Perfect Simulation of Hawkes Processes
Author(s): Jesper Møller and Jakob G. Rasmussen
Source: Advances in Applied Probability, Vol. 37, No. 3 (Sep., 2005), pp. 629-646
Published by: Applied Probability Trust
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Adv.Appl. Prob. (SGSA)37, 629-646 (2005)
Printed in Northern Ireland
© Applied Probability Trust 2005
PERFECT SIMULATION
HAWKES PROCESSES
OF
JESPER M0LLER * ** AND
JAKOB G. RASMUSSEN,* *** Aalborg University
Abstract
Ourobjectiveis to constructa perfectsimulationalgorithmfor unmarked
andmarked
Hawkesprocesses.The usualstraightforward
simulationalgorithmsuffersfromedge
algorithm
doesnot.By viewingHawkesprocesses
effects,whereasourperfectsimulation
as Poissonclusterprocessesandusingtheirbranchingandconditionalindependence
functionforthe lengthof a cluster
structures,
of thedistribution
usefulapproximations
arederived.Thisis usedto construct
upperandlowerprocessesfortheperfectsimulation
algorithm.A tail-lightness
forthe applicability
conditionturnsoutto be of importance
of theperfectsimulationalgorithm.Examplesof applications
andempiricalresultsare
presented.
Approximate
couplingfromthepast;edgeeffect;exact
simulation;
dominated
Keywords:
simulation;markedHawkesprocess;markedpointprocess;perfectsimulation;point
process;Poissonclusterprocess;thinning;upperprocess;lowerprocess
Primary60G55
2000Mathematics
SubjectClassification:
Secondary
68U20
1. Introduction
Unmarkedand markedHawkes processes [10], [11], [12], [14] play a fundamentalrole in
pointprocesstheoryandits applications(see, e.g. [8]), andtheyhave applicationsin seismology
[13], [22], [23], [27] andneurophysiology[7]. Therearemanyways to definea markedHawkes
process,butfor ourpurposesit is most convenientto defineit as a markedPoisson clusterprocess
X = {(ti, Zi)} with events (or times) ti E R and marks Zi defined on an arbitrary (mark) space
M equippedwith a probabilitydistributionQ. The cluster centres of X are given by certain
events called immigrants,while the otherevents are called offspring.
Definition 1. (Hawkes process with unpredictable marks.)
(a) The immigrantsfollow a Poisson process with a locally integrableintensity function
i(t), t e R.
(b) The marksassociatedto the immigrantsareindependentandidenticallydistributed(i.i.d.)
with distributionQ, and are independentof the immigrants.
(c) Eachimmigrantti generatesa cluster Ci, which consists of markedeventsof generations
of order n = 0, 1, ... with the following branching structure (see Figure 1). We first
have (ti, Zi), which is said to be of generation0. Given the 0,....
n generationsin
Received 3 December2004; revision received 13 April 2005.
* Postaladdress:Departmentof MathematicalSciences, AalborgUniversity,FredrikBajersVej7G, DK-9220 Aalborg,
Denmark.
** Email address:[email protected]
***Email address:[email protected]
629
630 * SGSA
J. M0LLER AND J. G. RASMUSSEN
2
0
1
2
11
232
FIGURE
1: The branchingstructureof the variousgenerationsof events in a cluster (ignoringthe marks)
(top), and the events on the time axis (bottom).
Ci, each (tj, Zj) e Ci of generationn recursivelygenerates a Poisson process 'j of
offspring of generation n + 1 with intensity function yj (t) = y (t - tj, Zj), t > tj. Here,
y is a nonnegativemeasurablefunctiondefinedon (0, oo). We referto cj as an offspring
process, and to yj and y as fertility rates. Furthermore,the mark Zk associated to any
offspring tk e Fj has distributionQ, and Zk is independentof tk and all (t, Zi) with
tl < tk. As in [8], we refer to this as the case of unpredictable marks.
(d) Given the immigrants,the clustersare independent.
(e) X consists of the union of all clusters.
The independenceassumptionsin (c) and (d) imply thatwe have i.i.d. marks. In the special
case where y (t, z) = y (t) does not dependon its second argument(or if
P(y(t, Z) = y(t) for Lebesgue almost all t > 0) = 1,
where Z denotes a generic mark),the events follow an unmarkedHawkesprocess. Apartfrom
in that case, the events and the marksare dependentprocesses. Anotherway of defining the
process is as follows (see, e.g. [8]): the marksare i.i.d. and the conditionalintensity function
.(t) at time t e R, for the events given the previous history {(tk, Zk): tk < t}, is given by
,(t) = Iz(t) +
y(t - ti, Zi).
(1)
ti <t
Simulation proceduresfor Hawkes processes are needed for various reasons: analytical
resultsareratherlimiteddue to the complex stochasticstructure;statisticalinference,especially
model checking and prediction, require simulations; and displaying simulated realizations
of specific model constructionsprovides a better understandingof the model. The general
approachto simulatinga (markedor unmarked)point process is to use a thinning algorithm
such as the Shedler-Lewis thinning algorithmor Ogata's modified thinning algorithm(see,
e.g. [8]). However, Definition 1 immediatelyleads to the following approximatesimulation
algorithm,where t_ e [-oo, 0] and t+ e (0, oo] are user-specifiedparametersand the output
consists of all marked points (ti, Zi) with ti e [0, t+).
Algorithm 1. The following steps (i)-(ii) generatean approximatesimulationof those marked
events (ti, Zi) E X with 0 < ti < t+.
(i) Simulatethe immigrantson [t_, t+).
(ii) For each such immigrant ti, simulate Zi and those (tj, Z )
e Ci with ti < tj < t+.
Perfectsimulationof Hawkesprocesses
SGSA* 631
In general,Algorithm 1 suffers from edge effects, since clusters generatedby immigrants
before time t_ may contain offspring in [0, t+). Bremaud et al. [5] studied the 'rate of
installation',i.e. they considereda coupling of X, aftertime 0, to the outputfromAlgorithm 1
when t+ = oo. Undera tail-lightnessassumption(see the paragraphafterProposition3, below)
and other conditions, they established an exponentiallydecreasingbound for the probability
P(t_, oo), say, that X, after time 0, coincides with the outputof the algorithm. Algorithm 1
was also investigatedin the authors'own work [18], where variousmeasuresfor edge effects,
includingrefinedresults for P(t_, oo), were introduced.
Ourobjectivein this paperis to constructa perfect (or exact) simulationalgorithm.Perfect
simulation has been a hot research topic since the seminal Propp-Wilson algorithm [24]
appeared,butthe areasof applicationhaveso farbeenratherlimitedandmanyperfectsimulation
algorithmsproposedin the literaturearetoo slow for real applications.As demonstratedin [18],
our perfect simulation algorithm can be practical and efficient. Moreover, apart from the
advantageof not suffering from edge effects, our perfect simulation algorithmcan also be
useful in quantifyingthe edge effects sufferedby Algorithm 1 (see [18]).
The perfect simulationalgorithmis derivedusing similarprinciplesas in Brix and Kendall
[6], but our algorithmis a nontrivialextension, since the Brix-Kendall algorithmrequiresthe
knowledge of the cumulativedistributionfunction (CDF) F for the length of a cluster, and
F is unknowneven for the simplest examples of Hawkes processes. By establishingcertain
monotonicityand convergenceresults,we are able to approximateF to any requiredprecision,
and, more importantly,to constructa dominatingprocess and upperand lower processes in a
similar fashion as in the dominated-coupling-from-the-past
algorithmof [16]. Under a taillightness condition, our perfect simulationalgorithmturnsout to be feasible in applications,
while, in the heavy-tailedcase, we can at least say somethingaboutthe approximateform of F
(see Example7).
The paperis organizedas follows. Section 2 contains some preliminaries,includingilluminatingexamples of Hawkes process models used throughoutthe paperto illustrateour results.
In Section 3, we describe the perfect simulationalgorithm,assuming that F is known, while
the above-mentionedconvergence and monotonicityresults are establishedin Section 4. In
Section 5, we complete the perfect simulationalgorithm,using dominatedcoupling from the
past. Finally, Section 6 contains a discussion of our algorithmand results, and suggestions on
how to extend these to more general settings.
2. Preliminaries and examples
2.1. The branching structure and self-similarity property of clusters
By Definition 1, we can view the markedHawkes process X = {(ti, Zi)} as a Poisson
cluster process with cluster centres given by the immigrants,where the clusters, given the
immigrants,are independent. In this section, we describe a self-similaritypropertyresulting
from the specific branchingstructurewithin a cluster.
For events ti < tj, we say that (tj, Zj) has ancestor ti of order n > 1 if there is a sequence
sl,...,sn
of offspring such that Sn = tj and sk, k = 1,.... n, is one of the offspring of
Sk-1, with so = ti. We then say that tj is an offspring of nth generation with respect to ti; for
convenience, we say that ti is of zeroth generationwith respect to itself. Now we define the
total offspring process Ci as all those (tj, Zj) such that tj is an event of generation n e N0
with respect to ti (note that (ti, Zi) E Ci). The clusters are defined as those Ci for which ti is
an immigrant(see Definition 1).
632 SGSA
J. M0LLER AND J. G. RASMUSSEN
The total offspringprocesses have the same branchingstructurerelativeto their generating
events. More precisely, since yi (t) = y (t - ti, Zi) for any event ti, we see by Definition 1 that,
conditionalon the events ti < tj, the translatedtotal offspringprocesses
Ci - ti := {(tl - ti, Zi): (t, ZI) e Ci),
Cj
- tj := {(tl - tj, Z) : (t, ZI) e Cj}
are identically distributed.
In particular,conditional on the immigrants,the clusters relative to their cluster centres
(the immigrants)are i.i.d. with distributionP, say. Furthermore,conditional on a cluster's
nth generation events 9n,, say, the translated total offspring processes Cj - tj with tj E
n
are i.i.d. with distributionP. We refer to this last propertyas the i.i.d. self-similarityproperty
of offspring processes or, for short, the self-similarityproperty. Note that the assumptionof
unpredictablemarksis essential for these propertiesto hold.
2.2. A basic assumption and some terminology and notation
Let F denote the CDF for the length L of a cluster, i.e. the time between the immigrant
and the last event of the cluster. Considerthe mean numberof events in any offspringprocess
a (ti), v := E v, where
v=
y (t, Z)dt
is the totalfertility rate of an offspringprocess and Z denotes a generic markwith distribution
Q. Henceforth,we assume that
(2)
0 < P < 1.
The conditionv < 1 appearscommonly in the literatureon Hawkesprocesses (see, e.g. [5],
[8], and [14]), and is essential to our convergenceresultsin Section 4.2. It implies that
F(0)=
Ee-v
> 0,
where F(0) is the probabilitythat a clusterhas no offspring. It is equivalentto assumingthat
E S < oo, where S denotes the numberof events in a cluster: by inductionon n = 0, 1,...,
because of the branchingandconditionalindependencestructureof each cluster,vn is the mean
numberof generationn events in a cluster,meaningthat
ES = 1 +
+v2+ -
--=
1
1-v
(3)
if v < 1, while E S = oo otherwise.
The other condition, v > 0, excludes the trivial case in which there are almost surely no
offspring. It is readily seen to be equivalentto
F < 1.
(4)
Furthermore,
h(t) = E[y(t, Z)/v],
t > 0,
h(t) = E[y(t, Z)/v],
t > 0,
and
(5)
Perfectsimulationof Hawkesprocesses
SGSA* 633
are well-defined densities (with respect to the Lebesgue measure). The density h will play a
key role laterin this paper;it can be interpretedas the normalizedintensityfunctionfor thefirst
generationof offspringin a cluster started at time 0. Note that h specifies the density of the
distance R from an arbitraryoffspringto its nearestancestor. In the sequel, since the clusters,
relativeto theirclustercentres,arei.i.d. (see Section 2.1), we assumewithoutloss of generality
that L, R, and S are definedwith respect to the same immigrantto = 0, with markZo = Z.
Clearly, if L > 0 then R > t implies that L > t, meaning the distribution of L has a thicker
tail thanthatof R. The probabilityfunctionfor S is given by
P(S = k) = P(Sn+I = k - 1
Sn = k)/k,
k e N,
where Sn denotes the numberof events of nth generationand n e N is arbitrary(see [9] or
Theorem2.11.2 of [15]). Thus,
P(S = k) = E[e-kv(kv)k-1/k!],
k e N.
(6)
2.3. Examples
Throughoutthe paper,we illustratethe resultswith the following cases.
Example 1. (Unmarkedprocess.) An unmarkedHawkesprocess with exponentiallydecaying
fertilityrate is given by
v = v = a,
h(t) = h(t) = fe-pt
where a, 0 < a < 1 and Bf > 0 are parameters.Here, 1/fl is a scale parameterfor both the
distributionof R and the distributionof L.
The left-handpanel of Figure 2 shows perfect simulationsof this process on [0, 10] when
gj(t) = 1 is constant,a = 0.9, and 1f = 10, 5, 2, 1. By (3), we expect to see about 10 clusters
(in total) and 100 events. The clustersof course become more visible as ,f increases.
The left-handpanelof Figure3 shows six simulationsof clusterswith a = 0.9. Here,a is an
inverse scaling parameter;1f is irrelevantsince, to obtain comparableresults for this example
andthe following two examples,we have omittedshowing the scale. All the clustershave been
simulatedconditionalon S > 1 to avoid the frequentand ratheruninterestingcase containing
only the immigrant.These few simulationsindicatethe generaltendencyof L to vary widely.
Example 2. (Birth-deathprocess.) Considera markedHawkes process with
y(t, Z) =
a_l(t < Z)
EZ
where a, 0 < a < 1, is a parameter,Z is a positive randomvariable, and 1(.) denotes the
indicatorfunction. Then X can be viewed as a birth-deathprocess with birth at time ti and
survivaltime Zi for the ith individual.The birthrate is
)(t) = g(t) + -
EZ
card{i: ti < t < ti + Zi},
t E
(cf. (1)). Moreover,
v
=
aZ
EZ
,V
=
a,
h(t)=E
F1(t<Z)
Z
,
h(t) =
P(Z > t)
EZ
Since v is random,the distributionof S is more dispersedthanin the unmarkedcase (cf. (6)).
634 * SGSA
J. M0LLER AND J. G. RASMUSSEN
on [0, 10] of theunmarked
2: Onthe left, we displayfourperfectsimulations
Hawkesprocess
FIGURE
a = 0.9, g = 1, and/ = 10,5, 2, 1 (toptobottom).Random
jitterhasbeen
(Example1)withparameters
addedin theverticaldirectionto helpdistinguisheventslocatedclosetogether.Ontheright,we display
threeperfectsimulationson [0, 10] of the birth-deathHawkesprocess(Example2) with parameters
of thelinesontothehorizontal
a = 0.9, fi = 1, and,8 = 5, 2, 1 (topto bottom),wheretheprojections
axisshowthesize of themarks.
of clustersstartedat0 andconditioned
on S > 1 in the
FIGURE
3: Ontheleft,we displaysix simulations
case,and,
unmarked
casewitha = 0.9. Inthecentre,we displaythesamesimulations,
in thebirth-death
case. Differentscalingsareusedin thethreecases.
on theright,in theheavy-tailed
The special case in which g(t) = /x is constantand Z is exponentiallydistributedwith mean
1/f3 was consideredin [5, p. 136]. In this case, X is a time-homogeneousMarkovbirth-death
process with birthrate 1 + apn and deathrate fn, wheren is the numberof living individuals.
Furthermore, h(t) = /e- t and h(t) = pE1(,8t), where
El(s) =
-
dt
is the exponentialintegralfunction. As in Example1, 1/1f is a scale parameterfor the distribution
of L. As discussed in Example8, below, the stationarydistribution(i.e. the distributionof X at
any fixed time) is known up to a constantof proportionality,and it is possible to simulatefrom
this by rejectionsampling.
The right-handpanel of Figure 2 shows threeperfect simulationson [0, 10] in the Markov
case with ,L = 1, a = 0.9, and fi = 5, 2, 1, where the marksare indicatedby line segments of
differentlengths. The centrepanel of Figure 3 shows six simulationsof clusters (with marks
excluded) with a = 0.9, simulatedconditionalon S > 1. These simulationsindicatethat L is
slightly more dispersedthanin Example 1, since the marksintroduceadditionalvariationin the
cluster lengths. In fact, the coefficient of variationestimatedfrom 10 000 perfect simulations
is 1.92 in Example 1 and 2.85 in the presentcase.
Perfectsimulationof Hawkesprocesses
SGSA* 635
Example 3. (Heavy-taileddistributionfor L.) Suppose that y(t, Z) = aZe-tz, where a E
(0, 1) is a parameterand Z is exponentiallydistributedwith mean 1//3. Then v = v = a
is constant, meaning that the distributionof S is the same as in the unmarkedcase (cf. (6)).
Furthermore,
h(t) = h(t) =
(t + )2
specifies a Paretodensity. This is a heavy-taileddistribution,as it has infiniteLaplacetransform
(£(O) = E eR = 00 for all 0 > 0). Moreover, it has infinite moments (E[RP] = 00 for all
p > 1). Consequently,L also has a heavy-taileddistributionwith infinitemomentsandinfinite
Laplacetransform.Note that 6 is a scale parameterfor the distributionof L.
The right-handpanel of Figure3 shows six simulationsof clusterswith a = 0.9 and 16= 1.
These indicatethatL is much more dispersedthanin Examples 1 and 2 (in fact, the dispersion
is infinitein the presentcase).
3. Perfect simulation
Assuming, for the moment, that F (the CDF for the length of a cluster) is known, the
following algorithmfor perfect simulation of the markedHawkes process is similar to the
algorithmfor simulationof Poisson clusterprocesses withoutedge effects given in [6] (see also
[17] and [21]).
Algorithm 2. Let II be the point process of immigrantson [0, t+), and let 12 be the point
process of immigrants ti < 0 such that {(tj, Zj) e Ci: tj e [0, 00)}
0.
1. Simulate II as a Poisson process with intensity function k1(t) = /(t) on [0, t+).
2. For each ti E II, simulate Zi and those (tj, Zj) E Ci with ti < tj < t+.
3. Simulate 12 as a Poisson process with intensity function X2(t) = (1 - F(-t))s(t)
(-oo, 0).
on
4. For each ti E 12, simulate Zi and {(tj, Zj) e Ci : tj e [0, t+)} conditional on the event
that {(tj, Zj) E Ci: tj E [0, 00)}) 0.
5. The outputis all markedpoints from steps 1, 2, and 4.
Remark 1. In steps 1 and 2 of Algorithm2, we use Algorithm 1 (with t_ = 0). In step 4, it
is not obvious how to constructan elegant approachensuringthat at least one point will fall
after0. Instead,we use a simple rejectionsampler:we repeatedlysimulateZi, from Q and the
successive generationsof offspringtj to ti (togetherwith theirmarksZ ), until thereis at least
one event of Ci aftertime 0.
The key point is how to simulate12 in step 3, since this requiresknowledge of F, which is
unknownin closed form (see Remark3, below). In Section 4, we addressthis problemand, in
Section 5, we constructan algorithmfor simulating12.
In practice,we must requirethat 12 is (almost surely) finite or, equivalently,that
f
(1 - F(-t))L(t)
dt <00.
(7)
In the case that /z(t) is bounded, (7) is satisfiedif supt>ogL(t)E L < 00. A condition for the
finitenessof E L is establishedin Lemma 1 and Remark2, below.
636 SGSA
J. M0LLER AND J. G. RASMUSSEN
Proposition 1. The output of Algorithm 2 follows the distributionof the marked Hawkes
process.
Proof The immigrantprocess minus II U 12 generatesclusters with no events in [0, t+).
Since II consists of the immigrantson [0, t+), it follows directly that II is a Poisson process
with intensity Ail(t) = /z(t) on [0, t+). Since 12 consists of those immigrants on (-oo, 0)
with offspring after 0, 12 can be viewed as an independentthinningof the immigrantprocess
with retentionprobabilityp(t) = 1 - F(-t) and, thus, 12 is a Poisson process with intensity
X2(t) = (1 - F(-t))itt(t). Since II and 12 are independent,it follows from Section 2.1 that
{Ci: ti E I1) and {Ci: ti E 12} are independent. Viewing the markedHawkes process as a
Poisson cluster process, it follows from Remark 1 that the clusters are generatedin the right
way in steps 2 and4 of Algorithm2 when we only wantto samplethose markedpoints (tj, Zj)
with tj E [0, t+). Thus,Algorithm2 producesrealizationsfrom the distributionof the marked
Hawkes process.
Using the notationof Section 2.2, the following lemma generalizes and sharpensa result
of [14] aboutthe mean length of a cluster.
Lemma 1. Wehave
1
E[(1-e-v)
Ee-V
E[R Z]]<EL<
v
--E.
1- v
(8)
Proof Consider a cluster startingwith an immigrantat time to = 0, with mark Zo = Z
(cf. Section 2.1). For tj E 1, let Rj denote the distance from tj to 0, and Lj the length of the
total offspring Cj process started by tj. Then L = max{Rj + Lj : tj e ~i }, so, if we condition
on Z and let Rj,z be distributedas Rj, conditionalon the event Z = z, then
[°e-v
EL = E[E[L | Z]] = E
v1
E[max{Rj,z + L: j = 1 ...,
e
]
i}] .
(9)
i=1
To obtainthe upperinequality,observe that
Ei!E
E
EL
i=1
(Rj,z + Lj)
=E[vE[R I Z]]+
vEL,
j=1
wherewe haveused the fact thatthe Lj areidenticallydistributedandhavethe same distribution
as L, because of the self-similarityproperty(see Section 2.1), and the fact that the Rj are
identicallydistributedwhen conditionedon Z. Hence,
EL< -
1
E[vE[R | Z]]=
1-v
1
-v
E
r
sy(s, Z) ds = ER,
1- v
which verifies the upperinequality.Finally,by (9),
EL>E
E. i
(E[R | Z]+EL)
i=1
which reduces to the lower inequality.
=E[(1 -e-)
E[R | Z]] + E[1 - e-] E L,
SGSA* 637
Perfect simulation of Hawkes processes
Remark 2. If either v or y/v is independentof Z (in other words, either the numberor the
locations of offspring in an offspring process are independentof the mark associated to the
generic event), then it is easily proventhath = h and, thus, (8) reducesto
1--1
Ee
-
ER<EL<
-
-ER.
-v
Consequently,E L < oo if and only if E R < oo. This immediatelyshows that E L < oo in
Example 1 and E L = oo in Example 3. In Example 2, when Z is exponentiallydistributed
with mean 1/f/, (8) becomes
a
a (a + 2) <EL<
2(a + 1)fl - 1(1 - a)'
so in this case E L < oo. Not surprisingly,apartfromfor small values of a e (0, 1), the bounds
are ratherpoor and of little use except in establishingthe finitenessof E L.
4. The distribution for the length of a cluster
In this section, we derivevariousdistributionalresultsconcerningthe length L of a cluster.
The results are needed in Section 5 to complete step 3 of Algorithm2; however,many of the
results are also of independentinterest.
4.1. An integral equation for F
Below, in Proposition2, an integralequationfor F is derived, and we discuss how to approximateF by numericalmethods,using a certainrecursion.Proposition2 is a generalization
of Theorem5 of [14], which was provedusing void probabilitiesobtainedfrom a generalresult
for the probability-generatingfunctionalfor an unmarkedHawkes process. However,as was
pointed out in [8], the probability-generatingfunctional for the markedHawkes process is
difficultto obtain. We give a directproof based on void probabilities.
For n E No, let In denote the CDF for the length of a clusterwhen all events of generations
n + 1, n + 2,... are removed (it will become clear in Section 4.2 why we use the notation
'ln'). Clearly, In is decreasingin n, In -> F pointwise as n -- oo, and
lo(t) = 1,
t >0.
Let C denote the class of Borel functionsf : [0, oo) -- [0, 1]. For f E e, define(p(f) e C by
q(f)(t)
= E exp -v +
f(t - s)y(s, Z)ds
,
t > 0.
(10)
Proposition 2. Wehave
In =
(In-1),
ne N,
(11)
and
F = (p(F).
(12)
Proof As in the proof of Lemma 1, we can consider a cluster startedat time to = 0,
with associated marks Zo = Z. For a fixed t > 0 and n E N, split 0(0) into three point
processes 01, F2, and 43: the process 41 consists of those first generationoffspring ti e
<c(0) n [0, t) thatdo not generateeventsof generationn - 1 or lower with respectto ti on [t, oo);
J. M0LLER AND J. G. RASMUSSEN
638 * SGSA
02 = (j (0) n [0, t)) \i consistsof the remainingfirstgenerationoffspringon [0, t); and I3 =
0 (0) n[t, oo) consists of the firstgenerationoffspringon [t, oo). Conditionalon Z, Q1, F2,and
(3 are independent Poisson processes with intensity functions
i1(s) = y(s, Z)ln-1(t
- s) on
[0, t), X2(s) = y(s, Z)(1 - ln-1(t - s)) on [0, t), and X3(s) = y(s, Z) on [t, oo), respectively.
This follows by an independentthinningargumentsince, conditionalon 8n (the nth generation
of offspringin Co), the processes Cj - tj with tj e n are i.i.d. and distributedas Co (this is
the self-similaritypropertyfrom Section 2.1). Consequently,
ln(t) = E[P(02 = 0
= Eexp(-
I Z) P(3
= 0 I Z)]
z(s, Z)ds -
which reduces to (11). Takingthe limit as n -
f
3(s, Z)ds) ,
oo on both sides of (11), we obtain (12) by
monotone convergence, since in (t) < ln-1 (t) for all t > 0 and n e N.
Remark 3. As illustratedin the following example, we have been unsuccessful in using (12)
to obtain a closed form expression for F even for simple choices of y. Fortunately,the
recursion(11) provides a useful numericalapproximationto F. As the integralin (10), with
f = 1n-1, quickly becomes difficult to evaluate analyticallyas n increases, we compute the
integralnumerically,using a quadraturerule.
Example 4. (Unmarkedprocess.) ConsiderExample 1 with B = 1. Then (12) is equivalentto
[t
e
F(s)es ds = -
n(eaF(t)),
which is not analyticallysolvable.
4.2. Monotonicity properties and convergence results
As established in Theorem 1 below, many approximationsof F other than In exist, and
their rates of convergence may be geometric with respect to different norms. Notice that
certainmonotonicitypropertiesare fulfilled by (p, where, for functions f: [0, oo) -+ [0, 1],
and set
n = 1, 2,...,
we recursively define o[](f)
= f and p[n](f) = (P[n-l](f)),
fn = W[n](f), n = 0, 1,.... Note that Fn = F for all n E N0. As In = q[n](1) is decreasing
towardsthe CDF F, cases in which G is a CDF and Gn increasesto F areof particularinterest.
Lemma 2. For any f, g e C, we have
f < g=
fn
gn,
ne N,
(13)
f < p(f) = fn is nondecreasingin n,
(14)
f > ((f) = fn is nonincreasingin n.
(15)
Proof Equation(13) follows immediatelyfrom (10) when n = 1, and then by inductionin
the remainingcases. Equations(14) and (15) follow from (13).
Theorem 1. Withrespectto the supremumnorm IlflII = supt>oIf (t) I, p is a contractionon
C, that is, for all f, g E C andn e N, we have fn, gn E C and
|i(f) - (p(g)I~\0 ivIIf - gll0.
(16)
SGSA* 639
Perfect simulation of Hawkes processes
Furthermore,F is the uniquefixpoint, i.e.
0 as n -+ oo
0iF - fnloo
and
(17)
-)n
(18)
lII(f) - flloo,
< -f
-1-v
where IIp(f) - flloo < 1. Furthermore,if f < qp(f) or f > (p(f), then fn convergesto F
from below or respectively,above.
l\F - foo
Proof Let f, g e C. Recall that,by the mean value theorem(e.g. Theorem5.11 of [1]), for
any real numbersx and y, we have ex - ey = (x - y)ez(x'y), where z(x, y) is a real number
between x and y. Thus, by (10),
I(f)
- p(g)00
=
sup Ee-ec(tf')
gl
(f(t
-
s) - g(t - s))y(s, Z)ds
,
wherec(t, f, g) is arandomvariablebetweenfo f(t - s)y(s, Z) ds andf g(t - s)y(s, Z) ds.
Since f, g _ 1, we obtainec(tfg) _ eV(see (2)). Consequently,
lio(f) - y(g)lloo < sup E
t>o
< E[
0
(f(t - s) - g(t - s))y(s, Z)ds
If - gllooy(s, Z) ds
= v||f - glloo.
Thereby,(16) is verified. Since C is complete (see, e.g. Theorem3.11 of [26]), it follows, from
the fixpointtheoremfor contractions(see, e.g. Theorem4.48 of [1]), thatthe contractionhas a
uniquefixpoint: by (12), this is F.
Since f e C implies that ((f) e C, we find that fn e C by induction. Hence, using (12),
(16), and induction,we have
lf- F|loo
- o(F)|loo _ v\\fn- - FIIoo _ Un
Ifn - Floo = IISP(fn-1)
(19)
for n e N. Since v < 1, we recover(17).
Similarlyto (19), we have
IIfn- fn-llloo _ Vn-1Ifi - flloo,
n E N.
(20)
Furthermore,by (17), we have
lim I|fm - fIoo.
||F - f ||oo= m-+oo
So, by the triangleinequalityand (20), we have
IIF- f|loo < m-+oo
lim (| fi - f||oo + II/2 - fllo +
lim
_ m--oo
"+ 1Ifm
- fm-llloo)
||ift- f|lloo(1 + v + -- - + vOm-1)
Ilfi - flloo
1-v
(see (2)). Combiningthis with (19), we obtain (18). Finally, if f < p(f) or f > (o(f) then
by (14) or, respectively,(15) and (17), fn convergesfrom below or, respectively,above.
640 * SGSA
J. M0LLER AND J. G. RASMUSSEN
Similarresults to those of Theorem 1, but for the L-norm, were establishedin [18]. The
following remarkand propositionshow how to find upper and lower bounds on F in many
cases.
Remark 4. Considera function f e G. The conditionsf < p(f) and f > yp(f) are satisfied
in the extremecases f = 0 and f = 1, respectively. The upperbound f = 1 is useful in the
following sections, but the lower bound f = 0 is too small a functionfor our purposes;if we
requirethat E L < oo (cf. Remark1) then f = 0 cannotbe used (in fact we use only f = 0
when producingthe right-handplot in Figure4, below). To obtaina more useful lower bound,
observe that f < p(f) implies f < F < 1 (cf. (4) andTheorem 1). If f < 1 then a sufficient
conditionfor f < yp(f) is
1 fo(1 - f(t - s))h(s) ds + ft° h(s) ds
- >
1 - f (t)
v
t > 0.
(21)
This follows readilyfrom (5) and (10), using the fact that ex > 1 + x.
The function f in (21) is closest to F when f is a CDF G and we have equality in (21).
Equivalently,G satisfies the renewalequation
G(t) = 1 - v +
G(t - s)h(s) ds,
t > 0,
which has the unique solution
G(t) = 1 -
+ y(1
- v)3n
*n(s) ds,
t > 0,
(22)
n=1
where *n denotes n-times convolution(see TheoremIV2.4 of [2]). In other words, G is the
CDF of R1 + . . . + RK (setting R i+
-
-+ RK = 0 if K = 0), where K, R, R2, ...
are
independentrandomvariables,each Ri has densityh, and K has a geometricdensity (1 - i)in.
Interestingly,this geometricdensity is equal to E Sn/ E S (see (3)).
The next propositionshows that, in many situations,G < p(G) when G is an exponential
CDFwith a sufficientlylargemean. In suchcases, F has no heaviertailsthansuchanexponential
distribution.
Denote by
£(0) =
efth(t) dt,
0 e R,
the Laplace transformof h.
Proposition 3. If G(t) = 1 - e-t for t > 0, where 0 > 0 and £(0) < 1/f, then G < qp(G).
Proof Upon insertingf = G into the right-handside of (21), we obtain
esh(s) ds + eJ
h(s) ds.
Since this is an increasingfunctionof t > 0, (21) is satisfiedif and only if L(0) < 1/v.
Note thatProposition3 always appliesfor sufficientlysmall 0 > 0, except in the case where
h is heavy tailed in the sense that L(0) = 00oo
for all 0 > 0. The condition d(0) < 1/v is
equivalentto the tail-lightnesscondition [5, Equation(2.1)].
Perfectsimulationof Hawkesprocesses
1.0
SGSA
1.0
0.30
0.8-
0.20
0.8-
0.6-
0.10
0.6
0.00
0.40 10 20 30 40 50 60
641
0.4
0 10 20 30 40 50 60
0
100 200 300 400 500
case with
4: Onthe left, we displayplotsof In andGn for n = 0, 5,..., 50 in the unmarked
FIGURE
a = 0.9 and 3 = 1 (see Example5); 150andGsoaredrawnsolidto illustratetheapproximate
formof
F, whereastheothercurvesaredashed.Inthecentre,we displayplotsof thedensity [1' /(1 - 1(0)) +
Gn/(1 - Gn(0))] when n = 50 (solid) andthe exponentialdensity with the same mean (dashed). On the
right,we displaythe sameplotsas on theleft,forExample7 witha = 0.9 and/ = 1, usingIn andOn
of F.
as approximations
4.3. Examples
In Examples 5 and 6 below, we let
G(t) = 1 - e-t,
t > 0O,
(23)
be the exponentialCDF with parameter0 > 0.
Example 5. (Unmarked process.) For the case in Example 1, £(0) = /3/(/3 - 0) if 0 < /,
and £(0) = oc otherwise. Interestingly, for the 'best choice', 0 =
-(l1/ii) = /3(1 - a),
(23) becomes the CDF for R times E S, which is easily seen to be the same as the CDF in (22).
The left-handpanel of Figure4 shows In and Gn when 0 = /3(1 - a) and (a, /3) = (0.9, 1).
The convergenceof In and Gn (with respect to || IIoo)and the approximateform of F are
clearly visible. Since G is a CDF and Gn+i > Gn, we find that Gn is also a CDF. The centre
panel of Figure4 shows the density F'(t)/(1 - F(0)) (t > 0) approximatedby
I(t)
1/
2 1 - I1 (0)
Gn(t0)
1 - Gn(O)
when n = 50 (in which case 1/n(t)/(1 - ln(0)) and Gn(t)/(1 - Gn(0)) are effectively equal).
As shown in the plot, the density is close to the exponentialdensity with the same mean, but
the tail is slightly thicker.
Example 6. (Birth-deathprocess.) For the case in Example 2,
foz e s
£(0) = E f-
ds =
0EZ
z(0) - 1
OEZ
,
where £z (0) = E eoz is the Laplace transformof Z. In the special case where Z is exponentially distributedwith mean 1//3, £(0) = £z(0) = //( - 0) is of the same form as in
Example 5. Plots of In, Gn, and
1
2-
1
In
Gn
~
1- Gn(0)
for n = 0, 5,..., 50 and (a, /3) = (0.9, 1) are similar to those in the centre and right-hand
panels of Figure4, and are thereforeomitted.
642 SGSA
J. M0LLER AND J. G. RASMUSSEN
Example 7. (Heavy-taileddistributionfor L.) For the case in Example 3, Proposition3 does
not apply, as £(0) = oo for all 8 > 0. The CDF in (22) is not known in closed form, since
the convolutionsare not tractable(in fact, this is the case when h specifies any known heavytailed distribution,including the Pareto, Weibull, log-normal, or log-gamma distributions).
Nonetheless, it is still possible to get an idea of what F looks like: the right-handpanel of
Figure 4 shows In and On := o[n](0) for n = 0, 5, ..., 50 in the case (a, /8) = (0.9, 1). As in
Examples5 and6, the convergenceof In andGn (where,now, G = 0) andthe approximateform
of F areclearlyvisible. However,as indicatedby the plots andverifiedin [18], limt-o Gn(t) <
1 when G = 0, meaningthat Gn is not a CDF.
5. Simulation of 12
To complete the perfect simulation algorithm (Algorithm 2), we need a useful way of
simulating 2. Ourprocedureis based on a dominatingprocess and the use of coupled upper
algorithm[16].
and lower processes, in the spiritof the dominated-coupling-from-the-past
Suppose that f e 2 is in closed form, with f < Op(f),and that (7) is satisfied when we
replaceF by f (situationsin which these requirementsarefulfilledareconsideredin Sections 3,
4.2, and4.3). Particularly,if 4/ is constantand f is a CDF, (7) implies that f has a finite mean.
Now, for n E No, let Un and Ln denote Poisson processes on (-oo, 0) with intensityfunctions
Xu(t)
=
(1
-
fn(-t))/z(t)
and
Xn(t) = (1 - In(-t)/z(t),
respectively. By Theorem 1, XAis nonincreasingand 4 is nondecreasingin n, and they both
convergeto A2(geometricallyfast with respectto the supremumnorm). Consequently,we can
use independentthinningto obtainthe following sandwiching/funnellingproperty(see [16]):
0 = Lo _c L
_c L2 c...
CI2 C... c U2 c U1 c U0.
(24)
The details are given by the following algorithm.
Algorithm 3. (Simulationof 2.)
1. Generate a realization {(tl, Z1),...
(tk, Zk)} of U0, where tl < - .. < tk.
2. If Uo = 0 then return 12 = 0 and stop; otherwise, generate independentuniform
numbersWi,..., Wkon [0, 1] (independentlyof Uo) and set n = 1.
3. For j = 1,..., k, assign (tj, Zj) to Ln or Un if WjX(tj)
Wj2(tj) < u(t).
<
An(t) or, respectively,
4. If Un = Ln thenreturn12 = Ln andstop;otherwise,increasen by 1 andrepeatsteps 3-4.
Proposition 4. Algorithm3 workscorrectlyand terminatesalmost surelywithinfinite time.
Proof To see this, imagine that, regardlessof whether Uo = 0 in step 2 or Un = Ln in
step 4, we continue to generate (U1, L1), (U2, L2), etc. Furthermore,add an extra step: for
j = 1, ..., k, assign (tj, Zj) to /2 if and only if
Wjh~(tj)l
h2(tj).
SGSA* 643
Perfect simulation of Hawkes processes
Then clearly, because of the convergencepropertiesof Auand A. (see the discussion above),
(24) is satisfied and, conditional on t,
, tk,
...
k
P(Ln # Un for alln e No) <
(t), W
(t)
lim P(W
(t) > n(t))
j=1
k
< W1j,~(tj) _ X2(tj)) = 0.
= EP(2(tj)
j=1
Thus,Algorithm3 almost surely terminateswithin finite time and the outputequals 12.
Remark 5. We compute In and fn numerically,using a quadraturerule (see Remark3). After
step 1 in Algorithm3, we let the last quadraturepoint be given by -ti (since we do not need to
calculate In(t) and fn(t) for t > -tl). Since we have to calculate In and fn recursivelyfor all
n = 0, 1, 2, ... untilAlgorithm3 terminates,thereis no advantagein using a doublingscheme
for n, as in the Propp-Wilsonalgorithm[24].
Example 8. (Birth-deathprocess.) We have checked our computercode for Algorithms 2
and 3 by comparingwith results producedby anotherperfect simulationalgorithm. Consider
the case in Example2 when /(t) = /z is constantand Z is exponentiallydistributedwith mean
1/ft. If N denotesthe numberof events alive at time 0, we have the following detailedbalance
conditionfor its equilibriumdensity lrn:
7n (l + afn) =
rn+fl(n
n
+ 1),
This density is well defined, since limn-+0 (n+l/n)
E NO.
= a < 1. Now, choose m e No and
e > 0 such that a = a + s < 1 and rn+1/rn < a whenever n > m. If/z < aft, we can take
e = m = 0; otherwise, we can use m >_ (t - af)/fe for some e > 0. Define an unnormalized
density xrn,n e No, by rn = Ttn/ro if n < m, and by Xr = an-m7tm/o otherwise. We can
easily sample from Jr by inversion(see [25]), since we can calculate
00
/
n-
---
.1
0zno
0
Then, since Trn >
m
-
a
1-a
n
ro
7n/rO, we can sample N from rn by rejection sampling (see [25]). Fur-
thermore, conditional on N = n, we generate n independentmarks Z, ..., Zn that are
exponentially distributedwith mean 1/fl (here, we exploit the memoryless propertyof the
exponential distribution). Finally, we simulate the markedHawkes process with events in
(0, t+], using the conditionalintensity
'(t) =
+a(
l(t < ti + Zi)).
l (t < Z) +
i=1
0<ti<t
We have implementedthis algorithmfor comparisonwith our algorithm. Not surprisingly,
it is a lot faster than our perfect simulationalgorithm(roughly 1200 times as fast in the case
a = 0.9, p = /z = 1, and t+ = 10), since it exploits the fact that we know the stationary
distributionin this special case.
J. M0LLER AND J. G. RASMUSSEN
644 SGSA
6. Extensions and open problems
Except in the heavy-tailedcase, ourperfectsimulationalgorithmis feasible in the examples
we have considered. However,simulationof the heavy-tailedcases is an unsolvedproblem. In
these cases, we can only say somethingaboutthe approximateform of F (see Example7).
For applicationssuch as those in seismology [23], extensions of our results and algorithms
to the heavy-tailed cases are important. The epidemic-type aftershock sequences (ETAS
model) [22], used for modellingtimes andmagnitudesof earthquakes,is a heavy-tailedmarked
Hawkes process. Its spatio-temporalextension, which also includes the locations of the
earthquakes(see [23]), can be applied to the problem of predictablemarks (the location of
an aftershockdependson the location of the earthquakethat causes it). This problemis easily
solved, though, since the times and magnitudesare independentof the locations and can be
simulatedwithout worryingaboutthese. This, of course, still leaves the unsolved problemof
the heavy tails.
Extensions to nonlinearHawkes processes [3], [8] would also be interesting. However,
things again become complicated, since a nonlinearHawkes process is not even a Poisson
clusterprocess.
Simulationsof Hawkesprocesses with predictablemarkscan, in some cases, be obtainedby
using a thinning algorithm,if it is possible to dominatethe Hawkes process with predictable
marksby a Hawkesprocesswith unpredictablemarks.We illustratethe procedurewith a simple
birth-deathexample.
Example 9. (Birth-deathprocess.) Considertwo birth-deathHawkesprocesses as definedin
Example2. Let Ii have unpredictablemarks,with Z1 ~ Exp(6), and let 2 have predictable
marks,with Z ~ Exp(f + 1/ZZ ), where Z is the markof the first-orderancestorof ti. Both
models have y(t, Z) = a l1(t < Z), with the same a and r, andthey also have the same (t).
The model tP2 has the intuitive appeal that long-lived individualshave long-lived offspring.
Note thatthe intensityof 'i dominatesthe intensityof 'I2 if the marksare simulatedsuch that
Z' > Z?
To simulate 'I2, we first simulate I1 using Algorithm 3, with the modificationsthat we
associate both marks Z\ and Z to the event ti, and we keep all events from the algorithm
whetherthey fall in or before [0, t+). Each markedevent (tj, ZJ) is then includedin '2 with
retentionprobability
z(t) + af
Eti<tj,
g (t) + a
tiE'2
l(tj - ti < Z2)
ti;<tj l(tj - ti <
Z)
'
andthe final outputis all markedevents from 'I2 falling in [0, t+). It is easily proventhatthese
retentionprobabilitiesresultin the correctprocess I2-.
Another process that would be interesting to obtain by thinning is the Hawkes process
withoutimmigrantsconsideredin [4]; this process has l(t) = 0 for all t. However,for this to
be nontrivial(i.e. not almost surely empty), it is necessary that v = 1, which means that any
dominatingHawkes process has v > 1 and, thus, cannotbe simulatedby Algorithm3.
Many of our results and algorithmscan be modifiedif we slightly extend the definition(in
Section 1) of a markedHawkesprocess, as follows. For any event ti with associatedmarkZi,
let ni denote the number of (first-generation)offspring generatedby (ti, Zi), and suppose
thatni, conditionalon Zi, is not necessarilyPoisson distributed,butthatni is still conditionally
Perfectsimulationof Hawkesprocesses
SGSA
645
independentof ti andthe previoushistory.A particularlysimple case occurswhen ni is either 1
or 0, and
p = E[P(ni = 1 I Zi)]
is assumedto be strictlybetween 0 and 1 (here p plays a role similarto thatof v, introducedin
Section 4). We then redefineq9by
p(f)(t) = 1 - p + p
f (t - s)h(s) ds,
where, now,
h(s) = E[p(Z)h(s,
Z)]/p.
Since (pis now linear, the situationis much simpler. For example, F is given by G in (22)
(with v replacedby p).
Anotherextensionof practicalrelevanceis to considera non-Poissonimmigrantprocess,e.g.
a Markovor Cox process. The results in Section 4 do not depend on the choice of immigrant
process, and the straightforwardsimulationalgorithm(Algorithm 1) applies providedthat it
is feasible to simulatethe immigrantson [t , t+). However,the perfect simulationalgorithm
relies heavily on the assumptionthatthe immigrantprocess is Poisson.
Finally,we notice thatit wouldbe interestingto extendourideas to spatialHawkesprocesses
(see [19] and [20]).
Acknowledgements
We aregratefulto S0renAsmussen, Svend Bemtsen, HoriaCornean,MartinJacobsen,Arne
Jensen, and GiovanniLuca Torrisifor helpful discussions. The researchof JesperM0ller was
supportedby the Danish NaturalScience ResearchCouncil and the Networkin Mathematical
Physics and Stochastics (MaPhySto), funded by grants from the Danish National Research
Foundation.
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