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ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΙΓΑΙΟΥ
Θεωρία των Κινδύνων Ι Ενότητα 1: Κλασική Μοντελοποίηση Κινδύνου
Δημήτριος Κωνσταντινίδης
Τμήμα Μαθηματικών
Σάμος, Οκτώβριος 2014
Άδειες Χρήσης
Το παρόν εκπαιδευτικό υλικό υπόκειται σε άδειες χρήσης Creative Commons.
Για εκπαιδευτικό υλικό, όπως εικόνες, που υπόκειται σε άλλου τύπου άδειας
χρήσης, η άδεια χρήσης αναφέρεται ρητώς.
Χρηματοδότηση
Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί στα πλαίσια του εκπαιδευτικού
έργου του διδάσκοντα.
Το έργο «Ανοικτά Ακαδημαϊκά Μαθήματα στο Πανεπιστήμιο Αθηνών» έχει
χρηματοδοτήσει μόνο τη αναδιαμόρφωση του εκπαιδευτικού υλικού.
Το έργο υλοποιείται στο πλαίσιο του Επιχειρησιακού Προγράμματος
«Εκπαίδευση και Δια Βίου Μάθηση» και συγχρηματοδοτείται από την
Ευρωπαϊκή Ένωση (Ευρωπαϊκό Κοινωνικό Ταμείο) και από εθνικούς πόρους.
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