Affine processes on $\mathbb{R}_+^n \times \mathbb{R}^n$ and multiparameter time changes

0.28 MB·English

by  Ma. Emilia Caballero

AFFINE PROCESSES ON Rm Rn AND MULTIPARAMETER TIME
+
×
CHANGES
MA.EMILIACABALLERO,JOSE´ LUISPE´REZGARMENDIA,ANDGERO´NIMOURIBEBRAVO
ABSTRACT. Wepresentatimechangeconstructionofaffineprocesseswithstate-space
Rm Rn. Theseprocessesweresystematicallystudiedin[DFS03]sincetheycontain
+×
interestingclassesofprocessessuchasLe´vyprocesses,continuousbranchingprocesses
with immigration, and of the Ornstein-Uhlenbecktype. The constructionis based on
a(basically)continuousfunctionalofamultidimensionalLe´vyprocesswhichimplies
5 that limit theorems for Le´vy processes (both almost sure and in distribution) can be
1 inheritedtoaffineprocesses. Theconstructioncanbeinterpretedasamultiparameter
0
timechangeschemeorasa(random)ordinarydifferentialequationdrivenbydiscontin-
2
uousfunctions. Inparticular,weproposeapproximationschemesforaffineprocesses

n
basedontheEulermethodforsolvingtheassociateddiscontinuousODEs, whichare
a
showntoconverge.
J

3
1

]
R 1. INTRODUCTION
P
. Affineprocessesonthestate-spaceE=Rm Rnareaclassofprocessesintroducedin
h +×
t [DFS03]fortworeasons. First,theycontainimportantclassesofMarkovprocesseslike
a
Le´vy processes, (multi-type)continuous branching processes with immigration, and of
m
theOrnstein-Uhlenbecktype. Thatis,theycontainthefundamentalexamplesofmodels
[
in (stochastic) population dynamics (as in [Lam08]) and mathematical finance (as has
1
been argued in [DFS03] and [Kal06]). Second, they are analytically tractable. Indeed,
v
2 they have been shown to be parametrized in a manner similar to Le´vy processes and
2
one can access their finite dimensional distributions by solving an ordinary differential
1
3 equationoftheRiccati type(cf. [DFS03]).
0 Todefine them,letZ =(Z ,t 0)denoteastochasticprocesson ameasurablespace
t
1. (W ,F) whose paths are ca´dlag≥functions from [0,¥ ) to E. The canonical filtration of
0 Z will be denoted F . Suppose that the measurable space is equipped with a family of
5 t◦
1 (sub)probability measures (Pz,z E) such that under each Pz the process Z starts at z.
v: Furthermore, we assume that Z i∈s stochastically continuous under Pz for any z E and
∈
i thatthesemeasures constituteaMarkovfamily:
X
r E ( f(Z ) F )=P f(Z ) where P f(z)=E (f(Z )).
a z t+s s◦ t s t z t
|
Definition. TheMarkovfamily(P ,z E) isaffineif
z
∈
(1) E euZt =F (t,u)ezy (t,u)
z · ·
for all u Rm iRn, where Rm(cid:0)denot(cid:1)es the set of elements of Rm whose coordinates
are negat∈ive.−× −
2010MathematicsSubjectClassification. 60J80,60F17.
Keywordsandphrases. Le´vyprocesses,Continuousbranchingprocesseswithimmigration,Ornstein-
Uhlenbeckprocesses,multiparametertimechange.
ResearchsupportedbyUNAM-DGAPA-PAPIITgrantno.IA101014.
1
AFFINEPROCESSESONRm Rn ANDMULTIPARAMETERTIMECHANGES 2
+×
Theseprocessesarepartofalargeroneofso-calledaffineprocessesongeneralstate-
spacesandmuchrecentworkhasbeenaimedatcharacterizingtheseMarkovprocesses,
for example by proving their Feller property and the precise form of their infinitesimal
generator. This work started in [DFS03] for regular affine processes on E and was
later extended in [KRST11], [KRST13] and [CT13] by proving that regularity already
follows from stochastic continuity and also by considering more general state-spaces
than wedohere.
Ourmainresultaimsatgivingapathwiseconstructionofaffineprocessesintermsof
a multiparameter time change of Le´vy processes, which are considered as more basic
buildingblocks.
Theorem1. LetX1,...,XmandY beindependentLe´vyprocessesonRm+n. Wesuppose
that the first m coordinates ofY are subordinators, that Xi,i has no negative jumps and
thatXi,j isasubordinatorfor1 i, j mandi= j. Furthermore,intheGaussianpart
≤ ≤ 6
of Xi,thei-thcoordinateis assumedindependentofcoordinatesm+1up ton.
Let b beann n real matrix. Then, foranyz Ethereexists auniquesolutionZ to
× ∈
Zj =z +(cid:229) m Xi,j Ci+Yj 1 j m
(2) t j i=1 ◦ t t ≤ ≤
Zm+j =z +(cid:229) m Xi,m+j Ci+Ym+j+(cid:229) n Cm+ib 1 j n
( t m+j i=1 ◦ t t i=1 t i,j ≤ ≤
with
t
Ci = Zids
t s
0
Z
whose first m coordinates are non-negative. If P denotes the law of Z, then (P ,z E)
z z
∈
is an affine Markov family on E and every affine Markov family E is obtained by this
construction.
Note that the non-negativecoordinates are more difficult to handle. Indeed, the non-
negativecoordinatesaloneconstituteanaffineprocesswithn=0,whichisthencalleda
multitypecontinuous-statebranchingprocesswithimmigration(CBI)introduced(with-
out immigrationand with m=2) in [Wat69]. Once we analyze the case n=0, we will
then get thegeneral case bysolvingalineardifferential equationdrivenby thesolution
when n = 0. These real-valued coordinates constitute the Ornstein-Uhlenbeck part of
the process, which is now not only driven by a Le´vy process but also by a sum of time
changed Le´vy processes.
Equation (2) represents a multiparameter time change equation proposed in [Kur80]
to generalize theclassical timechange constructionof Markov processes of Volkonskii
(cf. [Vol58], [Dyn65, Vol 1, Ch 10]). A multiparameter time change representation
of affine processes was first proposed (in a weak sense) in [Kal06]; in that paper, the
question of whether the affine process was adapted to the filtration of the Le´vy pro-
cess was left open. Recently, there have been a number of results concerning this time
change representation. For example, the PhD thesis [Gab14] (the relevant chapter is
found in [GT14]) proves existence (under additional but minor technical assumptions)
foratimechangerepresentationas in Theorem1. A discretespaceversionofTheorem
1 has also been recently studied. Indeed, a construction of Galton-Watson processes
(without immigration) in terms of multiparameter time changes of random walks is
found in [CL13] in discrete time and [Cha14] in continuous time. More generally, the
connection between timechanges and changes ofmeasure and the applicationto math-
ematical finance is explored in [BNS10]. The main contributionof our work is that we
AFFINEPROCESSESONRm Rn ANDMULTIPARAMETERTIMECHANGES 3
+×
prove uniqueness of the pathwise representation in (2) (as well as for an accompany-
ing inequality). Uniqueness is the main toolin theforthcomingstabilityanalysis of the
pathwiserepresentation.
We now state some continuity properties of the system of equations (2). Consider
a sequence, indexed by l 1, of m+1 stochastic processes X1,,l,...,Xm,,l,Yl which
· ·
≥
satisfy the upcoming hypothesis H of p. 6. Consider also any sequence of numbers
0 s 0. The number s is interpreted as the discretization parameter to be used
l l
in≤an Eu→ler type scheme as follows. When s >0, let Zj,l andCj,l, 1 j m+n, be
l
≤ ≤
defined recursivelybymeans of
+
m
(3) Cj,l =0, Zj,l = (cid:229) zj+Xi,j,l Ci,l +Yj,l and Cj,l =Cj,l +Zj,l s
0 s lk "i=1 l ◦ s lk s lk# s l(k+1) s lk s lk l
when 1 j m, whilefor1 j nweonlychangethedefinitionofZm+j,l to
≤ ≤ ≤ ≤
m n
(4) Zm+j,l =zj+(cid:229) Xi,m+j,l Ci,l +Ym+j,l+(cid:229) Cm+i,lb
s lk l ◦ s lk s lk s lk i,j
i=1 i=1
When s = 0, the forthcoming Lemma 3 asserts that (2), when driven by Xi,j,l,Yj,l,
l
admits a (global) solution (which could, in principle, explode). In that case, we let
Zj,l,Cj,l be any such solution. We recall in Subsection 6.2 thedefinitions of theSkoro-
hodJ topologyand oftheuniformJ topology.
1 1
Theorem 2. LetX1,...,Xm andY beasinTheorem1. LetZ,C betheuniqueprocesses
satisfying(2).
SupposethatX1,,l,...,Xm,,l,Y ,l arestochasticprocesseswhichsatisfyhypothesisH
· · ·
of p. 6 and such that Xi,,l converges to Xi, (and Y ,l converges to Y) as l ¥ . (The
· · ·
→
convergence canbeweakoralmostsurelyintheSkorohodJ topologywhen (2)hasno
1
explosionandin theuniformJ topologyin caseofexplosion. ) Assumethatzj zj.
1 l →
If Z ,l, C,l are any processes satisfying (3) and (4) when s > 0 or (2) with respect
· · l
to the driving processes X1,,l,...,Xm,,l,Y ,l when s = 0 then Cl C (with respect
· · · l
→
to the topology of uniform convergence on compact sets when there is no explosion
and pointwise in case of explosion) and Zi,l Zi for 1 i m (with respect to the
→ ≤ ≤
SkorohodJ topologywhenthereisnoexplosionandintheuniformJ topologyincase
1 1
of explosion) as l ¥ . (The convergence will be either weak or strong depending on
→
thetypeofconvergence oftheXi,,l andY.)
·
Note that the above limit theorem is either weak or strong, which follows from con-
tinuity properties of the multiparameter time change equations explored in Section 6.
Indeed, we believe this is one strength of the time change representation versus, for
example,theSDE representationwhich isfound intheone-dimensionalcase in[FL10]
and[Li14]. Indeed,eveninthecaseofcontinuoussamplepaths,itisknownthatsolving
SDEs is a discontinuous operation of the driving processes. A manifestation of this is
found in Wong-Zakai type phenomena (discovered in [WZ65]) and depending on the
type of approximation to the driving processes one obtains limits to different SDEs, as
has been argued in [FH14]. On the other hand, Theorem 2 does not depend on how
one approximates the driving processes. We are not advocating, though, the use of one
representation overanother. Theconstructionof[Li14] isuseful inthegenealogical in-
terpretation of continuousbranching processes, constructing directly some of the flows
in [BLG03].
AFFINEPROCESSESONRm Rn ANDMULTIPARAMETERTIMECHANGES 4
+×
From Theorem 2 we deduce a limit theorem concerning multi-type Galton-Watson
processesstatedas Corollary 1. In theone-dimensionalcase, Corollary1 includeslimit
theoremsfoundin [Gri74], [Li06]and[CPGUB13]. Themultidimensionalcase hasof-
ten been studied in the literature when the limit process is continuous, as in [JM86].
We state a version without immigration, just to illustrate the kind of statement one
can achieve as well as the technique. The technique can be adapted to the case of
immigration as in Corollary 7 of [CPGUB13]. Let (X1,,l,1 i m) be independent
·
≤ ≤
d-dimensional random walks. Suppose that Xi,i,l has jumps in Z greater than 1 and
−
that otherwise the coordinates have jumps in N. Let kl = kl,...,kl Nm be a se-
1 m ∈
quence of starting states and define recursively the sequences Cl = Cj,l,1 j m
(cid:0) (cid:1) ≤ ≤
and Zl = Zj,l,1 j m by
≤ ≤ (cid:0) (cid:1)
m
Cl (cid:0)=0, Zl =kl, (cid:1) Zj,l =kl+(cid:229) Xi,j,l Ci,l and Cl =Cl +Zl .
0 0 n+1 ◦ n n+1 n n+1
i=1
It is easy to see that for each l, Zl is a multitype Galton-Watson process such that the
quantity of descendants of type j of an individual of type i has the same law as Xi,j,l
when i= j and the law of Xi,i,l+1 in the remaining case. However, if Xl is extended
6
by constancy on intervals of the form [n,n+1) with n N, we see thatCl is the Euler
∈
type approximation of span 1 applied to Xl that we have just introduced and Zl is the
right-handderivativeofCl.
Corollary 1. Let X1,,l, 1 i m be independent d-dimensional random walks. Sup-
·
≤ ≤
pose that Xi,i,l has jumps in Z greater than 1 that otherwise the coordinates have
−
jumpsinN.
Assume that for each i in 1,...,m there are scaling constants a and bi for l 1
{ } l l ≥
suchthat
a
l i,j,l
X ,t 0,1 j m
bj bilt ≥ ≤ ≤ !
l
converges in Skorohod space (either almost surely or in distribution)to a Le´vy process
Xi,. Furthermore,a ¥ , bj/a ¥ andkj issuchthatkja /bj zj.
· l → l l → l l l l →
Then, thescaled Galton-Watsonprocesses
a
l j,l
Z ,t 0,1 j m
at
b l ≥ ≤ ≤
j
(cid:18) (cid:19)
j
started from (k ,1 j m) converge in Skorohod space (either almost surely or in
l ≤ ≤
distribution) to the unique CB process Z started from z and constructed from X and
Y =0 inTheorem 1.
We end this section with an application of Corollary 1. Note that the different pro-
cessesinCorollary1havescalingsthathavetobeadequatelybalancedinordertoobtain
a limit (with non-trivial reproduction and immigration components). In order to exem-
plifyhowthiscouldbedone,letusstartbyconsideringtheframeworkofTheorem4.2.2
of[JM86],givingalimittheoremfornearlycriticalmultitypeGalton-Watsonprocesses
under finite-variance assumptions. Indeed, consider a sequence of multitype Galton-
Watsonprocesses Z ,l suchthat pi,l isthelawoftheoffspringofan individualoftypei.
·
Wethen definethemean matrixM by meansof
Ml = (cid:229) k pi,l(k).
i,j j
k Nm
∈
AFFINEPROCESSESONRm Rn ANDMULTIPARAMETERTIMECHANGES 5
+×
Assumethat Ml =Id+C /l whereC C as l ¥ . Consideralso the variancematrix
l l
→ →
s l givenby
1/2
s l = (cid:229) k Ml 2pi,l(k) .
i,j j i,j
−
" k #
(cid:16) (cid:17)
Suposethats l s 1 as l ¥ andthat thefollowingLindeberg conditionholds:
i,j → i i=j →
(cid:229) k Ml 2pi,l(k) 0
i i,i
− →
ki e √n(cid:16) (cid:17)
≥
as l ¥ . Recall our construction of such a process in terms of random walks Xi,,l
·
→
for i=1,...,m. From our hypotheses, it follows that Xi,i,l/l converges to s Bi+C Id
l2 i i,i
where Bi is a standard Brownian motion. Indeed, the con·vergence of one-dimensional
distributions is deduced from the Lindeberg-Feller central limit theorem. Because of
independence and stationarity of the increments this implies the convergence of finite-
dimensionaldistributionsandtightnessiseasilydeducedfromtheAldouscriterion. For
i= j,oneseesthatXi,j,l/l C Idasl ¥ . Indeed,itsufficesagaintoestablishcon-
6 l2 → i,j →
vergence of one-dimen·sional distributions which follow from Chebyshev’s inequality.
TightnessagainfollowsfromtheAldouscriterion. Hence,Corollary1allowsustocon-
cludethatifZl/l z then Zl/l converges weaklyto acontinuousbranchingprocess Z
0 → l
with continuous sample paths·. One can then use the martingales associated to X, as in
[Kur80], to seethatthegeneratorofZ isgivenby
(cid:229)m zis i2 ¶ 2 + (cid:229) zC ¶ .
2 ¶ z2 i i,j¶ z
i=1 i 1 i,j m j
≤ ≤
ThisfactcanalsobededucedfromtheinfinitesimalparametersofZ thatareintroduced
in Section2 and fromtheproofofTheorem1.
Ourworkcontinuesandextendstheone-dimensionalsituationcoveredin[CPGUB13].
There are however, important differences with that work. First of all, the discussion of
uniqueness to (2) now relies on the concept of (lack of) spontaneous generation. This
is to be contrasted to the previous analysis based on taking inverses. The multipletime
changes make this one-dimensional approach unfeasible. On the other hand, we also
take the point of view of multiparameter time changes from [Kur80], providing a very
concrete(butgeneral)exampleofitsapplicability. Thishasledtoseveralsimplifications
when provingthatsolutionsto (2)areaffine processes.
The paper is organized as follows. We first consider a deterministic framework for
equation (2) when n = 0 and analyze existence, uniqueness, and basic measurability
questions. This is done in Section 3. We then undertake the proof of Theorem 1 when
n = 0, which reduces basically to establishing the Markov property and constructing
relevant martingales, in Section 4. The case of general n is taken up in Section 5.
Finally, we pass to the stability of equation 2, which contains the proofs of Theorem 2
and Corollary 1in Section 6.
2. PRELIMINARIES ON AFFINE PROCESSES
Let Z be an affine process with laws (P ,z E). Let F and y be defined as in Equa-
z
∈
tion(1); applyingtheMarkovproperty,wegetthesemi-flowproperty
(5) y (t+s,u)=y (s,y (t,u)) and F (t+s,u)=F (t,u)F (s,y (t,u)).
AFFINEPROCESSESONRm Rn ANDMULTIPARAMETERTIMECHANGES 6
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From Theorem 5.1 in [KRST11] or Theorem 3.3 in [KRST13], it is known that the
followingderivativesexistandare continuousas afunctionofu:
¶ ¶
F(u)= F (t,u) and R(u)= y (t,u) .
¶ t ¶ t
(cid:12)t=0 (cid:12)t=0
(cid:12) (cid:12)
From thesemi-flowproperty,wededucetheso calledRiccati equations
(cid:12) (cid:12)
(cid:12) (cid:12)
¶ ¶
(6) F (t,u)=F (t,u)F(y (t,u)) and y (t,u)=R(y (t,u))
¶ t ¶ t
withtheinitialconditions
F (0,u)=1 and y (0,u)=u.
If F were non-zero and f were continuous and satisfied ef =F , we would obtain the
morefamiliarequation
¶
f (t,u)=F(y (t,u))
¶ t
whichgives
t
f (t,u)= F(y (s,u))ds.
0
Z
Furthermore, Theorem 2.7 in [DFS03] asserts that F and R have the following very
specific form: if X1,...,Xm andY are Le´vy processes satisfyingthe conditionsofThe-
orem 1then F and R=(R ,...,R ) aretheuniquecontinuousfunctionssuchthat
1 m+n
(7) eF(u) =E eu·Y1 and eRi(u) =E eu·X1i
(cid:16) (cid:17)
for1 i m whileform+1 (cid:0)i n(cid:1)weset
≤ ≤ ≤ ≤
n
R (u)= (cid:229) b u .
m+i i,j m+j
j=1
Furthermore, Section 6 of [DFS03] discusses the (global) existence and uniqueness
ofthegeneralized Riccati equationsof(6).
3. PATHWISE ANALYSIS OF THE MULTIDIMENSIONAL TIME CHANGE EQUATION
Following [CPGUB13], we begin by considering a deterministic system of time
changeequationsappearinginTheorem1inthecaseofnon-negativeprocesses(n=0).
Consider m(m+1) ca`dla`g functions labeled fi,j,1 i, j m and gj,1 j m .
≤ ≤ ≤ ≤
Thesefunctionssatisfythefollowingrequirements:
(cid:8) (cid:9) (cid:8) (cid:9)
H1: fi,j has nonegativejumpsifi= j andis non-decreasingotherwise.
H2: gj is non-decreasing.
H3: gj(0)+(cid:229) m fi,j(0) 0for1 j m.
i=1 ≥ ≤ ≤
Theabovehypothesesarecollectivelydenoted H.
We seek a solution to the following system of equations for the ca`dla`g function h=
h1,...,hm :
m t
(cid:0)(8) hj(t)(cid:1)= (cid:229) fi,j ci(t)+gj(t) for1 j m where cj(t)= hj(s)ds.
◦ ≤ ≤ 0
i=1 Z
This system can also be thought of as an ordinary differential equation for c when one
notes that hj is the right-hand derivativeof cj. With this interpretation, we might want
AFFINEPROCESSESONRm Rn ANDMULTIPARAMETERTIMECHANGES 7
+×
to use other initial conditions for c rather than only zero. This amounts to shifting the
functions fi,j; notehowever,thattheshiftsmuststillsatisfyH3.
3.1. A basic monotonicity lemma and existence. Our approach to the study of (8) is
based on its monotonicity properties. We begin with a simple and useful case of this
and postponeanelaboration ofthisideawhich willbeusefultoobtainuniqueness.
Lemma 1. Suppose that we have two sets of functions P=(fi,j,gj) and P˜ =(f˜i,j,g˜j)
satisfying hypothesis H. Assume that fi,j f˜i,j and gj g˜j and that additionally, for
every j 1,...,m ,either fi,j < f˜i,j or g≤j <g˜j. If han≤d h˜ arenon-negativefunctions
thatsati∈sfy{(8)driv}enbyP andP˜ respectivelythenc c˜.
≤
Proof. Let
P= fi,j,gj,1 i, j m and P˜ = f˜i,j,g˜j,1 i, j m
≤ ≤ ≤ ≤
beasystemoffun(cid:0)ctionssatisfyingthe(cid:1)assumptionsof(cid:0)ourlemmaandh,h˜ (cid:1)theassociated
non-negativesolutions to (8). For any e >0 and any a >1, define cj(t)=c˜j(e +a t).
Hence cj has a ca`dla`g right-hand derivativehj given by hj(t)=a h˜j(e +a t). We then
define
t =inf t 0:cj(t)>cj(t) forsome j
≥
as well as the set J of indices(cid:8)j 1,...,m such that cj exc(cid:9)eeds cj strictly at some
point of any right neighbourhood∈of{t . If j }J then cj(t )=cj(t ) while ci(t ) ci(t )
for i = j, so that also f˜i,j ci(t ) f˜i,j ci(∈t ) for i = j. We deduce the follow≤ing for
6 ◦ ≤ ◦ 6
j J:
∈
0 hj(t )=(cid:229) fi,j ci(t )+gj(t )
≤ ◦
i
<(cid:229) f˜i,j ci(t )+g˜j(t )
◦
i
<a (cid:229) f˜i,j ci(t )+g˜j(e +at ) =hj(t ).
◦
" i #
(Notethattheright-handsideofthefirststrictinequalitycannotbezero,whichjustifies
the second strict inequality.) We deduce that cj remains below cj in a right neighbour-
hood of t which contradicts the definitions of t and J. We deduce that cj cj and,
lettinga goto 1,that cj c˜j. ≤ (cid:3)
≤
We now tackle existence for (8) in the case when only f j,j,1 j m are not piece-
≤ ≤
wise constant. The proof will be based on the observation that under the piecewise
constant hypotheses, the system (8) is one-dimensional on adequate intervals. The
piecewise constant case will allow us to prove existence for (8) in general through the
monotonicityprovedinLemma1.
Lemma 2. Let fi,j,gj,1 i, j m satisfyH andsupposethat fi,j and gj arepiece-
≤ ≤
wiseconstantif1 i, j mandi= j. Then,thereexistsasolutionh= hj :1 j m
(cid:8)≤ ≤ 6 (cid:9) ≤ ≤
to (8). This solutionexistson aninterval[0,t )andcj =¥ forsome j.
t (cid:8) (cid:9)
−
Thetimet istermedtheexplosiontimeofc.
AFFINEPROCESSESONRm Rn ANDMULTIPARAMETERTIMECHANGES 8
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Remark1. Fortheone-dimensionalcase,existencefollowsfromTheorem1in[CPGUB13]
which asserts that the problem IVP(f,0,x) consisting of a finding a function c with a
right-handderivativeh which satisfies
IVP(f,0,x): h= f c with c(0)=x
◦
admits, for any x 0 and any ca`dla`g function f such that f has no negative jumps, a
≥
uniquesolutionwhich lacks spontaneousgeneration. When f(x)=0, theonlysolution
lacking spontaneous generation is the function c(t) = x. When f(x) > 0, the unique
solutioncan beconstructedby aLampertitypetransformationobtainedbyfirst making
zero absorbingafter x; formally
f(t) t <T
T =inf t x: f(t)=0 and f˜(t)= .
{ ≥ } 0 t [T,¥ ]
(
∈
Wethen definei on[x,¥ ) bymeans of
y 1
i(y)= dt.
x f˜(t)
Z
Notethatiisstrictlyincreasingon[x,T)andinfiniteon(T,¥ ). Then, letcbetheright-
continuous inverse of i (in the sense of Lemma 0.4.8 of [RY99]). Note that c is strictly
increasing on [0,i(T )] and constant on [I(T ),¥ ] and by definition c(0)= x. Then,
− −
sincetheright-handderivativeofiexistsandequals1/f,thencalsoadmitsaright-hand
derivative(on[0,i(T ))),sayh, andwehaveh=1/(1/f i 1)= f c. Thefunctionc
−
− ◦ ◦
soconstructedfrom f iscalledtheLampertitransformof f absorbedatitsfirstzeroafter
x. Note that c(¥ )=T. In the one-dimensional setting, when X is a spectrally positive
Le´vy process, Proposition 2 of [CPGUB13] shows that there is a unique solutionC to
IVP(x+X,0,0) (with right-hand derivativeZ) which has zero as an absorbing state; if
T denotes the hitting time of zero of x+X, X˜ equals X stopped at T, then C is also
the unique solution IVP(x+X,0,0), so that C¥ = T. This one dimensional result is
importantin our proofof uniqueness of solutionsto 2. Since stoppinga ca`dla`g process
at a stopping time and looking at a ca`dla`g process at a random time are measurable
transformations,weseethattheLampertitransformationismeasurableontheSkorohod
space of ca`dla`g trajectories with the s -field generated by projections. This would hold
evenifwetaketheinitialvaluex toberandomand measurable.
Proof. Supposefirst that
¥ ¥
fi,j = (cid:229) xi,j1 gj = (cid:229) yj1 ,
k [ti,j ,ti,j) k [tj ,tj)
k=1 k−1 k k=1 k−1 k
i,j i,j i,j i,j i,j
wherex x ifi= j, 0=t t , thesequencet ,k 0 hasno accumula-
k 1 ≤ k 6 0 ≤ 1 ≤··· k ≥
tionpoin−ts(similarassumptionsholdforgj)and additionally,foreach j
f j,j(0)+(cid:229) xi,j+yj 0
1 1 ≥
i=j
6
so that assumptions H hold. Let T (resp. T ) denote the set of change points of the
i,j j
functions fi,j (resp. gj):
i,j j
T = t :0 k and T = t :0 k .
i,j k ≤ j k ≤
n o n o
AFFINEPROCESSESONRm Rn ANDMULTIPARAMETERTIMECHANGES 9
+×
Let t = 0 and, for any j = 1,...,m, let c˜j be the unique solution of the problem
0 1
j j
IVP f ,0,0 , wherethefunction f is givenby
1 1
(cid:16) (cid:17) f j(t)= f j,j(t)+(cid:229) fi,j(0)+gj(0).
1
i=j
6
Wenowdefine thetimes
t i =inf t >0:t T orc˜i(t) T and t =mint i.
1 i 1 j=i i,j 1 1
∈ ∈∪ 6 i
Setcj equaltoc˜j on(cid:8)[0,t ]andrecursivelydefinec˜j (cid:9)asthesolutiontoIVP f j ,0,cj(t )
1 1 n+1 n+1 n
wherethefunction f j isgivenby (cid:16) (cid:17)
n+1
f j (t)= f j,j(t)+(cid:229) fi,j ci(t )+gj(t ).
n+1 ◦ n n
i=j
6
Wethen define
t i =inf t >t :t T orc˜i(t t ) T and t =mint i
n+1 n i 1 n j=i i,j n+1 n+1
∈ − ∈∪ 6 i
and let cj(t) =(cid:8)c˜j (t t ) on [t ,t ]. We assert (cid:9)that c = c1,...,cm solves (8);
n+1 − n n n+1
the proof is by induction. However, note that the starting point of c˜ is chosen so that
n
c is continuous and has a ca`dla`g right-hand derivative. On [0,(cid:0)t ], fi,j c(cid:1)i and gj are
1
◦
constant and hence, equal to their value at zero. Hence, if we let hj stand for the right-
hand derivativeofcj, weobtainthefollowingequalitiesforanyt <t
1
hj(t)= f j cj(t)
1 ◦
= f j,j cj(t)+(cid:229) fi,j(0)+gj(0)
◦
i=j
6
= f j,j cj(t)+(cid:229) fi,j ci(t)+gj(t),
◦ ◦
i=j
6
which allowus to concludethatc solves(8)on [0,t ]. On theotherhand, ifweassume
1
that c solves (8) on [0,t ], then note that, by definition, fi,j ci and gj are constant on
n
[t ,t ]. Wededucethatfort [t ,t ]: ◦
n n+1 n n+1
∈
hj(t)= f j c˜j (t t )
n+1◦ n+1 − n
= f j,j c˜j (t t )+(cid:229) fi,j cj(t )+gj cj(t )
◦ n+1 − n ◦ n ◦ n
i=j
6
= f j,j cj(t)+(cid:229) fi,j ci(t)+gj(t)
◦ ◦
i=j
6
so thatcsolves(8)on[0,t ].
n+1
Since t increases in n, there are two possibilities: either t ¥ (in which case the
n n
solutionwehaveconstructedisaglobalsolution)orcj(t ) ¥ →forsome jbydefinition
n
→
of t j, t , and thefact that the sets T and T haveno accumulationpoints. In the latter
n n i,j j
case, c explodes. (cid:3)
Remark 2. As in Remark 1, we notethat if we apply theprocedure of the aboveproof
in the case of ca`dla`g stochastic processes (satisfying the conditions of Lemma 2) then
the solutions are measurable. This follows because on adequate intervals (which are
obtainedbystopping),thesolutionsareunidimensionalandareconstructedthroughthe
Lampertitransformation.
AFFINEPROCESSESONRm Rn ANDMULTIPARAMETERTIMECHANGES 10
+×
Wenowtackleexistencefor(8).
Lemma 3. Let fi,j,gj,1 i, j m satisfy H. Then, there exists t > 0 such that a
non-negative solution h to≤(8) ex≤ists on [0,t ). Furthermore this solution explodes at t
(cid:8) (cid:9)
andis maximal:
(1) If cj(t)= thj(s)ds thencj(t )=¥ forsome j.
0
−
(2) Ifh˜ isanothersolutionto(8)(withitscorrespondingc˜)thenc˜ contheinterval
of existencReofh˜. ≤
i,j j
Proof. For 1 i, j m with i= j consider a sequence of ca`dla`g functions f and g
n n
whicharepie≤cewise≤constant,a6restrictlybiggerthan fi,j andgj,anddecreaseasn ¥
towards fi,j and gj respectively. We then set f j,j = f j,j. Using Lemma 2, we→can
n
considerforanynasolutionh =(h1,...,hm)to(8)drivenby fi,j,gj . ByLemma1,
n n n n n
{ }
we see that the cumulative population of h exceeds the cumulative population of any
n
solutionto (8).
Fix any K >0 and use it to stop c at the instant t that any one of its coordinates
n n,K
reach K. Call the resulting function c˜ . Since c c then t t ; set t =
n n+1 n n,K n+1,K K
≤ ≤
lim t . Notethatc˜ has aca`dla`g derivativeh˜n givenby
n n,K n
h˜nj(t)=1t tn,Khnj(t).
≤
Hence, h˜j can be bounded on any interval [0,t] by mmax inf fi,j(x)+gj(t), and
n i,j x K n n
i,j j ≤
can then be bounded in n by construction of f and g . By the Arzela`-Ascoli theorem
n n
j
(which applies since c˜ (0) =0), c˜ is sequentially compact. We now show that every
n n
j
subsequentiallimitcoincides. Indeed,ifc˜isthe(uniform)limit(oncompactsets)ofc˜
n
k
as k ¥ , thentheboundedconvergencetheoremimpliesthatforanyt <t :
K
→
t t
c˜j(t)=limc˜j(t)=lim (cid:229) fi,j c˜i(s)+gj(s)ds= (cid:229) fi,j ci(s)+gj(s)ds.
k k k Z0 i n ◦ n n Z0 i ◦
We conclude that c˜ admits a right-hand derivative h˜ on [0,t ) which satisfies (8) on
K
[0,t ). However,c˜isthemaximalsolutionbyconstruction(sincewecanapplyLemma
K
1 to the approximations c ), so that all subsequential limits agree on [0,t ]. Finally,
n K
note that before t the coordinates of c˜ have to be smaller than K and that at t some
K K
coordinate equals K. Hence t coincides with the instant in which some coordinate of
K
c˜ reaches K. By uniqueness, one can construct a function c which coincides with c˜ on
[0,t ), sothatc isdefined and solves(8)on[0,t )wheret =lim t . By construction,
K K K
cexplodesat t andis maximalin theclass ofsolutionsto(8). (cid:3)
Remark3. Recallthattheapproximationsoftheaboveproofaremeasurableinthecase
ofapplyingthemtoca`dla`gstochasticprocessesthankstoRemark2. Then,applyingthe
construction to a ca`dla`g stochastic process X satisfying hypotheses H, we get another
pair of stochastic processes Z andC. Since Z andC are cadlag, then Xi,j Ci is also a
◦
stochasticprocess.
3.2. Spontaneous generation and minimal solutions. An interpretation for the one-
dimensional case of (8) was proposed in [CPGUB13] by noting that if f1,1 represents
the breadth-first walk on a (combinatorial) forest representing the genealogy of a pop-
ulation with immigrants along each generation and g1 codes the immigration to the
population then h1 is the population profile (that is, the sequence of generation sizes),
while c1 is the cumulative population. The multidimensional case of this discrete cod-
ing can be found in Subsection 2.2 of [CL13], when gj =0 for all j, and it shows that