Ruin probability in the three-seasonal discrete-time risk model

0.2 MB·

by  Andrius Grigutis

ModernStochastics:TheoryandApplications2(2015)421–441
DOI:10.15559/15-VMSTA45
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J Ruin probability in the three-seasonal discrete-time

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risk model

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AndriusGrigutis1,AgneškaKorvel,JonasŠiaulys∗
.
h
at FacultyofMathematicsandInformatics,VilniusUniversity,Naugarduko24,
m VilniusLT-03225,Lithuania
[
[email protected](A.Grigutis),[email protected](A.Korvel),
1 [email protected](J.Šiaulys)
v
3 Received:26November2015,Revised:14December2015,Accepted:14December2015,
5 Publishedonline:30December2015
1
1 Abstract Thispaperdealswiththediscrete-timeriskmodelwithnonidenticallydistributed
0 claims.Wesupposethattheclaimsrepeatwithtimeperiodsofthreeunits,thatis,claimdis-
. tributionscoincideattimes{1,4,7,...},attimes{2,5,8,...},andattimes{3,6,9,...}.We
1
presenttherecursiveformulastocalculatethefinite-timeandultimateruinprobabilities.We
0
6 illustratethetheoreticalresultsbyseveralnumericalexamples.
1
Keywords Inhomogeneousmodel,three-seasonalmodel,finite-timeruinprobability,
:
v ultimateruinprobability
i
X 2010MSC 91B30,60G50
r
a
1 Introduction
Thediscrete-timeriskmodelisaclassicalcollectiveriskmodelforinsurance.Inthe
homogeneousversion of this model, the insurer’s surplus at each time n ∈ N =
0
{0,1,2,...}isdefinedbythefollowingequality:
n
W (n)=u+n− Z , (1)
u i
i=1
X
∗Correspondingauthor.
1ThefirstauthorwassupportedbygrantNo.MIP-049/2014fromtheResearchCouncilofLithuania.
©2015TheAuthor(s).PublishedbyVTeX.OpenaccessarticleundertheCCBYlicense.
www.i-journals.org/vmsta
422 A.Grigutisetal.
whereu ∈ N isthe initialinsurer’ssurplus,andthe claim amountsZ ,Z ,... are
0 1 2
assumedtobeindependentcopiesofanonnegativeinteger-valuedrandomvariableZ.
This random variable and the initial surplus u generate the homogeneousdiscrete-
timeriskmodel.AtypicalpathofthesurplusprocessW (n)isshowninFig.1.
u
✻ • • •
Wu(n) • •
• •
• •
• • •
• • •
• • •
• • •
• •
u • •
n
✲
Fig.1.BehaviorofthesurplussequenceWu(n)
TheclaimamountgeneratorZcanbecharacterizedbytheprobabilitymassfunc-
tion(p.m.f.)
z =P(Z =k), k ∈N ,
k 0
orbythecumulativedistributionfunction(c.d.f.)
⌊x⌋
F (x)= z , x∈R,
Z k
k=0
X
where⌊x⌋denotestheintegerpartofx.
Thehomogeneousdiscrete-timeriskmodelhasbeenextensivelyinvestigatedby
DeVylderandGoovaerts[5,6],Dickson[7,8],Gerber[10],Seal[17],Shiu[19,18],
PicardandLefèvre[15,16],LefèvreandLoisel[11],LeipusandŠiaulys[12],Tang
[20],andotherauthors.Theruintime,theultimateruinprobability,andthefinite-time
ruinprobabilityarethemainextremalcharacteristicsofanyriskmodel.Thefirsttime
T whenthesurplusW (n)becomesnegativeornulliscalledtheruintime,thatis,
u u
inf{n∈N:W (n)60},
u
T =
u (∞ ifWu(n)>0forall n∈N.
TheruinprobabilityuntiltimeT ∈ Niscalledthefinite-timeruinprobabilityandis
definedby
ψ(u,T)=P(T 6T).
u
Theinfinite-timeorultimateruinprobabilityisdefinedby
ψ(u)=P(T <∞).
u
Ruinprobabilityinthethree-seasonaldiscrete-timeriskmodel 423
So,fortheultimatesurvivalprobability,wehave
ϕ(u)=1−ψ(u)=P(T =∞).
u
Thepresenteddefinitionsimplythat
T n n
ψ(u,T)=P u+n− Z 60 =P max (Z −1)>u ,
i i
( )! 16n6T !
n=1 i=1 i=1
[ X X
∞ n n
ψ(u)=P u+n− Z 60 =P sup (Z −1)>u ,
i i
n=1( i=1 )! n>1i=1 !
[ X X
∞ n
ϕ(u)=P u+n− Z >0 ,
i
( )!
n=1 i=1
\ X
lim ψ(u,T)=ψ(u).
Tր∞
Severalformulas and proceduresfor computingfinite-time ruin probability and
ultimateruinprobabilityhavebeenproposedintheliterature.Herewepresentsome
ofthemhavingtherecursiveform.
• Forthehomogeneousdiscrete-timerisk model,wehave(see,forinstance,[5,
8,9]):
ψ(u,1)=1−F (u), u∈N ,
Z 0
u
ψ(u,T)=ψ(u,1)+ ψ(u+1−k,T −1)z , u∈N , T ∈{2,3,...}.
k 0
k=0
X
• Ifmodel(1)isgeneratedbytheclaimgeneratorZ suchthatEZ <1,thenthe
ultimateruinprobabilitycanbecalculatedbytheformulas(see,forinstance,
[8,9,18]):
ψ(0)=EZ, (2)
u−1 ∞
ψ(u)= 1−F (j) ψ(u−j)+ 1−F (j) , u∈N. (3)
Z Z
j=1 j=u
X(cid:0) (cid:1) X(cid:0) (cid:1)
Ifthehomogeneousdiscrete-timeriskmodelisgeneratedbyZ satisfyingcondi-
tionEZ > 1,thenwesaythatthenetprofitconditiondoesnothold,and,insucha
case,wehavethatψ(u) = 1forallu ∈ N accordingtothegeneralrenewaltheory
0
(see,e.g.,[14]andthereferencestherein).
The formulas presented enable us to calculate ψ(u) and ψ(u,T) for u ∈ N
0
andT ∈ N.Nevertheless,thereexistmanyothermethodsthatallowustocalculate
or estimate the finite-time and the ultimate ruin probabilities.Some of them can be
foundin[1,13,16].
Theassumptionforclaimamounts{Z ,Z ,...}tobenonidenticallydistributed
1 2
random variables is a natural generalization of the homogeneous model. If r.v.s
424 A.Grigutisetal.
{Z ,Z ,...} are independent but not necessarily identically distributed, then the
1 2
modeldefinedbyEq.(1)is calledtheinhomogeneousdiscrete-timerisk model.For
such a model, a recursive procedurefor calculation of finite-time ruin probabilities
canbefoundin[2,3].Forthefinite-timeruinprobabilities
T n
ψ(j)(u,T)=P u+n− Z 60 , j ∈N ,
i+j 0
( )!
n=1 i=1
[ X
wehavethefollowingtheorem.
Theorem1. Letusconsidertheinhomogeneousdiscrete-timeriskmodeldefinedby
Eq.(1)inwhichu∈N ,z(j) =P(Z =k),k,j ∈N ,andF(j)(x)=P(Z 6
0 k 1+j 0 Z 1+j
x),x∈R.Then
ψ(j)(u,1)=1−F(j)(u),
Z
u
ψ(j)(u,T)=ψ(j)(u,1)+ ψ(j+1)(u+1−k,T −1)z(j)
k
k=0
X
forallu∈N andT ∈{2,3,...}.
0
According to this theorem, we can calculate the finite-time ruin probability
ψ(0)(u,T) of the initial modelfor all u ∈ N and T ∈ N. Unfortunately,it is im-
0
possibletogetformulasforψ(u)similartoformulas(2)and(3)inthegeneralcase
becausein the case ofnonidenticallydistributedclaims, the futureof modelbehav-
ior at each time can be completelynew. In paper [4], the generaldiscrete-timerisk
model was restricted to the model with two kinds of claims. In this model, there
aretwodifferentlydistributedclaimamountsthatarechangingperiodically.Wecall
suchamodelthebi-seasonaldiscrete-timeriskmodel.In[4](seeTheorem2.3),the
followingstatementisprovedforthecalculationoftheultimateruinprobability.
Theorem 2. Let us consider a bi-seasonal discrete-time risk model generated by
d
independentrandomclaimamountsX andY,thatis,Z =X fork ∈{0,1,...}
2k+1
andZ =d Y fork ∈ {1,2,...}.DenoteS = X +Y andx = P(X = n), y =
2k n n
P(Y =n),s =P(S =n)forn∈N ={0,1,...}.
n 0
• IfEX +EY <2,then
lim ψ(u)=0.
u→∞
• Ifs =x y 6=0,then:
0 0 0
b −b
ψ(0)=1−(2−ES) lim n+1 n,
n→∞an−an+1
1−ψ(u)=α 1−ψ(0) +β (2−ES), u∈N,
u u
where{α },{β },n ∈ N ,(cid:0)aretwose(cid:1)quencesofrealnumbersdefinedrecur-
n n 0
sivelybyformulas:
1
α =1, α =− ,
0 1
y
0
Ruinprobabilityinthethree-seasonaldiscrete-timeriskmodel 425
n−1
1
α = α − s α −x , n>2,
n n−2 i n−i n−1
s
0 !
i=1
X
1
β =0, β = ,
0 1
y
0
n−1
1
β = β − s β +x , n>2.
n n−2 i n−i n−1
s
0 !
i=1
X
• Ifs 6=0,then
0
ψ(1)=1− 1+ψ(0)−ES /y ,
0
u−1
(cid:0)1 (cid:1)
ψ(u)=1+ ψ(u−2)−1+ s 1−ψ(u−k)
k
s
0 !
k=1
X (cid:0) (cid:1)
x (1−ψ(1))
u−1
− , u∈{2,3,...}.
x
0
• Ifx =0,y 6=0,thens 6=0andψ(0)=1.
0 0 1
• Ifx 6=0,y =0,thens 6=0andψ(0)=ES−1.
0 0 1
• Ifs =0,then,foru∈N,
0
u
1
ψ(u)=1− 1−ψ(u−1)− s 1−ψ(u−k+1) .
k
s
1 !
k=2
X (cid:0) (cid:1)
In this paper, we consider the discrete-time risk model with three seasons. We
obtainalistofformulassimilartothoseinTheorem2tocalculatetheultimateruin
probabilityinsuchamodel.InSection2,wepresentaprecisedefinitionofthethree-
seasonal discrete-time risk model and our main statements, whereas in Sections 3
and4,we givedetailedproofs.Finally,Section5 dealswith somenumericalexam-
ples.
2 Mainresults
Wenowpresentthemodelunderconsideration.
Definition1. Wesaythattheinsurer’ssurplusW (n)followsthethree-seasonalrisk
u
modelifW (n)isgivenbyEq.(1)foreachn ∈ N andthefollowingassumptions
u 0
hold:
• theinitialinsurer’ssurplusu∈N ,
0
• therandomclaimamountsZ ,Z ,...arenonnegativeinteger-valuedindepen-
1 2
dentr.v.s,
• forallk ∈N ,wehaveZ =d Z ,Z =d Z ,andZ =d Z .
0 3k+1 1 3k+2 2 3k+3 3
426 A.Grigutisetal.
Letusdefinep.m.f.sandp.d.f.sbythefollowingequalities:
a =P(Z =k), b =P(Z =k),
k 1 k 2
c =P(Z =k), s =P(S =k), k ∈N ,
k 3 k 0
whereS =Z +Z +Z ,
1 2 3
⌊x⌋ ⌊x⌋
A(x)= a , B(x)= b ,
k k
k=0 k=0
X X
⌊x⌋ ⌊x⌋
C(x)= c , D(x)= s , x>0.
k k
k=0 k=0
X X
Itisnotdifficulttoseethatthedefinitionsofruintime,finite-timeruinprobabil-
ity,ultimateruinprobability,andultimatesurvivalprobabilityremainthe same.All
expressionsofthesequantitiesarethesameasinthehomogeneousdiscrete-timerisk
model.However,theprocedurestocalculatethefinite-timeortheultimateprobabil-
ities are more complex than in the homogeneous or be-seasonal discrete-time risk
models.
OurfirstresultimmediatelyfollowsfromTheorem1.Theobtainedformulasal-
low us to calculate the finite-time ruin probabilities ψ(u,T) = ψ(0)(u,T) in the
three-seasonalriskmodelforallu∈N andallT ∈N.
0
Theorem3. Inthethree-seasonaldiscrete-timeriskmodel,foreachu∈N ,wehave
0
ψ(0)(u,1)= a , ψ(1)(u,1)= b , ψ(2)(u,1)= c ,
k k k
k>u k>u k>u
X X X
andforallu∈N andT ∈{2,3,...},wehavethefollowingrecursiveformulas:
0
u
ψ(0)(u,T)=ψ(0)(u,1)+ ψ(1)(u+1−k,T −1)a ,
k
k=0
X
u
ψ(1)(u,T)=ψ(1)(u,1)+ ψ(2)(u+1−k,T −1)b ,
k
k=0
X
u
ψ(2)(u,T)=ψ(2)(u,1)+ ψ(0)(u+1−k,T −1)c .
k
k=0
X
Oursecondresultdescribesthemeaningofthenetprofitconditioninthethree-
seasonaldiscrete-timeriskmodel.TheproofofthetheoremispresentedinSection3.
Theorem4. Considerthethree-seasonaldiscretetimeriskmodelgeneratedbyinde-
pendentrandomclaimamountsZ ,Z ,andZ .If ES > 3,thenψ(u)=1foreach
1 2 3
initialsurplusu∈N .If ES =3,thenwehavethefollowingpossiblecases:
0
• ψ(0) = ψ(1) = ψ(2) = 1 andψ(u) = 0 for u ∈ {3,4,...} if {a = b =
3 0
c =1};
0
• ψ(0) = ψ(1) = 1andψ(u) = 0foru ∈ {2,3,...}if{a = b = c = 1},
0 3 0
{a =b =c =1},{a =b =c =1},or{a =b =c =1};
2 1 0 1 2 0 2 0 1
Ruinprobabilityinthethree-seasonaldiscrete-timeriskmodel 427
• ψ(0) = 1andψ(u) = 0forallu ∈ N if{a = b = c = 1},{a = b =
0 0 3 0 2
c =1},{a =b =c =1},{a =b =c =1},or{a =b =c =1};
1 0 1 2 1 0 2 1 1 1
• ψ(u)=1forallu∈N ={0,1,2,...}ifs <1.
0 3
Ourlaststatementproposesarecursiveprocedureforcalculationoftheultimate
survivalprobabilitiesϕ(u) = 1−ψ(u), u ∈ N .Theproofoftheformulasisgiven
0
inSection4.
Theorem5. Consider the three-seasonaldiscrete-time risk modelgeneratedby in-
dependent random claim amounts Z , Z , and Z . Denote S = Z + Z + Z ,
1 2 3 1 2 3
s = P(S = n) for n ∈ N , and suppose that ES < 3. Then the following state-
n 0
mentshold.
• lim ϕ(u)=1.
u→∞
• Ifs 6=0,then
0
ϕ(n)=α ϕ(0)+β ϕ(1)+γ (3−ES), n∈N , (4)
n n n 0
where
α =1, α =0, α =− 1 ,
0 1 2 b0c0
(αn = s10(αn−3− kn=−11skαn−k−an−2), n>3;
P
β =0, β =1, β =−c1 − 1,
0 1 2 c0 b0
(βn = s10(βn−3− kn=−11skβn−k−an−2c0+c0 kn=−01akbn−1−k), n>3;
γ =0,P γ =0, γ = 1 , P
0 1 2 b0c0
(γn = s10(γn−3− kn=−11skγn−k+an−2), n>3.
P
• If{a =0,b 6=0,c 6=0,a 6=0},then
0 0 0 1
ϕ(0)=0,
(5)
(ϕ(n)=βˆnϕ(1)+γˆn(3−ES), n∈N,
where
βˆ =β , βˆ =β , γˆ =γ , γˆ =γ ,
1 1 2 2 1 1 2 2
βˆn = s11(βˆn−2− nk=2skβˆn−k+1
 −an−1c0−Pc0ϕ(1) nk=1akbn−k), n>3,
γˆn = s11(γˆn−2−Pnk=2skPγˆn−k+1+an−1), n>3.
• If{a 6=0,b =0,c 6=0,b 6=0}then
0 0 0 1
ϕ(n)=α˜ ϕ(0)+γ˜ (3−ES), n∈N, (6)
n n
428 A.Grigutisetal.
where
α˜ =−1/c , α˜ =c /c2+1/(a b c ),
1 0 2 1 0 0 1 0
γ˜1 =1/c0, γ˜2 =−c1/c20,
α˜n = s11(α˜n−2− nk=2skα˜n−k+1− kn=−01akbn−k), n>3,
γ˜ = 1 (γ˜ − n s γ˜ + n−1a b ), n>3.
 n s1 n−2 PPk=2 k n−k+1 PPk=0 k n−k
• If{a 6=0,b 6=0,c =0,c 6=0},then
0 0 0 1
ϕ(n)=α˘ ϕ(0)+γ˘ (3−ES), n∈N , (7)
n n 0
where
α˘ =1, α˘ =−1/(b c ), γ˘ =0, γ˘ =1/(b c ),
0 1 0 1 0 1 0 1
α˘ = 1 (α˘ − n s α˘ − n−1a b ), n>2,
γ˘n= s11(γ˘n−2− nk=2skγ˘ n−k+1+ nk−=10a kb n−k), n>2.
n s1 n−2 Pk=2 k n−k+1 Pk=0 k n−k
• If{a=0,b =0,c 6=P0},thenϕ(0)=0,ϕP(1)=(3−ES)/c ,and
0 0 0 0
u−1
1
ϕ(u+1)= (1−s )ϕ(u)− ϕ(k)s
3 u+3−k
s
2
k=1
X
u+2
+c ϕ(1) a b , u∈N. (8)
0 k u+2−k
!
k=0
X
• If {a = 0,b 6= 0,c = 0},thenϕ(0) = s ϕ(1), ϕ(1) = (3−ES)/(s +
0 0 0 2 2
b c ),and
0 1
u−1
1
ϕ(u+1)= (1−s )ϕ(u)− ϕ(k)s
3 u+3−k
s
2
k=1
X
+a b c ϕ(1) , u∈N. (9)
u+1 0 1
!
• If{a 6=0,b =0,c =0},thenϕ(0)=3−ES,ϕ(1)=(3−ES)/s ,and
0 0 0 2
u−1
1
ϕ(u+1)= (1−s )ϕ(u)+ ϕ(k)s , u∈N. (10)
3 u+3−k
s
2 !
k=1
X
• If{a =a = 0,b 6= 0,c 6= 0},thenϕ(0) =0,ϕ(1) =(3−ES)/(1/a +
0 1 0 0 2
c ),andtherecursionformula(8)issatisfied.
0
• If{a 6= 0,b = b = 0,c 6= 0},thenϕ(0) = 0,ϕ(1) = (3−ES)/c ,and
0 0 1 0 0
thesamerecursionformula(8)holds.
Ruinprobabilityinthethree-seasonaldiscrete-timeriskmodel 429
• If{a 6=0,b 6=0,c =c =0},thenϕ(0)=3−ES,ϕ(1)=(3−ES)/s ,
0 0 0 1 2
andtherecursionformula(10)issatisfied.
We observe that all formulas presented in Theorem 5 can be used to calculate
numericalvaluesofsurvivalorruinprobabilitiesforanarbitrarythree-seasonalrisk
modelandforanarbitraryinitialsurplusvalueu.Thealgorithmsbasedonthederived
relationsworkquitequicklyandaccurately.Afewnumericalexamplesforcalculating
ruinprobabilityinthevariousversionsofthethree-seasonalriskmodelarepresented
inSection5.
3 Netprofitcondition
Inthissection,wepresentaproofofTheorem4.Werecallthatwedenotetheultimate
survivalprobabilitybyϕ(u).
ProofofTheorem4. Foranarbitraryu∈N ,wehave
0
∞ n
ϕ(u)=P u+n− Z >0
i
( )!
n=1 i=1
\ X
∞ n
=P u+n− Z >0 ∩{Z >u+1}∩{Z +Z >u+2}
i 1 1 2
( ) !
n=3 i=1
\ X
∞ n
−P u+n− Z >0 ∩{Z >u+1}
i 1
( ) !
n=3 i=1
\ X
∞ n
−P u+n− Z >0 ∩{Z +Z >u+2}
i 1 2
( ) !
n=3 i=1
\ X
∞ n
+P u+n− Z >0 . (11)
i
( )!
n=3 i=1
\ X
Sincethemodelisthree-seasonal,thelastprobabilityin(11)canbeexpressedbythe
sum
u+2 ∞ n u+2
s P u+n+3−k− Z >0 = s ϕ(u+3−k), (12)
k i k
( )!
k=0 n=1 i=1 k=0
X \ X X
where,asbefore,s =P(Z +Z +Z =k)fork ∈N .
k 1 2 3 0
Thesecondprobabilityin(11)isequals
∞ n
a P u+n−k−Z −Z − Z >0
k 2 3 i
( )!
k=u+1 n>3 i=4
X \ X
∞ n
=a P(Z +Z =0)P n−2− Z >0
u+2 2 3 i
( )!
n=4 i=4
\ X
430 A.Grigutisetal.
∞ n
+a P(Z +Z =0)P n−1− Z >0
u+1 2 3 i
( )!
n=4 i=4
\ X
∞ n
+a P(Z +Z =1)P n−2− Z >0
u+1 2 3 i
( )!
n=4 i=4
\ X
=a b c ϕ(1)+a b c ϕ(2)+a b c ϕ(1)+a b c ϕ(1). (13)
u+2 0 0 u+1 0 0 u+1 0 1 u+1 1 0
Similarly,thethirdprobabilityin(11)is
∞ n
P(Z +Z =u+2)P(Z =0)P n−2− Z >0
1 2 3 i
( )!
n=4 i=4
\ X
u+2
=c ϕ(1) a b , (14)
0 k u+2−k
k=0
X
and,finally,thefirstprobabilityin(11)is
∞ n
P(Z =u+1)P(Z =1)P(Z =0)P n−2− Z >0
1 2 3 i
( )!
n=4 i=4
\ X
∞ n
+P(Z =u+2)P(Z =0)P(Z =0)P n−2− Z >0
1 2 3 i
( )!
n=4 i=4
\ X
=a b c ϕ(1)+a b c ϕ(1). (15)
u+1 1 0 u+2 0 0
Substituting(12)–(15)into(11),wegetthat
u+2
ϕ(u)= s ϕ(u+3−k)−a b c ϕ(2)
k u+1 0 0
k=0
X
u+2
−a b c ϕ(1)−c ϕ(1) a b (16)
u+1 0 1 0 k u+2−k
k=0
X
forallu∈N .
0
Therefore,forv ∈N ,wehave
0
v v u+2
ϕ(u)= s ϕ(u+3−k)
k
u=0 u=0k=0
X XX
−b c ϕ(2) A(v+1)−a −b c ϕ(1) A(v+1)−a
0 0 0 0 1 0
v u+2
(cid:0) (cid:1) (cid:0) (cid:1)
−c ϕ(1) a b . (17)
0 k u+2−k
u=0k=0
XX
Weobservethatthesum
v u+2
a b
k u+2−k
u=0k=0
XX