Product formulas for multiple stochastic integrals associated with Lévy processes
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P oduc o mulas o mul iple s ochas ic in eg als associa ed wi h Lé y p ocesses
© The Au ho (s) 2024
Published e sion
Di Tella, Paolo; Geiss, Ch is el; S einicke, Alexande
Di Tella, P., Geiss, C., & S einicke, A. (2024). P oduc o mulas o mul iple s ochas ic in eg als
associa ed wi h Lé y p ocesses. Collec anea ma hema ica, Ea ly online.
h ps://doi.o g/10.1007/s13348-024-00456-6
2024
Collec anea Ma hema ica
h ps://doi.o g/10.1007/s13348-024-00456-6
P oduc o mulas o mul iple s ochas ic in eg als associa ed
wi h Lé y p ocesses
Paolo Di Tella1·Ch is el Geiss2·Alexande S einicke3
To he memo y o Hans-Jü gen Engelbe ( ∗1944–2021†)
Recei ed: 29 Oc obe 2023 / Accep ed: 23 Sep embe 2024
© The Au ho (s) 2024
Abs ac
In he p esen pape , we ob ain an explici p oduc o mula o p oduc s o mul iple in eg als
w. . . a andom measu e associa ed wi h a Lé y p ocess. As a building block, we use a
ep esen a ion o mula o p oduc s o ma ingales om a compensa ed-co a ia ion s able
amily. This enables us o conside Lé y p ocesses wi h bo h jump and Gaussian pa . I is
well known ha o mul iple in eg als w. . . he B ownian mo ion such p oduc o mulas exis
wi hou u he in eg abili y condi ions on he ke nels. Howe e , i a jump pa is p esen , his
is, in gene al, alse. The e o e, we p o ide he e su icien condi ions on he ke nels which
allow us o es ablish p oduc o mulas. As an applica ion, we ob ain explici exp essions o
he expec a ion o p oduc s o i e a ed in eg als, as well as o he momen s and he cumulan s
o s ochas ic in eg als w. . . he andom measu e. Based on hese exp essions, we show a
cen al limi heo em o he long ime beha iou o a class o s ochas ic in eg als. Finally,
we p o ide me hods o calcula e he numbe o summands in he p oduc o mula.
Keywo ds P oduc o mulas o mul iple s ochas ic in eg als ·Momen o mulas ·Lé y
p ocesses ·Cen al limi heo em
Ma hema ics Subjec Classi ica ion 60H05 ·60G51 ·60G44 ·60F05
1 In oduc ion
In he p esen wo k we conside mul iple B ownian–Poisson s ochas ic in eg als, ha is,
mul iple s ochas ic in eg als gene a ed by a andom measu e ha exhibi s bo h a Gaussian and
BAlexande S einicke
alexande [email p o ec ed]
Paolo Di Tella
[email p o ec ed]
Ch is el Geiss
[email p o ec ed]
1Ins i u e o Ma hema ical S ochas ics, TU D esden, D esden, Ge many
2Depa men o Ma hema ics and S a is ics, Uni e si y o Jy askyla, Jy askyla, Finland
3Depa men o Ma hema ics and In o ma ion Technology, Mon anuni e si ae Leoben, Leoben, Aus ia
123
P. Di Tella e al.
a Poisson pa . This kind o andom measu es a e associa ed wi h Lé y p ocesses. We show
in Theo em 4.8 a compac and accessible p oduc o mula o he p oduc o N≥2 mul iple
B ownian–Poisson s ochas ic in eg als, ex ending p e ious esul s ob ained o N=2(see
o ins ance [9,17,21,22,25,29,33]). In such o mulas, p oduc s o mul iple s ochas ic
in eg als a e exp essed as a sum o mul iple s ochas ic in eg als.
To s a e ou p oduc o mula, a c ucial s ep is he de ini ion o he s a ope a o l
lo(see
De ini ion 4.5). I gene alizes he iden i ica ion and con ac ion ules in oduced e.g. in [8]
o N=2, o an a bi a y numbe N≥2 o mul iple in eg als.
Mul iple s ochas ic in eg als play an impo an ole in s ochas ic analysis because o hei
use o he chaos expansion and o he de ini ion o he Mallia in de i a i e. Mo eo e , hei
p oduc s ind applica ions in ce ain limi heo ems and nume ical simula ions [29].
The classic example o such a p oduc o mula can be ound e.g. in Nuala [25, P oposi ion
1.1.3], whe e o symme ic unc ions ∈L2([0,T]n), g∈L2([0,T]m), he p oduc o
wo mul iple in eg als w. . . he Gaussian measu e gene a ed by he B ownian mo ion is gi en
by
In( )Im(g)=
n∧m
=0
!n
m
In+m−2 ( ⊗ g). (1)
He e, ⊗ gdeno es an ope a o which ’con ac s’ a iables o and o g(i.e., hey a e
iden i ied and in eg a ed ou ). Fo he ex ension o (1) oana bi a yN≥2, we e e o Majo
[23, Theo em 10.2]. I is well known ha o mul iple B ownian in eg als he p oduc o mula
always holds o any N≥2, wi h no need o u he in eg abili y condi ions on he ke nels.
Con a ily, o mul iple Poisson in eg als, i.e., o mul iple s ochas ic in eg als gene a ed by
a Poisson andom measu e, his is no ue in gene al and, as a minimal equi emen o he
p oduc o mula, one has o ensu e ha he p oduc o mul iple Poisson s ochas ic in eg als is
again squa e in eg able. E en in he elemen a y case o a compensa ed Poisson p ocess, he
la e condi ion may be iola ed: Le Nbe a s anda d Poisson p ocess and se L =N − .
Fo any ∈L2([0,T]), by I ô’s o mula, we ha e
⎛
⎝
T
0
( )dL ⎞
⎠
2
=2
T
0
( )⎛
⎝
−
0
(s)dLs⎞
⎠dL +
T
0
( )2dL +
T
0
( )2d ,
which is squa e in eg able i and only i ∈L4([0,T]). In e p e ing he le -hand side as
he p oduc o wo mul iple s ochas ic in eg als o o de one, we see ha his is no squa e
in eg able, and hence does no ha e a chaos decomposi ion, unless ∈L4([0,T]).
The i s p oduc o mula was es ablished by I ô [14] in 1951 and co esponds o (1)
o an a bi a y nbu o m=1. Kabano [17] ex ended his o mula o mul iple Poisson
s ochas ic in eg als (keeping m=1). Su gailis [33] p o ed a p oduc o mula o N≥2
mul iple Poisson in eg als in 1984 and asked whe he he squa e in eg abili y o hei p oduc
is a su icien condi ion o he p oduc o mula o hold. Recen ly, in [8, Theo em 2.2],
Döble and Pecca i ga e a posi i e answe o he case o wo mul iple Poisson in eg als.
Howe e , he case o an a bi a y numbe o ac o s seems o be s ill open and, in he p esen
pape , we o mula ed his p oblem in Conjec u e 4.9.In[32], Russo and Vallois ob ained a
ep esen a ion o mula o he p oduc o wo mul iple in eg als o a bi a y o de associa ed
wi h a no mal ma ingale.1A quan um e sion o his o mula has been ob ained by P i aul ,
1A ma ingale Xis called no mal i i is squa e in eg able and (X2
− ) ≥0is a ma ingale oo. No mal
ma ingales need no ha e independen inc emen s.
123
P oduc o mulas o mul iple s ochas ic...
Solé and Vi es in [31]. Lee and Shih we e he i s au ho s who in [21] uni ied he B ownian
and he Poisson case conside ing p oduc o mulas o B ownian–Poisson mul iple in eg als.
P oduc o mulas o mul iple in eg als ha e been ob ained in he li e a u e by se e al
me hods. Fo example, in [22,29,33]diag am o mulae a e used. In [25]and[18], he
o mulas a e ob ained by induc ion on he o de o he mul iple in eg als. In [24]and[21],
he ela ion be ween he boson Fock space and he ep esen a ion o he s ochas ic exponen ial
o I1( )is exploi ed. In [1], he p ope ies o he s ochas ic exponen ial a e used o de i e a
o mal exp ession o he p oduc o mula o N≥2 mul iple Poisson s ochas ic in eg als.
Some pape s, as e.g. [19]o [2], ob ain explici momen and cumulan o mulas o Poisson
in eg als elying on Mecke’s o mula a he han on p oduc o mulas. In [19], he au ho s
s udy mul iple Poisson in eg als and in [2] he main goal is o conside s ochas ic in eg ands
ins ead o de e minis ic ones (see also [30]). In pa icula , Theo em 3.1 o [2] p o ides a join
momen o mula o single Poisson s ochas ic in eg als wi h andom in eg ands, whe eas he e
he momen o mula in Sec .5.2 conside s de e minis ic in eg ands.
In he p esen pape we use, like in [7], he p ope ies o compensa ed-co a ia ion s able
amilies o ma ingales. This allows us o di ec ly show a p oduc o mula o N≥2mul iple
B ownian-Poisson s ochas ic in eg als associa ed wi h a bi a y Lé y p ocesses. In [7]a
p oduc o mula o i e a ed in eg als wi h espec o in eg a o s belonging o ce ain amilies
o Lé y ma ingales was shown. Since he mul iple in eg als conside ed he e exhibi be e
symme y p ope ies ha allow, e.g. he de ini ion o he s a ope a o l
lo(see Sec .4.2
below), o ou opinion i is mo e e ec i e o s udy mul iple in eg als independen ly, ha
is wi hou using he esul s om [7] on i e a ed in eg als. Mo eo e , o ob ain he p oduc
o mulas o he mul iple in eg als om he one ob ained in [7], one has o use a domina ed
con e gence heo em. Fo his, we ha e o equi e in eg abili y condi ions o he ke nels,
ha would u n ou o be mo e es ic i e han hose we use he e.
Ob iously, a momen o mula can be immedia ely ob ained by aking he expec a ion in
ou p oduc o mula.
The p esen pape has he ollowing s uc u e: Sec . 2is de o ed o mul iple B ownian-
Poisson s ochas ic in eg als. Sec ion3 ecalls a ep esen a ion o mula o p oduc s o
compensa ed-co a ia ion s able amilies o ma ingales ha we apply in Sec .4 o ob ain
a i s p oduc o mula o i e a ed in eg als. This p oduc o mula is hen ans o med in o
one o mul iple in eg als and ex ended o he gene al se ing o Theo em 4.8. The la e is
applied in Sec .5 o ob ain momen and cumulan o mulas o s ochas ic in eg als. These
o mulas a e hen exploi ed o show a cen al limi heo em ha desc ibes he long ime
beha iou o mul iple in eg als o o de one o a class o in eg ands. Finally, in Sec .6we
p o ide se e al me hods o algo i hmic in e es o coun he numbe o summands con ained
in he p oduc o mula om Sec .4.
2 Mul iple in eg als
Le L=(L ) ≥0be a càdlàg one-dimensional Lé y p ocess wi h cha ac e is ic iple
(γ, σ 2,ν)on a comple e p obabili y space (, F,P). The augmen ed na u al il a ion o L
will be deno ed by (F ) ≥0.Wenow ixT>0, es ic ou analysis o [0,T]and assume
ha F=FT.
123
P. Di Tella e al.
The Lé y-I ô decomposi ion o L eads
L =γ +σW +
(0, ]×{|x|≤1}
x˜
N(ds,dx)+
(0, ]×{|x|>1}
xN(ds,dx),
whe e γ∈R,σ ≥0, Wis a s anda d B ownian mo ion and N(˜
N) is he (compensa ed)
Poisson andom measu e co esponding o L. The andom measu e Mgi en by
M(d ,dx)=σdW δ0(dx)+˜
N(d ,dx),
whe e δ0deno es he Di ac measu e concen a ed in ze o, is used o de ine he ollowing
i e a ed in eg als:Wese R0:= R {0}and
m(d ,dx):= d (σ 2δ0(dx)+ν(dx)).
Fo k≥1 we pu
∈Lp
k:= Lp(([0,T]×R)k,B(([0,T]×R)k), m⊗k), o p≥1,
and deno e ·Lp
k he Lp
k-no m o .Wedeno eby ˆ
(( 1,x1),...,( k,xk)) he symme i-
sa ion o he ke nel (( 1,x1),...,( k,xk)) w. . . he kpai s o a iables:
ˆ
(( 1,x1),...,( k,xk)) =1
k!
π∈Sk
(( π(1),xπ(1)),...,( π(k),xπ(k))), (2)
whe e Skis he se o all pe mu a ions πo {1,...,k}.We in oduce he i e a ed in eg als
as ollows:
J0:= he iden i y map on R
J1( )T:=
(0,T]×R
( ,x)M(d ,dx), ∈L2
1,
J2(ˆ
)T:=
(0,T]×R
(0, 2)×R
ˆ
(( 1,x1), ( 2,x2))M(d 1,dx1)M(d 2,dx2), ∈L2
2,
Jk(ˆ
)T:=
(0,T]×R
Jk−1(ˆ
(...,( k,xk))) k−M(d k,dxk), ∈L2
k.
To de ine mul iple in eg als w. . . M, one conside s he linea space Eko simple unc ions
o he o m
g(z1,...,zk)=
n
j1,..., jk=1
aj1,..., jk1Aj1×···×Ajk(z1,...,zk), zi:= ( j,xj), (3)
whe e he A1,...,An∈B([0,T]×R)a e disjoin and such ha m(Aj)<∞ o all
j=1,...,n.Mo eo e , aj1,..., jkis ze o whene e wo o mo e o he indices j1,..., jk
coincide. We hen pu
Ik(g)T:=
n
j1,..., jk=1
aj1,..., jkM(Aj1)···M(Ajk).
I holds ha EM(Aj)=0andEM(Aj)2=m(Aj), o j=1,...,n,and he
M(A1),...,M(An)a e independen . In he nex lemma we ecall some p ope ies o Jkand
Ik.
123
P oduc o mulas o mul iple s ochas ic...
Lemma 2.1 Le k,m∈Nand ∈L2
k,g∈L2
m.We henha e:
1. The space Ekis dense in L2
k, and Ik:Ek→L2(P)is a linea ope a o . The map Ikhas a
unique ex ension Ik:L2
k→L2(P).
2. EIk( )T=0,Ik( )TL2(P)=k! ˆ
L2
k≤k! L2
k,and EIk( )TIm(g)T=0i
m= k.
3. Ik(ˆ
)T=Ik( )Tand Ik(ˆ
)T=k!Jk(ˆ
)T.
P oo (1) By [14, Theo em 2.1] we ha e ha Ekis dense in L2
k.In [14] i is also shown ha
(2) holds o all ∈Ek, and he e o e, by linea i y and con inui y, Ikhas a unique ex ension
o L2
k. Also (3) is i s shown o all ∈Ekand hen ex ended o L2
k. Fo con enience and
la e use we ecall he p oo o he ela ion be ween Ikand Jkin (3). We ollow [21]ands a
assuming g(z1,...,zk)=N
j=1aj1Aj
1×···×Aj
k
(z1,...,zk) o measu able and pai -
wise disjoin Aj
1,...,Aj
ksa is ying m(Aj
l)<∞ o l=1,...,k. Mo eo e , we assume
g(z1, ..., zk)=0 o (z1,...,zk)/∈T(k),whe e
T(k):= {(( 1,x1),...,( k,xk)) :0≤ 1<···< k<T,x1,...,xk∈R}(4)
so ha , i ( 1,x1)∈Aj
iand ( 2,x2)∈Aj
l, hen 1< 2 o i<l.By (2) and he de ini ion
o g,wege
Ik(ˆg)T=Ik⎛
⎝1
k!
π∈Sk
N
j=1
aj1Aj
π(1)×...×Aj
π(k)⎞
⎠T
=1
k!
π∈Sk
N
j=1
ajM(Aj
π(1))···M(Aj
π(k))=
N
j=1
ajM(Aj
1)···M(Aj
k)=Ik(g)T.
On he o he hand, he de ini ion o he i e a ed in eg als yields
Jk(ˆg)T=1
k!
N
j=1
ajM(Aj
1)···M(Aj
k)=1
k!Ik(g)T.
This ex ends o symme ic ke nels ˆ
∈L2
k,hence he i e a ed in eg als and he mul iple
in eg als a e ela ed by Ik(ˆ
)T=k!Jk(ˆ
)T.
We ecall I ô’s chaos expansion esul :
Theo em 2.2 ([15]) The e exis s o any ξ∈L2(, F,P)a unique chaos expansion
ξ=∞
k=0
Ik(ˆ
)T,
whe e ˆ
∈L2
k,and I0is he iden i y map on R.
Ou aim is o show a p oduc o mula, ha means we wan o ind condi ions on he ( (j))N
j=1,
whe e (j)∈L2
kj,such ha he p oduc
N
j=1
Ikj(ˆ
(j))T
can be w i en as a sum o squa e in eg able mul iple in eg als.
123
P. Di Tella e al.
3 Compensa ed-co a ia ion s abili y and ma ingale p oduc s
To s a e a ep esen a ion esul o p oduc s o ma ingales, we i s need o ecall he concep
o compensa ed co a ia ion s able amilies (see [6, De ini ion 3.1]): Le be an a bi a y
pa ame e se and X={Xα,α∈}a amily o squa e in eg able ma ingales. Se
Xα1:2:= [Xα1,Xα2]−Xα1,Xα2,α
1,α
2∈.
Then Xis called compensa ed-co a ia ion s able i Xα1:2∈Xholds o any α1,α
2∈.
Le Xbe such a amily. Fo α1,...,α
k∈we can ecu si ely de ine
Xαj1:k:= [Xαj1:(k−1),Xαjk]−Xαj1:(k−1),Xαjk.(5)
Fo a compensa ed-co a ia ion s able amily X he ollowing ep esen a ion o mula holds:
P oposi ion 3.1 ([6, P oposi ion 3.3]) Le {Xα,α∈},be a compensa ed-co a ia ion
s able amily o quasi-le con inuous squa e in eg able ma ingales. Fo e e y N ≥1and
α1,...,αN∈,weha e
N
i=1
Xαi
=
N
i=1
1≤j1<···<ji≤N
0N
k=1
k=j1,..., ji
Xαk
s−dXαj1:i
s
+
N
i=2
1≤j1<...< ji≤N
0N
k=1
k=j1,..., ji
Xαk
s−dXαj1:i−1Xαjis.(6)
Using he andom measu e M om §2, we cons uc compensa ed-co a ia ion s able amilies
o ma ingales: Fo α∈L2
1, we de ine he squa e in eg able ma ingale
Yα
:= σ
0
α(s,0)dWs+
[0, ]×R0
α(s,x)˜
N(ds,dx), ∈[0,T].
We will call hese p ocesses Engelbe ma ingales, hey a e a gene alisa ion o he p ocesses
in oduced in [5, Equa ion (39)]. I holds
[Yα1,Yα2] =σ2
0
α1(s,0)α2(s,0)ds +
[0, ]×R0
α1(s,x)α2(s,x)N(ds,dx)
and
Yα1,Yα2 =σ2
0
α1(s,0)α2(s,0)ds +
[0, ]×R0
α1(s,x)α2(s,x)dsν(dx).
Fo =p≥2Lp
1, i is clea ha he amily X:= {Yα:α∈}is compensa ed-
co a ia ion s able. No e ha , o Yα1,...,Yαk∈X,k≥2, we ge he ollowing iden i y
Yα1:k
:= [Yα1:(k−1),Yαk] −Yα1:(k−1),Yαk =
[0, ]×R0k
j=1
αj(s,x)˜
N(ds,dx).
123
P oduc o mulas o mul iple s ochas ic...
Lemma 3.2 The linea hull o he i e a ed in eg als
(Jk(ˆ
) ) ∈[0,T]: =α1⊗···⊗αkwi h α1,...,α
k∈
p≥2
Lp
1,k=0,1,...
is compensa ed-co a ia ion s able.
P oo Fo symme ic ke nels and gwe ha e
[Jn( ), Jk(g)]T=
(0,T]×R
Jn−1( (...,( ,x))) −Jk−1(g(...,( ,x))) −
×(σ2d δ0(dx)+N(d ,dx)) (7)
and
Jn( ), Jk(g)T=
(0,T]×R
Jn−1( (...,( ,x))) −Jk−1(g(...,( ,x))) −
×(σ2d δ0(dx)+d dν(x)) (8)
Now we use he p oduc o mula o wo ac o s om Lee and Shih [21, Theo em 3.5] (o
[9] o he pu e jump case) which s a es ha he p oduc o wo mul iple in eg als o o de n
and kis equal o a linea combina ion o mul iple in eg als o o de less o equal o n+k.
Mo eo e , om [21, equa ion (21)] i can be seen ha he ke nels o hese mul iple in eg als
a e buil om enso p oduc s o unc ions om p≥2Lp
1.
4 P oduc o mulas
4.1 P oduc o mula o i e a ed in eg als
The ep esen a ion o mula (6) will be used o de i e a p oduc o mula o N
j=1Jmj( (j))T.
Fi s we conside a special case assuming ha he ke nels a e gi en by enso p oduc s:
(j)=α⊗mj
jwi h αj∈p≥2Lp
1.We in oduce some no a ion. Le
S:= {s={j1,..., ji}:1≤j1<···<ji≤N o i=1,...,N}.(9)
Then S=2{1,...,N} {∅}has ηN:= 2N−1 elemen s which we deno e by
S={e1,...,eηN}.(10)
P oposi ion 4.1 (P oduc o mula o i e a ed in eg als) Le αj∈p≥2Lp
1 o j =
1,...,N. Then
N
j=1
Jmj(α⊗mj
j)T=
k≤m1+···+mN
(s1,...,sk)∈Ak
T(k)
α(s1)(z1)M(s1)(dz1)...α
(sk)(zk)M(sk)(dzk)
(11)
123
P. Di Tella e al.
whe e zn:= ( n,xn), and
α(sn)(zn):=
∈sn
α(zn),
Ak:= (s1,...,sk)∈Sk:
k
n=1
1sn(j)=mj,∀j=1, ..., N,(12)
and T(k)s ands o he ’simplex’ de ined in (4). Fo any sn∈S he in eg a o s a e gi en
by (he e |sn|deno es he ca dinali y o he se sn)
M(sn)(dzn):= ⎧
⎪
⎪
⎨
⎪
⎪
⎩
M(d n,dxn)i |sn|=1,
m(d n,dxn)+˜
N(d n,dxn)i |sn|=2,
d nν(dxn)+˜
N(d n,dxn)i |sn|≥3.
(13)
The spli ing in o he cases |sn|=1,|sn|=2and|sn|≥3in(13) can be explained by (7)and
(8). No e ha he e ms ela ed o he B ownian pa anish o |sn|≥3 due o he i e a ed
use o (5).
P oo We use (6) o
Xαj
T:= Jmjα⊗mj
jT=
(0,T]×R
Jmj−1α⊗(mj−1)
j −αj( ,x)M(d ,dx).
Then
Xαj1:2
T=
(0,T]×R0
Jmj1−1α⊗(mj1−1)
j1 −
Jmj2−1α⊗(mj2−1)
j2 −
αj1( ,x)α j2( ,x)˜
N(d ,dx).
(14)
Fu he
Xαj1,Xαj2T=
(0,T]×R
Jmj1−1α⊗(mj1−1)
j1 −
Jmj2−1α⊗(mj2−1)
j2 −
αj1( ,x)α j2( ,x)
×m(d ,dx), (15)
and o mo e han wo p ocesses we ha e
Xαj1:i
T=
(0,T]×R0
i
k=1Jmjk−1α⊗(mjk−1)
jk −αjk( ,x)˜
N(d ,dx),
(16)
Xαj1:(i−1),XαjiT=
(0,T]×R0
i
k=1Jmjk−1α⊗(mjk−1)
jk −αjk( ,x)d ν(dx). (17)
Then (6) implies ha
N
j=1
Jmj(α⊗mj
j)T=
2N−1
i=1
(0,T]×RN
n=1
n/∈ei
Jmnα⊗mn
n −
jk∈ei
Jmjk−1α⊗(mjk−1)
jk −α(ei)( ,x)M(ei)(d ,dx). (18)
123
P oduc o mulas o mul iple s ochas ic...
We no e ha M
i=11Rj,i∈Hmj,whe e Hmjdeno es he linea hull o he se
⊗mj
i=1αi:αi∈∩p≥2Lp
1.
Mo eo e , 0 ≤M
i=11Rj,i≤1. So we ha e shown ha he e exis s a sequence (j)
n∞
n=1⊆
Hmjwi h
| (j)
n|≤1and (j)
n→1Ajin L2
mj,as n→∞, o all j=1...,N.(29)
Fo each nwe ha e ha N
j=1Imj (j)
nT∈L2(P)and sa is ies (27). To show ha he p od-
uc N
j=1Imj (j)
nTis a Cauchy sequence in L2(P) o n→∞we es ima e (abb e ia ing
mN:= m1+···+mN)
&&&&&&
N
j=1
Imj (j)
nT−
N
j=1
Imj (j)
mT&&&&&&L2(P)
≤
N
=1&&&&&&
−1
j=1
Imj (j)
nTIm ( )
n− ( )T
N
j= +1
Imj (j)
mT&&&&&&L2(P)
≤
N
=1
k≤mN
|l|=k,
(l,lo)∈DN
m1!···mN!
l!lo!&&&Ik(!
l
lo (1)
n,..., ( −1)
n, ( )
n− ( )
m, ( +1)
m, ..., (N)
mT&&&L2(P)
=
N
=1
k≤mN
|l|=k,
(l,lo)∈DN
m1!···mN!
l!lo!k!&&&!
l
lo (1)
n, ..., ( −1)
n, ( )
n− ( )
m, ( +1)
m, ..., (N)
m&&&L2
k
≤
N
=1
k≤mN
|l|=k,
(l,lo)∈DN
m1!...mN!
l!lo!k!&&&!
l
lo| (1)
n|,...,| ( −1)
n|,| ( )
n− ( )
m|,| ( +1)
m|,...,| (N)
m|&&&L2
k
(30)
which con e ges o ze o as n,m→∞by domina ed con e gence since he in eg ands a e
bounded by !
l
lo(1A1,...,1AN)and con e ge in measu e o ze o. Hence N
j=1Imj( (j)
n)Tis
a Cauchy sequence in L2(P). On he o he hand, (29) implies ha Imj( (j)
n)T→Imj(1Aj)T
in L2(P) o j=1, ..., N,and consequen ly N
j=1Imj( (j)
n)T→N
j=1Imj(1Aj)Tin
p obabili y. This gi es N
j=1Imj(1Aj)T∈L2(P), as i is also he limi o he Cauchy
sequence in L2(P). I emains o show ha he .h.s. o (27) (used now o he ˆ
(1)
n,..., ˆ
(N)
n)
con e ges in L2(P)i (29) holds. This ollows om he con e gence
!
l
lo(ˆ
(1)
n,..., ˆ
(N)
n)−!
l
lo(ˆ
1A1, ..., ˆ
1AN)
in L2
k.
S ep 2. Le (i)∈L2
mi o i=1,...,Nsa is y (26). Especially, hen he .h.s. o (27)
is well-de ined and in L2(P). We show (27) by app oxima ion as ollows. Since (27)
holds o indica o unc ions, i holds o simple unc ions. We app oxima e any (j)
by simple unc ions (g(j)
n)∞
n=1such ha 0 ≤g(j),±
n↑ (j),±.Applying now (30) o
&&&N
j=1Imj(g(j)
n)T−N
j=1Imj(g(j)
m)T&&&L2(P)we ha e again con e gence o ze o by dom-
123
P. Di Tella e al.
ina ed con e gence which holds hanks o (26). Repea ing he a gumen s o S ep 1 o he
p esen si ua ion i ollows ha N
j=1Imj( (j))T∈L2(P)and ha (27) holds.
The ela ion 28 ollows immedia ely om 27 by aking k=0.
Simila o [9, Theo em 2.2.] which conce ns he p oduc o 2 mul iple in eg als, we expec
ha he e exis s an i and only i ela ion be ween N
j=1Imj( (j))∈L2(P)and he exis ence
o he p oduc o mula. This was al eady add essed by Su gailis in [33].
Conjec u e 4.9 Le (j)∈L2
mj o j =1, ..., N.
N
j=1
Imj( (j))Tis squa e in eg able
⇐⇒
|l|=k,
(l,lo)∈DN
1
l!lo!!
l
lo( (1), ..., (N))∈L2
k, o each k =1, ..., m1+... +mN,
!
l
lo( (1), ..., (N))is well de ined in he sense o De ini ion 4.5, and he exp ession is ini e
o k =0. Mo eo e , he p oduc o mula (27)holds.
5 Applica ions
5.1 The expec a ion o a powe o a s ochas ic in eg al
We apply now (27) o ob ain explici o mulas o momen s and cumulan s o I1( )T.The
app oach in [29, Chap e 7], conside s he B ownian and Poisson se ing sepa a ely and uses
diag am o mulae o ea all kinds o powe momen s and p oduc s. This p og am can o
cou se be conduc ed he e as well – howe e , o his applica ion, we ollow a mo e elemen a y
way.
To poin ou he connec ion be ween he se s enand he numbe s lnand lo
nmo e clea ly,
we will use he no a ion lso lenins ead o lni s=en, and simila ly o lo
n.
Fi s we obse e ha o compu e EI1( )N
T, o someN≥2 (which means we ha e
m1= ··· = mN=1), we need o ex ac he e m o k=0in(27). This has he
consequence ha :
•The iden i y !
0
lo( ,..., )=0
lo( ,..., )holds, since no z- a iables a e in ol ed
in hese exp essions.
•I is only necessa y o conside uples (0,...,0,lo
1,...,lo
sN)in DNso ha o each
j∈{1,...,N},weneedsjlo
s=mj=1.
The la e poin implies ha one can es ablish a bijec ion be ween he uples in D∗
N, i.e. hose
uples om DNsuch ha l1= ··· = lsN=lo
{1}= ··· = lo
{N}=0, and he pa i ions
o {1,...,N}o block size ≥2. (We call he elemen s o he pa i ion blocks and hei
ca dinali y block size.) We conside his se o pa i ions, since by (24), blocks o size 1 do
no yield a e m in he p oduc o mula o k=0. No e ha his bijec ion is one pa icula
case o hose ha a e analyzed and numbe ed by diag ams and mul ig aphs in [29].
123
P oduc o mulas o mul iple s ochas ic...
We hus ge
{Pa i ions o block size ≥2}→D∗
N,P→ 1P=
s∈P
1{s}.
Fo one such pa i ion consis ing o qse s excluding single ons, {s1,...,sq}, we deno e
he block sizes by pi:= |si|. To compu e 0
lo( ,..., ),weha e oin es iga e he a i-
ables (zen,0
1:lo
n) ha a ise in he p oduc N
j=1 (zen,0
1:lo
n)1≤n≤sN,j∈en. Fo a gi en pa i ion
{s1,...,sq},i jis con ained in si, hen he same a iable zsi,0
jappea s in pio he p oduc ’s
ac o s, and al oge he he e can be only qdi e en a iables. Hence we ob ain
0
lo( ,..., )=
q
i=1
(0,T]×R
(z)picsi(z)m(dz),
and, using (24) again o he se s o block size 2,
0
lo( ,..., )=
(0,T]×R
(z)2m(dz)|{i:pi=2}| ·
pi>2
(0,T]×R0
(z)pim(dz). (31)
To ind how many pa i ions deli e such a e m, we i s conside he numbe o all pa i ions
o {1,...,N}in o qblocks o size p1,...,pq( ha is, wi hou he es ic ion pi≥2).
I we assume ha he e a e jablocks o size ain he pa i ion, we ha e he ela ionships
j1+j2+···+jN−q+1=q(and he e canno be mo e han N−q+1), ja=|{i:pi=a}|
and p1+···+pq=j1+2j2+···+(N−q+1)jN−q+1=N. Then he numbe o
such pa i ions is gi en by he coe icien bN,( j1,..., jN−q+1)o he monomial Xj1
1···XjN−q+1
N−q+1
in he N- h pa ial exponen ial Bell polynomial BN,q(X1,...,XN−q+1)(see, o example,
[10, De ini ion 11.2] o [29, De ini ion 2.2.1]). In pa icula , he coe icien is
bN,( j1,..., jN−q+1)=N!
j1!(1!)j1j2!(2!)j2... jN−q+q!((N−q+1)!)N−q+1.
The polynomial con aining he in o ma ion abou all pa i ions i.e.wi h a bi a ily many
blocks is he (exponen ial) Bell polynomial BN(X1,...,XN):= N
q=1BN,q(X1,...,
XN−q+1).
Since we exclude pa i ions wi h block size 1, we only ha e o conside he polynomial
BN(0,X2,...,XN).In(31), each pi-in eg al ac o appea s jpi imes. The e o e, we ge he
ollowing ela ions:
P oposi ion 5.1 .
1. Le ∈L2
1∩LN
1.Then
EI1( )N
T=BN⎛
⎜
⎝0,
(0,T]×R
(z)2m(dz),
(0,T]×R0
(z)3m(dz),...,
(0,T]×R0
(z)Nm(dz)⎞
⎟
⎠,
123
P. Di Tella e al.
o mo e in de ail,
EI1( )N
T=
N
q=1
N−q+1
i=2ji=q
N−q+1
i=2ij
i=N
bN,( j1,..., jN−q+1)⎛
⎜
⎝
(0,T]×R
(z)2m(dz)⎞
⎟
⎠
j2
···⎛
⎜
⎝
(0,T]×R0
(z)N−q+1m(dz)⎞
⎟
⎠
jN−q+1
.
2. Fo he B ownian case we eco e he known ela ion
EI1( )N
T=BN⎛
⎜
⎝0,
(0,T]
(s)2σ2ds,0,...,0⎞
⎟
⎠
=(N−1)!!
(0,T]
(s)2σ2dsN
21{N∈2N}.
3. I ∈L2
1∩LN
1 o all N ∈N hen he cumulan s κNo I1( )Ta e gi en by
κ1=0,κ
2=
(0,T]×R
(z)2m(dz), κN=
(0,T]×R0
(z)Nm(dz), N≥3.
P oo (1) is clea om he abo e conside a ions, (2) is ob ious. (3) one ge s om (1) by he
o mula ela ing cumulan s and momen s (see [29, Co olla y 3.2.2] o [30]).
5.2 Expec a ions o p oduc s o s ochas ic in eg als
In he same way as be o e, we may conside he p oduc o in eg als o di e en unc ions,
EI1( (1))T···I1( (N))T. Using pa i ions as abo e, we can compu e 0
lo( (1),..., (N))
by ela ing lowi h he acco ding pa i ion {s1,...,sq}wi hou single ons and ge
0
lo( (1),..., (N))=
q
i=1
(0,T]×R
j∈si
(j)(z)m(dz).
Taking all pa i ions in o accoun , we ob ain
EI1( (1))T...I1( (N))T
=
P∈P∗(N)
P={sq,...,sq}
⎛
⎜
⎜
⎝
q
i=1
|si|=2
(0,T]×R
j∈si
(j)(z)m(dz)⎞
⎟
⎟
⎠⎛
⎜
⎜
⎝
q
i=1
|si|>2
(0,T]×R0
j∈si
(j)(z)m(dz)⎞
⎟
⎟
⎠.
123
P oduc o mulas o mul iple s ochas ic...
5.3 Long ime beha iou and limi heo ems
In his sec ion, which is inspi ed by [19, §4], we a e going o add ess he long ime beha iou o
in eg als o he o m I1( )Tand, combining (27) wi h he me hod o momen s and cumulan s
(see [26, §A.3]), we deduce a cen al limi heo em ( om now on CLT) o T→+∞in
some special cases.
Using he p ope ies o he cumulan s and se ing '
I1( )T:= I1( )T
E[I2
1( )T]1/2we ge om
P oposi ion 5.1 (3) ha
κN'
I1( )T=⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1,N=2,
(
(0,T]×R0
N(z)m(dz)
"(
(0,T]×R
2(z)m(dz)#N/2,N≥3.
So, i is measu able on [0,∞)×Rand ∈LN([0,T]×R,m) o all N≥1and o
e e y T>0 sa is ies also
lim
T→+∞ (
(0,T]×R0
N(z)m(dz)
"(
(0,T]×R
2(z)m(dz)#N/2=0,N≥3,(32)
hen, by [16, Theo em 1] we ge ha '
I1( )Tcon e ges in dis ibu ion o X∼N(0,1)as
T→+∞. No e ha o ≥0 condi ion (32) can be e o mula ed in e ms o no ms.
We now conside unc ions o he o m (z)= ( ,x)=g( )h(x),whe eg:
[0,∞)→Rsa is ies
T
(0|g( )|Nd <∞ o all T>0andh∈LN(ρ) o e e y N≥1.We se
ρ(dx)=σ2δ0(dx)+ν(dx). We assume hL2(ρ) =1 and deno e ν(hN):= (
R0
hN(x)ν(dx).
In his special case, we ha e
κN'
I1( )T=ν(hN)(T
0gN( )d
(T
0g2( )d N/2,N≥3.
The e o e, a su icien condi ion o he CLT o '
I1( )T,is
lim
T→+∞ (T
0gN( )d
(T
0g2( )d N/2=0,N≥3.(33)
Condi ion (33) is i ially sa is ied i gis a polynomial unc ion o i i is he p oduc o a
polynomial unc ion wi h a apidly decaying unc ion as, o example, i g( )= me− 2/2.
Mo eo e , i gis cons an , which means ha ('
I1( )T)T≥0is a Lé y p ocess, (33)isalways
sa is ied.
Mo e in e es ing is he case g( )=e α, wi h α>0, ha we illus a e in he ollowing
lemma.
123
P. Di Tella e al.
Lemma 5.2 Le g( )=e α,α>0. We hen ha e
lim
T→+∞ (T
0eN αd
(T
0e2 αd N/2=⎧
⎪
⎨
⎪
⎩
0,α<1,
2N/2
N,α=1,
+∞,α>1,
N≥3.
P oo We use he a iable ans o m u= α o ew i e he exp ession
(T
0eN αd
(T
0e2 αd N/2=
1
α(Tα
0eNuu1
α−1du
1
α(Tα
0e2uu1
α−1duN/2.
Since o all k∈Ni holds limx→+∞
k(x
0ekuu1
α−1du
ekxx1
α−1=1,we ge
lim
T→+∞
1
α(Tα
0eNuu1
α−1du
1
α(Tα
0e2uu1
α−1d N/2=lim
T→+∞
1
NαeNTαT1−α
1
2αe2TαT1−αN/2
=⎧
⎪
⎨
⎪
⎩
0,α<1,
2N/2
N,α=1,
+∞,α>1,
The ollowing heo em is an immedia e consequence o Lemma 5.2.
Theo em 5.3 Le ( ,x)=h(x)e α,whe eh∈LN(ρ) o e e y N ≥1and hL2(ρ) =1.
We ha e:
(i) I α<1,'
I1( )Tcon e ges in law as T →+∞ o a s anda d no mal dis ibu ed
andom a iable.
(ii) I α>1,'
I1( )Tcanno con e ge in law as T →+∞ o a andom a iable whose
dis ibu ion is de e mined by i s momen s.
Rema k 5.4
1. In [11, Thm 4.1 and Co olla y 4.3] es ima es o he Kolmogo o dis ance be ween mul i-
ple in eg als Im( )(i m≥2) wi h espec o he compensa ed Poisson andom measu e
and a s anda d Gaussian andom a iable Za e shown by he Mallia in–S ein me hod.
Fo ou simple case o i s o de chaos wi h he special se ing ( ,x)=h(x)e αwe ge
easily an es ima e in Wasse s ein dis ance om [27, Co olla y 3.4] which s a es ha ( o
σ=0)
dW('
I1( )T,Z)≤$$$$$$$
1−
T
0
R0
( ,x)2d ν(dx)$$$$$$$+
T
0
R0
( ,x)3d ν(dx)
=|1−ν(h2)|+ν(|h|3)(T
0eN αd
(T
0e2 αd N/2.
This p o ides us wi h a con e gence a e o T→∞i α<1andν(h2)=1.
123
P oduc o mulas o mul iple s ochas ic...
2. In [28, Theo em 2.6] condi ions on he ke nels a e gi en such ha mul iple in eg als
o o de m≥2 wi h espec o he compensa ed Poisson andom measu e con e ge
o a Gamma dis ibu ion. We will show in he nex p oposi ion ha ou i s o de chaos
exp ession ˜
I1( )Twi h he ke nel ( ,x)=h(x)e assuming h(0)=0 con e ges weakly
o a Poisson dis ibu ed andom a iable i T→∞.
P oposi ion 5.5 Le be a Poisson andom measu e on R+×R×R0wi h he compensa o
λ+⊗λ⊗ν(he e we deno e he Lebesgue measu e on Rand on R+by λand λ+, espec i ely)
and le he unc ion h ≥0be such ha ν(euh)<+∞ o e e y u ≤ε, o a ixed ε>0.We
se a(x,y):= h(x)2y/2,E:= [0,1
2ln 2]×(−∞,1]×R0and de ine
Y:= E
a(x,y)( −λ+⊗λ⊗ν)(d ,dy,dx).
We hen ha e:
(i) E[euY ]<+∞, o e e yu≤ε
√2. Hence he dis ibu ion PYo Y is de e mined by i s
momen s.
(ii) The sequence o he cumulan s o Y is gi en by kY
1=0and kY
N=ν(hN)2N/2
N,N≥2.
(iii) Le ( ,x)=h(x)e wi h h(0)=0.Then'
I1( )Tcon e ges in dis ibu ion o Y as
T→+∞.Fo h(0)>0i holds ha '
I1( )Tcon e ges in dis ibu ion o Y +Gas
T→+∞whe e G is a cen e ed Gaussian andom a iable wi h a iance σ2h(0)2,
independen om Y .
P oo Since he condi ion on he con e gence o he cumulan s is equi alen o he con-
di ion on he con e gence o he momen s, (iii) o h(0)=0 ollows om (i), (ii) and
Lemma 5.2 by [3, Theo em 30.2]. Fo h(0)>0 we simply ake in o conside a ion ha
one can decompose ˜
I1( )Tin o an independen sum o Gaussian pa and jump pa , and
h(0)σ (T
0g( )dW /(T
0g2( )d ∼G.To see (i) and (ii), we in oduce he nonnega i e an-
dom a iable Z:= (Ea(x,y)(d ,dy,dx)and no ice ha , o e e y 0 <u≤ε
√2, om
[20, Theo em 3.9 and Exe cise 3.4], we ge
E)euZ*=exp 1
2ln 2
0
1
−∞
R0
(eua(x,y)−1)ν(dx)dyd ≤exp ν(e√2uh −1)<+∞,
whe e, o ge he i s es ima e, we expand he in eg and as a powe se ies. This oge he wi h
he de ini ion o Yyields
+∞ >E)euY *=exp +∞
N=2
uN
N!ν(hN)2N/2
N,0<u≤ε
√2.
Thus, (i) and (ii) a e shown.
6 The numbe o elemen s in DNi |l|=k
Wi hou handling he enume a ion o appea ing e ms, i is no clea how o pe o m he
calcula ions on a compu e . The o mulas below allow o ecu si e as well as i e a i e
implemen a ions. He e we use again he no a ion lo
{j}o lo
en o lo
ni en={j}. Coun ing he
123
P. Di Tella e al.
numbe o con ibu ing e ms in (27) o agi en|l|=k≤m1+···+mN, no e ha o any
j∈{1,...,N}i holds lo
{j}=0. This is because o de ini ion (24) (s a ing ha he c{j}a e
ze o). Thus, we will omi hose componen s.
We will ackle he p oblem in he subsec ions below in 3 ways, allowing a a ie y o
app oaches o calcula e he numbe on a compu e : by de i ing a ecu sion o mula, by using
weak composi ions, and by applying gene a ing unc ions. O cou se, i is also possible o
coun and s udy he se DN h ough o he combina o ial app oaches like numbe ing diag ams
and mul ig aphs. This has been done in [29] o he B ownian and pu e-jump se ing sepa-
a ely. The same p og am as he e can be conduc ed o ou case whe e he B ownian pa
and he jump pa a e combined and is subjec o ongoing esea ch. S udies in combina o ical
de ails o his e y di ec ion ough o deli e u he insigh s in o he s uc u e o in eg ands
appea ing in he p oduc o mula.
6.1 The numbe o elemen s by a ecu sion o mula
P oposi ion 6.1 We ha e he ollowing ecu sion o mula o
|{(l,lo)∈DN:|l|=k and lo
{1}=···=lo
{N}=0}| =:
C
(k,N,(m1,...,mN,0,...)),
whe e
C
(k,˜
N,(m1,...,mN,0,...)):= 0i ˜
N= N,
C
k,N,(m1,...,mN,0,...)
:= 0i ∃k∈{1, ..., N}wi h mk<0.(34)
C
0,0,(0,0,...)
:= 1,
and
C
k,N,(m1,...,mN,0,...)
=
min{k,mN}
κ=0
l{N}+···+l{1,...,N}
=κ
lo
{1,N}+···+lo
{1,...,N}
=mN−κ
C
k−κ, N−1,(ˆm1(l,lo), ..., ˆmN−1(l,lo), 0,...)
whe e
ˆm (l,lo):= m −
s∈P({1,...,N−1})
∈{N}∪s
(l{N}∪s+lo
{N}∪s),
and P({1,...,N−1})s ands o he powe se o {1,...,N−1}.
P oo The de ini ion o DN(see (21)) equi es (we use he e len:= ln)
ηN
n=1
(len+lo
en)1en(j)=mj o all j=1, ..., N.
O de ing he index se such ha all encon aining Na e behind he o he se s ha do no
con ain N, we obse e he uples
(l{1},...,l{1,...,N−1},l{N},...,l{1,...,N},lo
eN+1,...,lo
eηN).
123
P oduc o mulas o mul iple s ochas ic...
We pu κ:= |(l{N},...,l{1,...,N})|and know ha
κ+|(lo
{1,N},...,lo
{N−1,N}...,lo
{1,...,N})|=mNand κ≤min{k,mN}.
Fo e e y choice o (l{N},...,l{1,...,N})and (lo
{1,N},...,lo
{N−1,N},...,lo
{1,...,N}) he numbe o
possible choices o he emaining (l{1},...,l{1,...,N−1},lo
{1,2},...,lo
{N−2,N−1},...,lo
{1,...,N−1})
is de e mined by
m1−
s∈P({1,...,N−1})
1∈{N}∪s
(l{N}∪s+lo
{N}∪s), ... mN−1−
s∈P({1,...,N−1})
N−1∈{N}∪s
(l{N}∪s+lo
{N}∪s)
=(ˆm1(l,lo), ..., ˆmN−1(l,lo))
wi h k eplaced by k−κ, which implies he claim. Clea ly, only hose (l,lo)can be chosen
which sa is y ˆm (l,lo)≥0 o all =1, ..., N−1 because ˆm (l,lo)is he numbe o
’unused’ a iables o he ke nel in he - h ac o o he p oduc . The e o e we use (34).
Conce ning ou special cases conside ed in Rema k 4.7 he ollowing holds
Rema k 6.2
1. Fo he jump case M(d ,dx)=˜
N(d ,dx) he numbe o appea ing e ms is
C(k,N,(m1,...,mN,0, ...)).
2. Fo he B ownian mo ion case M(d ,dx)=σdW δ0(dx) he numbe o appea ing possi-
ble nonze o e ms is signi ican ly less and can be – in he same way as be o e – ecu si ely
enume a ed by he o mula
CWk,N,(m1,...,mN,0,...)
=
min{k,mN}
κ=0
lo
{1,N}+···+lo
{N−1,N}=mN−κ
lo
{i,j}≥0,i=j,
CWk−κ, N−1,(m1−lo
{1,N}, ... , mN−1−lo
{N−1,N},0, ...),
se ing CW o ze o o one o all o he cases, as be o e.
6.2 The numbe o elemen s by coun ing weak composi ions
To coun hose uples wi hou using a ecu sion, he ollowing o mula o uples (lu)u∈U
whe e Uis a ini e se and lu∈{0,1,...}, in ol ing weak composi ions is a help. A weak
composi ion o ∈N∪{0}in o n∈Npa s is any ep esen a ion =a1+... +anwi h
a1, ..., an∈N∪{0}.I we deno e by
n he numbe o weak composi ions o in o n
pa s, i holds ha
n= +n−1
n−1. Fo example, he weak composi ions o =5
in o n=2 pa s a e gi en by
0+5,5+0,1+4,4+1,2+3,3+2.
Lemma 6.3 Le A1,...,ANbe subse s o a ini e se U such ha +N
i=1Ai=U, Aj∩
N
i=1
i=j
Ac
i=∅, o all j =1,...,N, and le numbe s m1,...,mN∈Nbe gi en. Fo
123
P. Di Tella e al.
=( 1, ..., N)∈{0,1}Nwe se
A( 1,..., N):= ⎛
⎝
N
i=1, i=1
Ai⎞
⎠∩⎛
⎝
N
j=1, j=0
Ac
j⎞
⎠
min
m:= min{mi: i=1,i=1, ..., N}.
Le n:{0,1}N {(0,...,0)}↔{1,...,2N}be a bijec ion such ha
n(1,...,1)=1,n(0,...,0,1)=2N
and n( )<n(s)whene e s> (whe e by s> we mean ha si≥ i o all i =1, ..., N,
and he e exis s a leas one i0such ha si0=1and i0=0).
Fo sho ness, wi h a sligh abuse o no a ion we iden i y qn( )and q (so e.g. q(1,...,1)=q1),
and we abb e ia e,
min
m−q>:= min ,mi−
n( )−1
ι=1
n−1(ι)(i)=1
qι: i=1,i=1, ..., N-.
Then he ca dinali y o
,(lu)u∈U:
a∈Ai
la=mi,i=1,...,N-
is gi en by
minn−1(1)m−q>
q1=0 q1
|An−1(1)|···
minn−1(j)m−q>
qj=0 qj
|An−1(j)|···
···
minn−1(2N−N)m−q>
q2N−N=0 q2N−N
|An−1(2N−N)|·
| |=1min m−q>
|A |,
and whene e a se Asis emp y, he acco ding summa ion does no appea in he abo e
o mula and qsis hen se o ze o.
P oo The o mula can be seen by pa i ioning Uin o all possible pieces eme ging om
in e sec ing he se s A1,...,AN. P o ided ha N
i=1Ai=∅, one summand con ained in
each o he sums a∈Ailais a∈N
i=1Aila=a∈A(1,...,1)la=q(1,...,1)=q1(an in e -
sec ion wi h all se s in ol ed). The possibili ies o q1a e 0, ..., min(1,...,1)m(which equals
min(1,...,1)m−q>), and he numbe o uples (la)a∈A(1,...,1) ha sum up o q1a e gi en by
q1
|A(1,...,1)|= q1
|An−1(1)|.Fo he second sum, as we al eady decided o a pa o ha
sum o be q1, o q2=q(1,...,1,0,1,...,1) he e a e possibili ies om 0 o min(1,...,1,0,1,...,1)m−q1,
which equals min(1,...,1,0,1,...,1)m−q>again. Fo q3and he acco ding uple n−1(3),weha e
he uppe limi min{mi−q1−q2·1{n−1(2)(i)=1}:n−1(3)(i)=1,i=1,...,N},whichis
minn−1(3)m−q>, and he numbe o uples is gi en by q3
|An−1(3)|. We p oceed nes ing he
sums un il we each he numbe 2N−N, which means ha om he eon, he co esponding
uples n−1(j), j>2N−N,ha e only one nonze o elemen .
123
P oduc o mulas o mul iple s ochas ic...
P oo o P oposi ion 4.3 We ha e by P oposi ion 4.1 and (19)
N
j=1
Jmj(α⊗mj
j)T
=
k
(s1,...,sk)∈Ak
i∈{0,1}k
T(k)
αs1,i1(z1)Mi1(dz1)...α
sk,ik(zk)Mik(dzk).
We deno e by [s1,i1, ..., sk,ik]/∼all equi alence classes [(s1,i1), ..., (sk,ik)]wi h espec
o pe mu a ions om Skwhe e (s1, ..., sk)∈Akand (i1, ..., ik)∈{0,1}k.Ou in en ion is
o eplace he abo e exp ession by he ollowing one (up o some ac o s which we wan o
de e mine nex )
k
[s1,i1,...,sk,ik]/∼
σ∈Sk
T(k)
k
j=1
αsσ(j),iσ(j)(zj)Miσ(1)(dz1)...Miσ(k)(dzk).
Whene e we ha e he equali y (de ining ∼)
((s1,i1), ..., (sk,ik)) =((sσ(1),iσ(1)), ..., (sσ(k),iσ(k))),
he same summand appea s. Like in he mul inomial heo em, also he e he mul iplici y o
he summands equals he numbe o hose pe mu a ions ha do no change a uple (in he
mul inomial heo em o (a1+···+aK)n, wi h pai wise disjoin ai, he mul iplici y o he
e m K
i=1aki
iis n!
k1!···kK!,whe e k1+...+kK=n). In ou case, o any (s1, ..., sk)∈Ak
and i∈{0,1}k, o coun he mul iplici y o he smin ((s1,i1), ..., (sk,ik)) pai ed ei he wi h
im=1o im=0, we de ine o j=1, ..., ηN
lj:= |{m:im=1andsm=ej}| and lo
j:= |{m:im=0andsm=ej}|.
Then
k
(s1,...,sk)∈Ak
i∈{0,1}k
T(k)
αs1,i1(z1)Mi1(dz1)...α
sk,ik(zk)Mik(dzk)
=
k
[s1,i1,...,sk,ik]/∼
1
l!lo!
σ∈Sk
T(k)
k
j=1
αsσ(j),iσ(j)(zj)Miσ(1)(dz1)...Miσ(k)(dzk).
We deno e by n:= |lo|=k−k
j=1ij he numbe o ze os in i∈{0,1}kand choose a
pe mu a ion π∈Sk o which
Miπ(1)(dz1)...Miπ(k)(dzk)=(m(dz1),...,m(dzn), M(dzn+1), ..., M(dzk)).
123
P. Di Tella e al.
Then we ha e by Lemma A.3 ha
σ∈SkT(k)
k
j=1
αsσ(j),iσ(j)(zj)Miσ(1)(dz1)...Miσ(k)(dzk)
=((0,T]×R)k
k
j=1
αsπ(j),iπ(j)(zj)m(dz1)···m(dzn)M(dzn+1)···M(dzk)
=
((0,T]×R)n
n
.
j=1
αsπ(j),0(z1, ..., zn)m(dz1)...m(dzn)Ik−n⎛
⎝
k
.
j=n+1
αsπ(j),1⎞
⎠T
.
This yields
N
j=1
Jmj(α⊗mj
j)T
=
k
[s1,i1,...,sk,ik]/∼
1
l!lo!
σ∈Sk
T(k)
k
j=1
αsσ(j),iσ(j)(zj)Miσ(1)(dz1)...Miσ(k)(dzk)
=
k
|lo|+|l|=k,
(l,lo)∈DN
1
l!lo!⎛
⎜
⎝
((0,T]×R)|lo|
α⊗lodm⊗|lo|⎞
⎟
⎠I|l|(α⊗l)T,
whe e
α⊗l=α⊗l1
e1,1⊗... ⊗α⊗lηN
eηN,1and α⊗lo=α⊗lo
1
e1,0⊗... ⊗α⊗lo
ηN
eηN,0.
Rew i ing he condi ion in Ak om P oposi ion 4.1 o his se ing leads o he se DNgi en
in 21. By ea anging he summands we ge
N
j=1
Jmj(α⊗mj
j)T=
k
|l|=k,
(l,lo)∈DN
1
l!lo!⎛
⎜
⎝
((0,T]×R)|lo|
α⊗lodm⊗|lo|⎞
⎟
⎠Ik(α⊗l)T.
Finally, o ge (23), we use Lemma 2.1 o w i e he i e a ed in eg als on he l.h.s. as mul iple
in eg als which implies on he .h.s. he ac o m1!···mN!.
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