An Analytic Expression for the Distribution of the Generalized Shiryaev-Roberts Diffusion

1.9 MB·

by  Aleksey S. Polunchenko

MethodolComputApplProbabmanuscriptNo.
(willbeinsertedbytheeditor)
An Analytic Expression for the Distribution of the Generalized
Shiryaev–Roberts Diffusion
TheFourierSpectralExpansionApproach
AlekseyS.Polunchenko · GrigorySokolov
Received:date/Accepted:date
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Abstract Weconsiderthequickestchange-pointdetectionproblemwheretheaimistode-
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tect the onset of a pre-specified drift in “live”-monitored standard Brownian motion; the

n change-pointisassumedunknown(nonrandom).Thetopicofinterestisthedistributionof
a the Generalized Shryaev–Roberts (GSR) detection statistic set up to “sense” the presence
J
ofthedrift.Specifically,wederiveaclosed-formformulaforthetransitionprobabilityden-
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sity function (pdf) of the time-homogeneous Markov diffusion process generated by the
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GSRstatisticwhentheBrownianmotionundersurveillanceis“drift-free”,i.e.,inthepre-

change regime; the GSR statistic’s (deterministic) nonnegative headstart is assumed arbi-
]
E trarilygiven.Thetransitionpdfformulaisfoundanalytically,throughdirectsolutionofthe
M respectiveKolmogorovforwardequationviatheFourierspectralmethodtoachievesepara-
tionofthespacialandtemporalvariables.Theobtainedresultgeneralizesthewell-known
.
t formulaforthe(pre-change)stationarydistributionoftheGSRstatistic:thelatter’sstation-
a
t ary distribution is the temporal limit of the distribution sought in this work. To conclude,
s
weexploittheobtainedformulanumericallyandbrieflystudythepre-changebehaviorof
[
the GSR statistic versus three factors:(a) drift-shift magnitude, (b) time, and (c) the GSR

1 statistic’sheadstart.
v
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Keywords GeneralizedShiryaev–Robertsprocedure·Kolmogorovforwardequation·
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Markovdiffusionprocesses·Quickestchange-pointdetection·Sequentialanalysis
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0 A.S.Polunchenko
1. DepartmentofMathematicalSciences
StateUniversityofNewYorkatBinghamton
0
Binghamton,NY13902–6000,USA
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Tel.:[phone]
1
Fax:[phone]
:
v E-mail:[email protected]
i G.Sokolov
X
DepartmentofMathematicalSciences
r StateUniversityofNewYorkatBinghamton
a Binghamton,NY13902–6000,USA
Tel.:[phone]
Fax:[phone]
E-mail:[email protected]
2 PolunchenkoandSokolov
MathematicsSubjectClassification(2000) MSC62L10·MSC60G10·MSC62M15·
MSC60J60
1 Introduction
Sequential(quickest)change-pointdetectionisconcernedwiththedevelopmentandevalu-
ationofreliablestatisticalproceduresforearlydetectionofunanticipatedchangesthatmay
(or may not) occur online in the characteristics of a “live”-monitored (random) process.
Specifically, the latter is observed continuously with the intent to “flag an alarm” in the
event (and as soon as) the behavior of the process starts to suggest the process may have
(been)statisticallychanged.Thealarmistobeflaggedasquicklyasispossiblewithinaset
tolerablelevelofthe“falsepositive”risk.See,e.g.,Shiryaev(1978),BassevilleandNiki-
forov(1993),PoorandHadjiliadis(2009),VeeravalliandBanerjee(2013),Tartakovskyetal
(2014)andthereferencestherein.
Achange-pointdetectionprocedureisidentifiedwithastoppingtime,T,thatisadapted
tothefiltration,(Ft)t(cid:62)0,generatedbytheobservedprocess,(Xt)t(cid:62)0;thesemanticsofT is
thatitconstitutesaruletostopanddeclarethatthestatisticalprofileoftheobservedprocess
mayhave(been)changed.A“good”(i.e.,optimalornearlyoptimal)detectionprocedureis
onethatminimizes(ornearlyminimizes)thedesireddetectiondelaypenalty-function,sub-
jecttoaconstraintonthefalsealarmrisk.Foranoverviewofthemajoroptimalitycriteria,
see,e.g.,TartakovskyandMoustakides(2010),PolunchenkoandTartakovsky(2012),Pol-
unchenkoetal(2013),VeeravalliandBanerjee(2013)or(Tartakovskyetal2014,PartII).
This work focuses on the popular minimax setup of the basic change-point detection
problemwheretheobservedprocess,(Xt)t(cid:62)0,isstandardBrownianmotionthatatanun-
known (nonrandom) time moment ν—referred to as the change-point—may (or may not)
experienceanabruptandpermanentchangeinthedrift,fromavalueofzeroinitially,i.e.,
E[dX ] = 0 for t ∈ [0,ν], to a known value µ (cid:54)= 0 following the change-point, i.e.,
t
E[dX ] = µ for t ∈ (ν,∞). This is illustrated in Figure 1. The goal is to find out—as
t
quicklyasispossiblewithinanapriorisetlevelofthe“falsepositive”risk—whetherthe
driftoftheprocessisnolongerzero.See,e.g.,PollakandSiegmund(1985),Shiryaev(1996,
2002),Moustakides(2004),FeinbergandShiryaev(2006),andBurnaevetal(2009).
Moreformally,undertheaboveBrownianmotionchange-pointscenario,theobserved
process,(Xt)t(cid:62)0,isgovernedbythestochasticdifferentialequation(SDE):
dX =µ1l dt+dB , t(cid:62)0, withX =0, (1)
t {t>ν} t 0
where(Bt)t(cid:62)0isstandardBrownianmotion(i.e.,E[dBt]=0,E[(dBt)2]=dt,andB0 =
0),µ(cid:54)=0istheknownpost-changedriftvalue,andν ∈[0,∞]istheunknown(nonrandom)
change-point;hereandonward,thenotationν =0(ν =∞)istobeunderstoodasthecase
whenthedriftisineffectabinitio(ornever,respectively).
Toperformchange-pointdetectionundermodel(1),thestandardapproachhasbeento
employ Page’s (1954) Cumulative Sum (CUSUM) “inspection scheme”. This choice may
bejustifiedbythefact(establishedbyBeibel1996,byShiryaev1996andbyMoustakides
2004)thattheCUSUMschemeisstrictlyminimax-optimalinthesenseofLorden(1971);
the discrete-time equivalent of this result was first established by Moustakides (1986), al-
thoughanalternativeproofwasalsolaterofferedbyRitov(1990)whousedagame-theoretic
argument.
However, when one is interested in minimax optimality as defined by Pollak (1985),
a sensible alternative to using the CUSUM scheme would be to devise the Generalized
ExactDistributionoftheGeneralizedShiryaev–RobertsDiffusion 3
Standard Brownian Motion + Drift Function
Drift Function
a
t
a
D
Change-Point(Unknown)
Standard Brownian Motion (No Drift)
Time
Fig.1:StandardBrownianmotiongainingapersistentdrift.
Shiryaev–Roberts(GSR)procedure.ThelatterisduetoMoustakidesetal(2011)andisa
headstarted version of the classical quasi-Bayesian Shiryaev–Roberts (SR) procedure that
emergedfromtheindependentworkofShiryaev(1961,1963)andthatofRoberts(1966).
WithPollak’s(1985)definitionofminimaxoptimalityinmind,themotivationtopreferthe
GSR procedure over the CUSUM chart stems from the results obtained (for the discrete-
timeanalogueoftheproblem)byTartakovskyandPolunchenko(2010);Polunchenkoand
Tartakovsky (2010) and then also by Tartakovsky et al (2012) who showed that the GSR
procedure with a carefully designed headstart may be faster (in Pollak’s 1985 sense) than
theCUSUMscheme;asamatteroffact,TartakovskyandPolunchenko(2010);Polunchenko
andTartakovsky(2010)provedtheGSRprocedure(witha“finetuned”headstart)tobethe
fastest(i.e.,thebestonecando)intwospecific(discrete-time)scenarios.Foranattemptto
extendtheseresultstotheBrownianmotionscenario,see,e.g.,Burnaev(2009).
Morespecifically,theGSRprocedurecallsforstoppingassoonastheGSRdetection
statistic,(Rtr)t(cid:62)0,hitsacriticallevelknownasthedetectionthreshold.Thelatterissetso
ashavethe“falsepositive”riskatadesired“height”.LetP (P )denotetheprobability
∞ 0
measure(distributionlaw)generatedbytheobservedprocess,(Xt)t(cid:62)0,undertheassump-
tionthatν = ∞(ν = 0);notethatP istheWienermeasure.Let P | (P | )bethe
restrictionofprobabilitymeasureP ∞(P )tothefiltrationF .Further,∞deFfitne 0 Ft
∞ 0 t
dP |
Λ (cid:44) 0 Ft , t(cid:62)0,
t dP |
∞ Ft
i.e.,theRadon–Nikodýmderivativeof P | withrespectto P | .Itiswell-knownthat
0 Ft ∞ Ft
fortheBrownianmotionscenariounderconsideration
(cid:26) µ2 (cid:27)
Λ =exp µX − t , sothatdΛ =µΛ dX , Λ =1;
t t t t t 0
2
cf. Shiryaev (1999) and Liptser and Shiryaev (2001). The process {Λt}t(cid:62)0 manifests the
likelihood ratio to test the hypothesis H : ν = 0 against the alternative H : ν = ∞,
0 ∞
4 PolunchenkoandSokolov
andisthekeyingredientoftheCUSUMstatisticaswellasoftheGSRstatistic,(Rtr)t(cid:62)0.
Specifically,tailoredtotheBrownianmotionscenarioathand,theGSRstatistic,(Rtr)t(cid:62)0,
isoftheform
(cid:90) t Λ
Rr (cid:44)rΛ + t ds
t t Λ
0 s (2)
(cid:26) µ2t(cid:27) (cid:90) t (cid:26) µ2(t−s)(cid:27)
=rexp µX − + exp µ(X −X )− ds, t(cid:62)0,
t t s
2 2
0
where Rr = r (cid:62) 0 is the headstart (a deterministic point selected so as to optimize the
0
GSRprocedure’sperformance;see,e.g.,TartakovskyandPolunchenko2010;Polunchenko
and Tartakovsky 2010; Moustakides et al 2011; Tartakovsky et al 2012; Polunchenko and
Sokolov2014).WhenRr = r = 0,itissaidthatthereisnoheadstart.TheGSRstatistic,
0
(Rtr)t(cid:62)0,withnoheadstartisequivalenttotheclassicalSRstatistic.Consequently,theGSR
procedurewhosestatistichasnoheadstartisequivalenttotheclassicalSRprocedure.Hence,
thelabels“GeneralizedSRstatistic”and“GeneralizedSRprocedure”,whichappeartohave
bothbeencoinedbyTartakovskyetal(2012).
Wearenowinapositiontoformulatethespecificproblemaddressedinthispaper:to
obtain an explicit closed-form formula for p (y,t|x,s) (cid:44) dP (Rr (cid:54) y|Rr = x)/dy,
∞ ∞ t s
0 (cid:54) s < t < ∞, x,y (cid:62) 0, i.e., for the P -transition probability density function (pdf)
∞
oftheGSRstatistic(Rtr)t(cid:62)0 withtheheadstart,R0r = r (cid:62) 0,assumedgiven.Thatis,in
this paper the GSR statistic, (Rtr)t(cid:62)0, is effectively let “run loose” over the entire space
(Rr,t) ∈ [0,∞)×[0,∞) with no detection threshold imposed, and the goal is to find
t
the exact transition pdf of the one-dimensional Markov diffusion process (Rtr)t(cid:62)0 under
probabilitymeasureP .Morespecifically,observethat,byItôformula,theP -differential
∞ ∞
of the GSR diffusion (Rtr)t(cid:62)0 is dRtr = dt + µRtrdBt, where R0r = r (cid:62) 0, whence
(Rtr)t(cid:62)0 is seen to be time-homogeneous. Therefore, p∞(y,t|x,s) depends on s and t
onlythroughthedifferencet−s (cid:62) 0,anditsufficestofindp (x,t|r) (cid:44) p (x,t|r,0),
∞ ∞
x,r,t(cid:62)0.Hence,weshallconcentrateonfindingp (x,t|r),x,r,t(cid:62)0.Also,tolighten
∞
thenotation,fromnowonweshalldropthe“∞”inthesubscriptofp (x,t|r)andsimply
∞
writep(x,t|r).
Theprincipalapproachweintendtoundertaketofindp(x,t|r)consistsinsolvingdi-
rectlytherespectiveKolmogorovforwardequation
∂ p(x,t|r)=− ∂ p(x,t|r)+ µ2 ∂2 (cid:2)x2p(x,t|r)(cid:3), x,t,r(cid:62)0, (3)
∂t ∂x 2 ∂x2
subjectto(a)the(natural)normalizationconstraint
(cid:90) ∞
p(x,t|r)dx=1 (4)
0
validforallr,t(cid:62)0,and(b)theinitialconditionlim p(x,t|r)=δ(x−r)validforall
t→0+
x; here and onward δ(z) denotes the Dirac delta function so that “lim p(x,t|r) =
t→0+
δ(x − r)” is to be understood as equality of distributions. We note also that naturally
p(x,t|r) (cid:62) 0forallr,x,t (cid:62) 0.Theinitialconditionandthenormalizationconstraint(4)
arealsotobecomplementedbytwoboundaryconditionsofspacial(i.e.,inthexvariable)
type.TheseconditionsareprovidedinSection3whichisalsothesectionwherewesolve(3)
explicitlyandthusobtainthemainresultofthispaper.
Morespecifically,torecoverp(x,t|r)from(3)wedevisetheso-calledFourierspectral
method, a separation-of-variables-type technique the general theory for which was devel-
oped by Weyl (1910) and by Titchmarsh (1962). See also, e.g., Levitan (1950), McKean
ExactDistributionoftheGeneralizedShiryaev–RobertsDiffusion 5
(1956),DunfordandSchwartz(1963),LevitanandSargsjan(1975).TheWeyl–Titchmarsh
theorywasthenappliedandsharpenedfurtherinthecontextsofprobabilityandstochastic
processes by Kac (1951, 1959), Feller (1952), Itô and McKean (1974). With the specifics
deferred to Section 3, the Fourier method allows to separate the spacial variable, x, and
the temporal variable, t, and thus reduce the original equation (3) to a (singular) Sturm–
Liouvilleproblem.Thefundamentalsolutionsofthelatter,inturn,allowtoexplicitlycon-
structtheGreen’sfunctionassociatedwiththecorrespondingSturm–Liouville(linear,dif-
ferential)operator.AstheresolventoftheSturm–Liouvilleoperator,theGreen’sfunction
providesanexhaustivecharacterizationoftheoperator’sproperties,includingitsspectrum.
Inparticular,theGreen’sfunctionisdirectlyconnectedtothetransitionpdfp(x,t|r):the
formeristheLaplacetransform(takenwithrespecttotime)ofthelatter.Therefore,getting
p(x,t|r)isamatterofinvertingits(temporal)LaplacetransformgivenbytheGreen’sfunc-
tion.TheinversioncanaccomplishedbyvirtueofCauchy’sResidueTheorem.Thepoles(or
branchcuts)oftheGreen’sfunctionyieldtheeigenvaluesoftheSturm–Liouvilleoperator
andthecorrespondingresiduesdeterminethecontributionsoftheeigenfunctionstothepdf
p(x,t|r).
Theformulaforp(x,t|r)foundusingtheFouriermethodinSection3isageneralization
oftheresultofShiryaev(1961,1963)whoobtainedthelimitρ(x) (cid:44) lim p(x,t|r),
t→+∞
i.e.,thestationarysolutionofequation(3).Thissolutioniseffectivelythestationarydistribu-
tionoftheGSRstatistic,(Rtr)t(cid:62)0,underthe“no-drift”hypothesis;seealso,e.g.,(Feinberg
andShiryaev2006,Remark4.7,pp.466–467)and(Burnaevetal2009,Remark2,p.529).
For a proof of existence of this limiting invariant distribution for any initial point Rr =
0
r(cid:62)0,see,e.g.,(PollakandSiegmund1985,Proposition3,p.271).Alltheseresultsarere-
viewedatgreaterlengthinSection2whichisintendedtosurveytherelevantpriorliterature.
Remarkably,theKolmogorovforwardequation(3)(possiblysubjecttodifferentconditions
andconstraints)hasariseninotherdisciplinesaswell,notablyinphysics(inparticular,in
quantummechanics),chemistry,andinmathematicalfinance.Itshouldthereforecomeas
no surprise that in these fields the respective solution has been obtained using completely
different techniques (e.g., Feynman path integrals; see Feynman 1948) and with no refer-
encetotheGSRstatistic.InSection2webrieflyreviewtheseresultsaswell,astheymay
castlightonapossibleinterpretationoftheGSRstatistic.
The plan for the remainder of the paper is as follows. First, in Section 2 we give a
brief account of the relevant prior literature. The main section of the paper is Section 3,
which is where we first set up the corresponding Kolmogorov forward equation (3) more
formally,andthensolveitexplicitly,i.e.,obtainanalyticallyaclosed-formformulaforthe
P -transitionpdfoftheGSRstatistic.Next,inSection4theobtainedformulaisreconciled
∞
with several previously published parallel results. In Section 5 we exploit the found pdf
p(x,t|r)numericallyandofferabriefnumericalstudyofthestatisticalbehaviorexhibited
bytheGSRstatisticinthe“drift-free”regime.Tocarryoutthestudy,weimplementedthe
obtainedpdfformulainMathematica1.Lastly,Section6ismeanttodrawalineunderthe
entirepaper.
2 OverviewoftheRelevantPriorLiterature
As can be gathered from the introduction, the centerpiece of this work is the solution of
the Kolmogorov forward equation (3). This problem—in one or another form, shape, and
1 MathematicaisapopularsoftwarepackagedevelopedbyWolframResearch,Inc.asaprogrammingen-
vironmentforscientificcomputing.SeeontheWebathttp://www.wolfram.com/mathematica/.
6 PolunchenkoandSokolov
context—hasbeenaddressedintheliteraturebefore,andthissection’sobjectiveistoprovide
abriefoverviewoftherelevantpriorresults.
One of the first to arrive at a closely related problem was Wong (1964). The actual
objective was to find the transition pdf of a stochastic process with a given stationary
(as t → +∞) distribution. Specifically, the point of departure for Wong (1964) was the
celebrated work of Kolmogoroff (1931), where it was noticed for the first time that it is
possible to construct a diffusion process whose stationary distribution is a member of the
Pearson (1895) distribution family2. However, since Kolmogoroff (1931) didn’t carry out
theactualconstruction,Wong(1964)effectivelypickedupwhereKolmogoroff(1931)left
off, and built on to the work of Karlin and McGregor (1960) to derive a number of pro-
cesseswithaPearson-typestationarydistribution;foranoverviewofanalyticallytreatable
Kolmogorov–Pearsondiffusionssee,e.g.,Avrametal(2012).OneofthePearsondistribu-
tions considered by Wong (1964) was the inverted Gamma distribution, which is a gener-
alizedextremevalueFréchet–Gumbel-typedistribution.Incidentally,thislatterdistribution
wasdemonstratedbyShiryaev(1961,1963)tobethestationarydistributionoftheclassical
SR statistic (special case of the GSR statistic with no headstart) set up for the Brownian
motion change-point scenario (1) when the observed process, (Xt)t(cid:62)0, is still drift-free;
seealso,e.g.,(FeinbergandShiryaev2006,Remark4.7,pp.466–467)and(Burnaevetal
2009,Remark2,p.529).Foraproofofexistenceofthisdistributionsee,e.g.,(Pollakand
Siegmund1985,Proposition3,p.271)and(Peskir2006,Section2.2,p.3).Hence,effec-
tively (Wong 1964, Section 2.F, p. 271) inadvertently discovered the SDE whose solution
(inaspecialcase)istheGSRdiffusion3.One“problem”withWong’s(1964)work,how-
ever, is that it is rather concise; in fact, it is so brief that it is, in essence, a “cook book”
of“ready-made”formulaewithpracticallynoderivationthereofoffered.Thatsaid,Wong
(1964)doesprovidethegeneralrecipetofindthetransitionpdfformulae,andtherecipeis
tosolvetherespectiveKolmogorovforwardequationdirectlythroughtheFourierspectral
methodwhichwementionedemergedfromtheWeyl–Titchmarshtheory;seeWeyl(1910)
andTitchmarsh(1962).ItisthisapproachandmethodthatweintendtoemployinSection3
below, but with no key details omitted. Another “issue” with Wong’s (1964) work is that
littleattentionispaidtotheamenabilityoftheobtainedtransitionpdfformulaetonumerical
evaluation,anaspectimportantforapplications.Weprovideashortnumericalstudyofthe
founddistributioncarriedoutwiththeaidofaMathematicascriptweprepared.
Amorerecentandmorerelevantexampleofpriorworkonourproblemwouldbethe
workofPeskir(2006).Justlikeusinthepresentpaper,Peskir(2006)alsofocusedon(effec-
2 Tobemoreprecise,Pearson(1895)onlyobtainedanddealtwithaspecialcaseofwhatlaterbecame
knownasthePearson(first-orderseparabledifferential)equation,thebasisforthePearsondistributionfamily.
3 Wealsowarnofatypoonpage271of(Wong1964,Section2.F,p.271)inthesecondformulafromthe
bottomofthepage(theformulaisunnumbered).Specifically,asgivenby(Wong1964,Section2.F,p.271),
theformula(intheoriginalnotation)is
(cid:18) α 1(cid:19)
2F0(−α−iµ,−α+iµ,−x)=xα+iµΨ − −iµ,1−2iµ, ,
2 x
√
wherei(cid:44) −1,2F0isthegeneralizedhypergeometricseries,andΨistheTricomiconfluenthypergeomet-
ricfunction.ThetypoisthatthefirstargumentoftheΨ-functionintheright-handsideshouldbe−α−iµ,
not−α/2−iµ.Thatis,thecorrectidentityis
(cid:18) 1(cid:19)
2F0(−α−iµ,−α+iµ,−x)=xα+iµΨ −α−iµ,1−2iµ, ,
x
ascanbeconfirmed,e.g.,with(AbramowitzandStegun1964,Section13.1,p.504).
ExactDistributionoftheGeneralizedShiryaev–RobertsDiffusion 7
tively)theGSRstatistic,andconsideredaproblemevenmoregeneralthantheonetreated
inthiswork.Specifically,Peskir(2006)wasafterthefundamentalsolutionofthefollowing
Kolmogorovforwardequation:
∂ p˜(x,t|y)=− ∂ (cid:2)(1+ax)p˜(x,t|y)(cid:3)+b ∂2 (cid:2)x2p˜(x,t|y)(cid:3), t(cid:62)0, (5)
∂t ∂x ∂x2
wherea∈Randb(cid:62)0aregivenconstants.Thisequationismoregeneralthanequation(3)
thatweareafter:thetwocoincidewhena=0andb=µ2/2.However,evenwhena(cid:54)=0,
theforegoingequationisstillcloselyconnectedtotheGSRdiffusion:ifa = θµ,thenthe
equationgovernsthetransitionpdfoftheGSRstatistic,(Rtr)t(cid:62)0,whenthedriftoftheob-
servedprocess(Xt)t(cid:62)0 isθforallt (cid:62) 0,buttheGSRstatisticissetupto“anticipate”the
drifttobeµ.Moreover,Peskir(2006)didnotrestrictattentiontothecasewhenx,y (cid:62)0(as
wedointhiswork),butallowedx,y ∈R.OnecanthereforearguethatPeskir(2006)was,
in effect, concerned with the transition pdf of the GSR statistic (with the headstart given
and arbitrary from the entire real line) in a regime general enough to include as special
cases(a)thepre-changeregime,(b)thepost-changeregime,and(c)thepost-changeregime
withdrift-shift-misspecification,i.e.,whentheactualdrift-shiftmagnitudeisdifferentfrom
theone“anticipated”bytheGSRstatistic;thislatterinterpretationcanbegivenbasedon,
e.g., the work of Pollak and Siegmund (1985), where the performance of the original SR
procedurewasstudiedintheminimaxBrownianmotioncontextwithapossibledrift-shift-
misspecification. Peskir’s (2006) approach to solving equation (5) involved obtaining the
Laplacetransform(theGreen’sfunction)oftheequation’sfundamentalsolution,andthen
invertingittogetthesoughttransitionpdfitself.However,althoughPeskir(2006)didsuc-
ceedinobtainingtheLaplacetransform(whichinthepre-changeregimetunedouttobethe
LaplacetransformoftheHartman–Watsondistribution;seeHartmanandWatson1974),he
wasabletoinvertitexplicitlyinthepre-changeregimeonlyandonlyforthezeroheadstart
(i.e.,fortheclassicalSRstatistic).AsperthehopeexpressedbyPeskir(2006)forsomeone
to“runthelastleg”andcompletehiswork,wegeneralizetheresultofPeskir(2006)forthe
pre-changeregimetoanarbitrary(nonnegative)headstart.
WenowrecalltheremarkwemadeintheintroductionthattheverysameKolmogorov
forward equation (3) that we are after in this paper has also appeared in such disciplines
asphysicsandmathematicalfinance.Specifically,inphysics,theequationisknownasthe
Fokker–Planckequation(afterFokker1914andPlanck1917),andthereisvoluminouslit-
eraturedevotedtothesolutionofthisequation(usingcompletelydifferentmethods).Tobe
more specific, equation (3) arises in physics (as well as in chemistry), e.g., in connection
withtheso-calledMorse(1929)potential.Thelatterisaharmonicoscillatormodelwidely
usedtodescribethevariationofenergywithrespecttotheinternucleardistanceinadiatomic
molecule.See,e.g.,Grosche(1988)andMorseandFeshbach(1953).Morespecifically,the
respective equation that governs these energy variations is equivalent to our Kolmogorov
forward equation (3) up to the initial and boundary conditions. It is known as one of the
veryfewequationsinquantummechanicsthatpermitanalyticalsolution.Thestandardso-
lutionstrategyusedinphysicstohandlethisequation(aswellasFokker–Plankequations
ingeneral)istoexploittheFeynman(1948)pathintegralframework.Onahigherlevel,the
physicalequivalentofourKolmogorovforwardequation(3)describesthepositionofapar-
ticlemovingaroundinaninhomogeneousenvironment,drivenbyacombinationofrandom
forces(e.g.,thermalnoise).Inthiscontext,equation(3)hasbeenstudied,e.g.,byMonthus
andComtet(1994)andbyComtetandMonthus(1996).
Another area to which the Kolmogorov forward equation (3) is no stranger is mathe-
maticalfinance.Thefinancialinterpretationoftheequationisthatitdescribesthepriceof
8 PolunchenkoandSokolov
acertainfinancialderivativeasafunctionoftimeandthederivativeparameters.Moreover,
Markovprocesses(especiallyindimensionone)areoftenusedinfinancetomodelthetime-
evolutionofthe“stateoftheeconomy”;thelatterisessentiallythefinancialequivalentofthe
randomenvironmentinsidewhichthephysicalparticleismoving,ifoneistorelatetothe
physicalanalogy.TheWeyl–TitchmarshspectraltheoryaswellastheFeynmanpathinte-
gralapproachhavebothbeenusedinmathematicalfinancetopricefinancialinstrumentsand
to describe the evolution of the underlying economic state. Excellent references assessing
thestate-of-the-artoffinancialderivativepricingmethodswouldbeBorodinandSalminen
(2002),Linetsky(2006,2007),andthereferencestherein.Bywayofexample,theequation
consideredbyPeskir(2006),i.e.,equation(5),arisesinfinanceinthecontextofpricingthe
so-calledAsianoptionsandMerton’s cashdividends;see,e.g.,Linetsky(2004a,b).Asan
economicmodel,equationsakinto(3)orto(5)ariseinconnectiontotheso-calledHull–
Whitevolatilitymodel(seeHullandWhite1987)whichpillarsonthesametypeofSDEs
that governs the GSR statistic. For a thorough mathematical treatment of the Hull–White
volatilitymodel,see,e.g.,Fatoneetal(2013).Moreover,diffusionprocessessimilartothe
GSRstatistichavealsobeenencounteredasstochasticinterestratemodels,andhavebeen
used to find the present value of annuities. See, e.g., Vanneste et al (1994); De Schepper
et al (1994); De Schepper and Goovaerts (1999). In the latter papers, the authors devised
theFeynmanpathintegralapproachtosolvetheequivalentofequation(3).Wenotethatin
Feynmanpathintegralframeworkthesolutionneednotintegratetoone,asisrequiredinour
casebythenormalizationconstraint(4).Asaresult,theobtainedsolutionsimplydiesoffto
zeroastimegoeson,andnonontrivialstationarydistributionisexhibited,whichisinstark
contrastwiththeaforementionedresultofShiryaev(1961,1963)concerningthe(nontrival)
stationarydistributionoftheGSRstatistic.
One more area where the Kolmogorov forward equation (3) has appeared is stochas-
tic processes. In the latter field, one of the key methods to deal with the equation is the
Feynman–Kac formula that emerged from the pioneering work of Feynman (1948) and
that of Kac (1949, 1951, 1959). The Feynman–Kac formula establishes a bridge between
parabolicpartialdifferentialequations(PDEs)andstochasticprocesses;wenotethatequa-
tion(3)isaparabolicPDE.Anotherpossibleapproachtofindingthetransitionpdfofthe
GSRdiffusion,(Rtr)t(cid:62)0,istoderiveitdirectlyfromthedefinitionoftheprocess,viz.from
formula(2).Specifically,theideaistoexploitthelinearconnectionbetween(Rtr)t(cid:62)0 and
thelikelihoodratioprocess{Λt}t(cid:62)0,andthencapitalizeonthefactthatΛt isageometric
(exponential)Brownianmotion,awell-studiedtypeofdiffusions.Theaforementionedwork
ofPeskir(2006)containsanumberofreferenceswherepreciselythisapproachwasdevised
to derive the distribution of effectively the GSR diffusion (although for obvious reasons
no connection to the GSR statistic was made). See, e.g., Yor (1992), Donati-Martin et al
(2001), Dufresne (2001) and Schröder (2003). However, the obtained distribution is only
forthecaseofnoheadstart,i.e.,assumingthestartingpointofthediffusioniszero.
Weconcludethissectionwitharemarkthattheaboveisjustasmallsampleofexamples
ofcontextswheretheKolmogorovforwardequationarises:the“footprint”oftheequation
ismuchlargerandspansmanymoredisciplines.Itwouldbeadauntingtasktoreviewallof
therelevantapplicationsoftheequationinaholisticmannerinasinglepaper.
ExactDistributionoftheGeneralizedShiryaev–RobertsDiffusion 9
3 TheMainResult
Thissectionisthecenterpieceofthiswork.Itisintendedtoprovideasolutiontothemain
problemofthispaper:toobtainaclosed-formformulafortheP -transitionpdfoftheGSR
∞
diffusion,(Rtr)t(cid:62)0,definedby(2)above.
Togetstarted,recallthatunderprobabilitymeasureP∞,theGSRdiffusion,(Rtr)t(cid:62)0,
solvestheSDE:dRr = dt+µRrdB ,t (cid:62) 0,whereRt = r (cid:62) 0.Sincethedriftfunc-
t t t r
tion a(x,t) ≡ 1 and the diffusion coefficient b(x,t) = µx are both independent of time
t (cid:62) 0,theGSRdiffusion,(Rtr)t(cid:62)0,istime-homogeneous.Consequently,accordingtothe
fundamentalworkofKolmogoroff(1931),thesoughttransitionpdf,p(x,t|r),x,r,t (cid:62) 0,
satisfiestheKolmogorovforwardequation:
∂ p(x,t|r)=− ∂ p(x,t|r)+ µ2 ∂2 (cid:2)x2p(x,t|r)(cid:3), x,r,t(cid:62)0, (6)
∂t ∂x 2 ∂x2
subjectto(a)the(natural)normalizationconstraint
(cid:90) ∞
p(x,t|r)dx=1, r,t(cid:62)0, (7)
0
and(b)theinitialconditionlim p(x,t|r)=δ(x−r)validforallx.Sinceequation(6)
t→0+
isaPDEoforderoneintimetandordertwoinspacex,andistoholdovertheentirespace
(x,t)∈[0,∞)×[0,∞),thetemporalinitialconditionandthenormalizationconstraintare
tobecomplementedbytwospacialboundaryconditions:oneatx=0(orasx→0+)and
oneasx→+∞.Toobtaintheseconditionslet
J(x,t|r)(cid:44)p(x,t|r)− µ2 ∂ (cid:2)x2p(x,t|r)(cid:3)
2 ∂x
betheprobabilitycurrent,sothatequation(6)canberewrittenmorecompactlyas
∂ ∂
p(x,t|r)+ J(x,t|r)=0, (8)
∂t ∂x
whichiseffectivelya(one-dimensional)continuityequation:itconstitutesthelawofcon-
servationofprobability,ananalogueofthewell-knownlawofconservationofenergyfrom
physics.Forfurthernotationalbrevity,weshallomittheheadstartrinp(x,t|r)andsimply
writep(x,t)throughoutthissection,unlessitisnecessarytoemphasizethedependenceon
r.
Toobtainthefirstboundarycondition,observethatsinceRr =r(cid:62)0byassumption,it
0
canbededucedfromdefinition(2)thatRr (cid:62)0almostsurelyunderanyprobabilitymeasure
t
andforanyt (cid:62) 0.IfRr isinterpretedasthepositionofahypothetical“particle”attime
t
instancet,thenthenonnegativityofRr forallt (cid:62) 0istosaythattheparticleistonever
t
leave the nonnegative half-plane. Put another way, no particle flow through the x-axis is
permitted.Asaresult,theprobabilitycurrentJ(x,t)throughthex-axismustbezeroatall
times,i.e.,J(0,t)=0forallt(cid:62)0,orexplicitly
p(x,t)− µ2 ∂ (cid:2)x2p(x,t)(cid:3)=0, x→0+, (9)
2 ∂x
whichisthefirstboundarycondition.
The second boundary condition can be obtained by integrating both sides of (8) with
respecttoxover[0,+∞)andthenusingthenormalizationconstraint(7).Specifically,this
10 PolunchenkoandSokolov
yieldsthatJ(+∞,t)−J(0,t)=0forallt(cid:62)0,.Therefore,theprobabilitycurrentthrough
x = +∞mustmatchtheprobabilitycurrentthroughx = 0atalltimes.Sinceinthelatter
casetheprobabilitycurrentiszero,thesecondboundaryconditionis:
p(x,t)− µ2 ∂ (cid:2)x2p(x,t)(cid:3)=0, x→+∞. (10)
2 ∂x
Tosolveequation(6),letusfirstconsidertheequation’sstationarysolution.Specifically,
thestationarityhereisinthetemporalsense,i.e.,p(x,t)isindependentoft,whichisthe
caseinthelimitast→+∞.Letρ(x)(cid:44)lim p(x,t)denotethecorrespondinglimit.
t→+∞
Foraproofthatρ(x)exists,see,e.g.,(PollakandSiegmund1985,Proposition3,p.271).
Sinceinthestationaryregime∂p(x,t)/∂t ≡ ∂ρ(x)/∂t ≡ 0forallx,equation(8)simpli-
fiesto∂J(x,t)/∂x = 0.Thelatterequation,inturn,istosaythattheprobabilitycurrent,
J(x,t),asafunctionoftime,isconstant,andinviewofthetwoboundaryconditionsestab-
lishedabove,thevalueofthatconstantiszero.Hence,weobtain
ρ(x)− µ2 d (cid:2)x2ρ(x)(cid:3)=0.
2 dx
Thesolutiontotheforegoingequationiswell-knownandisgivenbytheFréchet-type
distributiondensity
ρ(x)=e−µ22xµ22x2 1l{x(cid:62)0}. (11)
ThisresultwasfirstobtainedbyShiryaev(1961,1963)asthestationarydistributionof
theoriginalSRstatistic(withzeroheadstart).Seealso,e.g.,FeinbergandShiryaev(2006)
and(Burnaevetal2009,Remark2,p.529).Asastationarydistribution,notonlyisρ(x)
independentoft,butitisalsoindependentoftheheadstartr.Therefore,ρ(x)givenby(11)
is the stationary distribution of the GSR statistic, (Rtr)t(cid:62)0, as well. It is also noteworthy
thatthedistribution(11)isaspecialcaseoftheextremevalueinverseGammadistribution,
andinacompletelydifferentcontext(viz.finance)andusingdifferenttechniqueshasbeen
discovered,e.g.,by(Milevsky1997,Theorem1,p.224).Itisalsoaspecialcaseof(Comtet
etal1998,Formula(74),p.264).
Letusnowattackequation(6)forarbitrary(finite)time0 (cid:54) t < ∞.Tothatend,let
usfirstheuristicallyoutlineourstrategy.Themainideaistosupposethatp(x,t)isofthe
formp(x,t)=ρ(x)ψ(x)τ(t)whereρ(x)isthestationarydensitygivenby(11),andψ(x)
andτ(t)aretobefound.Inotherwords,theideaistoassumethatthespacialvariable,x,
and the temporal variable, t, can be separated. If that were the case, then the substitution
p(x,t)=ρ(x)ψ(x)τ(t)wouldbringequation(6)intothefollowingform:
τ(cid:48)(t) = 1 (cid:18)− d (cid:2)ρ(x)ψ(x)(cid:3)+ µ2 d2 (cid:2)x2ρ(x)ψ(x)(cid:3)(cid:19), x,t(cid:62)0.
τ(t) ρ(x)ψ(x) dx 2 dx2
Sincethexandtvariablesarenowondifferentsidesoftheequation,inorderforthe
twosidestobeequaltooneanotherirrespectiveofxandt,thetwosidesarebothtobeequal
tothesameconstant,sayλ.Therefore,thesubstitutionp(x,t)=ρ(x)ψ(x)τ(t)effectively
splitstheoriginalPDE(6)intotwoordinarydifferentialequations(ODEs):
τ(cid:48)(t) =λ and 1 (cid:18)− d (cid:2)ρ(x)ψ(x)(cid:3)+ µ2 d2 (cid:2)x2ρ(x)ψ(x)(cid:3)(cid:19)=λ, (12)
τ(t) ρ(x)ψ(x) dx 2 dx2