Likelihood inference for exponential-trawl processes

0.37 MB·English

by  Neil Shephard

Likelihood Inference for Exponential-Trawl
Processes
NeilShephardandJustinJ.Yang
5
1
0
2

n
a AbstractInteger-valuedtrawlprocessesareaclassofseriallycorrelated,stationary
J andinfinitelydivisibleprocessesthatOleE.Barndorff-Nielsenhasbeenworkingon

5 inrecentyears.InthisChapter,weprovidethefirstanalysisoflikelihoodinference

for trawl processes by focusing on the so-called exponential-trawl process, which
]
E is also a continuous time hidden Markov process with countable state space. The
M coreideasincludepredictiondecomposition,filteringandsmoothing,complete-data
analysisandEMalgorithm.Thesecanbeeasilyscaleduptoadapttomoregeneral
.
t trawlprocessesbutwithincreasingcomputationefforts.
a
t
s
[

1 1 Introduction
v
5
2 Inrecentyears,OleE.Barndorff-Nielsenhasbeenworkingonaclassofstochastic
9 models called integer-valued trawl processes. References include [2], [4] and [5].
0 These are flexible models whose core randomness is driven by Poisson random
0 measures. Trawl processes are related to the up-stairs processes of [25] and the
.
1 randommeasureprocessesof[24].Bothoftheseprocessesarestationary.[5]also
0 brings out the relationship between their processes and M/G/∞ queues (e.g. [16],
5
[18] and [8, Ch. 6.31]) and mixed moving average processes (e.g. [22]). Related
1
discretetimecountmodelsinclude[9],[13],[10],[12],[15],[17],[26],[14],[17]
:
v and[23].Trawlprocessesalsofallwithinthewideclassoftheso-calledambitfields
i
X
r
NeilShephard
a
DepartmentofEconomics,HarvardUniversity
1802CambridgeStreet,02138,Cambridge,MA,USA
e-mail:[e-mail]
JustinJ.Yang
DepartmentofStatistics,HarvardUniversity
1OxfordStreet,02138,Cambridge,MA,USA
e-mail:[e-mail]
1
2 NeilShephardandJustinJ.Yang
(e.g. [7] and [3]). Recently, [21] models high frequency financial data by using a
trawl process to allow for fleeting movements to prices in addition to an integer-
valuedLe´vyprocessproposedby[6].
As far as we know, there is no existing literature that directly and completely
addresses likelihood inference for these trawl processes—or equivalently the pre-
diction based upon it. Even though there are a large number of papers that focus
on likelihood inference for marked point processes (see [11] for a survey), it only
indirectly and partially describes trawl processes in terms of their jumps. A thor-
ough likelihood inference for trawl processes needs to include the information in
theinitialvalueoftheprocess.
In this Chapter, we provide a thorough analysis of likelihood inference for
integer-valuedtrawlprocessesanddemonstratethecoreideas—predictiondecom-
position,filtering,smoothingandEMalgorithm—byfocusingontheso-calledex-
ponentialtrawl.Itisnotonlyasimplificationofthemodellingframeworkbutalso
anintellectuallyinterestingspecialcaseofitsown,asinthisspecialcasetheresult-
ingtrawlprocessisacontinuoustimehiddenMarkovprocesswithcountablestate
space.Thetheoreticalanalysisforthefilteringandsmoothingproblemsforthistype
ofprocesshasbeendiscussedindetailsby[19]and[20],usingtheclassicaltheory
ofKolmogorov’sforwardandbackwarddifferentialequations.Weparticularyem-
phasizethattheresultingEMalgorithminthisspecialcaseisexactinthesensethat
therearenodiscretizationerrorsinitscomputation.
ThemajorgoalofthisChapteristoderivefilteringandsmoothingresultsinthe
frameworkoftrawlprocesses,sotheanalysisadoptedherecanbeeasilyscaledup
to adapt to the discussions of other general trawls or even the inclusion of a non-
stationarycomponentproposedin[21].Thesegeneraldiscussionswillbedealtwith
elsewhere, for they require a significantly more sophisticated particle filtering and
smoothingdevice.Wealsodiscussnon-negativetrawlprocesses,whichareparticu-
larlyeasytoworkwith.
The structure of this Chapter is as follows. In Section 2, we remind the reader
how to construct trawl processes using the exponential trawl. Section 3 includes
detailsofhowtocarryoutfilteringandsmoothingforthesemodels.InSection4,we
showlikelihoodinferenceforexponential-trawlprocessesbasedonthesefiltersand
smoothers.Section5discussestheimportantbutanalyticallytractablespecialcase
of non-negative trawl processes. We finally conclude in Section 6. The Appendix
containstheproofsandderivationsofvariousresultsgiveninthisChapter.
2 Exponential-TrawlProcesses
InthisSection,webuildournotation,definitionsandkeystructuresfortheexponential-
trawl process that will be focused on throughout this Chapter. We also provide its
log-likelihoodfunctionbasedonobserveddata.
LikelihoodInferenceforExponential-TrawlProcesses 3
2.1 Definition
Our model will be based on a homogeneous Le´vy basis on [0,1]×R(cid:55)−→Z\{0},
which models the discretely scattered events of integer size (with direction) y∈
Z\{0}ateachpointintimes∈Randheightx∈[0,1].Itisdefinedby
(cid:90) ∞
L(dx,ds)(cid:44) yN(dy,dx,ds), (x,s)∈[0,1]×R,
−∞
whereN isathree-dimensionalPoissonrandommeasurewithintensitymeasure
E(N(dy,dx,ds))=ν(dy)dxds.
Heredsmeansthearrivaltimesareuniformlyscattered(overR),dxmeanstheran-
domheightsarealsouniformlyscattered(over[0,1])andν(dy)isaLe´vymeasure
concentratedonthenon-zerointegersZ\{0}.Withoutanyconfusion,wewillabuse
thenotationν(y)todenotethemassoftheLe´vymeasurecenteredaty.Throughout
thisChapter,weassumethat
(cid:90) ∞
ν(dy)= ∑ ν(y)<∞.
−∞ y∈Z\{0}
Following [5], we think of dragging a fixed Borel measurable set A⊆[0,1]×
(−∞,0]throughtime
A (cid:44)A+(0,t), t≥0,
t
sothetrawlprocessisdefinedby
(cid:90)
Y (cid:44)L(A)= 1 (x,s−t)L(dx,ds).
t t A
[0,1]×R
ThroughouttherestofthisChapter,wewillfocusontheexponentialtrawl
(cid:110) (cid:111)
A(cid:44) (x,s):s≤0, 0≤x<d(s)(cid:44)exp(sφ) , φ >0,
to simplify our exposition of the key ideas. We will leave results on more general
trawlsinanotherstudy.
Example1.Supposethat
ν(dy)=ν+δ (dy)+ν−δ (dy), ν+,ν−>0,
{1} {−1}
whereδ (dy)istheDiracpointmassmeasurecenteredat±1.Thecorrespond-
{±1}
ing L(dx,ds) is called a Skellam Le´vy basis, while the special case of ν− =0 is
calledPoisson.TheupperpanelofFig.1showseventsinLusingν+=ν−=10,
takingsizeson1,−1withblackandwhitedotsrespectivelyandwithequalproba-
bility.ThelowerpanelofFig.1thenillustratestheresultingSkellamexponential-
trawlprocessY =L(A)usingφ =2,whichsumsupalltheeffects(bothpositive
t t
4 NeilShephardandJustinJ.Yang
Levy basis and moving trawl: A =A+(0,t)
t
0
1. l
l l
0.8 l ll l l
l l
ght 0.6 l l ll
ei l
H 0.4 l l l l
l
0.2 l l l l ll
l l l l
0
0.
−1.0 −0.5 0.0 0.5 1.0
Trawl process
5 Time
4
3
)At 2
(
L 1
0
1
−
2
−
−1.0 −0.5 0.0 0.5 1.0
c(0, fleetProcesTsi[mfleeetProcess$Time > 0, 1])
Fig.1 AmovingtrawlAt isjoinedbytheSkellamLe´vybasisL(dx,ds),wherethehorizontalaxis
sistimeandtheverticalaxisxisheight.Theshadedareaisanexampleoftheexponentialtrawl
A,whilewealsoshowtheoutlinesofAtwhent=1/2andt=1.Alsoshownbelowistheimplied
trawlprocessYt=L(At).Code:EPTprocess Illurstration.R
andnegative)capturedbytheexponentialtrawl.Dynamically,L(A)willmoveup
t
by1ifthemovingtrawlA eithercapturesonepositiveeventorreleasesanegative
t
one;conversely,itwillmovedownby1ifviceversa.NoticethatY =L(A )might
0 0
notbenecessarilyzeroandthepathofY atnegativetimeisnotobserved.
2.2 MarkovianCountingProcess
Fory∈Z\{0},letC(y)∈{0,1,2,...}bethetotalcountsofsurvivingeventsofsize
t
yinthetrawlattimet,whichalsoincludestheeventthatarrivesexactlyattimet,so
(y)
eachC mustbeca`dla`g(right-continuouswithleft-limits).Thenclearlythetrawl
t
processcanberepresentedas
Y = ∑ yC(y), t≥0. (1)
t t
y∈Z\{0}
LikelihoodInferenceforExponential-TrawlProcesses 5
(y)
NotethateachC isnotonlyaPoissonexponential-trawlprocesswith(differ-
t
ent)intensityofarrivalsν(y)(andsharingthesametrawl)butalsoaM/M/∞queue
(cid:110) (cid:111)
andhenceacontinuoustimeMarkovprocess.Hence,for C(y) beingthenatu-
t
t≥0
(cid:18)(cid:110) (cid:111) (cid:19)
ralfiltrationgeneratedbythecountingprocessC(y),i.e.,C(y)(cid:44)σ C(y) ,
t t s
0≤s≤t
ithas(infinitesimal)transitionprobabilities(orratesorintensities)
lim P(cid:16)Ct(y)−Ct(−y)dt = j(cid:12)(cid:12)(cid:12)Ct(−y)dt(cid:17) =φνC(y(y)),, ififjj==−11 . (2)
dt→0 dt  0t,− if j∈Z\{−1,1}
Thecasesof j=1or−1—whichcorrespondtothearrivalofaneweventofsizey
andthedepartureofanoldone—aretheonlytwopossibleinfinitessimalmovements
(y)
ofC duetothepointprocessnatureoftheLe´vybasis.Notethatthearrivalrate
t
anddepartureratearecontrolledbytheLe´vymeasureν andthetrawlparameterφ
respectively.Derivationof(2)canbefoundinmanystandardreferencesforqueue
theory(e.g.[1]).
Remark1.Let ∆X (cid:44)X −X denote the instantaneous jump of any process X at
t t t−
timet.Thenthetransitionprobability(2)canbeconvenientlywritteninadifferen-
tialform

ν(y)dt, if j=1
(cid:16) (cid:12) (cid:17) 
P ∆Ct(y)= j(cid:12)(cid:12)Ct(−y) = φCt(−y)dt, if j=−1 .
 0, if j∈Z\{−1,1}
Throughout this Chapter, our analysis will be majorly based on this infinitessimal
pointofviewfortheeaseofdemonstration.Allofourargumentscanberephrased
inamathematicallytighterway.
TheindependencepropertyoftheLe´vybasisimpliestheindependencebetween
eachC(y)fory∈Z\{0},sothejointcountprocess
t
(cid:16) (cid:17)
C (cid:44) ...,C(−2),C(−1),C(1),C(2),...
t t t t t
is also Markovian, which serves as the unobserved state process for the observed
hiddenMarkovprocessY andwillbethecentraltargetforthefilterandsmoother
t
wewilldiscussinamoment.LetC (cid:44)σ(cid:0){C } (cid:1)=(cid:95) C(y) bethejoin
t s 0≤s≤t y∈Z\{0} t
filtration.Clearly,from(2),C has(infinitesimal)transitionprobabilities
t
ν(y)dt, ifj=1(y)forsomey

P(∆Ct =j|Ct−)= φCt(−y)dt,ifj=−1(y)forsomey , (3)
 0, otherwise
where1(y)∈Z∞isthevectorthattakes1aty-thcomponentand0otherwise.
ThetrawlprocessY canbealsowrittenas
t
6 NeilShephardandJustinJ.Yang
∞
Y = ∑yY(y), Y(y)(cid:44)C(y)−C(−y),
t t t t t
y=1
(y) (y)
whereeachY isaSkellamexponential-trawlprocess.EachY isobservedfrom
t t
(y)
thepathofY uptoitsinitialvalueY ,forwecanexactlyobserveallthejumpsofY
t 0 t
(y)
andhenceallocatethemintotheappropriateY .Inotherwords,wecanregardthe
t
observedtrawlprocessas(i)amarkedpointprocess∆Y ∈Z\{0},whichconsists
t
(y) (y)
of several independent (given all theY ) marked point process ∆Y ∈{−1,1},
0 t
(y)
plus (ii) the initial valueY . The missing componentsY ’s will have some mild
0 0
(y)
effects on ∆Y . It is this initial value challenge that differentiates the likelihood
t
analysisoftrawlprocessesfromthatofmarkedpointprocesses.
ThespecialcasewhereY isalwaysnon-negativehasfurthersimplerstructure,as
t
(−y) (y) (y)
wemusthaveC =0forally=1,2,...andhenceC =Y isdirectlyobserved
t t t
(y)
up to its initial conditionC , which can be well-approximated if the observation
0
periodT islargeenough.WewillgothroughthesedetailsinSection5.
2.3 ConditionalIntensitiesandLog-likelihood
Let {F} be the natural filtration generated by the observed trawl process Y,
t t≥0 t
i.e.F (cid:44)σ(cid:0){Y} (cid:1).Definetheca`dla`gconditionalintensityprocessofthetrawl
t s 0≤s≤t
processY as
P(Y −Y =y|F )
λ(y)(cid:44) lim t t−dt t−dt , y∈Z\{0}, t>0 (4)
t− dt→0 dt
orconvenientlyinadifferentialform
λ(y)dt(cid:44)P(∆Y =y|F ). (5)
t− t t−
It means the (time-varying) predictive intensity of a size y move at time t of the
trawlprocess,conditionaloninformationinstantaneouslybeforetimet.
Remark2.ToemphasizetheF-predictabilityofλ(y),i.e.,beingadaptedtotheleft
t
naturalfiltrationF ,wewillkeepthesubscriptt−throughoutthisChapter.Thisis
t−
particularlyinformativeintheimplementationoflikelihoodcalculations,reminding
ustotaketheleft-limitoftheintensityprocesswheneverthereisajump.
For any two σ-fields F and G, let the Radon-Nikodym derivative over F|G
betweentwoprobabilitymeasuresPandQbe
LikelihoodInferenceforExponential-TrawlProcesses 7
(cid:18)dP(cid:19)
(cid:18)dP(cid:19) dQ
(cid:44) F∨G.
dQ (cid:18)dP(cid:19)
F|G
dQ
G
In particular, when G =σ(X) for any random variable X, we will simply write
thesubscriptasF|X.Thefollowingclassicalresultservesasthefoundationforall
likelihoodinferenceforjumpprocesses.
Theorem1.Let X be any integer-valued stochastic process and (cid:8)FX(cid:9) be its
t t t≥0
associatednaturalfiltration.Assumethat,underbothPandQ,(i)ithasfiniteex-
(y),P
pectednumberofjumpsduring(0,T],and(ii)theconditionalintensitiesλ and
t−
λ(y),Q arewell-definedusing(4)andFX.ThenP(cid:28)QoverFX|X ifandonlyif
t− t− T 0
(y),Q
λ is strictly positive. In this case, the logarithmic Radon-Nikodym derivative
t−
overFX|X is
T 0
(cid:18)dP(cid:19) (cid:32)λ(y),P(cid:33)
log = ∑ ∑ log t− 1
dQ (y),Q {∆Xt=y}
FTX|X0 0<t≤Ty∈Z\{0} λt−
−(cid:90) ∑ (cid:16)λ(y),P−λ(y),Q(cid:17)dt.
t− t−
t∈(0,T]y∈Z\{0}
Proposition14.4.Iof[11]providesacompleteandmathematicallyrigoroustreat-
mentforthisTheorem.Forcompleteness,wealsoprovideanintuitiveandheuristic
derivationintheAppendix.AdirectapplicationofTheorem1givesthefollowing
Corollary.
Corollary1.Thelog-likelihoodfunctionofthe(general)trawlprocessis(ignoring
theconstant)
(cid:90)
lFT(θ)= ∑ ∑ logλt(−y)1{∆Yt=y}− ∑ λt(−y)dt+lY0(θ), (6)
0<t≤Ty∈Z\{0} t∈(0,T]y∈Z\{0}
wheretheparametersofinterestθ includetheLe´vymeasureν(dy)(i.e.ν(y)’s)and
thetrawlparameterφ.
The study of likelihood inference for trawl processes then reduces to the cal-
culations of conditional intensities λ(y) for y ∈ Z\{0}. Now, by law of iterated
t−
expectationsandthefactthatC ⊇F forallt (becauseof(1)),wehave
t t
λ(y)dt =E(P(∆Y =y|C )|F )
t− t t− t−
(cid:16) (cid:16) (cid:12) (cid:17)(cid:12) (cid:17) (cid:16) (cid:16) (cid:12) (cid:17)(cid:12) (cid:17)
=E P ∆C =1(y)(cid:12)C (cid:12)F +E P ∆C =−1(−y)(cid:12)C (cid:12)F
t (cid:12) t− (cid:12) t− t (cid:12) t− (cid:12) t−
(cid:16) (cid:12) (cid:17)
=ν(y)dt+φE C(−y)(cid:12)F dt,
t− (cid:12) t−
8 NeilShephardandJustinJ.Yang
wherethesecondlinefollowsbecausetheevent∆Y =ymustcomefromeitheran
t
arrival of a new size y event or a departure of an old size −y event; the third line
followsfrom(3).Thus,
(cid:16) (cid:12) (cid:17)
λ(y)=ν(y)+φE C(−y)(cid:12)F , y∈Z\{0}. (7)
t− t− (cid:12) t−
In next Section, we will study an exact filtering scheme to numerically calculate
(cid:16) (cid:12) (cid:17)
E C(−y)(cid:12)F .
t− (cid:12) t−
The non-negative exponential-trawl process, where we always have positive
events,admitsafurthersimplification
(cid:16) (cid:12) (cid:17)
λ(y)=ν(y), λ(−y)=φE C(y)(cid:12)F , y=1,2,..., (8)
t− t− t−(cid:12) t−
solikelihoodinferenceforsuchacaseiseasier.InthePoissoncase,alltheimpacts
(1) (y)
areofsizeone,soinparticularC =Y isalsoobserved(asC =0forally(cid:54)=1),
0 0 0
whichallowsustobypasstheconditionalexpectationin(8)fory=1.
3 ExactFilterandSmootherforExponential-TrawlProcesses
3.1 Filtering
IngeneralweneedtosolvethefilteringproblemsforC toimplement(6)and(7).
t
Denotethefilteringprobabilitymassfunctionas
p (j)(cid:44)P(C =j|F ), j=(...,j ,j ,j ,j ,...), j =0,1,2,..., t,s≥0.
t,s t s −2 −1 1 2 y
Also,let(cid:107)j(cid:107)1(cid:44)∑y∈Z\{0} jyandDt (cid:44)(cid:107)Ct(cid:107)1=∑y∈Z\{0}Ct(y).
Ourgoalhereistosequentiallyupdate p (j),wheretheinitialdistributionis
t−,t−
derivedfrom
(cid:18) (cid:19)
C(y)in(cid:118)dep.Poisson ν(y) subjectto ∑ yC(y)=Y ,
0 φ 0 0
y∈Z\{0}
so,bylettingPoisson(x|λ)(cid:44)λxe−λ/x!,wehave
∏y∈Z\{0}Poisson(jy|ν(y)/φ)
p (j)= ,
0,0 (cid:16) (cid:17)
P ∑y∈Z\{0}yC0(y)=Y0
wherethedenominatorcanbenumericallycalculatedusingtheinversefastFourier
transform[21].
LikelihoodInferenceforExponential-TrawlProcesses 9
Noticethatthefilteringdistributionnotonlyupdatesatthetimeswhenthepro-
cess jumps but also at those inactivity periods. We discuss these two cases sepa-
rately.
Theorem2(ForwardFiltering).
1.[Update by inactivity] Assume that the last jump time is τ (or τ =0) and the
currenttimeist−,where∆Y =0forτ <s<t (and∆Y (cid:54)=0ifτ >0).Then
s τ
p (j)= e−φ(cid:107)j(cid:107)1(t−τ)pτ,τ(j) , (9)
t−,t− ∑ke−φ(cid:107)k(cid:107)1(t−τ)pτ,τ(k)
where p isthefilteringdistributionwehavealreadyknownattimeτ.
τ,τ
2.[Update by jump] Assume that the current time is τ− and ∆Y =y for some
τ
y∈Z\{0}.Then
1 (cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
p (j)= ν(y)p j−1(y) +φ(j +1)p j+1(−y) ,
τ,τ (y) τ−,τ− −y τ−,τ−
λ
τ−
(10)
where p isthefilteringdistributionwehavealreadyknownattimeτ−.
τ−,τ−
Overall,thefilteringprocedures(9)and(10)implythat p (j)canbeupdated
t−,t−
in continuous time without discretization errors at any set of finite discrete time
points,sowecallitanexactfilter.
Example2.ForSkellamexponential-trawlprocesswithLe´vyintensitiesν+andν−,
wealwayshave
(+) (−)
Y =C −C , t>0,
t− t− t−
(cid:16) (cid:12) (cid:17)
so knowing p (j) (cid:44) P C(−)= j(cid:12)F immediately gives us p (j,k).
t−,t− t− (cid:12) t− t−,t−
Hence,thefilteringupdatingschemereducestothefollowing:startingfromτ =0,
p (j)∝e−φ(2j+Yτ)(t−τ)p (j) if∆Y =0forτ <s<t,
t−,t− τ,τ s
p (j)∝ν+p (j)+φ(j+1)p (j+1) if∆Y =1,
τ,τ τ−,τ− τ−,τ− τ
p (j)∝ν−p (j−1)+φ(j+Y )p (j) if∆Y =−1.
τ,τ τ−,τ− τ− τ−,τ− τ
We then renormalize pt−,t−(j) such that ∑∞j=0pt−,t−(j)=1 in each step of the
updates.Knowingthefilteringdistributions p (j)allowsustocalculate
t−,t−
(cid:16) (cid:12) (cid:17) ∞ (cid:16) (cid:12) (cid:17) ∞
E C(−)(cid:12)F = ∑ jp (j), E C(+)(cid:12)F = ∑ jp (j)+Y .
t− (cid:12) t− t−,t− t− (cid:12) t− t−,t− t−
j=0 j=0
Usingthefollowingsettings,withtimeunitbeingsecond,
ν+=0.013, ν−=0.011, φ =0.034, T =21×602=75,600(sec.), (11)
10 NeilShephardandJustinJ.Yang
Yt Ct(+)
2 3.0 True
5 Filtering
2.
data$Price − V0 101 cplus_mean 1.01.52.0
−
5
0.
−2 0.0
ue 0 100 200 300 400 500 600 0 100 200 300 400 500 600
Val 2.0 True datCa$t(T−)ime 4 c_Dtimte True
Filtering an Filtering
e
mean 1.5 minus_m 3
nus_ 1.0 n + c 2
mi ea
c 0.5 s_m 1
u
pl
c
0.0 0
0 100 200 300 400 500 600 0 100 200 300 400 500 600
c_time Time (sec.) c_time
Fig. 2 Top left: A simulated path for the Skellam exponential-trawl process Yt. Top right,
Bottom left, Bottom right: Paths of the true hidden counting processes Ct(+), Ct(−) and Dt =
C(+)+C(−) of surviving events in the trawl along with their filtering estimations. Code:
t t
EPTprocess FilteringSmoothing Illustration.R
Fig. 2 shows a simulated path of the trawl process Y together with the filtering
t
(+) (−) (+) (−)
expectations of C , C and D =C +C , the total number of surviving
t t t t t
(bothpositiveandnegative)eventsinthetrawlattimet.
3.2 Smoothing
We now consider the smoothing procedure for the exponential-trawl process Y,
t
whichisnecessaryforthelikelihoodinferencebasedontheEMalgorithmwewill
seeinamoment.
RunningthefilteringprocedureuptotimeT,wethenstartfrom p toconduct
T,T
thesmoothingprocedure.
Theorem3(BackwardSmoothing).
1.[Updatebyinactivity]Assumethatthe(backward)lastjumptimeisτ (orτ=T)
and the current time ist, where ∆Y =0 fort ≤s<τ (and ∆Y (cid:54)=0 if τ <T).
s τ
Then