Медведев Г.А. Математические основы финансовой экономики. Часть2

Подождите немного. Документ загружается.



G_ljm^ghihdZaZlvqlh









W





d

W



Bkihevamy y\gu_ \ujZ`_gby \_ebqbg b T b k q_j_a iZjZf_lju

fh^_ebdjZldhkjhqghcklZ\dbihnhjfmeZfihemqZ_fke_^mx

sb_ij_^_evgu_khhlghr_gby

   

@>



@>





o

G

HHHH



 

@>



@>







G

o

DG





 

@>



@>



o





W



Ijbf_g_gb_wlbog_jZ\_gkl\bij_^_evguokhhlghr_gbc\\ujZ

`_gbb^eyeh]ZjbnfZhlghr_gbyINF^hdZau\Z_lg_h[oh^bfucj_

amevlZl

@OQ>OLP X00

 qlhhagZqZ_lkoh^bfhklvmkeh\ghc

iehlghklb\_jhylghkl_cdghjfZevghciehlghklb<aZdexq_gb_

aZf_lbf qlh fZl_fZlbq_kdh_ h`b^Zgb_? b ^bki_jkby 9bf_xlke_

^mxsb_ y\gu_ \ujZ`_gby q_j_a iZjZf_lju fh^_eb djZldhkjhqghc

ijhp_glghcklZ\db

? 





XT

 

> @5H H





9 





 



@>





DG



 

@>



G



LZdbfh[jZahf?g_aZ\bkblhl

Z



@>





H9 

Ml\_j`^_gb_

Ijb





kh\f_klgZyiehlghklv\_jhylgh

kl_c djZldhkjhqghc ijhp_glghc klZ\db  koh^blky d iehlghklb

^\mof_jgh]hghjfZevgh]hjZkij_^_e_gbyk\_dlhjhffZl_fZlbq_kdbo

h`b^Zgbc



bdh\ZjbZpbhgghcfZljbp_c



 





_



]^_

b'hij_^_eyxlkyke_^kl\b_f

>hdZaZl_evkl\hQlh[um[_^blvky\ijZ\bevghklbwlh]hml

\_j`^_gby ^hklZlhqgh ihdZaZlv qlh INF kh\f_klghc iehlghklb

\_jhylghkl_cijb





koh^blkydijhba\h^ys_cnmgdpbb

fhf_glh\^\mof_jgh]hghjfZevgh]hjZkij_^_e_gby

bY _oj

 





YXYX







_oj

bY

'b



bYY 





H[hagZqbf^eym^h[kl\Zo 



Kh]eZkghh[hagZq_gb

yf  b ke_^kl\bx  INF F b Y kh\f_klghc iehlghklb

\_jhylghkl_cfh`_l[ulvaZibkZgZ\\b^_

bY 

[









U

T

U



T

U



U







[

T

U

T





< khhl\_lkl\bb k h]jZgbq_gbyfb  gZ INF wlZ nmgdpby

hij_^_e_gZgZfgh`_kl\_i_j_f_gguom^h\e_l\hjyxsbog_jZ\_g

kl\m





T

U

T



[

YX 



GZf^hklZlhqghihdZaZlvqlh OLP OQ

> @0XY0 XY

 >ey

wlh]h aZf_lbf \gZqZe_ qlh ba ij_^klZ\e_gby



OQ ]





ijb_]_



ke_^m_lqlh^ey



]



bf_xlf_klhg_jZ\_gkl\Z





OQ



] 



]







 N

]]





]



]

]]











OQ] ]











 N

]]





]



]

]]











Hp_gbfl_i_jvOQF bY



OQF bY 



bYo













[

T

 

T

U

T





[

YX  



bYo



[

T

 

T

U

T





[

YX 







[



T

 



T

U

T



[

YX 







[

T



[

XYYX

[

T

U

T



 



bY







'b

bY

Y 



bYbY

U



[

T











bY



U

 '

T

[











[

XYYX

[

T

U

T







Ihwlhfm







T

U

T

U











[

'XY

[

YXXY

YX0









T

U

T



[

XYYX

[

' H

T

[



Hlkx^Zbke_^m_lqlh







OQOLP

YX0



l_kijZ\_^eb\hml\_j`^_gb_

Ml\_j`^_gb_

Ijb





iehlghklv\_jhylghkl_cdjZldh

kjhqghcijhp_glghcklZ\dbkoh^blkydiehlghklbghjfZevgh]h

jZkij_^_e_gbykfZl_fZlbq_kdbfh`b^Zgb_f

b^bki_jkb_c'hi

j_^_ey_fufbke_^kl\b_f

>hdZaZl_evkl\hQlh[um[_^blvky\wlhf^hklZlhqghihdZaZlv

qlhijhba\h^ysZynmgdpbyfhf_glh\iehlghklb\_jhylghkl_cdjZldh

kjhqghcijhp_glghcklZ\dbijb





koh^blkydINFghj

fZevgh]hjZkij_^_e_gby



 _oj

^TW

'





Ijhba\h^ysZy nmgdpby iehlghklb\_jhylghkl_c \ujZ`Z_lky \

\b^_  IhkdhevdmF



y\ey_lkyijhba\h^ys_cnmgdpb_c k^\b

gmlh]h ]ZffZjZkij_^_e_gby lh f_`^m fZl_fZlbq_kdbf h`b^Zgb_f

 ^bki_jkb_c ' b iZjZf_ljZfb wlh]h jZkij_^_e_gbyT k

 o 





bf_xlf_klhkhhlghr_gby

 







' 





k

 

[

T









[

T





INFF



fh`_l[ulvaZibkZgZq_j_aiZjZf_lju

b'ke_^mx

sbfh[jZahf



 



[



T











[

T





JZkkfhljbfeh]Zjbnfhlghr_gbyINFF 

F 





 

T



T

WWTW

[

[' OQ







 



T



T



T



T



WT















[

[  





















T



T



T



[



KmffZjy^Z\kdh[dZoe_]dhhp_gb\Z_lkyk\_jombkgbamK^_eZ_f

wlhlhevdh^eyiheh`bl_evguo

>eyhljbpZl_evguo

wlh^_eZ_lky

ZgZeh]bqgh

[

T

























T



T



[











[

T



Ihwlhfm



WT























  



T  W



Hlkx^Z ke_^m_l qlh ijb o







 bf__l f_klh koh^bfhklv





beb

OLP    00

W W

qlhblj_[h\ZehkvihdZaZlv

<k_ ml\_j`^_gby [ueb ^hdZaZgu ^ey fh^_eb FK J_amevlZlu

wlbo ml\_j`^_gbc mklZgZ\eb\Zxl qlh hljZ`ZxsZy ]jZgbpZ

G



 km

s_kl\_ggh \eby_l gZ \b^ iehlghklb \_jhylghkl_c djZldhkjhqghc

klZ\db UW < h[s_fkemqZ_wlZ iehlghklv \_jhylghkl_c±k^\bgmlZy

]ZffZiehlghklv?keb



  l_

 



lh iehlghklv

\_jhylghkl_cklZgh\blkyh[uqghc]ZffZiehlghklvx<lZdhfkemqZ_

fh^_ev FK ij_h[jZam_lky\fh^_evK,5GZdhg_p _keb hljZ`ZxsZy

]jZgbpZ

G





f

l_







Zkj_^g__agZq_gb_

b^bki_j

kby ' nbdkbjh\Zgu lh]^Z iehlghklv \_jhylghkl_c djZldhkjhqghc

klZ\dbkoh^blkydghjfZevghciehlghklbZfh^_evFKij_\jZsZ_lky

\fh^_ev<Zkbq_dZ



KnbgZgkh\hclhqdbaj_gbyfh^_evkhljbpZl_evghchljZ`Zxs_c

]jZgbp_cg_j_ZebklbqgZH^gZdhlZdZyfh^_evfh`_lhdZaZlvky[he__

ijZ\^hih^h[ghc ^ey g_dhlhjuo j_Zevguo \u[hjhqguo ^Zgguo ^ey

^hoh^ghklb GZ kZfhf ^_e_ _kebg_dhlhjZy fh^_ev khljbpZl_evghc

hljZ`Zxs_c]jZgbp_c^hklZlhqghlhqghhibku\Z_lj_Zevgucijhp_kk

b h[_ki_qb\Z_l hq_gv fZemx \_jhylghklv hljbpZl_evguo agZq_gbc

ijhp_glguo klZ\hd lh]^Z lZdZy fh^_ev fh`_l bkihevah\Zlvky gZ

ijZdlbd_kf]eHdhgqZl_evgh_j_r_gb_wlhcijh[e_fufh

`_l[ulvk^_eZghkihfhsvxklZlbklbq_kdh]hZgZebaZ

Qlh dZkZ_lky fh^_ebFK lhke_^m_lhlf_lblv qlhijbkmsZy_c

hljZ`ZxsZy]jZgbpZg_h[yaZl_evghhljbpZl_evgZDZd[uehmdZaZgh

\ur_mkeh\b_fkms_kl\h\ZgbyhljZ`Zxs_c]jZgbpuy\ey_lkyg_jZ

\_gkl\h

o

!'AgZq_gbyiZjZf_ljZoh[_ki_qb\Zxsb_iheh`b

l_evghklvijhp_glghcklZ\d_e_`Zl\bgl_j\Ze_

T

Ijbo mk

eh\b_ \uihegy_lky _keb klZg^Zjlgh_ hldehg_gb_ p_gu Zdlb\Z [m^_l

f_gvr_ kj_^g_]h __ agZq_gbyqlh gZijZdlbd_ h[uqgh \k_]^Zbf__l

f_klhIhwlhfmkms_kl\m_lg_dhlhjZy©gbrZª^eyijbf_g_gbyfh^_

ebFKhij_^_ey_fZyg_jZ\_gkl\hf



T







KI?PBNBD:PBYDHWNNBPB?GLH<

:NNBGGHCKLJMDLMJU

>EYJ?:EVGUOIJHP?KKH<

>ey\kydhcaZ^Zgghc\j_f_gghckljmdlmjubgl_j_kgh\uykgblv

gZkdhevdh ohjhrh __ bgl_jij_lbjm_l lZ beb bgZy fh^_ev Ijb Zi

ijhdkbfZpbbijhp_kkZdjZldhkjhqghcijhp_glghcklZ\dbhij_^_ey_

fh]hmjZ\g_gb_fjZaebqgufbfh^_eyfbh^gbbl_`_dhwnnb

pb_gluZnnbgghckljmdlmju^eyjZaguofh^_e_cg_h[yaZl_evghijb

gbfZxl h^bgZdh\u_ agZq_gby Ihwlhfm ijb kjZ\gbl_evghf ZgZeba_

m^h[g__hlijbgyluodhwnnbpb_glh\Znnbgghckljmdlmjui_j_clbd

iZjZf_ljZf oZjZdl_jbamxsbf kZf ijhp_kk djZldhkjhqghc ijhp_gl

ghcklZ\dbLZdbfbiZjZf_ljZfbfh]mlkem`blvmklZgh\b\rb_kyagZ

q_gby kj_^g_]h

 b ^bki_jkbb ' ijhp_kkZ b kdhjhklv mklZgh\e_gby

klZpbhgZjgh]hj_`bfZNIhnbabq_kdhfmkfukemwlbiZjZf_ljuy\

eyxlky iheh`bl_evgufb >ey hibkZgby fh^_eb FK ke_^m_l \\_klb

_s_h^bgiZjZf_lj



mjh\_gvhljZ`Zxs_c]jZgbpu



oo 



Lh]^Z

gZhkgh\Zgbbke_^kl\bykhhl\_lkl\b_f_`^mdhwnnbpb_glZfbZn

nbgghc kljmdlmju b iZjZf_ljZfb ijhp_kkZ djZldhkjhqghc ijhp_gl

ghcklZ\dbmklZgZ\eb\Z_lkyihlZ[e



LZ[ebpZ

Khhl\_lkl\b_f_`^mdhwnnbpb_glZfbZnnbgghckljmdlmju

bdhwnnbpb_glZfbfh^_eb

Dhwnnbpb_glu Fh^_ev9 Fh^_evK,5 Fh^_evFK



N



N



N



N

 N





N'

 N'

o



N' 

N'o

o



LZdbfh[jZahfmjZ\g_gb_ijhp_kkZdjZldhkjhqghcijhp_gl

ghcklZ\db^eyjZkkfZljb\Z_fuofh^_e_c\gh\uoh[hagZq_gbyoijb

h[j_lZ_l\b^^eyfh^_eb<Zkbq_dZ

GU N





UGW



G:W

^eyfh^_ebK,5

GU N





UGW

N'U G:W

^eyfh^_ebFK

GU N





UGW

[

T



 G:W

AZf_lbfqlhijb\_^_ggh_\ur_mkeh\b_













g_

^hklb`bfhklb gb`g_c hljZ`Zxs_c ]jZgbpu \ gh\uo h[hagZq_gbyo

ij_\jZsZ_lky\g_jZ\_gkl\h

o



'Djhf_lh]hg_ljm^ghaZf_

lblvqlhfh^_evFKijbo

ij_\jZsZ_lky\fh^_evK,5Zdh]^Z

o



 fh^_ev FK ij_h[jZam_lky\ fh^_ev <Zkbq_dZIhke_^g__kh

]eZkm_lkykml\_j`^_gb_fhkoh^bfhklbklZpbhgZjgh]hjZkij_^_

e_gby\_jhylghkl_cdjZldhkjhqghcijhp_glghcklZ\db\fh^_ebFKd

ghjfZevghfmjZkij_^_e_gbxijb







Dhwnnbpb_gluZnnbgghckljmdlmjuk\yaZggu_kjughqghcp_

ghcjbkdZhij_^_eyxlkykhhlghr_gb_fBkihevamyb

fh`ghgZibkZlvqlh

U 

U

K[

 





[GKJ

V

[





Hq_\b^gh_keb\heZlbevghklvdjZldhkjhqghcijhp_glghcklZ\db

U klj_fblky d gmex lh i_j_oh^bf d ^_l_jfbgbjh\Zgghfm jugdm

ihwlhfm jughqgZy p_gZ jbkdZ

U b ij_fby aZ jbkd ^he`gZ klj_

fblvkydgmexBanhjfmeuke_^m_lqlhwlhlnZdl[m^_lbf_lv

f_klhlh]^Zblhevdhlh]^Zdh]^ZkijZ\_^eb\hjZ\_gkl\h



 

l_



 

[



B\ujZ`_gb_^ey

Uij_h[jZam_lkyd\b^m



U 



[

 



[

U  





U 



>ey kjZ\g_gby ih\_^_gby nmgdpbc Znnbgghc kljmdlmju \ jZa

ebqguo fh^_eyo ijb ZgZeba_ h^ghc b lhc `_ j_ZebaZpbb ijhp_kkZ

djZldhkjhqghcijhp_glghcklZ\dbg_h[oh^bfhlZd`_mklZgh\blvagZ

q_gbydhwnnbpb_glh\

[

b

Znnbgghckljmdlmju\wlbofh^_eyolZd

qlh[u hgb h[_ki_qb\Zeb gZkdhevdh wlh \hafh`gh h^bgZdh\uc mjh

\_gv jughqghcp_gu jbkdZ ?kl_kl\_ggh h`b^Zlv qlh jbkdh\Zy ij_

fbyijbdjZldhkjhqghcijhp_glghcklZ\d_jZ\ghckj_^g_fm__mjh\

gx\mklZgh\b\r_fkyj_`bf_[m^_lh^bgZdh\hc^ey\k_okjZ\gb\Z_

fuofh^_e_cDjhf_lh]hqlh[ujbkdh\Zyij_fby[ueZiheh`bl_ev

ghcg_h[oh^bfh\uiheg_gb_mkeh\by

U



kfgZijbf_jK,5

Ihwlhfmihlj_[m_fqlh[u^ey\k_olj_ojZkkfZljb\Z_fuofh

^_e_c



 







DZdm`_[ueh\ur_kdZaZghfh^_evK,5ij_

h[jZam_lkybafh^_ebFKijbo 





Zfh^_evFK\fh^_ev<Z

kbq_dZ±ijbo 





l_







LZdbfh[jZahfbanhjfmeukmq_lhfwlbok\hckl\jughq

gZyp_gZjbkdZ

U b dhwnnbpb_glu Znnbgghc kljmdlmju

[

b

^ey

jZkkfZljb\Z_fuofh^_e_chij_^_eyxlkyihlZ[e

LZ[ebpZ

IZjZf_ljup_gujbkdZ

IZjZf_lju Fh^_ev9 Fh^_evK,5 Fh^_evFK

U





± 







[





[

















[























[







LZdbfh[jZahfbaijb\_^_gguolZ[ebfh`ghaZdexqblv

qlhdhwnnbpb_gluZnnbgghckljmdlmju



[

b

\aZbfghh^gh

agZqgh \ujZ`Zxlky q_j_a iZjZf_lju kemqZcgh]h ijhp_kkZ djZldh

kjhqghcijhp_glghcklZ\dbbjughqghcp_gujbkdZN

'ob



Ml\_j`^_gb_

>eyex[h]hnbdkbjh\Zggh]hgZ[hjZiheh`b

l_evguoagZq_gbciZjZf_ljh\N

'

nmgdpbbZnnbgghckljmdlmju

:

b<

ijb\kydbo



b

o





m^h\e_l\hjyxlke_^mxsbf

g_jZ\_gkl\Zf





: 



: 



N

<





< 



< 





>hdZaZl_evkl\hAZnbdkbjm_f g_dhlhjh_ agZq_gb_



b \u

qbkebfijhba\h^gu_nmgdpbc:

b <

hij_^_ey_fuonhjfmeZfb

 bih iZjZf_ljm o HdZau\Z_lky qlh ^ey ex[h]h gZ[hjZ

iheh`bl_evguoagZq_gbciZjZf_ljh\N

'

ijhba\h^gu_y\eyxlky

g_hljbpZl_evgufbWlhb^hdZau\Z_lml\_j`^_gb_ihkdhevdm^eyfh

^_ebK,5o Z^eyfh^_eb<Zkbq_dZo 

Y\gu_\ujZ`_gbyijh

ba\h^guo

:

ob

<

obaaZbo]jhfha^dhklba^_kvg_ijb\h^ylky

AZf_lbfqlh\fh^_ebFKmjh\_gvhljZ`Zxs_c]jZgbpufh`_l

[ulv b iheh`bl_evguf l_ o fh`_l ijbgbfZlv b hljbpZl_evgu_

agZq_gbyghlZdqlh[u\uihegyehkvg_jZ\_gkl\h

o

Lh_klv

ijb

o



 [m^ml \uihegylvky g_jZ\_gkl\Z :







 : 

 b







< 





J:AGHKLGU?<?JKBBKLHO:KLBQ?KDBO

>BNN?J?GPB:EVGUOMJ:<G?GBC

>HOH>GHKLB>HIH=:R?GBY

GZj_ZevghfnbgZgkh\hfjugd_djZldhkjhqgZyijhp_glgZyklZ\

dZUWh[uqghg_dhlbjm_lkyQZs_\k_]hgZZmdpbhgZoP;hij_^_ey

xlkybop_guJWUW

bebklZ\db^hoh^ghklbmWUW

Ihwlhfm

kl_i_gv Z^_d\Zlghklb lhc beb bghc fh^_eb hij_^_eblv g_ihkj_^kl

\_gghihdhlbjh\dZfg_m^Z_lky<qZklghklbDZagZq_ckl\hKR:\u

imkdZ_l ljb lbiZ P; [be_luELOOV \_dk_eyQRWHV b h[eb]Zpbb

ERQGV dhlhju_ ijh^Zxlky q_j_a ZmdpbhggZhkgh\_gZagZq_gby ^h

oh^ghklb\ujZ`Z_fhc\ijhp_glZoklhqghklvx^h\lhjh]h^_kylbq

gh]h agZdZ Ihwlhfm ^ey \uy\e_gby kl_i_gb Z^_d\Zlghklb lhc beb



bghcfh^_ebdjZldhkjhqghcijhp_glghcklZ\dbUWfh`gh\hkihevah

\ZlvkygZ[ex^_gbyfbaZklZ\dZfb^hoh^ghklbBanhjfmeubf__f

m>WUWW

@ mW

 >UW<





:

@



Ihwlhfm^eyex[h]hnbdkbjh\Zggh]hkjhdZ^hih]Zr_gby

kih

fhsvxmjZ\g_gbyihemqZ_fke_^mxs__klhoZklbq_kdh_^bnn_

j_gpbZevgh_mjZ\g_gb_hij_^_eyxs__ijhp_kkbaf_g_gbyklZ\db^h

oh^ghklb\h\j_f_gb

Gm N





mGW

[

T





G:W

]^_



>

<





:

@

o 

>:

o<

@

' 

'< 





KjZ\g_gb_ mjZ\g_gbyk mjZ\g_gb_fiha\hey_l k^_

eZlvaZdexq_gb_qlhk\hckl\Zijhp_kkZklZ\db^hoh^ghklbmW

[m

^ml kh\iZ^Zlv kh k\hckl\Zfb djZldhkjhqghc ijhp_glghc klZ\db UW

_kebaZf_gblvagZq_gby iZjZf_ljh\

'bogZagZq_gbyiZjZf_ljh\

' bo LZdbfh[jZahf_kebdjZldhkjhqgZyijhp_glgZyklZ\dZUW

ke_^m_lijhp_kkmlh]^ZklZ\dZ ^hoh^ghklb mW

ke_^m_lkem

qZcghfm ijhp_kkm k iehlghklvx \_jhylghkl_c kf_s_ggh]h ]ZffZ

jZkij_^_e_gby

mo _bY 



@H[S>



[\Y









o 

m







khkf_s_gb_f



iZjZf_ljhfnhjfub 

o  ' biZjZf_ljhf

fZkrlZ[ZY 

o ' Kj_^g__agZq_gb_klZ\db^hoh^ghklb[m^_l

jZ\gh

Z__^bki_jkby±' :gZeh]bqghwlhfmfh`ghgZibkZlvmk

eh\gmxbkh\f_klgmxiehlghklb\_jhylghkl_c^ey^hoh^ghkl_cmWb

mVV



WgZhkgh\_iehlghkl_c\_jhylghkl_c^eyUWbUV\nhjf_





bebqlhlh`_\nhjf_

AgZgb_fh^_ebkijZdlbq_kdhclhqdbaj_gbyfh`_lihfhqv

nbgZgkh\hfmZgZeblbdmijh]ghabjh\Zlv^hoh^ghklbh[eb]Zpbb\[m

^ms_fIhwlhfm\Z`ghjZkkfhlj_lvijh[e_fmb^_glbnbdZpbbfh^_eb

 H^gZdh dZd m`_ [ueh kdZaZgh \ur_ ^hoh^ghklb h[eb]Zpbb

\uykgyxlky\j_amevlZl_Zmdpbhgh\ijhbkoh^ysboq_j_ahij_^_e_g

gu_ i_jbh^u \j_f_gb Ihwlhfm iZjZf_lju fh^_ebfh]mlhp_

gb\ZlvkylhevdhgZhkgh\_g_dhlhjhc\u[hjdbgZ[ex^_gbc^hoh^gh

kl_c ^h ih]Zr_gby mW \ fhf_glu \j_f_gb khklZ\eyxsb_dhg_qgh_