Δημτριος Κωνσταντινδης
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Keflaio 1
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 uperspise
sto Panepist mio thc Ouylac (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. Mlista me thn bo jeia katllhlou
qronikoÔ metasqhmatismoÔ angetai h anlush tou analogistikoÔ montè-
lou sthn melèth thc omogenoÔc diadikasÐac Poisson.
Aut  h anakluyh 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, enswmtwse tic idèec tou Filip Lundberg sth jewrÐa twn
stoqastik¸n diadikasi¸n kai sunèbale kajoristik sthn jemelÐwsh thc
1
2 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
shmerin c analogistik c epist mhc twn genik¸n asfalÐsewn all kai sthn
anptuxh thc jewrÐac pijanot twn kai twn stoqastik¸n diadikasi¸n proc
thn kateÔjunsh thc jewrÐac kindÔnou.
Sthn analogistik  epist mh brÐskoume èna qarakthristikì pardeig-
ma gìnimhc allhlepÐdrashc metaxÔ jewrÐac kai prxhc. Prgmati, oi dÔo
autèc proseggÐseic sumblloun armonik sthn dhmiourgÐa miac pl rouc
kai austhr c je¸rhshc, qwrÐc thn kuriarqÐa thc miac pnw 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
uprqei apnthsh. '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 anabjmish 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 prospjeia
tou foitht  gia katanìhsh thc prosèggishc metaxÔ jewrÐac kai prxhc.
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 asflishc sth sÔgqronh
epiqeirhmatik  drasthriìthta eÐnai plèon pasÐdhlh. Wstìso, h prosfor
beltiwmènwn asfalistik¸n proðìntwn eÐnai to anamenìmeno b ma proìdou
apì pleurc 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 epishmnoume kai mia llh ptuq  thc jewrÐac kindÔnou
pou mporeÐ na enjousisei touc upoy fiouc analogistèc. EÐnai h epèktash
1.1. PIJANOTHTA QREOKOPIAS 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 epitugqnoume 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 emfnishc
apozhmi¸sewn pou emfanÐzontai sta plaÐsia kpoiou asfalistikoÔ qarto-
fulakÐou pou ja to onomzoume asfalistik  politik . Me thn bo jeia
twn stoqastik¸n diadikasi¸n mporoÔme na montelopoi soume tic tuqaÐec
diakumnseic tou apojèmatoc thc asfalistik c etaireÐac pou qrhsimopoieÐ-
tai gia thn plhrwm  twn apozhmi¸sewn.
Kje asfalistik  politik  stoqeÔei sthn elfrunsh 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 peltec se an-
tllagma katabloun sthn etaireÐa asflistra gia na exasfalÐsoun thn
biwsimìthta thc etaireÐac kai na sumbloun sthn dhmiourgÐa tou anagkaÐ-
ou apojèmatoc. Profan¸c ta asflistra ja prèpei na xepernoÔn to mèso
kìstoc twn apozhmi¸sewn se opoiod pote qronikì disthma, pou shmaÐnei
ìti oi peltec dèqontai ex' arq c kpoia jetik  epibrunsh asfaleÐac.
Ta analogistik montèla dÐnoun thn dunatìthta na melethjeÐ me kje
leptomèreia h sqèsh metaxÔ twn emplekomènwn megej¸n kai prosfèroun ta
ergaleÐa gia touc swstoÔc upologismoÔc twn asfalÐstrwn, tou kindÔnou
kai twn apojemtwn.
4 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
Ac upojèsoume ìti kpoia asfalistik  etaireÐa xekin drasthriìthta
thn stigm  0 me arqikì keflaio u ≥ 0 kai to sunolikì eisìdhma apì
asflistra pou katablletai apì touc peltec mèqri kai thn stigm  t
paristnetai me C(t). To eisìdhma apì asflistra C(t) eÐnai aÔxousa
sunrthsh tou qrìnou. Sun jwc jewroÔme thn prosdioristik  (nteter-
ministik ) grammik  sunrthsh C(t) = c t ìpou h stajer c onomzetai
rujmìc eÐspraxhc asfalÐstrou.
Oi qronikèc stigmèc {Tk , k ∈ N} to sÔnolo twn diadoqik¸n stigm¸n
emfnishc atuq matoc kai ta Ôyh twn apozhmi¸sewn {Zk , k ∈ N} apote- loÔn akoloujÐec tuqaÐwn metablht¸n. Tic stigmèc emfnishc atuq matoc
mporoÔme na tic parast soume me thn bo jeia twn apostsewn 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 emfnishc atuq matoc sto disthma [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) =
Zi .
Upojètoume ìti oi endimesoi qrìnoi {θk , k ∈ N} apoteloÔn ako-
loujÐa anexrthtwn isìnomwn tuqaÐwn metablht¸n me katanom  A(x) =
P[θ1 ≤ x]. Oi ropèc twn endimeswn qrìnwn k txhc, gia k = 0, 1, . . .,
en uprqoun sumbolÐzontai me
ak = E[θk1 ] =
yk A(dy) .
Upojètoume ìti ta Ôyh twn apozhmi¸sewn {Zk , k ∈ N} apoteloÔn
mia akoloujÐa anexrthtwn 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. PIJANOTHTA QREOKOPIAS 5
Deqìmaste ìti
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 en
uprqoun sumbolÐzontai me
yk B(dy) ,
gia k = 0, 1, . . .. Upojètoume ìti oi akoloujÐec {Tk , k ∈ N} kai
{Zk , k ∈ N} eÐnai anexrthtec metaxÔ touc. Me ta megèjh pou pa-
rousisame, 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 kje stigm  apoteleÐ stoqastik  diadikasÐa
kaj¸c sta emplekìmena megèjh perilambnontai 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 epibrunsh asfaleÐac kai dÐnei shmantik  plhroforÐa gia thn
axiopistÐa thc asfalistik c drasthriìthtac. Sthn prxh qrhsimopoioÔme
kurÐwc to ìrio
ρ = lim t→∞
E[C(t)− S(t)]
E[S(t)] , (1.1.1)
pou eÐnai gnwstì me to ìnoma sqetik  epibrunsh asfaleÐac. H sqetik 
epibrunsh asfaleÐac perigrfei to anamenìmeno eisìdhma thc asfalisti-
k c etaireÐac an monda apozhmÐwshc. 'Otan to ρ plhsizei sto mhdèn, h
6 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
asfalistik  etaireÐa mènei qwrÐc apìjema kai o kÐndunoc qreokopÐac thc
megal¸nei. 'Otan to ρ gÐnetai meglo h etaireÐa parousizei kerdoforÐa
all ta asflistr thc eÐnai apojarruntik gia touc upoy fiouc peltec.
Sth sunèqeia axi¸noume ìti aut  h sqetik  epibrunsh asfaleÐac u-
prqei kai eÐnai jetik , dhlad  isqÔei
ρ > 0 . (1.1.2)
Aut  h upìjesh eÐnai eurèwc apodekt  sthn analogistik  praktik  kai
onomzetai AxÐwma KajaroÔ Kèrdouc. Mlista to AxÐwma KajaroÔ Kèr-
douc praktik shmaÐnei ìti h diadikasÐa apojèmatoc U(t) èqei auxhtik 
tsh, pou eÐnai anagkaÐa proôpìjesh gia na elpÐzoume sthn kerdoforÐa
thc etaireÐac. H diadikasÐa apojèmatoc {U(t), t ≥ 0} perièqei thn plh-
roforÐa pou qreizetai gia thn axiolìghsh thc biwsimìthtac thc asfali-
stik c epiqeÐrhshc sta plaÐsia kpoiou montèlou kindÔnou, paristmenou
sun jwc apì thn trida (A, C, B).
En to apìjema prei arnhtik  tim  se kpoia qronik  stigm  t > 0,
lème ìti parousizetai qreokopÐa. H pijanìthta autoÔ tou endeqomènou
paÐrnei thn morf 
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
pernei 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 grfetai sthn morf  ψ(u) = P[τ(u) <
∞]. Sthn genik  perÐptwsh o qrìnoc qreokopÐac τ(u) eÐnai mia ellip c
1.1. PIJANOTHTA QREOKOPIAS 7
tuqaÐa metablht  (dhlad  h sunrthsh katanom c den teÐnei sthn monda)
kaj¸c mporeÐ na prei thn tim ∞ me jetik  pijanìthta P[τ(u) =∞] > 0.
Prgmati, diaisjhtik antilambanìmaste ìti ktw apì to AxÐwma KajaroÔ
Kèrdouc to apìjema U(t) teÐnei sto peiro kai gi' autì eÐnai pijanì na mhn
emfanisteÐ potè qreokopÐa.
φ(u) = P
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 ,
gia kje n = 0, 1, . . ..
EÔkola diapist¸noume ìti h akoloujÐa {Un, n = 0, 1, . . .} apoteleÐ omogen  markobian  alusÐda me timèc apì to sÔnolo twn pragmatik¸n
R. Kaj¸c to endeqìmeno qreokopÐac mporeÐ na emfanisteÐ mìno kat tic
stigmèc emfnishc apozhmi¸sewn {Tn, n ∈ N}, h pijanìthta qreokopÐac
paÐrnei diakrit  morf 
H pijanìthta qreokopÐac upologÐzetai me bsh 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 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
telik¸n apotelesmtwn 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 bsh 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} .
P[S(t) ≤ x] = ∞∑ n=0
P[S(t) ≤ x , N(t) = n]
= ∞∑ n=0
=
Bn∗(x)P[N(t) = n] ,
ìpou to Bk∗(x) sumbolÐzei thn k-txhc 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 TUQAIO AJROISMA S(T ) 9
X kai Y , en uprqoun 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
uprqoun oi ropèc twn apozhmi¸sewn bk = E[Zk] gia k ∈ N. Tìte
E[S(t) |N(t) = n] = E
[ 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 kje
apozhmÐwshc.
Parìmoia, qrhsimopoi¸ntac thn upìjesh ìti oi Zi , i ∈ N eÐnai ane-
xrthtec isìnomec tuqaÐec metablhtèc, paÐrnoume
var[S(t) |N(t) = n] = var
[ n∑ i=1
var [Zi] = n (b2 − b21) ,
opìte brÐskoume var[S(t) |N(t)] = N(t) (b2− b21) kai me bsh ta prohgoÔ- mena
var[S(t)] = E(var[S(t) |N(t)]) + var(E[S(t) |N(t)])
= E [ N(t) (b2 − b21)
= (b2 − b21)E [N(t)] + b21 var[N(t)] .
Se autìn ton tÔpo blèpoume pli ìti h diakÔmansh thc sunolik c apozh-
mÐwshc ekfrzetai sunart sei twn dÔo pr¸twn rop¸n tou pl jouc twn
10 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
proserqìmenwn apozhmi¸sewn sto disthma [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
= [PZ(y)]n .
[PZ(y)]N(t) )
= PN(t) [PZ(y)].
To Ðdio paÐrnoume gia thn ropogenn tria sunrthsh 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} sto disthma [0, t], apoteleÐ diadikasÐa Poisson me parmetro λt, paÐrnoume
thn sunolik  apozhmÐwsh S(t) san sÔnjeth Poisson tuqaÐa metablht .
Eidikìtera, efarmìzontac touc gnwstoÔc tÔpouc thc katanom c Poisson
brÐskoume
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)
Prgmati, 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
1.2. TO TUQAIO 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 anexrthtwn all ìqi a-
nagkastik isìnomwn sÔnjetwn Poisson tuqaÐwn metablht¸n brÐskoume
pli mia sÔnjeth Poisson tuqaÐa metablht . Prgmati, èstw {Xi(t) , i =
1, . . . , n} to sÔnolo twn anexrthtwn sÔnjetwn Poisson me paramètrouc
{λi t , i = 1, . . . , n} antÐstoiqa. Tìte h tuqaÐa metablht 
n∑ i=1
Λn = n∑ i=1
Gia na to deÐxoume, paÐrnoume thn ropogenn tria sunrthsh
M∑n i=1Xi
MXi(t)(s) ,
apì thn anexarthsÐa twn {Xi(t) , i = 1, . . . , n}. Epomènwc apì ton tÔpo
12 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
(1.2.5)
)} ,
pou eÐnai pli sthn morf  thc sÔnjethc Poisson, arkeÐ na jewr soume mia
nèa katanom  apozhmi¸sewn mèsa apì thn èkfrash
n∑ i=1
Λn P[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.−→ b1 a1 , (1.2.6)
kaj¸c t → ∞. En h katanom  A eÐnai mh arijmhtik , tìte gia kje
h ≥ 0, isqÔei
kaj¸c t→∞.
Apìdeixh. ParathroÔme ìti apì ton isqurì nìmo twn meglwn
arijm¸n paÐrnoume
1.2. TO TUQAIO AJROISMA S(T ) 13
kaj¸c t→∞, diìti N(t)→∞. Apì ed¸ kai me bsh to [3, Je¸rhma 15]
brÐskoume
N(t)
t
Gia thn deÔterh sqèsh qrhsimopoioÔme thn tautìthta Wald (blèpe [3,
Je¸rhma 6]) gia na broÔme
E
= 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
] =
] An∗(dy)
] E[N(dy) + 1]
ìpou g(x) := E [ Z 1[θ>x]
] . ParathroÔme ìti g(0) = E[Z], g(∞) = 0 kai
14 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
N(0) = 0 gia na sumpernoume
E[S(t)] = g(0)E[N(t) + 1]− ∫ t
0 g(t− y)E[N(dy) + 1]
= g(0)E[N(t) + 1] +
= g(t)− ∫ t
= g(0)− ∫ t
T¸ra èqoume
0 {E[N(t+ h− y)]−E[N(t− y)]} g(dy)
− ∫ t+h
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 proume 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→∞.
'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 anexrthtwn isìnomwn tuqaÐwn metablht¸n me thn
ex c ekjetik  katanom 
A(x) = 1− e−λx .
1.3. KLASIKO MONTELO KINDUNOU 15
Epomènwc, gia kje qronik  stigm  t ≥ 0, h tuqaÐa metablht  N(t) ako-
loujeÐ thn katanom  Poisson me parmetro 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 onomzetaiKlasikì Montèlo KindÔnou (KMK) kai parÐstatai
apì thn trida (λ, c, B).
Sto KMK, ìpwc eÐdame apì ton tÔpo tou Wald (blèpe sqèsh (1.2.1)),
h mèsh sunolik  apozhmÐwsh sto disthma [0, t] eÐnai Ðsh me
E[S(t)] = E[N(t)]E[Z1] = λ t b1 ,
kaj¸c to S(t) apoteleÐ tuqaÐo jroisma anexrthtwn isìnomwn tuqaÐwn
metablht¸n ìpou to pl joc twn prosjetèwn N(t) den exarttai apì touc
ìrouc tou ajroÐsmatoc. Epomènwc mporoÔme na parathr soume ìti to
klsma
E[S(t)] ,
den exarttai apì ton qrìno t, opìte sto KMK o tÔpoc (1.1.1) gia th
sqetik  epibrunsh 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 anexrthtwn isìnomwn mh arnhtik¸n tuqaÐwn metablht¸n.
16 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
3. Oi dÔo akoloujÐec {Tk, k = 0, 1, . . .} kai {Zk, k ∈ N} eÐnai ane- xrthtec metaxÔ touc.
Ja exetsoume 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 trida (λ, c, µ).
Sthn perÐptwsh tou montèlou (λ, c, µ) upologÐzontai eÔkola oi ropèc
bs = Γ(s+ 1)
Γ(s) =
∫ ∞ 0
ys−1 e−y dy .
Parapèra o tÔpoc gia th sqetik  epibrunsh 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. EKJETIKES APOZHMIWSEIS 17
U1. 'Etsi me bsh ton tÔpo thc olik c pijanìthtac brÐskoume
φ(u) = 1− ψ(u) = P
]
=
]
=
] P [(θ1, Z1) ∈ (dt, dz)] ,
epomènwc
φ(u) =
∫ θ,Z
P
] ·P [(θ1, Z1) ∈ (dt, dz) | u+ c θ1 − Z1 ≥ 0]
= Eθ,Z
{ P
] U1 ≥ 0
=
) λ e−λt dt .
T¸ra me allag  metablht c (y antÐ u+ c t), paÐrnoume
φ(u) = λ
) 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
) . (1.4.2)
Sth sunèqeia, oloklhr¸nontac thn (1.4.2) sto disthma [0, x], brÐ-
skoume
) dy
) . (1.4.3)
18 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
T¸ra knoume ton upologismì tou diploÔ oloklhr¸matoc. Me allag 
metablht c (v antÐ y − z) brÐskoume∫ x
0
(∫ y
) dy =
∫ x
) dv =
∫ x
Bzontac thn teleutaÐa èkfrash sth sqèsh (1.4.3), katal goume
φ(x)− φ(0) = λ
) =
λ
c
∫ x
'Ara, met apì antikatstash sthn pijanìthta qreokopÐac, èqoume
ψ(u) = ψ(0)− λ
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
µ − c
λ < 0 ,
kai ìti ta {Xk = Zk − c θk, k ∈ N} apoteloÔn akoloujÐa anexrthtwn
isìnomwn tuqaÐwn metablht¸n me jroisma
Sn =
Xk = n∑ k=1
Zk − c θk , (1.4.5)
opìte apì ton isqurì nìmo twn meglwn arijm¸n brÐskoume
P [
Sn = −∞ ]
= 1 .
Apì tic idiìthtec tou tuqaÐou periptou (blèpe [3, Prìtash 11]) prokÔptei
ìti
All apì thn sqèsh (1.1.5) jumìmaste ìti
Un = u− Sn , (1.4.7)
lim u→∞
P[sup n≥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) = lim u→∞
ψ(u) + lim u→∞
= λ
c
∫ ∞ 0
λ
c
∫ u
= λ
− lim u→∞
all apì thn (1.4.8)
(u 2
0 e−µy dy → 0 ,
kaj¸c u→∞ kai paÐrnontac up' ìyin ìti h ψ(u) eÐnai mh aÔxousa sunr-
thsh, blèpoume ìti
∫ u/2
ψ(0) = λ
c µ =
20 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
Tèloc, me antikatstash tou ψ(0) sthn sqèsh (1.4.4) katal goume
ψ(u) = λ
= λ
c
∫ ∞ u
ψ(u) = λ
0 ψ(y) eµy dy . (1.4.10)
Apì ed¸ xekinei h anlush thc pijanìthtac qreokopÐac. Sugkekri-
mèna, paÐrnontac parag¸gouc sta dÔo mèlh thc exÐswshc, èqoume
ψ′(u) = −λµ c e−µu
(∫ u
ψ(u) = ψ(0) exp
c
) u
} kai me bsh 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
} . (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. KLASIKO MONTELO ME GENIKES APOZHMIWSEIS 21
Uprqei ìmwc kai lloc trìpoc epÐlushc thc exÐswshc (1.4.10). Ton
parajètoume sunoptik, epishmaÐnontac ìti en¸ eÐnai suntomìteroc den
eÐnai pnta leitourgikìc, me thn ènnoia ìti se efarmogèc ìpou h antistrof 
twn metasqhmatism¸n gÐnetai polÔplokh, den dÐnei mesa apotelèsmata.
SumbolÐzoume me
ton metasqhmatismì Laplace miac sunrthshc B(x). PaÐrnontac touc me-
tasqhmatismoÔc Laplace sta dÔo mèlh thc (1.4.10) èqoume thn algebrik 
exÐswsh
antistrof  dÐnei
ψ(u) = λ
seic
T¸ra ja exetsoume pli 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 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
Pli xekinme 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
] U0 = u, U1 ≥ 0
=
) λ e−λt dt
) e−λ(y−u)/c dy ,
ìpou sthn teleutaÐa isìthta kname allag  metablht c (y antÐ u + c t).
Ed¸ èqoume thn pijanìthta epibÐwshc sthn morf  ginomènou thc ekjetik c
sunrthshc eλu/c kai tou oloklhr¸matoc
λ
c
∫ ∞ u
(∫ y
pou eÐnai paragwgÐsima wc proc u. ParagwgÐzontac kai qrhsimopoi¸ntac
thn prohgoÔmenh èkfrash thc pijanìthtac qreokopÐac paÐrnoume
φ′(u) = λ
) .
T¸ra oloklhr¸noume ta dÔo mèlh gia na katal xoume
φ(x)− φ(0) = λ
0
∫ y
0 φ(x− v)B(v) dv .
1.5. KLASIKO MONTELO ME GENIKES APOZHMIWSEIS 23
T¸ra antikajist¸ntac thn teleutaÐa èkfrash sthn exÐswsh (1.5.1)
èqoume
pou dÐnei telik
ψ(0)− ψ(u) = λ
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
0 ≤ lim u→∞
u→∞
∫ u/2
) = 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
= 1
1 + ρ . (1.5.4)
T¸ra gurnme pÐsw sthn exÐswsh (1.5.3), ìpou antikajistme to apotè-
lesma sth sqèsh (1.5.4)
24 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
Ac sumbolÐsoume ton metasqhmatismì Laplace-Stieltjes miac sunrth-
shc B(x) me
thc (1.5.5) èqoume
+ λ
c
∫ ∞ 0
B(y)
∫ ∞ y
= λ
c
∫ ∞ 0
kai parapèra
c
∫ ∞ 0
∫ x
+ λ
c
∫ ∞ 0
ψ(s) = λ b1 s− λ+ λ B(s)
c s− λ+ λ B(s) . (1.5.6)
'Estw h katanom  thc oloklhrwmènhc ourc
B0(u) := 1
b 1
0 B(y) dy .
1.5. KLASIKO MONTELO ME GENIKES APOZHMIWSEIS 25
Opìte h ananewtik  exÐswsh (1.5.2) gÐnetai
φ(u) = ρ
1 + ρ +
kai
φ(s) = c− λ b1
tÔpo Pollaczeck-Khinchin, pou eÐnai gnwstìc apì thn jewrÐa our¸n
anamon c (blèpe [1])
Aut  h anaparstash thc pijanìthtac qreokopÐac sto klasikì montèlo
kindÔnou sthn morf  sÔnjethc gewmetrik c katanom c diathreÐtai kai sto
ananewtikì montèlo kindÔnou, ìpwc ja doÔme paraktw.
26 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
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
0 B(y) dy ,
antistoiqeÐ se ellip  katanom , dhlad  se katanom  pou èqei ìrio kaj¸c
to x → ∞ mikrìtero thc mondac. Prgmati, apì to AxÐwma KajaroÔ
Kèrdouc (blèpe sqèsh (1.1.2)) èqoume
λ
c
∫ ∞ 0
< 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
sunrthsh prèpei na eÐnai tèleia.
Gia na antimetwpÐsoume aut  thn duskolÐa, pollaplasizoume ta dÔo
mèlh thc exÐswshc (1.5.5) me kpoion pargonta eRx, ìpou emfanÐzetai
kpoia stajer R > 0, pou onomzetai rujmistikìc suntelest c, kai ika-
nopoieÐ thn ex c sunj kh Cramer
λ
c
∫ ∞ 0
(1− eRy)B(dy) = 1 . (1.5.10)
Prgmati, me aut  thn epilog  tou rujmistikoÔ suntelest  R paÐrnoume
pl rh ananewtik  exÐswsh me gnwsth th sunrthsh ψR(u) = eRu ψ(u)
kai thn tèleia katanom  sth sunèlixh
BR(x) = λ
ψR(u) = λ
c eRu
= z(u) + ψR ∗BR(u) , (1.5.11)
B(y) dy .
1.5. KLASIKO MONTELO ME GENIKES APOZHMIWSEIS 27
T¸ra ac sumbolÐsoume
∫ ∞ 0
Je¸rhma-kleidÐ, ¸ste na broÔme
lim u→∞
ψR(u) = 1
bR1
∫ ∞ 0
z(y)dy ,
ap' ìpou prokÔptei h ex c asumptwtik  sqèsh gia thn pijanìthta qreo-
kopÐac
kaj¸c u→∞. 'Etsi paÐrnoume sto akìloujo apotèlesma.
Je¸rhma 3. JewroÔme to KMK (λ, c, B) kai upojètoume ìti uprqei
kpoioc rujmistikìc suntelest c R > 0, tètoioc ¸ste na isqÔei h sunj kh
Cramer (1.5.10).
ψ(u) ∼ ρ
kaj¸c u→∞.
lim u→∞
λ
c
∫ x
apoteleÐ tèleia katanom  pijanot twn me puknìthta
λ
28 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
kai h upìjesh bR1 <∞ exasfalÐzei thn Ôparxh thc antÐstoiqhc mèshc ti-
m c, h bohjhtik  sunrthsh 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 uprqei to ìrio
lim u→∞
(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  sunrthsh kai anadeiknÔei th
shmasÐa tou rujmistikoÔ suntelest  R.
1.6 Sunrthsh proexoflhtik c poin c
T¸ra ja melet soume ìqi mìno thn pijanìthta qreokopÐac all kai tic
sunj kec ktw apì tic opoÐec h qreokopÐa emfanÐzetai. Sthn ergasÐa [2]
protjhke 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) kpoia jetik  metr simh sunrthsh 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. SUNARTHSH PROEXOFLHTIKHS POINHS 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 sunrthsh proexoflhtik c poin c Gerber-Shiu kai apoteleÐ lÔsh
thc akìloujhc ananewtik c exÐswshc
Je¸rhma 4. Sto KMK (λ, c, B) h sunrthsh proexoflhtik c poin c
ikanopoieÐ thn ananewtik  exÐswsh
ìpou to R apoteleÐ thn mègisth (jetik ) lÔsh thc exÐswshc
λ+ α− cx = λ B(x) . (1.6.3)
Apìdeixh. Prgmati, upojètoume to klasikì montèlo kindÔnou
kai gia kpoio arket mikrì h > 0 jewroÔme to endeqìmeno emfnishc
apozhmÐwshc sto disthma (0, h). H pijanìthta na emfanisteÐ h pr¸th
apozhmÐwsh sto apeirostì disthma (t, t + dt) isoÔtai me λ e−λt dt kai
h pijanìthta dÔo apozhmi¸sewn sto disthma (0, h) jewreÐtai amelhtèa,
opìte apì thn sqèsh (1.6.1) èqoume
mα(u) =
∫ h
0
] λ e−(λ+α) t dt
+
] 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
∫ u
30 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
ìpou qrhsimopoi same to sumbolismì
a(u) :=
∫ ∞ u
w(u, y)B(u+ dy) .
'Estw gia kpoio pragmatikì R ≥ 0 h bohjhtik  sunrthsh mR α (u) =
e−Rumα(u). Pollaplasizontac thn sqèsh (1.6.4) me e−Ru èqoume
cmR α ′ (u) = (λ+ α− cR)mR
α (u) (1.6.5)
0 mR α (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 sunrthsh wc proc thn parmetro
α me arqik  tim  sto mhdèn. En epilèxoume gia to R aut  thn lÔsh thc
(1.6.3), tìte h exÐswsh (1.6.5) gÐnetai
c
∫ u
= B(R)mR α (u)− e−Ru a(u)−
∫ u
0 mR α (v) e−R(u−v)B(u− dv) .
T¸ra me olokl rwsh sto disthma (0, y) brÐskoume
c
∫ y
= B(R)
∫ y
∫ y
0
∫ u
=
− ∫ y
=
1.6. SUNARTHSH PROEXOFLHTIKHS POINHS 31
kai af nontac to y na teÐnei sto peiro, h teleutaÐa èkfrash gÐnetai mhdèn
kai isqÔei mR α (y)→ 0, opìte èqoume
mR α (0) =
mR α (y)
e−Rz B(dz) dv .
T¸ra gurn¸ntac pÐsw sth sunrthsh proexoflhtik c poin c paÐrnoume
thn exÐswsh
c
∫ y
pou mac dÐnei pl rh ananewtik  exÐswsh sthn morf  (1.6.2).
ShmeÐwsh 2. En jèsoume α = 0 kai w(t, s) = 1 gia kje t ≥ 0 , s ≥ 0,
h sunrthsh (1.6.1) sumpÐptei me thn pijanìthta qreokopÐac m0(u) =
ψ(u) = P[τ(u) <∞].
32 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
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 anexrth-
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  epibrunsh 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 disthma [0, t]
prokÔptei apì to pl joc apozhmi¸sewn N(t) me arnhtik  diwnumik  ka-
tanom 
1.7. ASKHSEIS 33
gia m = 0, 1, . . ., me paramètrouc r > 0 kai 0 < p < 1, kai thn logarij-
mokanonik  katanom  apozhmÐwshc me puknìthta
b(x) = 1
gia x > 0. UpologÐste ta E[S(t)] kai var[S(t)].
Upìdeixh:
p ,
kai
p2 .
'Ara
MN(t)(s) =
( p
'Askhsh 5. Ac upojèsoume ìti èqoume to klasikì montèlo kindÔnou
me ekjetikèc apozhmi¸seic pou parÐstatai me thn bo jeia thc tridac
(λ, c, µ) = (0.1, 1, 0.2). Na elègxete to AxÐwma KajaroÔ Kèrdouc kai
na upologÐsete thn pijanìthta qreokopÐac ψ(100) gia arqikì keflaio
u = 100. Na prosdiorÐsete to arqikì keflaio 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 keflaio apì u se r u, me r > 1 kai na mei¸soume antÐstoiqa ton
rujmì asflistrou 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
'Askhsh 7. BreÐte ton rujmistikì suntelest  R gia ekjetik  katanom 
apozhmi¸sewn me parmetro µ kai sqetik  epibrunsh asfaleÐac ρ kai
twn paramètrwn λ, c.
34 KEFALAIO 1. KLASIKH MONTELOPOIHSH KINDUNOU
'Askhsh 8. BreÐte ton rujmistikì suntelest  R gia ekfulismènh ka-
tanom  apozhmi¸sewn me stajer  tim  Ðsh me thn monda kai sqetik 
epibrunsh 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  epibrunsh
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
epibrunsh asfaleÐac, 3
Klasikì Montèlo KindÔnou, 15
stigm  emfnishc atuq matoc, 4
36