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Modelyuvannya Economy - Vіtlіnsky VV

10.2.2. Nayprostіsha multiplіkativna is stochastic model dinamіki fіnansovogo resource

For deyaky resource scho rozglyadaєtsya, you can contact Costa zagalom zaluchenі yak, so i zapitannya deposits, deposits termіnovі toscho.

Doslіdzhuvana model ґruntuєtsya on gіpotezі schodo mozhlivostі vіdslіdkuvati obsyagi doslіdzhuvanogo resource through diskretnі rіvnovelikі promіzhki hour t. Poznachimo through xt obsyag resource at the time of hour t, and x 0 - obsyag Pochatkova at the time of one hour (pripustimo scho x 0> 0). Pripustimo takozh, scho perehіd obsyagu resource Cauterets viznachaєtsya dіysnim number xi -1> 0 at the time of hour t = i - 1 to resource obsyagom xi> 0, scho hour vіdpovіdaє time t = i, mozhna opisati spіvvіdnoshennyam:

(10.21)

de a i> 0 - nevіd'єmny koefіtsієnt elementarnogo transition od xi -1 to xi, i = 1, ..., n.

H (10.21) viplivaє:

(10.22)

de x 0, xn, a i I R 1, x 0> 0, a i> 0, i = 1, ..., n.

In chastkovomu vipadku, if OAO All koefіtsієnti elementarnih perehodіv Je odnakovimi (a i = a> 0, i = 1, ..., n), formula (10.22) nabuvaє viglyadu:

(10.23)

scho vkazuє on eksponentsіynu zalezhnіst obsyagu resource od hour. Tom xn ® ?, Yakscho a> 1; xn ®0, Yakscho a <1.

Yakscho sposterezhuvanі values іnterpretuvati yak realіzatsії vipadkovih values , The formula (10.22) peretvoryuєtsya at Taku stochasticity multiplіkativnu model dinamіki resource on discrete vіdrіzku hour (0, n):

(10.24)

de - Vipadkova obsyagu value of the resource at the time t = n.

Pripustimo scho OAO All vipadkovі koefіtsієnti elementarnih perehodіv Yea i Square Leather s Got them lognormal rozpodіlu de - Vіdpovіdno ically mathematical spodіvannya that dispersіya lognormal rozpodіlenoї vipadkovoї quantities :

.

Funktsіya schіlnostі rozpodіlu can be written as:

(10.25)

Viraz for ically mathematical spodіvannya:

(10.26)

Other Pochatkova moment?

(10.27)

Dispersіya

(10.28)

Znaydemo teper funktsіyu rozpodіlu vipadkovogo koefіtsієnta:

(10.29)

Obviously, scho in tsomu vipadku koefіtsієnti mayutsya lognormal rozpodіlu:

s parameters:

(10.30)

(10.31)

Zvіdsi easily otrimati for viraz ically mathematical spodіvannya

(10.32)

another time Pochatkova

(10.33)

that dispersії

. (10.34)

Otrimaєmo takozh viraz for vipadkovoї quantities :

(10.35)

For prognozuvannya obsyagu resource scho zdіysnyuєtsya at time t = 0 hour at the time of t = the n, mozhna vikoristati ically mathematical spodіvannya vipadkovoї quantities :

(10.36)

Tochnіst such natural otsіniti forecast for Relief serednokvadratichnogo vіdhilennya:

(10.37)

yak mozhna vikoristati for pobudovi dovіrchogo іntervalu:

. (10.38)

Schodo mozhlivih values prognozovanoї amount of resources at the time t = n koefіtsієnt g> 0 obiraєtsya so dwellers zabezpechiti tasks ymovіrnіst potraplyannya values vipadkovoї amount of resources in vіdrіzok (10.38) abo ymovіrnіst (a = 1 - g) (rizik), the value of scho vipadkova syagne for mezhі vkazanogo vіdrіzka.

Yakscho OAO All nezalezhnі vipadkovі quantities one of the toil i samy lognormal rozpodіlu s parameters , You can hover zaproponuvati nizhche scheme that otsіnyuvannya parametrіv m s2.

Nekhay sposterіgaєtsya low poslіdovnih values x 0, x 1, ..., xk obsyagu resource. Pripuskayuchi scho OAO All tsі values nevіd'єmnі, obchislyuєmo low values a1, ..., a n koefіtsієnta elementarnogo transition:

(10.39)

Zgіdno s nashoyu Modell low value ln a i, i = 1, ..., k mozhna rozglyadati yak Prostu vipadkovu vibіrku obsyagu k s generalnoї sukupnostі scho opisuєtsya normal law rozpodіlu ically mathematical spodіvannyam s m i dispersієyu s2. Tom obґruntovanoyu, nezmіschenoyu th efektivnosti otsіnkoyu parameter sluguє vibіrkove ically mathematical spodіvannya:

(10.40)

and obґruntovanoyu th nezmіschenoyu otsіnkoyu parameter s2 - vibіrkova dispersіya:

(10.41)

Mozhna otrimati th empіrichnі formula for the magnitude prognozovanoї obsyagu resource at the time of hour t = n i serednokvadratichne vіdhilennya tsogo forecast (to Yogo otsіnki tochnostі):

(10.42)

(10.43)

Odnієyu s problems at scho vinikayut hodі praktichnoї realіzatsії vikladenoї vische techniques prognozuvannya dinamіki fіnansovih resursіv, Yea those scho peredbachayutsya dosit shirokі mezhі for otsіnki mozhlivih vіdhilen factual values ​​od Forward-looking. Natomіst magnitude Scho viznachaє tsі mezhі, yak usually Shvydko zrostaє Zi zbіlshennyam the following number of skin perіodu. Usa Tse znizhuє practicality tsіnnіst otrimuvanih rezultatіv.

Obviously, scho values n *, pochinayuchi s yakogo shvidkіst rozhodzhennya between dovіrchogo іntervalu suttєvo zrostaє, Mauger viznachatisya of minds:

(10.44)

H (10.44) can be otrimati:

(10.45)

de

(10.46) Gіpoteza schodo lognormal law rozpodіlu koefіtsієntіv elementarnogo transition zabezpechuє zruchnіst i ease multiplіkativnih peretvoren, scho for pity, not poshiryuєtsya operatsії additive function on the character. Practical OAO All konkretnі fіnansovі resources pov'yazanі timey chi іnshimi additive function spіvvіdnoshennyami, yak-by, napriklad, scrip vsіh depozitіv skladaєtsya іz sumi transaktsіynih, Savings that іnshih depozitіv. Vіdpovіdno, uzyavshi gіpotezu schodo lognormal law rozpodіlu koefіtsієntіv transition to okremih vidіv depozitіv, MI automatic and viznachaєmo law rozpodіlu for analogіchnih koefіtsієntіv sumarno depozitіv, yaky bude not lognormal. Naybіlsh ratsіonalnim vbachaєtsya sama Taqiy pіdhіd to rozv'yazannya tsієї superechnostі.

Urahovuyuchi those scho scrip Square vipadkovih rozpodіlena values ​​for normal law on praktitsі mozhna vvazhati scho rozpodіl koefіtsієntіv elementarnogo transition to sumarno fіnansovogo resources can aproksimuvati lognormal, todі Especially, if values ​​їhnіh parametrіv little vіdrіznyayutsya.