLRQR RJyOQ LGH Z\NRU]\VWDQLD

Z]PDFQLDF]DUy*QLFRZHJRMDNRNOXF]DDQDORJRZHJR

T

T

2

1

+

8

&&

5

&

5

&

X

67

,

−

8

((

X

2

X

,

D

,

(

5\V =DVDGD G]LDáDQLD

Z]PDFQLDF]D Uy*QLFRZHJR MDNR

NOXF]D

-H*HOL QDSLFLH VWHUXMFH MHVW XMHPQH WR GLRGD ' MHVW

VSRODU\]RZDQD ]DSRURZR L Z]PDFQLDF] Uy*QLFRZ\ ]DVLODQ\ MHVW RG

VWURQ\HPLWHUyZ]H(UyGáDSUGRZHJR

I

I

E

=

:WHG\Uy*QLFRZHQDSLFLH

Z\MFLRZHZ\QRVL

u

g R u

O

m

C

I

=

F]\OL QD Z\MFLH SU]HQRV]RQH MHVW QDSLFLH ZHMFLRZH ]H

ZVSyáF]\QQLNLHPZ]PRFQLHQLD

g R

m

C

-H*HOLQDSLFLHVWHUXMFHVWDQLH

VL GRGDWQLH WR SUG

I

SRSá\QLH SU]H] GLRG D WUDQ]\VWRU\

T

T

1

2

L

]RVWDQ RGFLWH 3RWHQFMDá\ NROHNWRUyZ Z]URVQ GR ZDUWRFL

U

CC

MHGQDNUy*QLFRZHQDSLFLHZ\MFLRZH

u

O

EG]LHUyZQH]HUX=DWHPWHQ

VWDQSUDF\Z]PDFQLDF]DUy*QLFRZHJRRGSRZLDGDVWDQRZLZ\áF]RQHJR

NOXF]DDQDORJRZHJR