Simulation of stochastic Volterra equations driven by space--time L\'evy noise

0.31 MB·English

by  Bohan Chen

Simulation of Stochastic Volterra Equations
Driven by Space–Time Le´vy Noise
BohanChen,CarstenChong,andClaudiaKlu¨ppelberg
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AbstractInthispaperweinvestigatetwonumericalschemesforthesimulationof

stochastic Volterra equationsdrivenby space–timeLe´vy noise of pure-jumptype.
]
R The first one is based on truncating the small jumps of the noise, while the sec-
P ond one relies on series representation techniques for infinitely divisible random
. variables. Under reasonable assumptions, we prove for both methods Lp- and al-
h
t most sure convergence of the approximations to the true solution of the Volterra
a
equation.WegiveexplicitconvergenceratesintermsoftheVolterrakernelandthe
m
characteristicsofthe noise.A simulationstudyvisualizesthemostimportantpath
[
propertiesoftheinvestigatedprocesses.

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v Key words: Simulation of SPDEs, Simulation of stochastic Volterra equations,
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Space–time Le´vy noise, Stochastic heat equation, Stochastic partial differential
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equation
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0 Mathematics Subject Classifications (2010): 60H35, 65C30, 60H20, 60H15,
. 60G57,60G51
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a BohanChen·CarstenChong·ClaudiaKlu¨ppelberg
TechnischeUniversita¨tMu¨nchen
Boltzmannstraße3,85748Garching,Germany
BohanChen
e-mail:[email protected]
CarstenChong
e-mail:[email protected]
ClaudiaKlu¨ppelberg
e-mail:[email protected]
1
2 BohanChen,CarstenChong,andClaudiaKlu¨ppelberg
1 Introduction
Theaimofthispaperistoinvestigatedifferentsimulationtechniquesforstochastic
Volterraequations(SVEs)oftheform
t
Y(t,x)=Y (t,x)+ G(t,x;s,y)s (Y(s,y))L (ds,dy), (t,x)∈R ×Rd, (1)
0 +
0 Rd
Z Z
whereGisadeterministickernelfunction,s aLipschitzcoefficientandL aLe´vy
basisonR ×Rd ofpure-jumptypewithnoGaussianpart.Inthepurelytemporal
+
casewherenospaceisinvolvedandthekernelGissufficientlyregularonthediag-
onal{(t;s)∈R ×R : t=s},theexistenceanduniquenessofthesolutionY to(1)
+ +
areestablishedforgeneralsemimartingaleintegratorsin[14].Thespace–timecase
(1) is treated in [5] for quite generalLe´vy bases. In particular,G is allowed to be
singularonthediagonal,whichtypicallyhappensinthecontextofstochasticpartial
differentialequations (SPDEs) where G is the Green’s function of the underlying
differentialoperator.MoredetailsontheconnectionbetweenSPDEsandtheSVE
(1)arepresentedinSect.2,orcanbefoundin[2,5,20].
SinceinmostcasesthereexistsnoexplicitsolutionformulafortheSVE(1), it
is a natural task to develop appropriate simulation algorithms. For SPDEs driven
byGaussiannoise,researchonthistopicisratherfaradvanced,seee.g.[7,9,21].
However,forSPDEsdrivenbyjumpnoisessuchasnon-GaussianLe´vybases,the
related literature is considerably smaller, see [3] and the work of Hausenblas and
coauthors[8, 10, 11]. The case s ≡1 hasbeentreated in [4]. The contributionof
ourpapercanbesummarizedasfollows:
• We proposeandanalyzetwo approximationschemesfor(1), eachofwhichre-
placestheoriginalnoisebya truncatednoisethatonlyhasfinitelymanyatoms
oncompactsubsetsofR ×Rd.Forthefirstscheme,wesimplycutoffalljumps
+
whosesizeissmallerthanaconstant.Forthesecondscheme,weuseseriesrep-
resentationtechniquesforthenoiseasin[17]suchthatthejumpstobedropped
offare chosenrandomly.Both methodshave alreadybeenappliedsuccessfully
tothesimulationofLe´vyprocesses,cf.[1,18].
• Inthe casewhereG originatesfromanSPDE,the crucialdifferenceofournu-
merical schemes to the Euler or finite element methods in the referencesmen-
tionedbeforeisthatwedonotsimulatesmallspace–timeincrementsofthenoise
butsuccessivelythetruejumpsoftheLe´vybasis,whichisaneasiertaskgiven
thatoneusuallyonlyknowstheunderlyingLe´vymeasure.Itisimportanttorec-
ognizethat this is only possible because the noiseL is of pure-jumptype, and
containsneitheraGaussianpartnoradrift.WeshallpointoutinSect.6howto
relaxthisassumption.
Theremainingarticleisorganizedasfollows:Section2givesthenecessaryback-
ground for the SVE (1). In particular, we present sufficient conditions for the ex-
istence and uniqueness of solutions, and address the connection between (1) and
SPDEs.InSect.3weconstructapproximationstothesolutionY of(1)bytruncat-
ing the small jumpsof the Le´vy basis. We provein Thm. 1 their Lp-convergence,
SimulationofStochasticVolterraEquationsDrivenbySpace–TimeLe´vyNoise 3
andinsomecasesalsotheiralmostsure(a.s.)convergencetothetargetprocessY.
InSect.4weapproximatethedrivingLe´vybasisusingseriesrepresentationmeth-
ods. This leads to an algorithmthat producesapproximationsagain convergingin
theLp-sense,sometimesalsoalmostsurely,toY,seeThm.2.Inboththeorems,we
findexplicitLp-convergenceratesthatonlydependonthekernelGandthecharac-
teristicsofL .Section5presentsasimulationstudyforthestochasticheatequation
which highlights the typical path behaviour of stochastic Volterra equations. The
final Sect. 6 compares the two simulation algorithms developed in this paper and
discussessomefurtherdirectionsofthetopic.
2 Preliminaries
Westartwithasummaryofnotationsthatwillbeemployedinthispaper.
R theset[0,¥ )ofpositiverealnumbers
+
N thenaturalnumbers{1,2,...}
B a stochastic basis (W ,F,F = (Ft)t∈R+,P) satisfying the usual hy-
pothesesofcompletenessandright-continuity
W¯,W˜ W¯ :=W ×R andW˜ :=W ×R ×Rd whered∈N
+ +
B(Rd) theBorels -fieldonRd
B˜ thecollectionofallboundedBorelsetsofR ×Rd
b +
P thepredictables -fieldonBorthecollectionofallpredictableprocesses
W¯ →R
P˜ theproductP⊗B(Rd)or thecollectionallP⊗B(Rd)-measurable
processesW˜ →R
P˜ the collection of sets in P˜ which are a subset of W ×[0,k]×[−k,k]d
b
forsomek∈N
p∗ p∨1
Lp thespaceLp(W ,F,P), p∈(0,¥ ],endowedwiththetopologyinduced
bykXkLp :=E[|X|p]1/p∗
L0 thespaceL0(W ,F,P)ofallrandomvariablesonBendowedwiththe
topologyofconvergenceinprobability
Blpoc the set of all Y ∈P˜ for which kY(t,x)kLp is uniformly bounded on
[0,T]×Rd forallT ∈R (p∈(0,¥ ])
+
Ac the complement of A within the superset it belongs to (which will be
clearfromthecontext)
A−B {x−y: x∈A,y∈B}
−A {−x: x∈A}
Leb theLebesguemeasureonRd (dshouldbeclearfromthecontext)
k·k theEuclideannormonRd.
C,C(T) twogenericconstantsinR ,onedependentandoneindependentofT,
+
whose values we do not care of and may therefore change from one
placetotheother
4 BohanChen,CarstenChong,andClaudiaKlu¨ppelberg
WesupposethatthestochasticbasisBsupportsaLe´vybasis,thatis,amapping
L : P˜ →L0 withthefollowingproperties:
b
• L (0/)=0a.s.
• Forallpairwisedisjointsets(Ai)i∈N⊂P˜bwith ¥i=1Ai∈P˜b wehave
¥ ¥ S
L A =(cid:229) L (A) inL0. (2)
i i
i=1 ! i=1
[
• (L (W ×Bi))i∈N is a sequence of independent random variables if (Bi)i∈N are
pairwisedisjointsetsinB˜ .
b
• ForeveryB∈B ,L (W ×B)hasaninfinitelydivisibledistribution.
b
• L (A)isF-measurablewhenA∈P˜ andA⊆W ×[0,t]×Rd fort∈R .
t b +
• Foreveryt∈R ,A∈P˜ andW ∈F wehavea.s.
+ b 0 t
L (A∩(W 0×(t,¥ )×Rd))=1W 0L (A∩(W ×(t,¥ )×Rd)).
Just as Le´vy processes are semimartingales and thus allow for an Itoˆ integra-
tiontheory,Le´vybasesbelongtotheclassofL0-valueds -finiterandommeasures.
Therefore,itispossibletodefinethestochasticintegral
H(s,y)L (ds,dy)
ZR+×Rd
forH∈P˜ thatareintegrablewithrespecttoL ,see[6]forthedetails.
Similarlyto Le´vyprocesses, thereexisttwo notionsof characteristicsforLe´vy
bases:onegoingbackto[15,Prop.2.1]thatisbasedontheLe´vy-Khintchinefor-
mulaandisindependentofF,andafiltration-basedonethatisusefulforstochastic
analysis[6,Thm.3.2].Forthewholepaper,wewillassumethatbothnotionscoin-
cidesuchthatL hasacanonicaldecompositionunderthefiltrationFoftheform
L (dt,dx)=B(dt,dx)+L c(dt,dx)+ z1 (m −n )(dt,dx,dz)
{|z|≤1}
R
Z
+ z1 m (dt,dx,dz),
{|z|>1}
R
Z
whereBisas -finitesignedBorelmeasureonR ×Rd,L c aLe´vybasissuchthat
+
L (W ×B) is normallydistributed with mean 0 and varianceC(B) for all B∈B˜ ,
b
andm aPoissonmeasureonR ×Rd relativetoFwithintensitymeasuren (cf.[12,
+
Def.II.1.20]).Thereexistsalsoas -finiteBorelmeasurel onR ×Rd suchthat
+
B(dt,dx)=b(t,x)l (dt,dx), C(dt,dx)=c(t,x)l (dt,dx) and
n (dt,dx,dz)=p (t,x,dz)l (dt,dx) (3)
withtwofunctionsb: R ×Rd →Randc: R ×Rd →R aswellasatransition
+ + +
kernelp from(R ×Rd,B(R ×Rd))to (R,B(R)) suchthatp (t,x,·)isaLe´vy
+ +
measureforeach(t,x)∈R ×Rd.
+
SimulationofStochasticVolterraEquationsDrivenbySpace–TimeLe´vyNoise 5
Wehavealreadymentionedintheintroductionthatwewillassume
C=0 (4)
throughout the paper. For simplicity we will also make two further assumptions:
first,thatthereexistb∈RandaLe´vymeasurep suchthatforall(t,x)∈R ×Rd
+
wehave
b(t,x)=b, p (t,x,·)=p and l (dt,dx)=d(t,x); (5)
second,that
L ∈S ∪V , (6)
0
whereS isthecollectionofallsymmetricLe´vybasesandV istheclassofLe´vy
0
baseswithlocallyfinitevariationandnodrift,definedbythepropertythat
|z|1 p (dz)<¥ , and b :=b− z1 p (dz)=0.
{|z|≤1} 0 {|z|≤1}
R R
Z Z
Furthermore,ifp hasafinitefirstmoment,thatis,
|z|1 p (dz)<¥ , (7)
{|z|>1}
R
Z
wedefine
B (dt,dx):=b d(t,x), b :=b+ z1 p (dz),
1 1 1 {|z|>1}
R
Z
M(dt,dx):=L (dt,dx)−B (dt,dx)= z(m −n )(dt,dx,dz).
1
R
Z
Next,letussummarizethemostimportantfactsregardingtheSVE (1).Allde-
tailsthatarenotexplainedcanbefoundin[5].First,manySPDEsofevolutiontype
drivenbyLe´vynoisecanbewrittenintermsof(1),whereGistheGreen’sfunction
of the corresponding differential operator. Most prominently, taking G being the
heatkernelinRd,(1)istheso-calledmildformulationofthestochasticheatequa-
tion (with constant coefficients and multiplicative noise). Typically for parabolic
equations, the heat kernel is very smooth in general but explodeson the diagonal
t=sandx=y.Infact,itisonlyp-foldintegrableon[0,T]×Rdforp<1+2/d.In
particular,assoonasd≥2,itisnotsquare-integrable,andasaconsequence,noso-
lutiontothestochasticheatequationintheform(1)willexistforLe´vynoiseswith
non-zeroGaussiancomponent.Thisisanotherreasonforincludingassumption(4)
inthispaper.
Second, let us address the existence and uniqueness problem for (1). By a so-
lution to this equation we mean a predictable process Y ∈ P˜ such that for all
(t,x)∈R ×Rd, the stochastic integral on the right-hand side of (1) is well de-
+
finedandtheequationitselfforeach(t,x)∈[0,T]×Rd holdsa.s.Weidentifytwo
solutionsassoonastheyaremodificationsofeachother.Givenanumberp∈(0,2],
thefollowingconditionsguaranteeauniquesolutionto(1)inBp by[5,Thm.3.1]:
loc
6 BohanChen,CarstenChong,andClaudiaKlu¨ppelberg
A1. Y ∈Bp isindependentofL .
0 loc
A2. s : R→RisLipschitzcontinuous,thatis,thereexistsC∈R suchthat
+
|s (x)−s (y)|≤C|x−y|, x,y∈R. (8)
A3. G: (R ×Rd)2→RisameasurablefunctionwithG(t,·;s,·)≡0fors>t.
+
A4. L satisfies(3)–(6)and
|z|pp (dz)<¥ . (9)
R
Z
A5. Ifwedefinefor(t,x),(s,y)∈R ×Rd
+
G˜(t,x;s,y):=|G(t,x;s,y)|1{p>1,L ∈/S}+|G(t,x;s,y)|p, (10)
thenwehaveforallT ∈R
+
T
sup G˜(t,x;s,y)d(s,y)<¥ . (11)
(t,x)∈[0,T]×RdZ0 ZRd
A6. Foralle >0andT ∈R thereexistk∈Nandapartition0=t <...<t =T
+ 0 k
suchthat
sup sup ti G˜(t,x;s,y)d(s,y)<e . (12)
(t,x)∈[0,T]×Rdi=1,...,kZti−1ZRd
ApartfromA1–A6,wewilladdanotherassumptionsinthispaper:
A7. Thereexistsasequence(UN)N∈NofcompactsetsincreasingtoRd suchthatfor
allT ∈R andcompactsetsK⊆Rd wehave,asN→¥ ,
+
t
r1N(T,K):= sup |G(t,x;s,y)|1{p>1,L ∈/S}d(s,y)
(t,x)∈[0,T]×K Z0 Z(UN)c
t 1/p∗
+ |G(t,x;s,y)|pd(s,y) →0. (13)
(cid:18)Z0 Z(UN)c (cid:19) !
ConditionsA6andA7areautomaticallysatisfiedif|G(t,x;s,y)|≤g(t−s,x−y)
forsomemeasurablefunctiongandA5holdswithGreplacedbyg.ForA6see[5,
Rem.3.3(3)];forA7chooseUN :={x∈Rd: kxk≤N}suchthatfor p>1
t
sup |G(t,x;s,y)|pd(s,y)
(t,x)∈[0,T]×KZ0 Z(UN)c
t
≤ sup gp(t−s,x−y)d(s,y)
(t,x)∈[0,T]×KZ0 Z(UN)c
T T
= sup gp(s,y)d(s,y)≤ gp(s,y)d(s,y)
x∈KZ0 Zx−(UN)c Z0 ZK−(UN)c
→0 as N→¥
SimulationofStochasticVolterraEquationsDrivenbySpace–TimeLe´vyNoise 7
by the fact thatK−(UN)c ↓0. A similar calculationappliesto the case p∈(0,1]
andthefirstterminrN(T,K).
1
Example1.WeconcludethissectionwiththestochasticheatequationinRd,whose
mildformulationisgivenbytheSVE(1)with
exp(−kxk2/(4t))
G(t,x;s,y)=g(t−s,x−y), g(t,x)= 1 (s) (14)
(4p t)d/2 [0,t)
for(t,x),(s,y)∈R ×Rd.WeassumethatY ands satisfyconditionsA1andA2,
+ 0
respectively.Furthermore,wesupposethat(3)–(6)arevalid,andthat(9)holdswith
some p∈(0,1+2/d). Itis straightforwardto show thatthen A3–A6are satisfied
withthesamep.LetusestimatetheraterN(T,K)forT ∈R ,K:={kxk≤R}with
1 +
R∈N, andUN :={kxk≤N}. We first consider the case p≤1 orL ∈S. Since
K−(UN)c=(UN−R)cforN≥R,thecalculationsafterA7yield(G (·,·)denotesthe
upperincompletegammafunctionand p(d):=1+(1−p)d/2)
(rN(T,K))p∗ ≤ T gp(t,x)d(t,x)= T ¥ exp(−pr2/(4t))rd−1drdt
1 Z0 Z(UN−R)c Z0 ZN−R (4p t)pd/2
T d p(N−R)2
=C tp(d)−1G , dt
0 2 4t
Z (cid:18) (cid:19)
d p(N−R)2 p(N−R)2 p(d)
=C(T) (p(d))−1G , −
2 4T 4T
(cid:18) (cid:19) (cid:18) (cid:19)
d p(N−R)2
×G −p(d),
2 4T
(cid:18) (cid:19)!
p(N−R)2
≤C(T)exp − (N−R)d−2, (15)
4T
(cid:18) (cid:19)
whichtendsto0exponentiallyfastasN→¥ .If p>1andL ∈/S,itfollowsfrom
formula(13)thatweneedanextrasummandforrN(T,K),namely(15)with p=1.
1
3 Truncation ofSmall Jumps
Inthissectionweapproximateequation(1)bycuttingoffthesmalljumpsofL .To
thisend,wefirstdefineforeachN∈N
GN(t,x;s,y):=G(t,x;s,y)1 (y), (t,x),(s,y)∈R ×Rd , (16)
UN +
wherethemeaningofthesetsUN isexplainedinA7.Furthermore,weintroduce
1/p∗
r2N := |z|pp (dz) , r3N := z1{p>1,L ∈/S}p (dz) , (17)
(cid:18)Z[−eN,eN] (cid:19) (cid:12)Z[−eN,eN] (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8 BohanChen,CarstenChong,andClaudiaKlu¨ppelberg
where (e N)N∈N ⊆ (0,1) satisfies e N →0 as N →¥ . Condition A4 implies that
rN,rN →0asN→¥ .Next,definingtruncationsoftheLe´vybasisL by
2 3
L N(dt,dx):= zm (dt,dx,dz), (18)
Z[−eN,eN]c
ourapproximationschemeforthesolutionY to(1)isgivenas:
t
YN(t,x):=Y (t,x)+ GN(t,x;s,y)s (YN(s,y))L N(ds,dy) (19)
0
0 Rd
Z Z
for(t,x)∈R ×Rd.Indeed,YN canbesimulatedexactlybecauseforallT ∈R the
+ +
truncationL N only hasa finite (random)numberRN(T) of jumpson [0,T]×UN,
say at the space–time locations(t N,x N) with sizes JN. This implies that we have
i i i
thefollowingalternativerepresentationofYN(t,x)for(t,x)∈[0,T]×Rd:
RN(T)
YN(t,x)=Y (t,x)+ (cid:229) G(t,x;t N,x N)s (YN(t N,x N))JN1 . (20)
0 i i i i i {tN<t}
i
i=1
What remains to do is to simulateYN(t N,x N), i=1,...,RN(T), iteratively, from
i i
whichthevaluesY(t,x)forallother(t,x)∈[0,T]×Rd canbecomputed.
Thefollowingalgorithmsummarizesupthesimulationprocedure:
Algorithm1.ConsiderafinitegridG thatisasubsetof[0,T]×Rd.Foreachstep
N proceedasfollows:
1.DrawaPoissonrandomvariableRN(T)withintensity
T
RN(T):= 1{z∈[−eN,eN]c}n (dt,dx,dz)=TLeb(UN)p [−e N,e N]c .
0 UN R
Z Z Z
(cid:0) (cid:1)
2.Fori=1,...,RN(T):
a.Drawapair(t N,x N)withuniformdistributionfrom[0,T]×UN.
i i
b.DrawJN from[−e N,e N]cwithdistributionp /p [−e N,e N]c .
i
3.Foreachi=1,...,RN(T)and(t,x)∈G simulateY0((cid:0)tiN,x iN)and(cid:1)Y0(t,x).
4.Foreachi=1,...,RN(T)set
i−1
YN(t N,x N):=Y (t N,x N)+(cid:229) G(t N,x N;t N,x N)s (YN(t N,x N))JN .
i i 0 i i i i j j j j j
j=1
5.Foreach(t,x)∈G defineYN(t,x)via(20).
The nexttheoremdeterminesthe convergencebehaviourof the scheme (19)to
thetruesolutionY to(1).
Theorem1.GrantassumptionsA1–A7underwhich the SVE (1)hasa uniqueso-
lutioninBp .ThenYN asdefinedin(19)belongstoBp forallN∈N,andforall
loc loc
SimulationofStochasticVolterraEquationsDrivenbySpace–TimeLe´vyNoise 9
T ∈R andcompactsetsK⊆Rd thereexists aconstantC(T)∈R independent
+ +
ofN andKsuchthat
sup kY(t,x)−YN(t,x)kLp ≤C(T)(r1N(T,K)+r2N+r3N). (21)
(t,x)∈[0,T]×K
Furthermore,if(cid:229) ¥ (rN(T,K)+rN+rN)p∗ <¥ isfulfilled,thenwealsohavefor
n=1 1 2 3
all(t,x)∈[0,T]×KthatYN(t,x)→Y(t,x)a.s.asN→¥ .
Proof. Itisobviousthat|GN|≤|G|pointwiseandthatwehaven N≤n forthethird
characteristicn N ofL N.Thus,A1–A6arestillsatisfiedwhenGandn arereplaced
byGN andn N (ifL ∈S,alsoL N∈S).SoYN asasolutionto(1)withGN andL N
instead of G andL belongsto Bp as well. Moreover,forall T ∈R there exists
loc +
C(T)∈R independentofN∈Nsuchthat
+
sup kYN(t,x)kLp ≤C(T), N∈N. (22)
(t,x)∈[0,T]×Rd
Weonlysketchtheproofforthisstatement.Infact,using[5,Lem.6.1(1)]itcanbe
shown thatthe left-handside of (22)satisfies an inequality of the same type as in
Lem.6.4(3)ofthesamepaper.Inparticular,itisboundedbyaconstantCN(T)that
dependsonN onlythrough|GN|andn N,andthatthisconstantisonlyincreasedif
we replace |GN| and n N by the larger |G| and n . In this way, we obtain an upper
boundC(T)thatdoesnotdependonN.
Next,weprovetheconvergenceofYN toY asstatedin(21).Wehave
t
Y(t,x)−YN(t,x)= [G(t,x;s,y)−GN(t,x;s,y)]s (Y(s,y))L (ds,dy)
0 Rd
Z Z
t
+ GN(t,x;s,y)[s (Y(s,y))−s (YN(s,y))]L (ds,dy)
0 Rd
Z Z
t
+ GN(t,x;s,y)s (YN(s,y))(L −L N)(ds,dy)
0 Rd
Z Z
=:IN(t,x)+IN(t,x)+IN(t,x), (t,x)∈R ×Rd. (23)
1 2 3 +
If p>1, we have by (8), Ho¨lder’s inequality and the Burkholder-Davis-Gundy-
inequality
t
kI2N(t,x)kLp ≤ GN(t,x;s,y)[s (Y(s,y))−s (YN(s,y))]B1(ds,dy)
(cid:13)Z0 ZRd (cid:13)Lp
(cid:13) t (cid:13)
+(cid:13)(cid:13) GN(t,x;s,y)[s (Y(s,y))−s (YN(s,y))]M(ds,dy(cid:13)(cid:13))
(cid:13)Z0 ZRd (cid:13)Lp
(cid:13)(cid:13) t p−1 (cid:13)(cid:13)
≤C(cid:13) |G(t,x;s,y)||B |(ds,dy) (cid:13)
1
(cid:18)Z0 ZRd (cid:19)
1/p
t
× |G(t,x;s,y)|kY(s,y)−YN(s,y)kp |B |(ds,dy)
Lp 1
Z0 ZRd !
10 BohanChen,CarstenChong,andClaudiaKlu¨ppelberg
t 1/p
+C |G(t,x;s,y)|pkY(s,y)−YN(s,y)kp d(s,y) . (24)
Lp
0 Rd
(cid:18)Z Z (cid:19)
If p∈(0,1],wehaveL ∈V by(6)and(9),andthusJensen’sinequalitygives
0
t p
kI2N(t,x)kLp =E GN(t,x;s,y)[s (Y(s,y))−s (YN(s,y))]zm (ds,dy,dz)
0 Rd
(cid:20)(cid:18)Z Z (cid:19) (cid:21)
t
≤E |GN(t,x;s,y)[s (Y(s,y))−s (YN(s,y))]z|pn (ds,dy,dz)
0 Rd
(cid:20)Z Z (cid:21)
t
≤C |G(t,x;s,y)|pkY(s,y)−YN(s,y)kLpd(s,y). (25)
0 Rd
Z Z
Inserting(24)and(25)backinto(23),wehaveforvN(t,x):=kY(t,x)−YN(t,x)kLp
t 1/p
vN(t,x)≤C(T) |G(t,x;s,y)|1{p>1,L ∈/S}(vN(s,y))pd(s,y)
(cid:18)Z0 ZRd (cid:19)
t 1/p∗
+ |G(t,x;s,y)|p(vN(s,y))p∗d(s,y)
(cid:18)Z0 ZRd (cid:19) !
+kI1N(t,x)+I3N(t,x)kLp , (t,x)∈[0,T]×Rd.
ByaGronwall-typeestimate,whichispossiblebecauseofA5(seetheproofof[5,
Thm.4.7(3)]foranelaborationofanargumentofthistype),weconclude
sup vN(t,x)≤C(T) sup kI1N(t,x)+I3N(t,x)kLp .
(t,x)∈[0,T]×K (t,x)∈[0,T]×K
whereC(T)doesnotdependonKbecauseof(11).ForIN(t,x)wehavefor p>1
1
t
kI1N(t,x)kLp ≤ [G(t,x;s,y)−GN(t,x;s,y)]s (Y(s,y))B1(ds,dy)
(cid:13)Z0 ZRd (cid:13)Lp
(cid:13) t (cid:13)
+(cid:13)(cid:13) [G(t,x;s,y)−GN(t,x;s,y)]s (Y(s,y))M(ds,dy(cid:13)(cid:13))
(cid:13)Z0 ZRd (cid:13)Lp
(cid:13) (cid:13)
(cid:13) t (cid:13)
≤C(cid:13)1+ sup kY(t,x)kLp |G(t,x;s,y)||(cid:13)B1|(ds,dy)
(t,x)∈[0,T]×Rd ! Z0 Z(UN)c
t 1/p
+ |G(t,x;s,y)z|pn (ds,dy,dz)
(cid:18)Z0 Z(UN)c (cid:19) !
≤C(T)rN(T,K), (26)
1
uniformlyin(t,x)∈[0,T]×K.Insimilarfashiononeprovestheestimate(26)for
p∈(0,1],perhapswithadifferentC(T).Next,when p>1,(22)implies