ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΙΓΑΙΟΥ

Θεωρία των Κινδύνων Ι Ενότητα 1: Κλασική Μοντελοποίηση Κινδύνου

Δημήτριος Κωνσταντινίδης

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Σάμος, Οκτώβριος 2014

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Kef�laio 1

Klasik  montelopoÐhsh

kindÔnou

H JewrÐa SullogikoÔ KindÔnou prwtoemfanÐsjhke to 1903 apì ton Fi-

lip Lundberg ìtan prìteine to klasikì montèlo asfalistik¸n kindÔnou.

Sugkekrimèna o Filip Lundberg sthn didaktorik  diatrib  pou uper�spise

sto Panepist mio thc Ouy�lac (SouhdÐa), èbale ta jemèlia thc analogi-

stik c jewrÐac kindÔnou ìpwc thn gnwrÐzoume s mera. Sthn diatrib  aut 

mporoÔme na diakrÐnoume mèsa sta montèla twn genik¸n asfalÐsewn, thn

parousÐa thc diadikasÐac Poisson. M�lista me thn bo jeia kat�llhlou

qronikoÔ metasqhmatismoÔ an�getai h an�lush tou analogistikoÔ montè-

lou sthn melèth thc omogenoÔc diadikasÐac Poisson.

Aut  h anak�luyh thc barÔthtac thc diadikasÐac Poisson sta analo-

gistik� montèla, parallhlÐzetai me thn eisagwg  thc kÐnhshc Brown san

basikì ergaleÐo kataskeu c qrhmatooikonomik¸n montèlwn apì ton Louis

Bachelier mìlic trÐa qrìnia nwrÐtera, to 1900.

H skandinabik  sqol  thc analogistik c epist mhc me epikefal c ton

Harald Cramer, enswm�twse tic idèec tou Filip Lundberg sth jewrÐa twn

stoqastik¸n diadikasi¸n kai sunèbale kajoristik� sthn jemelÐwsh thc

1

2 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

shmerin c analogistik c epist mhc twn genik¸n asfalÐsewn all� kai sthn

an�ptuxh thc jewrÐac pijanot twn kai twn stoqastik¸n diadikasi¸n proc

thn kateÔjunsh thc jewrÐac kindÔnou.

Sthn analogistik  epist mh brÐskoume èna qarakthristikì par�deig-

ma gìnimhc allhlepÐdrashc metaxÔ jewrÐac kai pr�xhc. Pr�gmati, oi dÔo

autèc proseggÐseic sumb�lloun armonik� sthn dhmiourgÐa miac pl rouc

kai austhr c je¸rhshc, qwrÐc thn kuriarqÐa thc miac p�nw sthn �llh,

all� kai me sten  sÔndesh metaxÔ touc, ètsi ¸ste na gÐnetai asaf c o

diaqwrismìc touc. Sto er¸thma pwc prèpei na proseggÐzei o sÔgqronoc

ereunht c thn jewrÐa kindÔnou apì praktik    apì jewrhtik  pleur�, den

up�rqei ap�nthsh. 'Ena ai¸na met� thn gènnhsh thc jewrÐac kindÔnou su-

neqÐzontai oi enallagèc metaxÔ praktik¸n kai jewrhtik¸n proseggÐsewn,

me apotèlesma na jewreÐtai aparaÐthto gia thn katanìhsh thc jewrÐac h

makroqrìnia kai epÐponh epaf  me ta probl mata thc analogistik c pra-

ktik c, ìpwc anadÔontai mèsa sthn asfalistik  biomhqanÐa en¸ thn Ðdia

stigm  gia thn anab�jmish thc kajhmerin c enasqìlhshc me ta analogi-

stik� eÐnai aparaÐthth h parapomp  sthn antÐstoiqh jewrÐa.

O stìqoc aut¸n twn shmei¸sewn eÐnai na enjarrÔnei thn prosp�jeia

tou foitht  gia katanìhsh thc prosèggishc metaxÔ jewrÐac kai pr�xhc.

QwrÐc na periorÐzetai sta tetrimmèna probl mata pou sunant� o analo-

gist c kajhmerin�, prospajeÐ na apod¸sei èna sÔnolo ide¸n pou èqoun

san telikì skopì thn beltÐwsh thc poiìthtac twn uphresi¸n pou prosfè-

rontai sthn asfalistik  agor�. H shmasÐa thc asf�lishc sth sÔgqronh

epiqeirhmatik  drasthriìthta eÐnai plèon pasÐdhlh. Wstìso, h prosfor�

beltiwmènwn asfalistik¸n proðìntwn eÐnai to anamenìmeno b ma proìdou

apì pleur�c twn asfalismènwn. Kai h beltÐwsh aut  mporeÐ na prokÔyei

mìno me akribèstero kai kurÐwc axiìpisto upologismì twn metablht¸n pou

upeisèrqontai sta montèla kindÔnou.

Tèloc axÐzei na epishm�noume kai mia �llh ptuq  thc jewrÐac kindÔnou

pou mporeÐ na enjousi�sei touc upoy fiouc analogistèc. EÐnai h epèktash

1.1. PIJAN�OTHTA QREOKOP�IAS 3

thc jewrÐac kindÔnou me mia poikilÐa fusik¸n kai koinwnik¸n fainomènwn

pou sqetÐzontai me tic aitÐec tou kindÔnou. QwrÐc na emplekìmaste stic

eidikìterec pleurèc twn fainomènwn, mporoÔme na melet soume to stoi-

qeÐo thc abebaiìthtac kai me autì to trìpo epitugq�noume thn bajÔterh

katanìhs  touc. Aut  h poreÐa odhgeÐ sthn plhrìthta plhrofìrhshc

pou kaleÐtai me sunduasmì me �lla, diaforetik� gnwstik� antikeÐmena na

katal xei telik� se axiìpistouc upologismoÔc.

1.1 Pijanìthta qreokopÐac

H jewrÐa sullogikoÔ kindÔnou sthrÐzetai sthn ènnoia thc stoqastik c

diadikasÐac pou epitrèpei thn swst  perigraf  thc diadoqik c emf�nishc

apozhmi¸sewn pou emfanÐzontai sta plaÐsia k�poiou asfalistikoÔ qarto-

fulakÐou pou ja to onom�zoume asfalistik  politik . Me thn bo jeia

twn stoqastik¸n diadikasi¸n mporoÔme na montelopoi soume tic tuqaÐec

diakum�nseic tou apojèmatoc thc asfalistik c etaireÐac pou qrhsimopoieÐ-

tai gia thn plhrwm  twn apozhmi¸sewn.

K�je asfalistik  politik  stoqeÔei sthn el�frunsh twn pelat¸n thc

apì ton fìbo tou kindÔnou pou endeqìmena ja sunant soun kai touc dieu-

kolÔnei na antimetwpÐsoun apotelesmatik� tic sunèpeiec touc, kalÔpton-

tac tic apozhmi¸seic pou prokaloÔntai apì atuq mata. Oi pel�tec se an-

t�llagma katab�loun sthn etaireÐa asf�listra gia na exasfalÐsoun thn

biwsimìthta thc etaireÐac kai na sumb�loun sthn dhmiourgÐa tou anagkaÐ-

ou apojèmatoc. Profan¸c ta asf�listra ja prèpei na xepernoÔn to mèso

kìstoc twn apozhmi¸sewn se opoiod pote qronikì di�sthma, pou shmaÐnei

ìti oi pel�tec dèqontai ex' arq c k�poia jetik  epib�runsh asfaleÐac.

Ta analogistik� montèla dÐnoun thn dunatìthta na melethjeÐ me k�je

leptomèreia h sqèsh metaxÔ twn emplekomènwn megej¸n kai prosfèroun ta

ergaleÐa gia touc swstoÔc upologismoÔc twn asfalÐstrwn, tou kindÔnou

kai twn apojem�twn.

4 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

Ac upojèsoume ìti k�poia asfalistik  etaireÐa xekin� drasthriìthta

thn stigm  0 me arqikì kef�laio u ≥ 0 kai to sunolikì eisìdhma apì

asf�listra pou katab�lletai apì touc pel�tec mèqri kai thn stigm  t

parist�netai me C(t). To eisìdhma apì asf�listra C(t) eÐnai aÔxousa

sun�rthsh tou qrìnou. Sun jwc jewroÔme thn prosdioristik  (nteter-

ministik ) grammik  sun�rthsh C(t) = c t ìpou h stajer� c onom�zetai

rujmìc eÐspraxhc asfalÐstrou.

Oi qronikèc stigmèc {Tk , k ∈ N} to sÔnolo twn diadoqik¸n stigm¸n

emf�nishc atuq matoc kai ta Ôyh twn apozhmi¸sewn {Zk , k ∈ N} apote-loÔn akoloujÐec tuqaÐwn metablht¸n. Tic stigmèc emf�nishc atuq matoc

mporoÔme na tic parast soume me thn bo jeia twn apost�sewn metaxÔ dia-

doqik¸n stigm¸n atuq matoc θk = Tk − Tk−1, gia k ∈ N, ìpou jewroÔme

T0 = 0. 'EstwN(t) = min{k ∈ N0 : Tk+1 > t} = max{k ∈ N0 : Tk ≤ t}o arijmìc twn qronik¸n stigm¸n emf�nishc atuq matoc sto di�sthma [0, t].

To Ôyoc thc k apozhmÐwshc sumbolÐzetai me Zk. Epomènwc h sunolik 

apozhmÐwsh mèqri kai thn stigm  t, dÐnetai apì to tuqaÐo �jroisma

S(t) =

N(t)∑i=1

Zi .

Upojètoume ìti oi endi�mesoi qrìnoi {θk , k ∈ N} apoteloÔn ako-

loujÐa anex�rthtwn isìnomwn tuqaÐwn metablht¸n me katanom  A(x) =

P[θ1 ≤ x]. Oi ropèc twn endi�meswn qrìnwn k t�xhc, gia k = 0, 1, . . .,

e�n up�rqoun sumbolÐzontai me

ak = E[θk1 ] =

∫ ∞0

yk A(dy) .

Upojètoume ìti ta Ôyh twn apozhmi¸sewn {Zk , k ∈ N} apoteloÔn

mia akoloujÐa anex�rthtwn isìnomwn tuqaÐwn metablht¸n me katanom 

B(x) = P[Z1 ≤ x] kai sumbolÐzoume thn our� thc katanom c me B(x) =

P[Z1 > x] = 1−B(x).

1.1. PIJAN�OTHTA QREOKOP�IAS 5

Deqìmaste ìti

supx<0

B(x) = 0 ,

kai B(0) < 1, pou shmaÐnei ìti ta Ôyh twn apozhmi¸sewn paÐrnoun mh

arnhtikèc timèc kai den ekfullÐzontai sthn mhdenik  prosdioristik  tuqaÐa

metablht , dhlad  P[Z1 > 0] > 0. Oi ropèc twn apozhmi¸sewn e�n

up�rqoun sumbolÐzontai me

bk = E[Zk1 ] =

∫ ∞0

yk B(dy) ,

gia k = 0, 1, . . .. Upojètoume ìti oi akoloujÐec {Tk , k ∈ N} kai

{Zk , k ∈ N} eÐnai anex�rthtec metaxÔ touc. Me ta megèjh pou pa-

rousi�same, eÐmaste se jèsh na upologÐsoume thn diadikasÐa apojèmatoc

sthn morf 

U(t) = u+ C(t)− S(t) = u+ c t−N(t)∑i=1

Zi .

To Ôyoc tou apojèmatoc se k�je stigm  apoteleÐ stoqastik  diadikasÐa

kaj¸c sta emplekìmena megèjh perilamb�nontai oi tuqaÐec metablhtèc

{Tk, k ∈ N} kai {Zk, k ∈ N}. H diafor�

C(t)− S(t) = c t−N(t)∑i=1

Zi ,

dhl¸nei thn epib�runsh asfaleÐac kai dÐnei shmantik  plhroforÐa gia thn

axiopistÐa thc asfalistik c drasthriìthtac. Sthn pr�xh qrhsimopoioÔme

kurÐwc to ìrio

ρ = limt→∞

E[C(t)− S(t)]

E[S(t)], (1.1.1)

pou eÐnai gnwstì me to ìnoma sqetik  epib�runsh asfaleÐac. H sqetik 

epib�runsh asfaleÐac perigr�fei to anamenìmeno eisìdhma thc asfalisti-

k c etaireÐac an� mon�da apozhmÐwshc. 'Otan to ρ plhsi�zei sto mhdèn, h

6 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

asfalistik  etaireÐa mènei qwrÐc apìjema kai o kÐndunoc qreokopÐac thc

megal¸nei. 'Otan to ρ gÐnetai meg�lo h etaireÐa parousi�zei kerdoforÐa

all� ta asf�listr� thc eÐnai apojarruntik� gia touc upoy fiouc pel�tec.

Sth sunèqeia axi¸noume ìti aut  h sqetik  epib�runsh asfaleÐac u-

p�rqei kai eÐnai jetik , dhlad  isqÔei

ρ > 0 . (1.1.2)

Aut  h upìjesh eÐnai eurèwc apodekt  sthn analogistik  praktik  kai

onom�zetai AxÐwma KajaroÔ Kèrdouc. M�lista to AxÐwma KajaroÔ Kèr-

douc praktik� shmaÐnei ìti h diadikasÐa apojèmatoc U(t) èqei auxhtik 

t�sh, pou eÐnai anagkaÐa proôpìjesh gia na elpÐzoume sthn kerdoforÐa

thc etaireÐac. H diadikasÐa apojèmatoc {U(t), t ≥ 0} perièqei thn plh-

roforÐa pou qrei�zetai gia thn axiolìghsh thc biwsimìthtac thc asfali-

stik c epiqeÐrhshc sta plaÐsia k�poiou montèlou kindÔnou, parist�menou

sun jwc apì thn tri�da (A, C, B).

E�n to apìjema p�rei arnhtik  tim  se k�poia qronik  stigm  t > 0,

lème ìti parousi�zetai qreokopÐa. H pijanìthta autoÔ tou endeqomènou

paÐrnei thn morf 

ψ(u) = P

[inft>0

U(t) < 0 |U(0) = u

]. (1.1.3)

H pijanìthta qreokopÐac ψ(u) qrhsimeÔei san deÐkthc poiìthtac thc asfa-

listik c drasthriìthtac. Dhlad  ìso mikrìterh pijanìthta qreokopÐac

brÐskoume, tìso kalÔterh asfalistik  etaireÐa èqoume apì �poyh biwsi-

mìthtac. To epÐpedo apojèmatoc met� to opoÐo jewroÔme ìti h etaireÐa

pern�ei se qreokopÐa, paÐrnetai sun jwc Ðso me to mhdèn. O qrìnoc qre-

okopÐac sumbolÐzetai me

τ(u) = inf{t ≥ 0 : U(t) < 0 | U(0) = u} ,

opìte h pijanìthta qreokopÐac gr�fetai sthn morf  ψ(u) = P[τ(u) <

∞]. Sthn genik  perÐptwsh o qrìnoc qreokopÐac τ(u) eÐnai mia ellip c

1.1. PIJAN�OTHTA QREOKOP�IAS 7

tuqaÐa metablht  (dhlad  h sun�rthsh katanom c den teÐnei sthn mon�da)

kaj¸c mporeÐ na p�rei thn tim ∞ me jetik  pijanìthta P[τ(u) =∞] > 0.

Pr�gmati, diaisjhtik� antilambanìmaste ìti k�tw apì to AxÐwma KajaroÔ

Kèrdouc to apìjema U(t) teÐnei sto �peiro kai gi' autì eÐnai pijanì na mhn

emfanisteÐ potè qreokopÐa.

H sumplhrwmatik  sun�rthsh

φ(u) = 1− ψ(u) , (1.1.4)

onom�zetai pijanìthta epibÐwshc kai paÐrnei antÐstoiqa thn morf 

φ(u) = P

[inft>0

U(t) ≥ 0 |U(0) = u

]= P[τ(u) =∞] .

Gia ton upologismì thc pijanìthtac qreokopÐac mporoÔme na jewr -

soume thn tuqaÐa metablht  Un = U(Tn), pou sumbolÐzei to apìjema

akrib¸c met� thn plhrwm  thc n apozhmÐwshc, opìte brÐskoume to akì-

loujo diakritì montèlo anagwgik¸n exis¸sewn

U0 = u ,

Un+1 = Un + c θn+1 − Zn+1 , (1.1.5)

gia k�je n = 0, 1, . . ..

EÔkola diapist¸noume ìti h akoloujÐa {Un, n = 0, 1, . . .} apoteleÐomogen  markobian  alusÐda me timèc apì to sÔnolo twn pragmatik¸n

R. Kaj¸c to endeqìmeno qreokopÐac mporeÐ na emfanisteÐ mìno kat� tic

stigmèc emf�nishc apozhmi¸sewn {Tn, n ∈ N}, h pijanìthta qreokopÐac

paÐrnei diakrit  morf 

ψ(u) = P[ infn≥1

Un < 0 |U0 = u] . (1.1.6)

H pijanìthta qreokopÐac upologÐzetai me b�sh tic poikÐlec paramè-

trouc tou montèlou kindÔnou kai gi' autì h petuqhmènh epilog  tou mo-

ntèlou epibebai¸netai me tic sugkrÐseic metaxÔ arqik¸n dedomènwn kai

8 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

telik¸n apotelesm�twn sthn leitourgÐa thc etaireÐac. 'Eqontac aut  thn

prooptik  sto mualì mac, ja anaptÔxoume majhmatikèc mejìdouc gia ton

upologismì   thn ektÐmhsh thc pijanìthtac qreokopÐac.

1.2 To tuqaÐo �jroisma S(t)

MporoÔme na upologÐsoume thn katanom  tou S(t) parathr¸ntac ìti to

endeqìmeno {S(t) ≤ x} analÔetai se ènwsh xènwn metaxÔ touc endeqomè-

nwn me b�sh ton arijmì apozhmi¸sewn N(t) = n pou emfanÐsjhkan mèqri

thn stigm  t. Dhlad 

{S(t) ≤ x} =∞⋃n=0

{S(t) ≤ x , N(t) = n} .

Opìte apì ton tÔpo thc olik c pijanìthtac èqoume

P[S(t) ≤ x] =∞∑n=0

P[S(t) ≤ x , N(t) = n]

=∞∑n=0

P[S(t) ≤ x |N(t) = n]P[N(t) = n]

=

∞∑n=0

P

[n∑i=1

Zi ≤ x

]P[N(t) = n]

=∞∑n=0

Bn∗(x)P[N(t) = n] ,

ìpou to Bk∗(x) sumbolÐzei thn k-t�xhc sunèlixh thc katanom c B(x) me

ton eautì thc kai me B0∗(x) = 1[x≥0]. Autìc eÐnai ènac tÔpoc gia ton

upologismì thc katanom c thc sunolik c apozhmÐwshc ìtan gnwrÐzoume

thn katanom  B(x) kai tic pijanìthtec P[N(t) = n].

Gia tic ropèc kai tic ropogenn triec ja qrhsimopoi soume tic desmeu-

mènec mèsec timèc. JumÐzoume ìti gia dÔo opoiesd pote tuqaÐec metablhtèc

1.2. TO TUQA�IO �AJROISMA S(T ) 9

X kai Y , e�n up�rqoun oi antÐstoiqec ropèc, èqoume

E[Y ] = E[E(Y |X)] ,

var[Y ] = E[var(Y |X)] + var[E(Y |X)] .

San sunèpeia paÐrnoume E[S(t)] = E[E(S(t) |N(t))]. 'Estw t¸ra ìti

up�rqoun oi ropèc twn apozhmi¸sewn bk = E[Zk] gia k ∈ N. Tìte

E[S(t) |N(t) = n] = E

[n∑i=1

Zi

]=

n∑i=1

E [Zi] = n b1 ,

gia n = 0, 1, . . ., ap' ìpou prokÔpteiE[S(t) |N(t)] = N(t) b1 kai parapèra

E[S(t)] = E [N(t)] b1 . (1.2.1)

'Etsi katal goume ston tÔpo tou Wald (blèpe [3, Je¸rhma 6]), ìpou

blèpoume ìti h anamenìmenh sunolik  apozhmÐwsh eÐnai to ginìmeno tou

anamenìmenou pl jouc apozhmi¸sewn me thn mèsh tim  tou Ôyouc thc k�je

apozhmÐwshc.

Parìmoia, qrhsimopoi¸ntac thn upìjesh ìti oi Zi , i ∈ N eÐnai ane-

x�rthtec isìnomec tuqaÐec metablhtèc, paÐrnoume

var[S(t) |N(t) = n] = var

[n∑i=1

Zi

]=

n∑i=1

var [Zi] = n (b2 − b21) ,

opìte brÐskoume var[S(t) |N(t)] = N(t) (b2− b21) kai me b�sh ta prohgoÔ-mena

var[S(t)] = E(var[S(t) |N(t)]) + var(E[S(t) |N(t)])

= E[N(t) (b2 − b21)

]+ var[N(t) b1]

= (b2 − b21)E [N(t)] + b21 var[N(t)] .

Se autìn ton tÔpo blèpoume p�li ìti h diakÔmansh thc sunolik c apozh-

mÐwshc ekfr�zetai sunart sei twn dÔo pr¸twn rop¸n tou pl jouc twn

10 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

proserqìmenwn apozhmi¸sewn sto di�sthma [0, t] kai tou Ôyouc twn apo-

zhmi¸sewn.

T¸ra proqwroÔme sth melèth thc genn triac PS(y) = E[yS ]. 'Eqoume

PS(t)(y) = E(E[yS(t) |N(t)

]). Apì thn anexarthsÐa twn {Zi} prokÔptei

E[yS(t) |N(t) = n] = E[y∑n

i=1 Zi

]=

n∏i=1

E[yZi].

Parapèra apì thn isonomÐa twn {Zi} paÐrnoume

E[yS(t) |N(t) = n] =(E[yZi])n

= [PZ(y)]n .

'Etsi odhgoÔmaste sthn PS(t)(y) = E(

[PZ(y)]N(t))

= PN(t) [PZ(y)].

To Ðdio paÐrnoume gia thn ropogenn tria sun�rthsh MS(y) = E[eyS ],

opìte o antÐstoiqoc tÔpoc gÐnetai:

MS(t)(s) = MN(t) [ln(MZ(s))] = PN(t) [MZ(s)] . (1.2.2)

Sthn perÐptwsh pou to pl joc twn apozhmi¸sewn {N(t) , t ≥ 0} stodi�sthma [0, t], apoteleÐ diadikasÐa Poisson me par�metro λt, paÐrnoume

thn sunolik  apozhmÐwsh S(t) san sÔnjeth Poisson tuqaÐa metablht .

Eidikìtera, efarmìzontac touc gnwstoÔc tÔpouc thc katanom c Poisson

brÐskoume

E[S(t)] = λ t b1 ,

var[S(t)] = λ t b2 . (1.2.3)

Parapèra h trÐth kentrik  rop  eÐnai

E[(S(t)− λ t b1)3] = λ t b3 . (1.2.4)

Pr�gmati, apì thn èkfrash (1.2.2) kai ton tÔpo thc genn triac thc Pois-

son (blèpe [3, sqèsh (1.4.19)]) brÐskoume gia thn ropogenn tria thc sÔn-

jethc Poisson ton tÔpo

MS(t)(s) = PN(t) [MZ(s)] = exp [−λ t(1−MZ(s))] . (1.2.5)

1.2. TO TUQA�IO �AJROISMA S(T ) 11

Parapèra, paÐrnoume tic parag¸gouc wc proc thn metablht  s

M′S(t)(s) = λ tM′Z(s)MS(t)(s) ,

M′′S(t)(s) = λ tM′′Z(s)MS(t)(s) + λ tM′Z(s)M′S(t)(s) ,

M′′′S(t)(s) = λ tM′′′Z (s)MS(t)(s) + 2λ tM′′Z(s)M′S(t)(s)

+λ tM′Z(s)M′′S(t)(s) .

Jètontac t¸ra s = 0 brÐskoume touc tÔpouc gia tic treic pr¸tec ropèc.

Sthn perÐptwsh pou èqoume to �jroisma n anex�rthtwn all� ìqi a-

nagkastik� isìnomwn sÔnjetwn Poisson tuqaÐwn metablht¸n brÐskoume

p�li mia sÔnjeth Poisson tuqaÐa metablht . Pr�gmati, èstw {Xi(t) , i =

1, . . . , n} to sÔnolo twn anex�rthtwn sÔnjetwn Poisson me paramètrouc

{λi t , i = 1, . . . , n} antÐstoiqa. Tìte h tuqaÐa metablht 

n∑i=1

Xi(t) ,

akoloujeÐ epÐshc thn sÔnjeth Poisson katanom  me par�metro

Λn =n∑i=1

λi t .

Gia na to deÐxoume, paÐrnoume thn ropogenn tria sun�rthsh

M∑ni=1Xi

(s) = E

[exp

(s

n∑i=1

Xi(t)

)]=

n∏i=1

E[exp(sXi(t))]

=n∏i=1

MXi(t)(s) ,

apì thn anexarthsÐa twn {Xi(t) , i = 1, . . . , n}. Epomènwc apì ton tÔpo

12 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

(1.2.5)

M∑ni=1Xi

(s) =n∏i=1

exp[−λi t(1−MZi(s))]

= exp

{−

n∑i=1

λi t(1−MZi(s))

}

= exp

{−Λn

(1−

n∑i=1

λi tMZi(s)

Λn

)},

pou eÐnai p�li sthn morf  thc sÔnjethc Poisson, arkeÐ na jewr soume mia

nèa katanom  apozhmi¸sewn mèsa apì thn èkfrash

n∑i=1

λi t

ΛnP[Zi ≤ x] .

Parapèra me thn bo jeia thc jewrÐac ananèwshc brÐskoume tic akì-

loujec qarakthristikèc sugklÐseic.

Je¸rhma 1. Upojètoume ìti A(0) = 0, a1 < ∞ kai b1 < ∞. Tìte

isqÔei

S(t)

t

a.s.−→ b1a1, (1.2.6)

kaj¸c t → ∞. E�n h katanom  A eÐnai mh arijmhtik , tìte gia k�je

h ≥ 0, isqÔei

E[S(t+ h)− S(t)]→ hb1a1, (1.2.7)

kaj¸c t→∞.

Apìdeixh. ParathroÔme ìti apì ton isqurì nìmo twn meg�lwn

arijm¸n paÐrnoume

1

N(t)

N(t)∑i=1

Zia.s.−→ E[Z] = b1 ,

1.2. TO TUQA�IO �AJROISMA S(T ) 13

kaj¸c t→∞, diìti N(t)→∞. Apì ed¸ kai me b�sh to [3, Je¸rhma 15]

brÐskoume

N(t)

t

a.s.−→ 1

a1,

kaj¸c t→∞, kai paÐrnoume th sqèsh (1.2.6).

Gia thn deÔterh sqèsh qrhsimopoioÔme thn tautìthta Wald (blèpe [3,

Je¸rhma 6]) gia na broÔme

E

N(t)+1∑i=1

Zi

= E[Z]E[N(t) + 1] .

Epomènwc E[S(t)] = E[Z]E[N(t) + 1]− E[ZN(t)+1]. Gia na ektim soume

thn teleutaÐa mèsh tim , paÐrnoume thn dèsmeush wc proc ton arijmì twn

apozhmi¸sewn mèqri th stigm  t

E[ZN(t)+1] =∞∑n=0

E[ZN(t)+1 1[N(t)=n]

]=∞∑n=0

E[Zn+1 1[Tn≤t<Tn+1]

]=

∞∑n=0

∫ t

0E[Zn+1 1[θn+1>t−y]

]An∗(dy)

=

∫ t

0E[Z 1[θ>t−y]

]E[N(dy) + 1]

=

∫ t

0g(t− y)E [N(dy) + 1] ,

ìpou g(x) := E[Z 1[θ>x]

]. ParathroÔme ìti g(0) = E[Z], g(∞) = 0 kai

14 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

N(0) = 0 gia na sumper�noume

E[S(t)] = g(0)E[N(t) + 1]−∫ t

0g(t− y)E[N(dy) + 1]

= g(0)E[N(t) + 1] +

∫ t

0g(w)E[N(t− dw) + 1]

= g(t)−∫ t

0E[N(t− w) + 1] g(dw)

= g(0)−∫ t

0E[N(t− w)] g(dw) .

T¸ra èqoume

E[S(t+ h)− S(t)] = −∫ t

0{E[N(t+ h− y)]−E[N(t− y)]} g(dy)

−∫ t+h

tE[N(t+ h− y)] g(dy) .

Sth sunèqeia qrhsimopoioÔme to ananewtikì Je¸rhma Blackwell (blèpe

[3, Je¸rhma 22]), pou mac epitrèpei na efarmìsoume to je¸rhma thc ku-

riarqhmènhc sÔgklishc sto pr¸to olokl rwma gia na p�roume to ìrio.

Qrhsimopoi¸ntac thn monotonÐa thc E[N(t)], o deÔteroc ìroc kuriarqeÐ-

tai apì thn èkfrash E[N(h)] [g(t) − g(t + h)], pou sugklÐnei sto mhdèn

kaj¸c t→∞.

1.3 Klasikì montèlo kindÔnou

'Estw N(t) mia shmeiak  diadikasÐa katamètrhshc apozhmi¸sewn, pou an-

tistoiqeÐ se mia diadikasÐa Poisson me rujmì λ. Autì shmaÐnei ìti oi

tuqaÐoi qrìnoi, metaxÔ diadoqik¸n proseleÔsewn apozhmi¸sewn, sqhma-

tÐzoun mia akoloujÐa anex�rthtwn isìnomwn tuqaÐwn metablht¸n me thn

ex c ekjetik  katanom 

A(x) = 1− e−λx .

1.3. KLASIK�O MONT�ELO KIND�UNOU 15

Epomènwc, gia k�je qronik  stigm  t ≥ 0, h tuqaÐa metablht  N(t) ako-

loujeÐ thn katanom  Poisson me par�metro λ t pou dÐnetai apì ton tÔpo

P[N(t) = k] =(λ t)k

k!e−λt ,

gia k = 0, 1, . . .. Me autèc tic dÔo paradoqèc paÐrnoume èna arket� aplì

montèlo pou onom�zetaiKlasikì Montèlo KindÔnou (KMK) kai parÐstatai

apì thn tri�da (λ, c, B).

Sto KMK, ìpwc eÐdame apì ton tÔpo tou Wald (blèpe sqèsh (1.2.1)),

h mèsh sunolik  apozhmÐwsh sto di�sthma [0, t] eÐnai Ðsh me

E[S(t)] = E[N(t)]E[Z1] = λ t b1 ,

kaj¸c to S(t) apoteleÐ tuqaÐo �jroisma anex�rthtwn isìnomwn tuqaÐwn

metablht¸n ìpou to pl joc twn prosjetèwn N(t) den exart�tai apì touc

ìrouc tou ajroÐsmatoc. Epomènwc mporoÔme na parathr soume ìti to

kl�sma

E[C(t)]−E[S(t)]

E[S(t)],

den exart�tai apì ton qrìno t, opìte sto KMK o tÔpoc (1.1.1) gia th

sqetik  epib�runsh asfaleÐac paÐrnei thn morf 

ρ =c

λ b1− 1 , (1.3.1)

kai to AxÐwma KajaroÔ Kèrdouc (blèpe sqèsh (1.1.2) ) gÐnetai c > λ b1.

SunoyÐzontac, sto klasikì montèlo kindÔnou deqìmaste ìti:

1. Oi qronikèc stigmèc twn apozhmi¸sewn {Tk, k = 0, 1, . . .} sqhma-tÐzoun omogen  diadikasÐa Poisson me rujmì λ kai T0 = 0.

2. H k apozhmÐwsh, pou emfanÐzetai thn stigm  Tk, èqei Ôyoc Zk. To

sÔnolo twn uy¸n twn apozhmi¸sewn {Zk, k ∈ N} apoteleÐ akolou-jÐa anex�rthtwn isìnomwn mh arnhtik¸n tuqaÐwn metablht¸n.

16 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

3. Oi dÔo akoloujÐec {Tk, k = 0, 1, . . .} kai {Zk, k ∈ N} eÐnai ane-x�rthtec metaxÔ touc.

Ja exet�soume paradeÐgmata sugkekrimènwn katanom¸n gia ta Ôyh

twn apozhmi¸sewn {Zk, k ∈ N}. Merik� apì aut� proèkuyan mèsa apì

jewrhtikèc proseggÐseic, en¸ �lla emfanÐzontai stic praktikèc efarmo-

gèc.

1.4 Ekjetikèc apozhmi¸seic

JewroÔme thn ekjetik  katanom  gia ta Ôyh twn apozhmi¸sewn me par�-

metro µ > 0, sthn morf  B(x) = 1− e−µx kai puknìthta b(x) = µ e−µx.

H ekjetik  katanom  eÐnai h aploÔsterh perÐptwsh kai autì mac epitrè-

pei na fèroume se pèrac ìlouc touc upologismoÔc kai na broÔme komy�

apotelèsmata. To montèlo autì parÐstatai me thn tri�da (λ, c, µ).

Sthn perÐptwsh tou montèlou (λ, c, µ) upologÐzontai eÔkola oi ropèc

bs =Γ(s+ 1)

µs

ìpou ed¸ qrhsimopoi same thn sun�rthsh G�mma

Γ(s) =

∫ ∞0

ys−1 e−y dy .

Parapèra o tÔpoc gia th sqetik  epib�runsh asfaleÐac (1.1.1) gÐnetai

ρ =c µ

λ− 1 . (1.4.1)

'Otan X1 := Z1−c θ1 > u apì tic sqèseic (1.1.5) paÐrnoume san qrìno

qreokopÐac τ(u) = T1. T¸ra ac jumÐsoume thn akoloujÐa {Un} sth sqèsh(1.1.5). 'Otan Z1− c θ1 ≤ u, tìte U1 ≥ 0 kai apì thn markobian  idiìthta

thc akoloujÐac {Un, n = 0, 1, . . .} ta U2, U3, . . . exart¸ntai mìno apì to

1.4. EKJETIK�ES APOZHMI�WSEIS 17

U1. 'Etsi me b�sh ton tÔpo thc olik c pijanìthtac brÐskoume

φ(u) = 1− ψ(u) = P

[infn≥2

Un ≥ 0 , U1 ≥ 0∣∣ U0 = u

]

=

∫ ∞0

P

[infn≥2

Un ≥ 0 , U1 ∈ dx∣∣ U0 = u

]

=

∫U1≥0

P

[infn≥2

Un ≥ 0∣∣ U1 ≥ 0 , U0 = u

]P [(θ1, Z1) ∈ (dt, dz)] ,

epomènwc

φ(u) =

∫θ,Z

P

[infn≥2

Un ≥ 0∣∣ U1 = u+ c θ1 − Z1

]·P [(θ1, Z1) ∈ (dt, dz) | u+ c θ1 − Z1 ≥ 0]

= Eθ,Z

{P

[infn≥2

Un ≥ 0∣∣U1 = u+ c θ1 − Z1

] ∣∣∣U1 ≥ 0

}= Eθ,Z [φ(u+ c θ1 − Z1) | u+ c θ1 ≥ Z1]

=

∫ ∞0

(∫ u+ct

0φ(u+ c t− z)µ e−µz dz

)λ e−λt dt .

T¸ra me allag  metablht c (y antÐ u+ c t), paÐrnoume

φ(u) =λ

c

∫ ∞u

eλ(u−y)/c(∫ y

0φ(y − z)µ e−µz dz

)dy .

H èkfrash aut  faner¸nei ìti h pijanìthta epibÐwshc φ(u) paragwgÐze-

tai wc proc u. PaÐrnontac parag¸gouc sta dÔo mèlh èqoume thn ex c

oloklhrodiaforik  exÐswsh

φ′(u) =λ

c

(φ(u)−

∫ u

0φ(u− z)µ e−µz dz

). (1.4.2)

Sth sunèqeia, oloklhr¸nontac thn (1.4.2) sto di�sthma [0, x], brÐ-

skoume

φ(x)− φ(0) =λ

c

(∫ x

0φ(y) dy

−∫ x

0

(∫ y

0φ(y − z)µ e−µz dz

)dy

). (1.4.3)

18 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

T¸ra k�noume ton upologismì tou diploÔ oloklhr¸matoc. Me allag 

metablht c (v antÐ y − z) brÐskoume∫ x

0

(∫ y

0φ(y − z)µ e−µz dz

)dy =

∫ x

0µ e−µy

(∫ y

0φ(v) eµv dv

)dy

=

∫ x

0φ(v) eµv

(∫ x

vµ e−µy dy

)dv =

∫ x

0φ(v)[1− e−µ(x−v)] dv .

B�zontac thn teleutaÐa èkfrash sth sqèsh (1.4.3), katal goume

φ(x)− φ(0) =λ

c

(∫ x

0φ(v) dv −

∫ x

0φ(v) [1− e−µ(x−v)] dv

)=

λ

c

∫ x

0φ(v) e−µ(x−v) dv =

λ

c

∫ x

0φ(x− y) e−µy dy .

'Ara, met� apì antikat�stash sthn pijanìthta qreokopÐac, èqoume

ψ(u) = ψ(0)− λ

c

∫ u

0[1− ψ(u− y)] e−µy dy . (1.4.4)

Parapèra gia ton upologismì tou ψ(0) paÐrnoume to ìrio tou ψ(u) ka-

j¸c to u→∞. Apì to AxÐwma KajaroÔ Kèrdouc (blèpe sqèsh (1.1.2))

èqoume ìti

E[Xk] = E(Zk − c θk) =1

µ− c

λ< 0 ,

kai ìti ta {Xk = Zk − c θk, k ∈ N} apoteloÔn akoloujÐa anex�rthtwn

isìnomwn tuqaÐwn metablht¸n me �jroisma

Sn =

n∑k=1

Xk =n∑k=1

Zk − c θk , (1.4.5)

opìte apì ton isqurì nìmo twn meg�lwn arijm¸n brÐskoume

P[

limn→∞

Sn = −∞]

= 1 .

Apì tic idiìthtec tou tuqaÐou perip�tou (blèpe [3, Prìtash 11]) prokÔptei

ìti

P[supn≥1

Sn <∞] = 1 . (1.4.6)

1.4. EKJETIK�ES APOZHMI�WSEIS 19

All� apì thn sqèsh (1.1.5) jumìmaste ìti

Un = u− Sn , (1.4.7)

kai apì thn (1.4.6) katal goume sto ex c

limu→∞

ψ(u) = limu→∞

P[supn≥1

Sn > u] = 0 . (1.4.8)

T¸ra, xanagurn¸ntac sth sqèsh (1.4.4) kai qrhsimopoi¸ntac to te-

leutaÐo ìrio, paÐrnoume

ψ(0) = limu→∞

ψ(u) + limu→∞

λ

c

∫ u

0[1− ψ(u− y)] e−µy dy

=λ

c

∫ ∞0

e−µy dy − limu→∞

λ

c

∫ u

0ψ(u− y) e−µy dy

=λ

c µ− limu→∞

λ

c

∫ u/2

0ψ(u− y) e−µy dy

− limu→∞

λ

c

∫ u

u/2ψ(u− y) e−µy dy ,

all� apì thn (1.4.8)

0 ≤∫ u/2

0ψ(u− y) e−µy dy ≤ ψ

(u2

) ∫ u/2

0e−µy dy → 0 ,

kaj¸c u→∞ kai paÐrnontac up' ìyin ìti h ψ(u) eÐnai mh aÔxousa sun�r-

thsh, blèpoume ìti

0 ≤∫ u

u/2ψ(u− y) e−µy dy ≤ e−µu/2

∫ u/2

0ψ(z) dz

≤ e−µu/2 ψ(0)u

2→ 0 ,

kaj¸c u→∞ opìte apì thn sqèsh (1.4.1) prokÔptei

ψ(0) =λ

c µ=

1

1 + ρ. (1.4.9)

20 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

Tèloc, me antikat�stash tou ψ(0) sthn sqèsh (1.4.4) katal goume

ψ(u) =λ

c

∫ ∞0

e−µy dy − λ

c

∫ u

0e−µy dy +

λ

c

∫ u

0ψ(y) e−µ(u−y) dy

=λ

c

∫ ∞u

e−µy dy +λ

ce−µu

∫ u

0ψ(y) eµy dy ,

pou dÐnei thn oloklhrwtik  exÐswsh

ψ(u) =λ

c µe−µu +

λ

ce−µu

∫ u

0ψ(y) eµy dy . (1.4.10)

Apì ed¸ xekin�ei h an�lush thc pijanìthtac qreokopÐac. Sugkekri-

mèna, paÐrnontac parag¸gouc sta dÔo mèlh thc exÐswshc, èqoume

ψ′(u) = −λµce−µu

(∫ u

0ψ(y) eµy dy +

1

µ

)+λ

cψ(u)

=

(λ

c− µ

)ψ(u) < 0 .

T¸ra oloklhr¸nontac ta dÔo mèlh brÐskoume

ψ(u) = ψ(0) exp

{−(µ− λ

c

)u

}kai me b�sh ton tÔpo (1.4.9) katal goume sto akìloujo apotèlesma.

Je¸rhma 2. Sto klasikì montèlo kindÔnou (λ, c, µ) me ekjetik� kata-

nemhmènec apozhmi¸seic h pijanìthta qreokopÐac dÐnetai apì ton tÔpo

ψ(u) =λ

c µexp

{−(µ− λ

c

)u

}=

1

1 + ρexp

{−λ ρ

cu

}

=1

1 + ρexp

{−µ ρ

1 + ρu

}. (1.4.11)

Se aut  thn perÐptwsh èqoume lÔsh thc exÐswshc (1.4.10) se kleist 

morf . Me �lla lìgia, mporoÔme na upologÐsoume thn pijanìthta qreo-

kopÐac ψ(u) gnwrÐzontac tic treic paramètrouc tou montèlou (λ, c, µ).

1.5. KLASIK�O MONT�ELO ME GENIK�ES APOZHMI�WSEIS 21

Up�rqei ìmwc kai �lloc trìpoc epÐlushc thc exÐswshc (1.4.10). Ton

parajètoume sunoptik�, epishmaÐnontac ìti en¸ eÐnai suntomìteroc den

eÐnai p�nta leitourgikìc, me thn ènnoia ìti se efarmogèc ìpou h antistrof 

twn metasqhmatism¸n gÐnetai polÔplokh, den dÐnei �mesa apotelèsmata.

SumbolÐzoume me

B(s) :=

∫ ∞0

e−sxB(x) dx , (1.4.12)

ton metasqhmatismì Laplace miac sun�rthshc B(x). PaÐrnontac touc me-

tasqhmatismoÔc Laplace sta dÔo mèlh thc (1.4.10) èqoume thn algebrik 

exÐswsh

ψ(s) =λ

c µ (µ+ s)+

λ

c (µ+ s)ψ(s) .

LÔnontac wc proc ψ(s)

ψ(s) =λ

c µ

(µ− λ

c+ s

) ,

èqoume ton metasqhmatismì Laplace thc pijanìthtac qreokopÐac, pou me

antistrof  dÐnei

ψ(u) =λ

c µexp

{−(µ− λ

c

)u

}.

1.5 Klasikì montèlo me genikèc apozhmi¸-

seic

T¸ra ja exet�soume p�li to Ðdio prìblhma upologismoÔ thc pijanìthtac

qreokopÐac, all� se pio genik  morf , ìtan h katanom  twn uy¸n twn a-

pozhmi¸sewn B(x) den paÐrnei ekjetik  morf , opìte eÐmaste sto montèlo

(λ, c, B).

22 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

P�li xekin�me apì thn sqèsh (1.1.6), ton tÔpo thc olik c pijanìthtac

kai paÐrnontac up' ìyin ìti ψ(u) = 1 gia u < 0, brÐskoume ìpwc sthn

perÐptwsh ekjetik c katanom c, ìti gia Z1 − c θ1 ≤ u, tìte

φ(u) = Eθ,Z

{P

[infn≥2

Un ≥ 0∣∣U1 = u+ cθ1 − Z1

] ∣∣∣U0 = u, U1 ≥ 0

}= Eθ,Z [φ(u+ c θ1 − Z1) | u+ cθ1 ≥ Z1]

=

∫ ∞0

(∫ u+ct

0φ(u+ c t− z)B(dz)

)λ e−λt dt

=λ

c

∫ ∞u

(∫ y

0φ(y − z)B(dz)

)e−λ(y−u)/c dy ,

ìpou sthn teleutaÐa isìthta k�name allag  metablht c (y antÐ u + c t).

Ed¸ èqoume thn pijanìthta epibÐwshc sthn morf  ginomènou thc ekjetik c

sun�rthshc eλu/c kai tou oloklhr¸matoc

λ

c

∫ ∞u

(∫ y

0φ(y − z)B(dz)

)e−λy/c dy ,

pou eÐnai paragwgÐsima wc proc u. ParagwgÐzontac kai qrhsimopoi¸ntac

thn prohgoÔmenh èkfrash thc pijanìthtac qreokopÐac paÐrnoume

φ′(u) =λ

c

(φ(u)−

∫ u

0φ(u− z)B(dz)

).

T¸ra oloklhr¸noume ta dÔo mèlh gia na katal xoume

φ(x)− φ(0) =λ

c

(∫ x

0φ(y)dy −

∫ x

0

[∫ y

0φ(y − z)B(dz)

]dy

). (1.5.1)

Ac doÔme pwc upologÐzetai to diplì olokl rwma∫ x

0

∫ y

0φ(y − z)B(dz) dy =

∫ x

0

∫ x−z

0φ(w) dwB(dz)

=

∫ x

0φ(w)B(x− w) dw =

∫ x

0φ(x− v)B(v) dv .

1.5. KLASIK�O MONT�ELO ME GENIK�ES APOZHMI�WSEIS 23

T¸ra antikajist¸ntac thn teleutaÐa èkfrash sthn exÐswsh (1.5.1)

èqoume

φ(x)− φ(0) =λ

c

∫ x

0φ(x− y)B(y) dy , (1.5.2)

pou dÐnei telik�

ψ(0)− ψ(u) =λ

c

∫ u

0[1− ψ(u− y)]B(y) dy . (1.5.3)

T¸ra mènei na upologÐsoume to ψ(0). 'Opwc eÐdame apì th sqèsh

(1.4.8), gia u→∞ h pijanìthta qreokopÐac mhdenÐzetai. Opìte h sqèsh

(1.5.3) dÐnei

ψ(0) = limu→∞

λ

c

∫ u

0B(y) dy − lim

u→∞

λ

c

∫ u/2

0ψ(u− y)B(y) dy

− limu→∞

λ

c

∫ u

u/2ψ(u− y)B(y) dy .

All� blèpoume ìti apì thn upìjesh b1 <∞

0 ≤ limu→∞

∫ u/2

0ψ(u− y)B(y) dy ≤

∫ ∞0

B(y) dy limu→∞

ψ(u

2

)= 0 ,

0 ≤ limu→∞

∫ u

u/2ψ(u− y)B(y) dy ≤ lim

u→∞

∫ u/2

0ψ(w) dwB

(u2

)= 0 ,

ìpou to teleutaÐo ìrio eÐnai orjì diìti apì tic (1.5.1) - (1.5.3) prokÔptei∫ ∞0

ψ(y) dy <∞ .

'Etsi èqoume

ψ(0) =λ

c

∫ ∞0

B(y) dy =λ b1c

=1

1 + ρ. (1.5.4)

T¸ra gurn�me pÐsw sthn exÐswsh (1.5.3), ìpou antikajist�me to apotè-

lesma sth sqèsh (1.5.4)

ψ(u) =λ

c

∫ ∞u

B(y) dy +λ

c

∫ u

0ψ(u− y)B(y) dy . (1.5.5)

24 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

Ac sumbolÐsoume ton metasqhmatismì Laplace-Stieltjes miac sun�rth-

shc B(x) me

B(s) =

∫ ∞0

s e−sxB(x) dx =

∫ ∞0

e−sxB(dx) .

PaÐrnontac t¸ra touc metasqhmatismoÔc Laplace-Stieltjes sta dÔo mèlh

thc (1.5.5) èqoume

ψ(s) :=

∫ ∞0

e−sx dψ(x) =λ

c

∫ ∞0

B(y)

∫ y

0s e−s x dx dy

+λ

c

∫ ∞0

B(y)

∫ ∞y

s e−s x ψ(x− y) dx dy

=λ

c

∫ ∞0

B(y) dy − λ

c

∫ ∞0

B(y) e−sy dy

+λ

c

∫ ∞0

B(y) e−sy∫ ∞0

s e−sw ψ(w) dw dy ,

kai parapèra

ψ(s) =λ b1c− λ

c

∫ ∞0

∫ x

0e−s y dy B(dx)

+λ

c

∫ ∞0

e−s y∫ ∞y

B(dx) dy ψ(s)

=λ

c

(b1 −

1− B(s)

s

)+ λ

1− B(s)

c sψ(s) ,

ap' ìpou lÔnontac wc proc ψ(s)

ψ(s) =λ b1 s− λ+ λ B(s)

c s− λ+ λ B(s). (1.5.6)

'Estw h katanom  thc oloklhrwmènhc our�c

B0(u) :=1

b 1

∫ u

0B(y) dy .

1.5. KLASIK�O MONT�ELO ME GENIK�ES APOZHMI�WSEIS 25

Opìte h ananewtik  exÐswsh (1.5.2) gÐnetai

φ(u) =ρ

1 + ρ+

1

1 + ρ

∫ u

0φ(u− y)B0(dy) .

ParathroÔme ìti o metasqhmatismìc Laplace-Stieltjes ekfr�zetai san

φ(s) = s φ(s) = 1− s ψ(s) = 1− ψ(s) ,

kai

B0(s) =1− B(s)

b1 s. (1.5.7)

'Etsi apì thn (1.5.6) kai (1.3.1) paÐrnoume ìti

φ(s) =c− λ b1

c− λ

s

[1− B(s)

] =ρ

1 + ρ− B0(s)=

ρ

1 + ρ

∞∑k=0

(B0(s)

1 + ρ

)k,

ψ(s) = 1− ρ

1 + ρ

∞∑k=0

(1

1 + ρ

)kB0

k(s) (1.5.8)

=ρ

1 + ρ

∞∑k=1

(1

1 + ρ

)k [1− B0

k(s)].

T¸ra antistrèfontac ton metasqhmatismì Laplace-Stieltjes brÐskoume ton

tÔpo Pollaczeck-Khinchin, pou eÐnai gnwstìc apì thn jewrÐa our¸n

anamon c (blèpe [1])

φ(u) =ρ

1 + ρ

∞∑k=0

(1

1 + ρ

)kBk∗

0 (u) ,

ψ(u) =ρ

1 + ρ

∞∑k=1

(1

1 + ρ

)kBk∗

0 (u) . (1.5.9)

Aut  h anapar�stash thc pijanìthtac qreokopÐac sto klasikì montèlo

kindÔnou sthn morf  sÔnjethc gewmetrik c katanom c diathreÐtai kai sto

ananewtikì montèlo kindÔnou, ìpwc ja doÔme parak�tw.

26 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

Parapèra, parathroÔme ìti h sqèsh (1.5.5) èqei thn morf  elleimma-

tik c ananewtik c exÐswshc diìti to mètro olokl rwshc

λ

c

∫ x

0B(y) dy ,

antistoiqeÐ se ellip  katanom , dhlad  se katanom  pou èqei ìrio kaj¸c

to x → ∞ mikrìtero thc mon�dac. Pr�gmati, apì to AxÐwma KajaroÔ

Kèrdouc (blèpe sqèsh (1.1.2)) èqoume

λ

c

∫ ∞0

B(y) dy =λ b1c

< 1 ,

kai gi' autì den efarmìzetai to Je¸rhma-kleidÐ thc jewrÐac ananèwshc

(blèpe [3, Je¸rhma 23]), ìpou h katanom  sth sunèlixh me thn �gnwsth

sun�rthsh prèpei na eÐnai tèleia.

Gia na antimetwpÐsoume aut  thn duskolÐa, pollaplasi�zoume ta dÔo

mèlh thc exÐswshc (1.5.5) me k�poion par�gonta eRx, ìpou emfanÐzetai

k�poia stajer� R > 0, pou onom�zetai rujmistikìc suntelest c, kai ika-

nopoieÐ thn ex c sunj kh Cramer

λ

c

∫ ∞0

B(y) eRy dy =λ

cR

∫ ∞0

(1− eRy)B(dy) = 1 . (1.5.10)

Pr�gmati, me aut  thn epilog  tou rujmistikoÔ suntelest  R paÐrnoume

pl rh ananewtik  exÐswsh me �gnwsth th sun�rthsh ψR(u) = eRu ψ(u)

kai thn tèleia katanom  sth sunèlixh

BR(x) =λ

c

∫ x

0B(y) eRy dy =

λ b1c

∫ x

0eRy B0(dy) ,

opìte h ananewtik  exÐswsh paÐrnei thn morf 

ψR(u) =λ

ceRu

∫ ∞u

B(y) dy +λ

c

∫ u

0ψR(u− y)B(y) eRy dy

= z(u) + ψR ∗BR(u) , (1.5.11)

ìpou

z(x) =λ

ceRx

∫ ∞x

B(y) dy .

1.5. KLASIK�O MONT�ELO ME GENIK�ES APOZHMI�WSEIS 27

T¸ra ac sumbolÐsoume

bR1 :=

∫ ∞0

y BR(dy) =λ b1c

∫ ∞0

y eRy B0(dy) . (1.5.12)

opìte mporoÔme na efarmìsoume sthn ananewtik  exÐswsh (1.5.11) to

Je¸rhma-kleidÐ, ¸ste na broÔme

limu→∞

ψR(u) =1

bR1

∫ ∞0

z(y)dy ,

ap' ìpou prokÔptei h ex c asumptwtik  sqèsh gia thn pijanìthta qreo-

kopÐac

ψ(u) ∼ e−Ru

bR1

∫ ∞0

z(y)dy ,

kaj¸c u→∞. 'Etsi paÐrnoume sto akìloujo apotèlesma.

Je¸rhma 3. JewroÔme to KMK (λ, c, B) kai upojètoume ìti up�rqei

k�poioc rujmistikìc suntelest c R > 0, tètoioc ¸ste na isqÔei h sunj kh

Cramer (1.5.10).

E�n bR1 <∞, tìte isqÔei o asumptwtikìc tÔpoc Cramer-Lundberg

ψ(u) ∼ ρ

(1 + ρ) bR1 Re−Ru , (1.5.13)

kaj¸c u→∞.

E�n bR1 =∞, tìte isqÔei h asumptwtik  sqèsh

limu→∞

ψ(u) eRu = 0 . (1.5.14)

Apìdeixh. Kaj¸c h èkfrash

λ

c

∫ x

0eRy B(y) dy , (1.5.15)

apoteleÐ tèleia katanom  pijanot twn me puknìthta

λ

ceRxB(x) , (1.5.16)

28 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

kai h upìjesh bR1 <∞ exasfalÐzei thn Ôparxh thc antÐstoiqhc mèshc ti-

m c, h bohjhtik  sun�rthsh ψR(u) brÐsketai san lÔsh thc pl rouc ana-

newtik c exÐswshc (1.5.11). Apì thn èkfrash gia thn puknìthta (1.5.16)

prokÔptei epÐshc ìti h katanom  den eÐnai arijmhtik . Epomènwc, efarmì-

zontac to ananewtikì Je¸rhma-kleidÐ, brÐskoume ìti up�rqei to ìrio

limu→∞

ψR(u) =

∫∞0 z(y)dy∫∞

0 y BR(dy)=

λ

c bR1

∫ ∞0

eRy∫ ∞y

B(z) dz dy .

All�

λ

c

∫ ∞0

eRy∫ ∞y

B(z) dz dy =λ

c

∫ ∞0

B(z)

∫ z

0eRy dy dz

=λ

cR

∫ ∞0

B(z) (eRz − 1) dz =ρ

(1 + ρ)R

ìpou h teleutaÐa isìthta ofeÐletai sthn sunj kh (1.5.10).

ShmeÐwsh 1. Apì ton asumptwtikì tÔpo (1.5.13) prokÔptei ìti h pi-

janìthta qreokopÐac fjÐnei san ekjetik  sun�rthsh kai anadeiknÔei th

shmasÐa tou rujmistikoÔ suntelest  R.

1.6 Sun�rthsh proexoflhtik c poin c

T¸ra ja melet soume ìqi mìno thn pijanìthta qreokopÐac all� kai tic

sunj kec k�tw apì tic opoÐec h qreokopÐa emfanÐzetai. Sthn ergasÐa [2]

prot�jhke o akìloujoc metasqhmatismìc Esscher

mα(u) = E[e−α τ(u)w(U [τ(u)−], |U [τ(u)]|)1{τ(u)<∞}

∣∣U(0) = u],

(1.6.1)

ìpou α ≥ 0 kai w(t, s) k�poia jetik  metr simh sun�rthsh dÔo metablh-

t¸n t ≥ 0 , s ≥ 0. Ed¸ to U [τ(u)−] sumbolÐzei to apìjema prin ton

qrìno qreokopÐac kai to −U [τ(u)] deÐqnei th drimÔthta qreokopÐac kai

1.6. SUN�ARTHSH PROEXOFLHTIK�HS POIN�HS 29

isoÔtai me to èlleimma thc etaireÐac kat� ton qrìno qreokopÐac. To apì-

jema prin th qreokopÐa kai h drimÔthta thc qreokopÐac dÐnoun pl rh eikìna

twn oikonomik¸n sunjhk¸n kat� ton qrìno qreokopÐac. H (1.6.1) p re to

ìnoma sun�rthsh proexoflhtik c poin c Gerber-Shiu kai apoteleÐ lÔsh

thc akìloujhc ananewtik c exÐswshc

Je¸rhma 4. Sto KMK (λ, c, B) h sun�rthsh proexoflhtik c poin c

ikanopoieÐ thn ananewtik  exÐswsh

mα(u) =λ

ceRu

∫ ∞u

e−Rx∫ ∞x

w(x, z − x)B(dz) dx

+λ

c

∫ u

0mα(u− x)

∫ ∞x

e−R (z−x)B(dz) dx , (1.6.2)

ìpou to R apoteleÐ thn mègisth (jetik ) lÔsh thc exÐswshc

λ+ α− cx = λ B(x) . (1.6.3)

Apìdeixh. Pr�gmati, upojètoume to klasikì montèlo kindÔnou

kai gia k�poio arket� mikrì h > 0 jewroÔme to endeqìmeno emf�nishc

apozhmÐwshc sto di�sthma (0, h). H pijanìthta na emfanisteÐ h pr¸th

apozhmÐwsh sto apeirostì di�sthma (t, t + dt) isoÔtai me λ e−λt dt kai

h pijanìthta dÔo apozhmi¸sewn sto di�sthma (0, h) jewreÐtai amelhtèa,

opìte apì thn sqèsh (1.6.1) èqoume

mα(u) =

∫ h

0

[∫ ∞u+c t

w(u+ c t, z − u− c t)B(dz)

]λ e−(λ+α) t dt

+

∫ h

0

[∫ u+c t

0mα(u+ c t− z)B(dz)

]λ e−(λ+α) t dt

+e−(λ+α)hmα(u+ c h) .

ParagwgÐzontac wc proc h kai af nontac to h na teÐnei sto mhdèn, brÐ-

skoume

(λ+ α)mα(u) = cm′α(u) + λ a(u) + λ

∫ u

0mα(u− z)B(dz) , (1.6.4)

30 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

ìpou qrhsimopoi same to sumbolismì

a(u) :=

∫ ∞u

w(u, z − u)B(dz) =

∫ ∞0

w(u, y)B(u+ dy) .

'Estw gia k�poio pragmatikì R ≥ 0 h bohjhtik  sun�rthsh mRα (u) =

e−Rumα(u). Pollaplasi�zontac thn sqèsh (1.6.4) me e−Ru èqoume

cmRα′(u) = (λ+ α− cR)mR

α (u) (1.6.5)

−λ e−Ru a(u)− λ∫ u

0mRα (u− z) e−Rz B(dz) .

JewroÔme thn sqèsh (1.6.3), h opoÐa èqei mia kai monadik  mh arnhtik 

lÔsh. H lÔsh aut  apoteleÐ aÔxousa sun�rthsh wc proc thn par�metro

α me arqik  tim  sto mhdèn. E�n epilèxoume gia to R aut  thn lÔsh thc

(1.6.3), tìte h exÐswsh (1.6.5) gÐnetai

c

λmRα′(u)

= B(R)mRα (u)− e−Ru a(u)−

∫ u

0mRα (u− z) e−Rz B(dz)

= B(R)mRα (u)− e−Ru a(u)−

∫ u

0mRα (v) e−R(u−v)B(u− dv) .

T¸ra me olokl rwsh sto di�sthma (0, y) brÐskoume

c

λ[mR

α (y)−mRα (0)] +

∫ y

0e−Ru a(u) du

= B(R)

∫ y

0mRα (u) du−

∫ y

0

∫ u

0mRα (v) e−R (u−v)B(u− dv) du

=

∫ ∞0

e−RzB(dz)

∫ y

0mRα (u) du

−∫ y

0mRα (v)

∫ y

ve−R (u−v)B(du− v) dv

=

∫ y

0mRα (v)

∫ ∞0

e−RzB(dz)dv −∫ y

0mRα (v)

∫ y−v

0e−RzB(dz)dv

=

∫ y

0mRα (v)

∫ ∞y−v

e−RzB(dz)dv =

∫ ∞0

e−Rz∫ y

y−zmRα (v)dvB(dz)

1.6. SUN�ARTHSH PROEXOFLHTIK�HS POIN�HS 31

kai af nontac to y na teÐnei sto �peiro, h teleutaÐa èkfrash gÐnetai mhdèn

kai isqÔei mRα (y)→ 0, opìte èqoume

mRα (0) =

λ

c

∫ ∞0

e−Ru a(u) du .

Met� apì antikat�stash sthn prohgoÔmenh sqèsh brÐskoume

mRα (y)

=λ

c

∫ ∞y

e−Rva(v) dv +λ

c

∫ y

0mRα (v)

∫ ∞y−v

e−Rz B(dz) dv .

T¸ra gurn¸ntac pÐsw sth sun�rthsh proexoflhtik c poin c paÐrnoume

thn exÐswsh

mα(y)

=λ

c

∫ ∞y

e−R(v−y)a(v) dv +λ

c

∫ y

0mα(v)

∫ ∞y−v

e−R (z−y+v)B(dz)dv ,

pou mac dÐnei pl rh ananewtik  exÐswsh sthn morf  (1.6.2).

ShmeÐwsh 2. E�n jèsoume α = 0 kai w(t, s) = 1 gia k�je t ≥ 0 , s ≥ 0,

h sun�rthsh (1.6.1) sumpÐptei me thn pijanìthta qreokopÐac m0(u) =

ψ(u) = P[τ(u) <∞].

32 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

1.7 Ask seic

'Askhsh 1. DeÐxte touc tÔpouc (1.2.3) - (1.2.4) qrhsimopoi¸ntac tic

parag gouc thc genn triac tou tuqaÐou ajroÐsmatoc S(t).

'Askhsh 2. BreÐte ton tÔpo thc genn triac ajroÐsmatoc dÔo anex�rth-

twn kai isìnomwn sÔnjetwn Poisson S1(t) +S2(t). DeÐxte ìti eÐnai epÐshc

sÔnjeth Poisson.

'Askhsh 3. BreÐte thn diadikasÐa apojèmatoc mèqri th stigm  T19 kai

proseggÐste thn sqetik  epib�runsh asfaleÐac ρ, ìtan dÐnontai ta akì-

louja stoiqeÐa: u = 100.00 , c = 0.4 (  0.3),

θ1 = 1.09594, θ2 = 3.60119, θ3 = 6.21095, θ4 = 0.0951223 ,

θ5 = 3.57556, θ6 = 1.54286, θ7 = 0.262182, θ8 = 2.73497 ,

θ9 = 0.073235, θ10 = 3.32073, θ11 = 0.776819, θ12 = 0.426323 ,

θ13 = 3.13747, θ14 = 0.577684, θ15 = 2.11186, θ16 = 0.90246 ,

θ17 = 0.638577, θ18 = 8.45489, θ19 = 5.95426 ,

Z1 = 2.53284, Z2 = 1.55106, Z3 = 11.1218, Z4 = 0.408499 ,

Z5 = 2.63738, Z6 = 9.82471, Z7 = 1.58155, Z8 = 4.79439 ,

Z9 = 29.193, Z10 = 11.4883, Z11 = 5.61012, Z12 = 2.28808 ,

Z13 = 5.16965, Z14 = 24.8462, Z15 = 13.2114, Z16 = 0.543624 ,

Z17 = 75.3512, Z18 = 4.19297, Z19 = 13.6481 .

'Askhsh 4. H sunolik  apozhmÐwsh S(t) sto qronikì di�sthma [0, t]

prokÔptei apì to pl joc apozhmi¸sewn N(t) me arnhtik  diwnumik  ka-

tanom 

P[N(t) = m] =(r +m− 1)!

r! (m− 1)!pr (1− p)m ,

1.7. ASK�HSEIS 33

gia m = 0, 1, . . ., me paramètrouc r > 0 kai 0 < p < 1, kai thn logarij-

mokanonik  katanom  apozhmÐwshc me puknìthta

b(x) =1

2x√

2πexp

[−(lnx− 1)2

8

],

gia x > 0. UpologÐste ta E[S(t)] kai var[S(t)].

Upìdeixh:

E[N(t)] =r (1− p)

p,

kai

var[N(t)] =r (1− p)

p2.

'Ara

MN(t)(s) =

(p

1− (1− p) es

)r.

'Askhsh 5. Ac upojèsoume ìti èqoume to klasikì montèlo kindÔnou

me ekjetikèc apozhmi¸seic pou parÐstatai me thn bo jeia thc tri�dac

(λ, c, µ) = (0.1, 1, 0.2). Na elègxete to AxÐwma KajaroÔ Kèrdouc kai

na upologÐsete thn pijanìthta qreokopÐac ψ(100) gia arqikì kef�laio

u = 100. Na prosdiorÐsete to arqikì kef�laio v pou mac dÐnei pijanì-

thta qreokopÐac ψ(v) = 5 · 10−5. DÐnontai ln 10−4 = −9.21 kai e−10 =

4.54 · 10−5.

'Askhsh 6. Sto klasikì montèlo (λ, c, µ) mporoÔme na aux soume to

arqikì kef�laio apì u se r u, me r > 1 kai na mei¸soume antÐstoiqa ton

rujmì asf�listrou apì c se c′. DeÐxte ìti me thn upìjesh ìti h pijanìthta

qreokopÐac paramènei Ðdia ta c kai c′ ikanopoioÔn thn sqèsh

ln c− ln c′ = µu

[r − 1− λ

µ

(r

c′− 1

c

)].

'Askhsh 7. BreÐte ton rujmistikì suntelest  R gia ekjetik  katanom 

apozhmi¸sewn me par�metro µ kai sqetik  epib�runsh asfaleÐac ρ kai

twn paramètrwn λ, c.

34 KEF�ALAIO 1. KLASIK�H MONTELOPO�IHSH KIND�UNOU

'Askhsh 8. BreÐte ton rujmistikì suntelest  R gia ekfulismènh ka-

tanom  apozhmi¸sewn me stajer  tim  Ðsh me thn mon�da kai sqetik 

epib�runsh asfaleÐac ρ = 0.2.

'Askhsh 9. BreÐte ton rujmistikì suntelest  R gia kanonik  katano-

m  apozhmi¸sewn me mèso µ kai diakÔmansh σ2 kai sqetik  epib�runsh

asfaleÐac ρ.

'Askhsh 10. DeÐxte ìti h exÐswsh (1.6.3) wc proc x, èqei dÔo lÔseic,

mia jetik  kai mia arnhtik .

BibliografÐa

[1] Asmussen, S. (1987) Applied Probability and Queues. Wiley, New

York.

[2] Gerber, H.U., Shiu, E.S.W. (1998) On the time value of ruin.

N. Amer. Actuar. J., 2, 48–78.

[3] Kwnstantinidhc, D.G. (2009) JewrÐa Stoqastik¸n Diadikasi¸n,

Mèroc A. Ekdìseic StamoÔlhc, Aj na.

35

Euret rio

apìjema etaireÐac, 3

arqikì kef�laio, 4

asf�listra, 3

asfalistik  politik , 3

AxÐwma KajaroÔ Kèrdouc, 6

eisìdhma apì asf�listra, 4

epib�runsh asfaleÐac, 3

Klasikì Montèlo KindÔnou, 15

pijanìthta qreokopÐac, 6

qreokopÐa, 6

rujmìc eÐspraxhc asfalÐstrou, 4

rujmistikìc suntelest c, 26

sqetik  epib�runsh asfaleÐac, 5

stigm  emf�nishc atuq matoc, 4

36