Passage times of perturbed subordinators with application to reliability

0.25 MB·English

by  Christian Paroissin

Passage times of perturbed subordinators with
application to reliability
2
1
Christian Paroissin1 and Landy Rabehasaina2
0
2
1UniversitédePauetdesPaysdel’Adour,LaboratoiredeMathématiquesetdeleursApplications-UMRCNRS5142,Avenue
n
a del’Université,64013Paucedex,France,[email protected]
J 2 UniversitédedeFranche-Comté, LaboratoiredeMathématiques - UMRCNRS6623, 16RoutedeGray, 25030 Besançon
5 cedex,France,[email protected]

]
R
WeconsiderawideclassofincreasingLévyprocessesperturbedbyanindependentBrownianmotionasadegradationmodel.
P Suchfamilycontainsalmostallclassicaldegradationmodelsconsidered intheliterature. Classicallyfailuretimeassociated
h. tosuchmodel isdefined asthehittingtimeor thefirst-passagetimeofafixedlevel. Sincesamplepathsarenot ingeneral
t increasing,weconsideralsothelast-passagetimeasthefailuretimefollowingarecentworkbyBarkerandNewby[4]. We
a
addressheretheproblemofdeterminingthedistributionofthefirst-passagetimeandofthelast-passagetime.Inthelastsection
m
weconsideramaintenancepolicyforsuchmodels.
[

Keywords:First-passagetime,last-passagetime,scalefunction,failuretime,Lévyprocess,gammaprocess,compoundPoisson
1
process,Brownianmotionwithdrift
v
AMS2000Classification:60K10,60J75
9
9
1
1
1 Introduction and Model
.
1
0 Forseveraldecades,degradationdatahavebeenmoreandmoreusedtounderstandageingofadevice,insteadof
2
onlyfailuredata. ThemostwidelyusedstochasticprocessesfordegradationmodelsbelongtotheclassofLévy
1
processes. More precisely, the three main models are the following ones: (a) Brownian motion with (positive)
:
v drift;(b)gammaprocesses;(c)compoundPoissonprocesses. MoregenerallyweconsiderabroadclassofLévy
i
X processescorrespondingtosubordinatorsperturbedbyanindependentBrownianmotion:
r
a t 0, D =G +σB
t t t
∀ ≥
where G , t 0 is a subordinator, i.e. a Lévy process with non decreasing sample paths. Since jumps of
t
{ ≥ }
D , t 0 areissuedfrom G , t 0 andarepositive,werecallthatwesaythat D , t 0 isspectrally
t t t
{ ≥ } { ≥ } { ≥ }
positive. ThisprocesscanbecharacterizedintermsofLévyexponents:
1
u R, exp(tφ (u)) =E[eiuDt]=exp(tφ (u))exp(tφ (u))=exp(tφ (u))exp( tu2σ2)
D G B G
∀ ∈ −2
φ (u) =iµˆu+ [eiux 1 iuxI (x)]Q(dx)
G [ 1,1]
ZR\{0} − − −
Exponent φ is associated to the Brownian motion and φ to G , which is in all generality a jump process.
B G t
It follows that the Lévy measure of D , t 0 is the same as that of G , t 0 that we will denote by
t t
{ ≥ } { ≥ }
ν (dx) = Q(dx). FurthermorewewillsupposethatmeasureQ(.)admitsadensitywithrespecttotheLebesgue
D
measure,i.e. thatQ(dx)=q(x)dxforsomedensityq(.). Inthefollowingwewillalsoneed
1
ϕ (u)=φ (iu)=ϕ (u)+ u2σ2,
D D G
2
i.e. ϕD(u)issuchthatE[e−uDt]=etϕD(u). Werecall,since Gt, t 0 isasubordinator,thatmaywriteinthis
{ ≥ }
caseϕ (u)inthefollowingway
D
ϕ (u)= µu+ ∞[e ux 1]Q(dx)+ 1u2σ2,
D −
− − 2
Z
2 ChristianParoissinandLandyRabehasaina
forsomeµ 0. Weconsiderinthispaperseveralapproachesformodellingdegradationofadeviceanditsfailure
≥
time. Failuretime cantraditionallybe derivedfroma degradationmodelbyconsideringthe firsthittingtime T
b
ofacriticallevelb>0. Thefirst-passagetimedistributionhasbeenalreadyderivedfortheparticularcaseoftwo
sub-models. InthecaseofBrownianmotionwithdrift(correspondingtoG = µt, µ > 0),itisthewell-known
t
inverse Gaussian distribution, see [15] for instance. For the pure gammaprocess (i.e. σ = 0 and G , t 0
t
{ ≥ }
is a gamma process), it has been studied by Park and Padgett [23]. Moreover they proposed an approximation
for the cumulative distribution function of the hitting time based on Birnbaum-Saunders and inverse Gaussian
distributions.
Recently a new approach to define the failure time was proposed by Barker and Newby [4] that consists in
consideringthelastpassagetimeofdegradationprocess D , t 0 aboveb. Asexplainedinthatpaper,thisis
t
{ ≥ }
motivatedbythefactthat,evenif D , t 0 reachesandgoesbeyondb,resultinginatemporarily"degraded"
t
{ ≥ }
stateofthedevice,itcanstillalwaysrecoverbygettingbackbelowbprovidedthiswasnotthelastpassagetime
throughb. Ontheotherhand,ifthisisthelastpassagetimethennorecoveryispossibleafterwardsandwemay
thenconsideritasa"real"failuretime. Ofcourse,thisdiscussionaboutmodellingfailuretimebythefirstorlast
passagetimebecomesirrelevantwheneverprocess D , t 0 hasnondecreasingpaths(whichisnotthecase
t
{ ≥ }
e.g. oftheBrownianmotion)sinceinthatcasebothquantitiescoincide.
Inthispaperwetheninvestigatethesequantitiesforaratherwideclassofso-calledperturbedprocess.InSection
2weprovidetheLaplacetransformofthefirstpassagetimeT withpenaltyfunctioninvolvingthecorresponding
b
under and overshoot of the process. We then confront this approach to related recent existing results on such
passagetimesdistributionsin thegeneraltheoryofLévyprocesses, thatintroducesthenotionofso-calledscale
functions. The case of severalsub-modelsis reviewed (or revisited) : in these cases the probabilitydistribution
function(pdf)and/orthecumulativedistributionfunction(cdf)canbecomputedexplicitly,oratleastnumerically.
Inconclusionofthissectionweproposeanalternativedegradationprocessthattakesintoaccountthefactthatthe
processcannotbeintheorynegativeandsuggeststhat D , t 0 bereflectedatzero. Inthatsettingweusethe
t
{ ≥ }
aforementionedrecentresults in the theoryof Lévy and reflected Lévy processes to obtain the jointdistribution
of the first passage time jointly to the overshootdistribution. In Section 3 we study the case where failure time
correspondstothelastpassagetimeL abovebandderiveitsdistributioninthenonreflectedandreflectedcase.
b
Finallywe considerin Section4a maintenancepolicyprobleminspiredby[4]andderivedistributionofrelated
quantities.
To conclude this introduction, we make precise where in the present paper previously published results are
reviewed and what is actually novel. Proposition 2.2 in Section 2.1 is new, but its proof is similar to the one
correspondingtoproofofRemark4.1aswellasExpressions(4.4)and(4.5)ofGarridoandMorales[16]. Section
2.3 recalls facts (with shortproofs)previouslyestablished in the literature thatare usefullater on. On the other
handandtothebestofourknowledge,Theorems3.2and3.3inSection3concerninglastpassagetimesmaybe
linkedtoChiuandYin[11],Baurdoux[5]andrecentpaperKyprianouetal.[21]butareotherwisegenuinelynew.
SimilarlySection4dealswithdeterminingreliabilityquantitiesfeaturesunheard-ofresults.
2 First-passage time as failure time
Weconsiderherethehittingtimedistributionofafixedlevelb>0bytheperturbedprocess D , t 0 :
t
{ ≥ }
T =inf t 0; D b
b t
{ ≥ ≥ }
whichweremindisa.s. finite. Westudybelowthedistributionof(T ,D ,D )bydeterminingthefollowing
b Tb− Tb
quantity
φw(δ,b)=E(e−δTbw(DTb−,DTb)) (1)
whereδ 0andw(.,.) isan arbitrarycontinuousboundedfunctionthatwillbereferredto aspenaltyfunction.
≥
Inthefollowingwewilldropthesubscriptwhenthereisnoambiguityonw(.,.)andthenwriteφ(δ,b)insteadof
φ (δ,b). We thendetermine(1)inthegeneralcaseandthenillustrateourresultstosub-models,someofwhich
w
distributionofT hasalreadybeenobtained.
b
2.1 General case
Weareinterestedinthecasewhereprocess G , t 0 isgeneral. Tothisend,weuseawellknowntechnique
t
{ ≥ }
thatconsistsinapproachingthejumppartprocessin G , t 0 byacompoundPoissonprocesseswhich,assaid
t
{ ≥ }
intheIntroduction,issimilartotheoneusedin[16](formoredetailsseeAppendixA.1in[16]). Moreprecisely
thisprocesscanbepointwiseapproximatedbyasequenceofcompoundPoissonprocesses((S(t,n))t 0)n Nsuch
that: ≥ ∈
Passagetimesofperturbedsubordinatorswithapplicationtoreliability 3
1. (S(t,n))n N isincreasingforallt 0,
∈ ≥
2. µt+lim S(t,n)=G forallt 0,
n t
→∞ ≥
3. foralln,(S(t,n)) hasintensityλ andjumpsizewithc.d.f.P (x)with
t 0 n n
≥
λ = Q¯(1/n) (2)
n
Q¯(1/n) Q¯(x)
P (x) = − I (3)
n Q¯(1/n) {x≥1/n}
whereQ¯(x):=Q([x,+ )). NotethatQ¯ definesmeasuresuchthatQ¯(dx)= Q(dx).
∞ −
Note that this approach is particularly interesting when λ = Q¯(1/n) Q¯(0) = Q([0,+ )) = + as
n
−→ ∞ ∞
n , i.e. when process has infinitely many jumpson any interval. Intuitively S(t,n), t 0 is obtained
→ ∞ { ≥ }
from G , t 0 bydiscardingalljumpsthatareofsizelessthan1/n.Since S(t,n), t 0 increasestowards
t
{ ≥ } { ≥ }
D , t 0 ,wehavethat
t
{ ≥ }
Tn T , n , a.s., (4)
b ց b →∞
whereTnisthehittingtimeoflevelbofthetruncatedprocess Dn, t 0 definedbyDn = S(t,n)+σB for
anyt b0andanyn N. WeremindthatTn isalsoa.s. finite{. Itnfac≥tTn}maybedescritbedasaruintimet(i.e.
≥ ∈ b b
thefirsthittingtimeof0ofastochasticprocess)inthefollowingway:
Tn =inf t 0; b µt S(t,n)+σB <0
b { ≥ − − t }
andweareinterestedintheLaplacetransformφn(δ) := E(e−δTbnw(DTnn ,DTnn))ofTbn with penaltyfunction
b− b
w(.,.)forallδ 0. Letρ =ρ (δ)bethepositivesolutiontothefollowingequation:
n n
≥
λn ∞e−ρnxdPn(x)=λn+δ− σ22ρ2n+µρn (5)
Z0
thatwewillcallgeneralizedLundbergequation.Westartbyshowingconvergenceofρ asn .
n
→∞
Proposition2.1 ρ converges as n to the unique solution ρ > 0 to the following generalized Lundberg
n
→ ∞
equation:
σ2
δ ρ2 =ϕ (ρ) δ =ϕ (ρ) (6)
G D
− 2 ⇐⇒
Proof:ThankstoExpressions(2)and(3)ofλ andc.d.f.P ,wemayrewrite(5)inthefollowingway
n n
∞e−ρnxQ(dx)=Q¯(1/n)+δ− σ22ρ2n+µρn ⇐⇒ ∞e−ρnxQ(dx)= ∞Q(dx)+δ− σ22ρ2n+µρn
Z1/n Z1/n Z1/n
⇐⇒ δ− σ22ρ2n+µρn+ ∞ 1−e−ρnx Q(dx)=0.
Z1/n
(cid:0) (cid:1)
Thusρnistheonlypositivesolutiontoequationfn(z)=0wherefn(z):=δ−σ22z2+µz+ 1∞/n(1−e−zx)Q(dx).
Letusnotethat(fn)n N increasinglyconvergespointwisetowards R
∈
f(z)=δ σ2z2+µz+ ∞ 1 e−zx Q(dx)=δ ϕD(z),
− 2 − −
Z0
(cid:0) (cid:1)
so that ρ convergesincreasingly towards ρ := sup ρ . Besides one can verify that f(z) = 0 admits an
n ∗ n N n
uniquesolutionon(0,+ ),whichissolutionρtoEqua∈tion(6).Thusρ islessthanorequaltosolutionρandwe
∗
∞
provethatweinfacthaveequalityρ =ρwhichisachievedbyshowingthatf(ρ )=0. Indeed,usinginequality
∗ ∗
0 1 e zx zxforallz,x 0andsincef (ρ )=0,wehave
− n n
≤ − ≤ ≥
f(ρ ) = f(ρ ) f (ρ ) f(ρ ) f(ρ ) + f(ρ ) f (ρ )
∗ ∗ n n ∗ n n n n
| | | − |≤| − | | − |
1/n 1/n
f(ρ∗) f(ρn) + 1 e−ρnx Q(dx) f(ρ∗) f(ρn) +ρn xQ(dx)
≤ | − | − ≤| − |
Z0 Z0
1/n(cid:0) (cid:1)
f(ρ ) f(ρ ) +ρ xQ(dx) sinceρ ρ ρ. (7)
∗ n n ∗
≤ | − | ≤ ≤
Z
4 ChristianParoissinandLandyRabehasaina
We recallthatthefactthat{Gt, t ≥ 0}is a subordinator(a nondecreasingLévyprocess)impliesthat 0∞(1∧
1/n
x)Q(dx) < + (see e.g. (2) p.72 of [6]), hence xQ(dx) 0. Remembering that f is a continuous
function,thisim∞pliesthat(7)tendstozeroasn + 0,hencef(ρ−)→=0. R ✷
∗
→ R∞
The Laplace transform φ (δ) with penalty function w(.,.) of Tn is given through the following which is a
n b
particularcaseofTheorem2of[29]adaptedtoourcontext:
Theorem2.1 Letw(.,.)beaboundedcontinuousfunctionanddefine
∞
ω (x)= w(x,y x)dP (y).
n n
−
Zx
Thenb7→φn(δ,b):=E(e−δTbnw(DTnbn−,DTnbn))satisfiestherenewalequation
φ (δ,b)=φ (δ, )⋆g (δ, )(b)+h (δ,b) (8)
n n n n
· ·
wherefunctionsg (, )andh (, )aredefinedby
n n
· · · ·
gn(δ,y) = 2σλ2n ye−[−2µ/σ2+ρn(δ)](y−s) ∞e−ρn(δ)(x−s)dPn(x)ds (9)
Z0 Zs
hn(δ,y) = e−[−2µ/σ2+ρn(δ)]y+ 2σλ2n ye−[−2µ/σ2+ρn(δ)](y−s) ∞e−ρn(δ)(x−s)ωn(x)dxds. (10)
Z0 Zs
Proof:Withnotationsof[29],wehaveg (δ,y)expressedasin(1.10)thereinwithb:=b(δ)= 2µ/σ2+ρ (δ),
n n
−
λ:=λ ,P():=P ()andD =σ2/2.Stillwithnotationsof[29],andinTheorem2therein,weseethatfunction
n n
· ·
y hn(δ,y) is the sum of e−[−2µ/σ2+ρn(δ)]y and some functiongw() defined in Expression(2.8)of [29] that
de7→pendsonω . Itiseasytoverifythatthisfunctionisthelasttermont·heright-handsideof(10). ✷
n
Passingonthelimitn + inTheorem2.1yieldsthefollowingrenewalequationforfunction(1):
→ ∞
Proposition2.2 Letω(x):= x∞w(x,y−x)Q(dy). Functionφ(δ,·)=φw(δ,·)satisfiestherenewalequation
R φ(δ,b)=φ(δ, )⋆g(δ, )(b)+h(δ,b) (11)
· ·
wherefunctionsg(, )andh(, )aredefinedby
· · · ·
g(δ,y) = 2 ye [ 2µ/σ2+ρ(δ)](y s) ∞e ρ(δ)(x s)Q(dx)ds (12)
−− − − −
σ2
Z0 Zs
h(δ,y) = e [ 2µ/σ2+ρ(δ)]y + 2 ye [ 2µ/σ2+ρ(δ)](y s) ∞e ρ(δ)(x s)ω(x)dxds. (13)
−− −− − − −
σ2
Z0 Zs
Henceφ(δ,b)isgivenbythePollaczek-Kinchinelikeformula
∞
φ(δ,b)= g⋆k(δ,.)⋆h(δ,.)(δ,b). (14)
k=0
X
Notethat(14)isanalogoustoExpression(4.2)in[16].
Proof: Letusprovethatλ ω convergestoω. Thisiseasilyseenbyrememberingthatλ = Q¯(1/n)andthus
n n n
that,by(3),
λ ω (x)= ∞w(x,y x)I dQ¯(y)
n n y 1/n
−Zx − { ≥ }
whichconvergestothedesiredexpression,rememberingthat dQ¯(y) =dQ(y). Convergenceofh tohfollows
n
−
from(10).Inthesameway,λn s∞e−ρn(δ)(x−s)dPn(x)convergesto s∞e−ρ(δ)(x−s)Q(dx),yieldingconvergence
ofg togthanksto(9). ✷
n
R R
As announcedin the Introduction, it is also possible to use the theory of Lévy processes to propose a differ-
ent approachfor determiningthe joint distribution of the hitting time T jointly to the state of D , using scale
b Tb
functions.Moreprecisely,wehavethefollowingpropositionfrome.g. KyprianouandPalmowski[20]:
Passagetimesofperturbedsubordinatorswithapplicationtoreliability 5
Proposition2.3(Theorem1(4)[20]) Let us define for all δ 0 the scale function W(δ), through its Laplace
≥
transform,andZ(δ) by
∞e−λxW(δ)(x)dx = 1 , λ>ρ(δ) (15)
ϕ (λ) δ
Z0 D −
x
Z(δ)(x) = 1+δ W(δ)(y)dy, (16)
Z0
wherewerecallthatρ(δ)issolutiontotheLundbergequationϕ (λ)=δ. ThenfromExpression(4)p.19of[20]
D
onehasthat
δ
E[e−δTb]=Z(δ)(b) W(δ)(b). (17)
− ρ(δ)
Just to be clear on notations, we emphasize that [20] deals with spectrally negative processes. To apply it here
(hence to obtain Expressions (15), (16) and (17)), we thus need to consider hitting time of 0 of process D˜ :=
t
D starting from D˜ = b. In particular, Laplace exponentψ(.) of D˜ as defined in Expression (2) of [20] by
t 0 t
−
E[eλD˜t] = etψ(λ) doescoincidewithfunctionϕD(.), andΦ(δ) = sup λ 0 ψ(λ) = δ , alsodefinedin[20],
{ ≥ | }
coincideswithρ(δ).
Remark2.2(scalefunctionregularity) AnecessaryconditionforfunctionW(δ) definedintheProposition2.3
tobedifferentiableisthat D , t 0 hasunboundedvariation,whichisthecaseheresinceithasaGaussian
t
{ ≥ }
component (i.e. σ > 0). In fact it is shown in [9] the stronger fact that σ > 0 implies that W(δ) is twice
differentiable.
Remark2.3(boundaryvalueofscalefunction) Still in the present case where process D , t 0 has un-
t
{ ≥ }
boundedvariation,wehavethatW(δ)(0)=0byLemma8.6p.222of[19].
As a complementto (17), it is interesting to note that Remark3 of [20] givesan explicitexpressionof the joint
Laplacetransformof(T ,D ).
b Tb
The approach in Proposition 2.3 has however a cost, which is that a Laplace Transform inversion of (15) is
requiredto obtainthescale function. HoweverrecentresultshavebeenfoundconcerningexpressionofW(δ) in
particularcases,seeHubalekandKyprianou[18]aswellasEgamiandYamazaki[14]inthecasewhere G , t
t
{ ≥
0 is a compound Poisson process with jumps following phase-type distribution. In fact the following result
}
combinesbothapproachesgiveninPropositions2.2and2.3,andtheoreticallygivesaclosedformexpressionof
scalefunctionW(δ) ofanyspectrallypositiveLévyprocess:
Proposition2.4 ScalefunctionW(δ)uniquelydefinedbyLaplacetransform(15)satisfiesthefollowingfirstorder
differentialequation
ρ(δ) ∞
W(δ)′(x) ρ(δ)W(δ)(x)= g⋆k(δ,.)⋆h′(δ,.)(δ,x):=H(δ,x) (18)
− − δ
k=0
X
whereg(δ,.)isgivenby(12)andh(δ,.)isderivativeofh(δ,.)givenin(13)withw 1,i.e.
′
≡
h(δ,y)= [ 2µ/σ2+ρ(δ)]e [ 2µ/σ2+ρ(δ)]y + 2 ∞e [ 2µ/σ2+ρ(δ)](x y)Q¯(x)dx
′ −− −− −
−− σ2
Zy
[ 2µ/σ2+ρ(δ)] 2 ye [ 2µ/σ2+ρ(δ)](y s) ∞e ρ(δ)(x s)Q¯(x)dxds. (19)
−− − − −
− − σ2
Z0 Zs
ThusW(δ)(x)hasthefollowingexplicitexpression
x
W(δ)(x)= e ρ(δ)(x y)H(δ,y)dy. (20)
− −
Z0
Proof:Differentialequation(18)simplycomesfrom(17)thatonedifferentiateswithrespecttob(whichispossible
sinceW(δ) isdifferentiableinlightofRemark2.2),usingexpressionE[e−δTb]=φw(δ,b)wherepenaltyfunction
w(.)isidenticallyequalto1,andfinallyusingExpression(14). Notethatdifferentiationof(14)isdonebyusing
thewellknownpropertyofderivationofconvolutedfunctions(f ⋆g) =f ⋆g =f ⋆g ,explainingwhyH(δ,.)
′ ′ ′
6 ChristianParoissinandLandyRabehasaina
featuresderivativeoffunctionh(δ,.).
SincebyRemark2.3onehasthatW(δ)(0)=0,Equation(20)isthenobtainedbysolvingthestandardfirstorder
differentialequation(18). ✷
NotehoweverthatFormula(20)requirestocomputetheinfiniteseriesappearingin(18),whichinpracticemay
not be handy. However, since such scale functionsare importantin the theory of Lévy processes (in particular,
these functions will be useful in Sections 2.3 and 3 for determining quantities related to first passage times of
reflectedprocessesandlastpassagetimes),anyexpressioncanbeconsideredaswelcome.
Remark2.4 AsymptoticbehaviourofT asb + maybeobtainedthroughRoynetteetal[27].Moreprecisely,
b
→ ∞
itwasprovedthat(T +b/ϕ (0))/√bconvergesindistributiontoan (0, ϕ (0)/ϕ (0)3)distribution. One
b ′D N − ′D′ ′D
can also compute from [27] asymptotic behaviourof triplet T +b/ϕ (0))/√b, D b, b D that we
b ′D Tb − − Tb−
didnotincludeherebutthatinvolvetechnicalexpressions. (cid:16) (cid:17)
2.2 Examples
Weillustratethepreviousstudywithexamplesandreviewsomefamousexamplesrelatedtodegradationmodels.
Puregammaprocess Hereweassumethatσ =0andthat G , t 0 isagammaprocesswithshapeparameter
t
{ ≥ }
αandscaleparameterξ. WerecallthatitsLévyexponentandLévymeasurearegivenby
ϕ (u) = ϕ (u)= αlog(1+u/ξ)
G D
−
νD(dx) = Q(dx)=x−1e−xξαdx.
ConsideringthisspecialcaseintothegeneralizedLundbergequation,itfollowsthatthisequationhasnopositive
solution. It appearsthat the presence of the perturbationin the degradationmodelis importantfor applyingthe
result obtained by Tsai and Wilmott [29] as we did in Proposition 2.2. However, in this first special case, the
degradation process reduces to a pure stationary gamma process and so D , t 0 has increasing paths. It
t
{ ≥ }
followsthat:
t 0, P[T >t]=P[D <b].
b t
∀ ≥
ConsequentlyitissufficienttostudythedistributionofD foranyt 0.Thehittingtimedistributionwasalready
t
≥
givenforinstancep.517ofParkandPadgett[23]:
Proposition2.5(ParkandPadgett[23]) Thecumulativedistributionfunction(cdf)ofT is:
b
Γ(αt,b/ξ)
t 0, F(t)= ,
∀ ≥ Γ(αt)
whereΓ(, )istheupperincompleteGammafunction. Theprobabilitydistributionfunction(pdf)ofT is,forany
b
· ·
t 0:
≥
αt
b γ(αt,b/ξ) α b
f(t)=α Ψ(αt) log + F (αt,αt;αt+1,αt+1; b/ξ),
− ξ Γ(αt) (αt)2Γ(αt) ξ 2 2 −
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
whereΨisthedi-gammafunction(orlogarithmicderivativeoftheGammafunction),γ(, )=Γ() Γ(, )isthe
· · · − · ·
lowerincompleteGammafunctionand F thegeneralizedhypergeometricfunctionoforder(2,2).
2 2
Ithasbeenproved(see[1]orSection5of[28]forinstance)thatT hasanincreasingfailurerate.
b
PerturbedgammaProcess Statisticalinferenceinaperturbedgammaprocesshasbeenstudiedin[8]usingonly
degradationdata. Howeversometimesbothdegradationdataandfailuretimedataareavailable(see[22]forsuch
problemfor a related model). In addition, fromparametersestimation (based on degradationdata for instance),
one can obtain an estimation of the failure time distribution. Hence the distribution of T can be of interest. In
b
thatcase, G , t 0 is a gammaprocesswith shape parameterα andscale parameterξ. We recallthatLévy
t
{ ≥ }
exponentandLévymeasureofprocess D , t 0 arethengivenby
t
{ ≥ }
ϕ (u) = αlog(1+u/ξ)+ 1u2σ2
ν D(dx) = Q−(dx)=x−1e−xξαd2x. (21)
Passagetimesofperturbedsubordinatorswithapplicationtoreliability 7
Thus, Proposition 2.2 gives joint distribution of (T ,D ,D ) through expression of φ(δ,b) where ω(x) :=
x∞w(x,y − x)e−yy/ξdy and g(δ,y) = σ22 0ye−ρ(bδ)(y−Tbs−) s∞Teb−ρ(δ)(x−s)e−xx/ξdxds, w(.,.) being an arbitrary
boundedfunction.AlsonotethatfromRemark2.4onehasthanksto[27]theCentralLimitTheorem
R R R
T ξb/α α/ξ2+σ2
b
− D 0, , b + .
√b −→N α3/ξ3 → ∞
(cid:18) (cid:19)
Finally,expressionofthescalefunctionisthengivenby(20)withϕ (.)andQ(.)definedin(21).Thiswillcome
D
inhandyinSection3.
Brownianmotion with positive drift We consider the case where G = µt, i.e. D , t 0 is a Brownian
t t
{ ≥ }
motion with drift. In such case, the distribution of the hitting time of the constant boundary b is known and is
calledtheinverseGaussiandistribution.Itspdfisgivenby:
b (b µt)2
t 0,f(t)= exp − .
∀ ≥ √σ2t3 − 2tσ2
(cid:18) (cid:19)
Proofofthisresultisgenerallybasedonthesymmetricprinciplefull-filledbytheBrownianmotionwhenµ =0,
orcanbeshowedwithmartingalemethodsinthecaseµ > 0. Alternativelythepdfcanbeobtainedbyinverting
theLaplacetransformofT :
b
(γ µ)b
φ(δ)=E[e−δTb]=exp δ− , (22)
− σ2
(cid:18) (cid:19)
with γ = µ2+2δσ2. Note that the expression of this Laplace transform is standard and can be found e.g.
δ
in Expression (38) p. 212 of [12] (see also [1], page 19). Also note that (22) is compatible with Expression
p
(14). Indeed in the context of Proposition 2.2 we have here g 0 and h 0, thus (14) reduces to φ(δ,b) =
≡ ≡
e [ 2µ/σ2+ρ(δ)]y whereρ(δ)satisfies(6) 0= σ2ρ(δ)2 µρ(δ) δ,givingtheexactsameexpression(22).
−− ⇐⇒ 2 − −
Expressionofthescalefunctionforthiscaseisthengivene.g.p.121in[18]by
2 x 2 x
W(δ)(x)= e µx/σ2sinh 2δσ2+µ2 = e µx/σ2sinh γ . (23)
2δσ2+µ2 − σ2 γδ − σ2 δ
(cid:16) p (cid:17) (cid:16) (cid:17)
Note thatthereseems topbea smallmistake in [18] ofexpressionof W(δ)(x) (wherethereare some µ’sinstead
of µ2’s), that we correctedhere. As provedby Chhikara and Folks [10], the failure rate of an inverse Gaussian
distributionisnon-monotone,butitisinitiallyincreasingandthendecreasing.
PerturbedcompoundPoissonprocesswithphase-typejumps Letussupposethat G , t 0 isacompound
t
Poisson process of intensity λ whose jumps are phase-type distributed with represen{tation (≥m,α},T). Let t :=
T1where1isacolumnvectorofwhichentriesareequalto1’sofappropriatedimension(seee.g.ChapterVIII
−
ofAsmussen[2]foranextensiveaccountonsuchdistributions).Inthatcaseϕ isgivenby
D
1
ϕ (u)= u2σ2+λ(α(uI T) 1t 1).
D −
2 − −
EgamiandYamazaki[14]givetheexpressionoftheLaplacetransformE(e−δTb)bydeterminingaclosedformula
forthescalefunctionsWδandusingresultsinProposition2.3.Morepreciselyfollowing[14],letusdenoteforall
δ > 0thecomplexsolutions(ξ ) (resp. (η ) )ofEquationϕ (u) = δ (resp. δ/(δ ϕ (u)) = 0),u C. We
i,δ i i i D D
− ∈
supposethattheξ ’saredistinctroots. Weset
i,δ
:= i ϕ ( ξ )=δand (ξ )>0 ,
δ D i,δ i,δ
I { | − ℜ }
:= i δ/(δ ϕ ( η ))=0and (η )>0 ,
δ D i i
J { | − − ℜ }
(u+η ) ξ
ϕ−δ (u) = j∈Jδ η j i∈(Iuδ+i,ξδ ).
Q j∈Jδ j iQ∈Iδ i,δ
Onpage4of[14]itisstatedthatCard( )=QCard( )+Q1(thisresultsinfactcomesfromLemma1(1)of[3]),
δ δ
I J
sothatϕ−δ (∞)existsandisequalto0. Wethendefine
ξ
(Ai,δ)i∈Iδ s.t. ϕ−δ (u)−ϕ−δ (∞)=ϕ−δ (u)= Ai,δξi,δi,+δ u,
iX∈Iδ
̺ := A ξ .
δ i,δ i,δ
X
8 ChristianParoissinandLandyRabehasaina
Then Proposition 2.1 of [14] gives expression of the Laplace transform φ(δ) = E(e−δTb) =
scail∈eIfδuAnci,tδioen−ξi,δ(x−a)and Proposition 3.1 of [14] yields the following interesting and useful expression of the
P 2 ξ
W(δ)(x)= σ2̺ Ai,δρ(δ)i+,δξ eρ(δ)x−e−ξi,δx . (24)
δ i,δ
iX∈Iδ h i
Furthermore,aspointedoutin[14],expressionsofW(δ)aremorecomplicatedbutavailablewhenrootsξ ’shave
i,δ
multiplicitym >1.
i
2.3 Reflected processes
Thepreviousmodelmaynotbetoorealistic ifwe considertheBrownianmotionasa meansofmodellingsmall
repairs, as the degradationprocess D , t 0 maythen be negative. An alternativefor thisis to considerthe
t
{ ≥ }
reflectedversionof D , t 0 definedinthefollowingway
t
{ ≥ }
t 0, D :=D inf (D 0).
∀ ≥ t∗ t−0 s t s∧
≤ ≤
The hitting time distribution T of D , t 0 jointly to the overshoot and undershoot pdf is given by the
b∗ { t∗ ≥ }
followingtheorem
Theorem2.1 Let ussuppose that D , t 0 is non monotone,i.e. that σ > 0. Let W(δ) be definedby (15)
t
{ ≥ }
wherewerecallthatρ=ρ(δ)issolutiontotheLundbergequationϕ (z)=δ. Then
D
E[e−δTb∗I{DT∗b∗−∈dy,DT∗b∗∈dz}]=νD(dz−y)rˆb(δ)(b,y)dy (25)
W(δ)(b)W(δ)′(y)
whererˆ(δ)(b,y):= W(δ)(y).
b W(δ)′(b) −
Proof: We apply results from Doney[13] and we write, following notationstherein, X := D , so that Lévy
t t
measure of X , t 0 is Π(dx) := ν ( dx)and process Yˆ(t) := sup (X 0) X− is equal to D .
Followingte{rmitnolo≥gyo}f[13],W(δ)istheDδ-−scalefunctionof X , t 0 an0≤dsis≤dtefinse∨dby(−15)wtithϕ insteat∗d
t D
ofϕ . Remark4p.14of[13]givesexpression(25)whererˆ(δ{) isgive≥nby}Pistorius[24](seealsoExpre−ssion(15)
D b
inTheorem1of[13])withx:=0anda:=b,notingthatfunctionW(δ) isdifferentiablebyRemark2.2. ✷
Note againthatTheorem2.1is especially interestingwhenfunctionW(δ) admitsclosed formexpressions,as
in[18,14]. ForexampleinthecaseofaperturbedcompoundPoissonprocesswithphase-typedistributedjumps
(andusing the same notationsas in Section 2.2)we haveν (dz) = λαeTzt and W(δ) givenby (24)(of which
D
derivative is easily available), which, plugged in (25), easily yields the Laplace transform of the corresponding
hittingtimeT jointlytotheovershootandundershootdistribution.
b∗
WenowstateafamouslemmathatlinksdistributionofD tothecumulativedistributionfunctionofT forall
t∗ b
b 0:
≥
Lemma2.1 Wehaveforallbandt 0,P(D >b)=P(T t).
≥ t∗ b ≤
Proof: Thisisasimpleconsequencefrome.g. Lemma3.5p.74ofKyprianou[19]thatimpliesthatP(D > b)=
t∗
P sup D >b whichinturnisequaltoP(T t). ✷
0≤s≤t s b ≤
(cid:0) (cid:1)
3 Last-passage time as failure time
WeletL andL bethelastpassagetimesofprocesses D , t 0 and D , t 0 belowlevelbdefinedas
b ∗b { t ≥ } { t∗ ≥ }
Lb :=sup{0≤u|Du ≤b} and L∗b :=sup{0≤u|Du∗ ≤b}
whicharewelldefinedasprocesses D , t 0 and D , t 0 satisfylim D =lim D =+ .
t ∗ t + t t + ∗
{ ≥ } { ≥ } → ∞ → ∞ ∞
Passagetimesofperturbedsubordinatorswithapplicationtoreliability 9
3.1 General case
Letusintroducethefollowingbivariatemeasures and ˆon[0,+ )2throughtheirdoubleLaplacetransforms
U U ∞
∞ ∞e−αs−βx (ds,dx)= ρ(α)−β , β >ρ(α), ∞ ∞e−αs−βxˆ(ds,dx)= 1 , β,α 0.
U α ϕ (β) ∀ U ρ(α)+β ∀ ≥
Z0 Z0 − D Z0 Z0 (26)
Expressions(26)maybefoundinExpressions(12)and(13)of[7],orp.154andp.170inChapter6of[19](note
that the latter reference considers spectrally negative processes, hence roles for and ˆ are swapped therein).
Furthermore,from(26)of[7]onehasthatUˆδ(dx):= s∞=0e−δsUˆ(ds,dx)hastheUexpressUion
R
ˆ (dx)=e ρ(δ)xdx, (27)
δ −
U
henceUˆδ([0,+∞)) = 1/ρ(δ). Inthesamespirit,wedefineUδ(dx) := s∞=0e−δsU(ds,dx). (26)thenreadsthat
oxn∞=.0e−βxUδ(dx) = δρ−(ϕδ)D−(ββ) forallβ > ρ(δ). WethenhavethefollowRingidentity,thatwillbeofinterestlater
R
Lemma3.1 Onehas
(dx)=[ ρ(δ)W(δ)(x)+W(δ)′(x)]dx. (28)
δ
U −
Proof:From(15)wegetthefollowing
∞ e βx (dx)= ρ(δ) ∞e βxW(δ)(x)dx+ ∞βe βxW(δ)(x)dx (29)
− δ − −
U −
Zx=0 Z0 Z0
where β > ρ(δ). We recallfrom Remark 2.3that W(δ)(0) = 0. As to behaviourat + of the scale function,
∞
we have, thanks to Lemma 8.4 p.222 of [19], relation W(δ)(x) = ecxWc(δ−ϕD(c))(x), for any c R such that
∈
δ ϕD(c) 0,whereWc(δ−ϕD(c)) isascalefunctiondefinedunderadifferentprobabilitymeasure. Bypicking
− ≥
c=ρ(δ)thenonegetsδ ϕ (c)=0and
D
−
W(δ)(x)=eρ(δ)xW(0) (x) (30)
ρ(δ)
(see e.g. Second Remark p.32 of [25] for this identity as well as details on this other probability measure). At
theendofProofofCorollary8.9p.227of[19],itisshownthatW(0) (+ ) = 1 whereϕ (q) :=
ρ(δ) ∞ ϕ′D,ρ(δ)(0+) D,ρ(δ)
ϕ (q+ρ(δ)) ϕ (ρ(δ))=ϕ (q+ρ(δ)) δ,hence
D D D
− −
1
W(0) (+ )= <+ . (31)
ρ(δ) ∞ ϕ (ρ(δ)) ∞
′D
ThusinviewofW(δ)(0)=0,(30)and(31),andsinceβ >ρ(δ),thefollowingintegrationbypartsmakessense:
∞βe−βxW(δ)(x)dx= e−βxW(δ)(x) ∞+ ∞e−βxW(δ)′(x)dx=0+ ∞e−βxW(δ)′(x)dx, (32)
Z0 h− i0 Z0 Z0
rememberingthatW(δ) isindeeddifferentiablebyRemark2.2. ComparingLaplacetransforms(29)and(32),we
thenobtain(28). ✷
Letusalsonotethat,accordingtoDefinition6.4p.142of[19], thefactthatσ > 0entailsthat0isregularfor
sets ( ,0)and(0,+ )(inparticular,Theorem6.5p.142of[19] applieshere). With thatinmind, andsince
−∞ ∞
D , t 0 is spectrally positive and drifts to + , we may recall the following important recent result from
t
{ ≥ } ∞
Kyprianouetal[21].
Theorem3.1(Corollary2ofKyprianou,PardoandRivero[21]) Letusdefine
D := inf D , D = infD , G :=sup s 0 D D =0 ,
∞ u 0 u s t s s ∞ { ≥ | s− s }
≥ ≤
D := infD , :=inf s>t D D =0 .
t s t s t
D { | − }
−→ −→ −→
10 ChristianParoissinandLandyRabehasaina
Thendistributionof(G ,D , L ,L ,D b,b D ,D b)isgivenbythefollowingidentityfor
t,b,v >0,s>r >0,0∞ y∞<−→bD+Lbv−,wb ub>−→0L:b− − Lb− Lb−
≤ ≥
P[G dr, D dv, L dt, L ds, D b du,b D dy, D b dw]
∞ ∈ − ∞ ∈ −→DLb − b ∈ b ∈ −→Lb − ∈ − Lb− ∈ Lb − ∈
= ˆδ([0,+ ))−1ˆ(dr,dv) (ds r,b+v dy)ˆ(dt,w du)Q(dw+y), (33)
U ∞ U U − − U −
where ˆ ([0,+ )) 1 =ρ(0)from(27).
δ −
U ∞
Itisclearthatdistributionof(L ,b D ,D b)maybetheoreticallyobtainedfromthistheorem.Infact,our
goalistoproposeexpressionsobfth−isdiLstbr−ibutioLnb−thatonlyinvolvesquantitiesthatweredeterminedinSection2.1,
e.g.scalefunctions,whichwesawcanbeavailableinmanysituations,asopposedtomeasures and ˆappearing
U U
in(33)which,asseenin(26),areavailableonlythroughdoubleLaplacetransforms. Moreprecisely,wehavethe
followingresults.
Theorem3.2 Wehaveforallt 0anda R,
≥ ∈
P(L <t) = ∞E[D ]W(a b)f (a)da (34)
b 1 − Dt
Zb
P(L t, D da) = [1 E[D ]W(a b)]f (a)da (35)
b ≥ t ∈ − 1 − Dt
wheref (.)isdensityofr.v. D andW(.) = W(δ)(.)definedin(15)withδ = 0. Besides,forallδ 0,andfor
Dt t ≥
b>y 0,w>0,theLaplacetransformofL jointlytodensityoftheunderandovershootisgivenby
b
≥
1
E[e−δLbI{b−DLb−∈dy,DLb−b∈dw}]=(cid:20)eρ(δ)(b−y)ϕ′D(ρ(δ)) −W(δ)(b−y)(cid:21)dy.[1−e−ρ(0)w]Q(dw+y). (36)
Let us compare results given in Theorem 3.2 with existing ones in the literature concerning last passage times
of Lévy processes. References [11] and [5] give distributions of respectively last exit times and last exit times
beforeanexponentiallydistributedtime,intermsoftheirLaplacetransform,forasimilarclassofLévyprocesses;
howeverTheorem3.2ismoreadaptedhereasitdirectlygivesitscdfjointlytothedensityoftheovershoot,thus
avoiding an inverse Laplace transform. As said before, the slight advantageof Formula (36) over(33) is that it
onlyinvolvesthescalefunction.
Proof: Let us start by showing (34) and (35). Let t > 0. By definition of L we note that for all a b
b
≥
event[L <t,D da]isequalto[D da, D willnothitlevelbanymoreaftert]. HenceusingtheMarkov
b t t s
∈ ∈ { }
property:
P[L <t,D da] = P [T =+ ]P[D da]
b t a b 0 t
∈ − ∞ ∈
whereP [T =+ ]istheprobabilitythatprocess D , t 0 startingfroma bwillneverhit0andisgiven
a b 0 t
e.g.throu−ghFormula∞(4)p.19of[20]byP [T =+{ ]=E≥[D}]W(a b)andP−[D da]=f (a)dawhere
f isthedensityofr.v. D andW(.)=Wa−(b0)(.0)in(1∞5). Byinteg1rating−afrombto+t ∈onegetsD(3t4). Equation
(3D5t)stemsfromthebasicetqualityP(L t, D da)=P(D da) P(L <t, D∞ da).
b t t b t
≥ ∈ ∈ − ∈
Wenowturnto(36),anduseTheorem3.1tothisend.SincebyFubinitheoremwehave
E[e−δLbI{b−DLb−∈dy,DLb−b∈dw}]=Zs=∞0e−δsP[Lb ∈ds,b−DLb− ∈dy, DLb −b∈dw],
andinviewof(33),onejustneedstocomputethefollowingintegral:
w
∞ ∞ e δsP[G dr, D dv, L dt, L ds,
Zv=0Zt=0Zs>r>0Zu=0 − ∞ ∈ − ∞ ∈ −→DLb − b ∈ b ∈
D b du,b D dy, D b dw]
Lb − ∈ − Lb− ∈ Lb − ∈
−→ w
=ρ(0) ∞ ˆ(dr,dv)e δs (ds r,b+v dy). ∞ ˆ(dt,w du).Q(dw+y), (37)
−
U U − − U −
Zv=0Zs>r>0 Zu=0Zt=0