Time regularity of solutions to linear equations with Lévy noise in infinite dimensions

Transcription

1 Tim rgularity of solutios to liar quatios with Lévy ois i ifiit dimsios S. Pszat Faculty of Applid Mathmatics, AG Uivrsity of Scic ad Tchology, Kraków, Polad, adrss: apszat@cyf-kr.du.pl. J. Zabczyk Istitut of Mathmatics, Polish Acadmy of Scics, Śiadckich 8, -95 Warszawa, Polad, adrss: zabczykt@impa.pl. Abstract Th xistc of strog ad wak càdlàg vrsios of a solutio to a liar quatio i a ilbrt spac, driv by a Lévy procss takig valus i a ilbrt spac U is stablishd. Th so-calld cylidrical càdlàg proprty is ivstigatd as wll. A spcial mphasis is put o ifiit systms of liar quatios driv by idpdt Lévy procsss. Kywords: Càdlàg ad cylidrical càdlàg trajctoris; Path proprtis; Orsti Uhlbck procsss; Liar volutio quatios; Lévy ois. 1 Mathmatics Subjct Classificatio: Primary: 615; Scodary: 6G5. 1. Itroductio This papr is cocrd with th tim rgularity of a solutio to th followig liar volutio quatio dx = AXdt + dz, X) =, 1) whr A grats a C -smigroup S o a ilbrt spac ad Z is a Lévy procss takig valus i a ilbrt spac U. Prprit submittd to Elsvir Novmbr 13, 1

2 Not that 1) icluds liar parabolic ad hyprbolic quatios prturbd by a ifiit dimsioal stochastic procss. To work with th solutio havig Markov proprty o has to assum s [19], Chaptr 1) that Z is a Lévy procss. A good thory for liar quatios is cssary to study th oliar problms of th typ dx = AX + F X)) dt + dz, X) = x. Ths problms ar v mor importat from th applicatio poit of viw. It is ot difficult to formulat cssary ad sufficit coditios udr which th solutio to 1) xists i ad/or is ma squar cotiuous; s Propositio.6. owvr for a dpr study of th solutio it is cssary to fid out how rgular ar its trajctoris ad this is th mai purpos of th prst study. Sic th cas of Z big a Wir procss is rathr wll udrstood s [7, 8, 11, 3]) w assum that Z is without th Gaussia part. Morovr if th procss Z taks valus i th spac U, th, udr rathr mild assumptios o th smigroup S, th solutio has a càdlàg modificatio du to th classical Kotlz rsults s [14, 15], ad [1]). This is obviously th bst possibl tim rgularity rsult for a quatio driv by a jump ois. It is howvr importat to cosidr th cas wh Z livs i a biggr spac U ; s.g. [1]. For istac to stablish th strog Fllr proprty of th corrspodig trasitio smigroup th iclusio U is oft vry hlpful. W will limit our cosidratios to this cas oly. Obviously to hav th solutio takig valus i w will hav to assum that th smigroup S xhibits som rgularisig proprty. Namly w assum that for ach t >, St) has a uiqu) xtsio to a boudd liar oprator, dotd also by St), from th spac i which Z livs to. W assum that th solutio X taks valus i ad is giv i th so-calld mild form: Xt) = St s)dzs), t. Path proprtis of X dpd havily o th jump masur ν of Z. Namly if νu \) >, th rgardlss th grator A, v if X is squar itgrabl i, thr is a o mpty st of vctors z such that for ay T >, with o zro, or v i may cass with probability 1, th trajctoris of

3 th ral-valud procss Xt), z, t [, T ], ar uboudd; s [4] ad [19], Propositio 9.5. Thrfor w will assum that th Lévy masur ν of Z satisfis νu \ ) =. ) O obtais a itrstig class of quatios, icludig th prturbd hat quatio, assumig that A is a gativ dfiit oprator with igvctors ), formig a orthoormal ad complt basis i, ad with th corrspodig, positiv igvalus γ ). If i additio Z = Z, 3) whr Z ) ar idpdt ral-valud Lévy procsss ach with a Lévy masur µ, th quatios of this form will b calld of th diagoal typ. I fact th Xt) = X t), whr dx = γ X dt + dz, X ) =. O ca thus idtify X with th squc of procsss X ) ad th spac with l. Du to th idpdc of th procsss Z, th Lévy masur ν of Z is always portd o th st sum R ad obviously hypothsis ) is satisfid. Coditio ) imposd o th Lévy masur of th ois, dos ot guarat that X has càdlàg trajctoris i. I fact it was show rctly, for a larg class of quatios satisfyig ), icludig a subclass of diagoal os, that thir solutios liv i but do ot allow a càdlàg modificatio; s [], ad also Thorm.3, Propositios 4.1, 4.3, ad Corollary 4. of th prst papr. Morovr, it was show by Liu ad Zhai [17], s also Rmark 3.5, for th diagoal systms with Z big idpdt α-stabl procsss that if X is càdlàg i th cssarily th procss Z taks valus i. Ths rsults ar i sharp cotrast with what o could xpct from th rsults o th quatios with Gaussia ois. Th càdlàg proprty for th quatios with Lévy prturbatios is much lss frqut tha th cotiuity of th trajctoris i th Gaussia cas; s [7, 8, 11, 3]. This is o of th rasos that i th jump cas mor sophisticatd cocpts of rgularity might b usful. I fact w will dal with th followig proprtis. 3

4 Dfiitio 1.1. i) W say that a procss X has a càdlàg modificatio if thr is a modificatio X of X with càdlàg trajctoris; that is right cotiuous ad havig lft limit at ay poit. ii) W say that a -valud procss X has a wakly càdlàg modificatio if thr is a modificatio X of X such that for ay z, th ral-valud procss X ), z has càdlàg trajctoris. iii) A -valud procss X is cylidrical càdlàg if for ay z, th ralvalud procss X ), z has a càdlàg modificatio. iv) Lt V. A V -valud procss X is V -cylidrical càdlàg if for all v V th ral-valud procss v X ) has a càdlàg modificatio. W hav th followig obvious implicatios i) = ii) = iii) = iv) Th càdlàg proprty is fudamtal for stablishig th strog Markov proprty of th solutio ad for various localisatio procdurs. It allows to formulat ad study th xit tim τ D from a giv st D. Th xistc of a wakly càdlàg modificatio surs at last th local bouddss of trajctoris; s Thorm.3ii). For solutios which ar cylidrically càdlàg or V -cylidrically càdlàg, th xit tims ar maigful for a larg class of cylidrical sts of th form D = {x : Πx G} or D = {x V : Πx G} whr Π is a fiit projctio i or i V, rspctivly. W will dscrib ow th mai rsults of th papr. Rsults o th càdlàg proprty ar formulatd ad discussd i Sctio 3. Our mai rsult is Thorm 3.1. W rstrict our atttio to th cas wh th oprator A grats a aalytic smigroup ad thus to parabolic typ of quatios. Th cas of hyprbolic quatios lads to th group, rathr tha to smigroup S, ad thrfor rquirs th ois takig valus i th stat spac. W show that th class of quatios havig càdlàg solutios i but with th ois procss volvig outsid is o mpty ad of som itrst; s.g. Exampl 3.4. Thus, th phomo coutrd i th cas of α-stabl prturbatios by Liu ad Zhai [17], ad dscribd abov, is ot valid i gral. Our rsults rly o th classical Chtsov work [5] or [9] ad o formula for momts of itgrals with rspct to Poissoia radom masurs gathrd i th prlimiaris. As a byproduct w obtai also th stimats of th typ E Xt) p <. t T 4

5 W ot that th classical rsult of Kiy s [13] or [19], Thorm 3.3, o càdlàg vrsio of a Markov procss would rquir that t ) lim P St)x + St s)dzs) x > r =, r >. 4) t x Obviously 4) is satisfid if ad oly if A =. Assum 4). Th puttig x = w obtai t ) lim P St s)dzs) t > r =, r >. Thrfor St)x x has to covrg to uiformly i x from th whol spac. It is impossibl as x St)x x = if St) is ot th idtity. W do ot trat i this papr a itrstig qustio udr which coditios th solutio X has a càdlàg or wakly càdlàg modificatio i a biggr spac. For som aswrs to ths qustios w rfr th radr to [16]. Rsults o th thr rmaiig càdlàg proprtis ar cotaid i Sctios 4 6 ad ar stablishd oly for th diagoal systms. Thy ar rathr tchical ad ivolv log formula. Extsios to th gral cas howvr would b possibl if dd. Th mai rsult o wak càdlàg proprty is formulatd as Propositio 4.1 ad is of gativ charactr. It stats sufficit coditios for o-xistc of wakly càdlàg solutios i trms of th tails of th masurs µ. Its proof is basd o th gral if ad oly if coditios for wak càdlàg proprty, formulatd i trms of th orthogoal xpasios ad stablishd i th prlimiaris as Thorm.3. It turs out that also wak càdlàg proprty implis svr rstrictios o th paramtrs of th quatios. I particular th cas of idpdt ad idtically distributd coordiats of th ois procss cylidrical ois) is xcludd. At this momt w do ot hav ay xampl of liar systm 1) whos solutio has o càdlàg but oly a wakly càdlàg modificatio. Fortuatly th cylidrical càdlàg proprty taks plac udr much wakr assumptios ad i particular th coordiats of th ois ca b idtically distributd. Sufficit coditios for this proprty ar formulatd i Thorms 5.1 ad 5.. It is worth otig that to show th cylidrical càdlàg proprty of X o caot apply dirctly th Chtsov rsult s Rmark 8.1) but o ds a w way of circumvtig that obstacl. 5

6 Rsults o V -cylidrical càdlàg proprty ar prstd i Sctio 6. Th mai rsult, Thorm 5., provids sufficit coditios for that proprty. As a cosquc of Thorm 5. ad Propositio 4.3 w s that v if th mbddig V is compact, th V -cylidrical càdlàg proprty dos ot sur th xistc of a wakly càdlàg modificatio. Thrfor th implicatio ii) = iv) dos ot hold i gral. W complt this itroductio with som op qustios. Qustio 1 W hav coditios for th xistc of càdlàg modificatios ad coditios which xclud th xistc of a wakly càdlàg modificatio. Fid applicabl coditios ladig to wak but ot strog càdlàg proprty. I particular fid xampl of a liar quatio whos solutio admits a wakly càdlàg but ot càdlàg modificatio. Qustio I our ivstigatio w rly o th Chtsov critria with th xpot p = ; s Corollary.. Fid xtsios of our rsults for p or mor grally usig th fuctio g apparig i Thorm.1 ot of th powr typ. Qustio 3 Fid coditios for cylidrical càdlàg proprtis i th odiagoal cas. Qustio 4 Fid coditios for th càdlàg or cylidrical càdlàg proprty i cas of th ois with tails s Corollaris 3. ad 5.3). I particular i diagoal cas assum that Z = σ L whr L ar idpdt symmtric α-stabl procsss ad σ α <. γ Is it tru that th procss X is cylidrical càdlàg?. Prlimiary rsults.1. Càdlàg critria i mtric spacs W first rcall som basic facts o path rgularity of stochastic procsss i mtric spacs. Thy ca b attributd to N.N. Chtsov, but th xpositio is basd o th book of Gihma ad Skorohod [9]. 6

7 Lt ξ = ξt), t [, T ]) b a sparabl procss takig valus i a mtric spac U, ρ). W xtt ξ o R puttig ξt) = ξ) for t < ad ξt) = ξt ) for t T. Th followig rsult holds s [9], Lmma 3 ad Thorm 1 of Chaptr 3). Thorm.1. Assum that thr ar a icrasig fuctio g:, ), ) ad a fuctio q:, ), ), ) such that for all C, h >, P {[ρξt), ξt h)) > Cgh)] [ρξt), ξt + h)) > Cgh)]} qc, h), ad G := gt ) <, QC) := qc, T ) <. Th with probability 1, ξ has o discotiuitis of th scod kid, ad for ay N >, { } { P ρξt), ξs)) > N P ρξ), ξt )) > N } ) N + Q. t,s [,T ] G G Corollary.. Assum that thr ar p, r, K > such that for all t [, T ] ad h >, E [ρ ξt), ξt h)) ρ ξt), ξt + h))] p Kh 1+r. 5) Th with probability 1, ξ has o discotiuitis of th scod kid. Morovr, for ay 1 q < p, E t,s [,T ] ρξt), ξs))) q G) q E ρξt ), ξ))) q + R, 6) whr < r < r, G = T ) r /p) <, 7) ad R := 1 + q KG) p T 1+r r p q 1 r r. 7

8 Proof Lt < r < r. Th th assumptios of Thorm.1 hold with gh) := h r /p) ad qc, h) := K C p h1+r r. By Chbyshv s iquality P {[ρξt), ξt h)) > Cgh)] [ρξt), ξt + h)) > Cgh)]} P { ρξt), ξt h))ρξt), ξt + h)) > C g h) } Kh1+r C p g p h) = Kh1+r r C p. Not that i this cas G is giv by 7), ad ) N Q = KG)p T ) 1+r r = KG)p T 1+r r G N p N p = KG)p T 1+r r 1 r r N p. To show 6) tak q 1. Sic E t,s [,T ] Thorm.1 yilds E t,s [,T ] ρξt), ξs))) q = q P { ρξt), ξs)) N t,s [,T ] ρξt), ξs))) q G) q E ρξt ), ξ))) q +1+q ad cosqutly 6). 1 } r r ) N q 1 dn, ) N Q N q 1 dn, G.. Critria for càdlàg proprtis i ilbrt spacs Lt ow X ) b a squc of ral-valud càdlàg procsss dfid o a fiit tim itrval [, T ], ad lt ) b a orthoormal basis of a ilbrt spac. Assum that for ach t [, T ], Xt) = X t), whr th sris covrgs i probability or quivaltly P-a.s. Th first part of th thorm blow is tak from th papr by Liu ad Zhai [17]. It will ot b usd i th papr but it is icludd for a mor complt pictur. I th proofs w follow suggstios by A. Jakubowski [1]. 8

9 Thorm.3. i) Procss X has a càdlàg modificatio if ad oly if ) P lim Xt) = = 1. 8) N t [,T ] =N ii) Procss X has a wakly càdlàg modificatio if ad oly if ) P Xt) < = 1. 9) t [,T ] Proof Sic ach X is càdlàg, 8) implis that X has a càdlàg modificatio. Thrfor w d to show that 8) follows from th xistc of a càdlàg modificatio X of X. To s this ot that o a ds st Q [, T ], X = X, P-a.s. Cosqutly, sic ach X is càdlàg, Xt), = X t) = Xt),, t [, T ], P a.s. Thrfor thr is a Ω Ω such that PΩ ) = 1 ad for ay ω Ω, [, T ] t Xt; ω) is càdlàg ad Xt; ω), = X t; ω). By càdlàg proprty, for ay ω Ω, th st {Xt; ω): t [, T ]} is compact i. Thrfor th dsird coclusio follows from th followig critrio for a rlativ compactss of a boudd st K i ; lim N x K =N x, =. To s that 9) implis th wakly càdlàg proprty of X tak a z. O has to show that [, T ] t Xt), z R is càdlàg P-a.s. Lt Ω b th st of all ω Ω such that Th t [,T ] Xt; ω) <. Xt; ω) = X t; ω) 9

10 is a boudd -valud mappig of t [, T ]. Morovr, Xt; ω), k = X k t; ω), t [, T ], ar càdlàg fuctios. Lt t [, T ) ad lt t m t. Th z Xt m ; ω), z, m = 1,,..., is a squc of liar fuctioals, covrgig o a ds st. Sic thir orms ar boudd Xt m ; ω), z, m = 1,,..., covrgs for ay z. Sic lim m Xt m; ω), z = Xt; ω), z holds o a ds st of z it holds for ay z. Thrfor X ; ω) is wakly right cotiuous. Now lt t, T ] ad lt t m t. Th lim Xt m; ω), k = X k t ; ω), k N. m Bouddss of Xt m ; ω), m N, implis that th wak limit of Xt m ; ω) xists. Sic it holds for ay squc t m t, th wak lft limit Xt ; ω) xists. Assum ow that X is a wakly càdlàg modificatio of X. W will show 9). To do this obsrv that X t; ω) = Xt; ω),, t [, T ], = 1,,..., ar càdlàg fuctios. Morovr, for ay t [, T ], P ω Ω: X ) t; ω) = X t; ω) = 1. Sic both procsss X ad X ar càdlàg, thrfor thr is a st Ω Ω such that P Ω) = 1 ad X t; ω) = X t; ω), t [, T ],, ω Ω. Sic 9) holds. t [,T ] X t; ω) <, P a.s., 1

11 .3. Momts of stochastic itgral I th propositio blow π is a compsatd Poisso radom masur o a masurabl spac E with th itsity masur ν. Propositio.4. Assum that dtrmiistic ral-valud masurabl fuctios f 1,..., f 4 hav fiit forth momts with rspct to ν. Th w hav E f 1 x) πdx) f x) πdx) f 3 x) πdx) f 4 x) πdx) E E E E = f 1 x)f x)νdx) f 3 x)f 4 x)νdx) E + E f 1 x)f 3 x)νdx) f x)f 4 x)νdx) E + E f 1 x)f 4 x)νdx) f x)f 3 x)νdx) E E + f 1 x)f x)f 3 x)f 4 x)νdx). E A simpl proof of th propositio abov ca b obtaid by coscutiv diffrtiatio of th charactristic fuctio { 4 } F x 1, x, x 3, x 4 ) = E xp i x j f j x) πdx) = xp { E j=1 E i 4 j=1 x jf j x) i ) } 4 x j f j x) 1 νdx). I th papr w will also d th followig spcial cas of Propositio.4. Namly, lt L b a ral-valud purly jump Lévy procss with a Lévy masur µ. W assum that µ has fiit momts up to ordr 4. Writ m j := y j µdy), j = 1,..., 4. R Th Lt), t, ca b writt as th sum of a drift trm mt ad a pur jump Lévy martigal with th Lévy xpot Ψz) = ) 1 + izy izy µdy). 1) R For our purposs w ca assum that th drift trm vaishs ad cosqutly that th Lévy xpot of L is giv by 1). 11 j=1

12 Propositio.5. For ay T < ad cotiuous dtrmiistic fuctios f 1, f, T ) T ) E f 1 s)dls) f s)dls) T ) = m f 1 s)f s)ds + m T +m 4 f1 s)f s)ds..4. Critria for volutio of OU procsss i T T f1 s)ds f s)ds I th propositio blow w ar cocrd with X giv by 1). Lt π b th radom jump masur of Z, ad lt πds, dz) := πds, dz) νdz)ds b th compsatd radom masur. By th Lévy Khichi rprstatio formula s.g. [19], Thorm 6.8) th procss Z ca b writt as follows Zt) = mt+ { z U >1} zπds, dz)+ { z U 1} z πds, dz), t. 11) Propositio.6. Assum that Z is a Lévy procss i a ilbrt spac U with rprstatio 11). Assum that th Lévy masur ν of Z satisfis νu \ ) = ad that th drift trm m =. Th w hav th followig. i) Procss Z taks valus i if ad oly if z 1νdz) <, t. ii) Procss X taks valus i if ad oly if for ay t >, whr B := {x U: x U 1}. U Ss)z 1dsνdz) <, [χ B v) χ B Ss)v)] Ss)vνdv)ds, 1

13 iii) Assum that z { z U >1} Uνdz) < ad that Ss)a ds, t >, whr a = zνdz). { z U >1} Th X has fiit scod momt i if ad oly if Ss)z dsνdz) <, t. 1) Morovr, if 1) holds th X is ma squar cotiuous i. Proof Th first part follows dirctly from th Lévy Khichi thorm s.g. [19], Thorms 4.3 ad 6.8). I ordr to show th scod part ot that th radom variabl Xt) is ifiitly divisibl with th Lévy masur ad th drift m t = U ν t := Ss)ds ν [χ B v) χ B Ss)v)] Ss)vνdv)ds, for mor dtails s [6]. W hav to chck oly that m t, ad that z 1ν tdz) <. Sic z 1ν tdz) = th dsird coclusio holds. Lt Z t) = U Ss)z 1νdz), z πdz, ds), t. Clarly 1) is a if ad oly if coditio udr which th procss St s)dz s), t, 13

14 is squar itgrabl i. O th othr had Zt) = Z t) + at, whr a U ad St s)ads. Thrfor th dsird quivalc holds. To s th cotiuity i probability assum that t > s. Without ay loss of grality w ca assum that Zt) = z πds, dz), t. Th U It, s) := E Xt) Xs) s = E St r)dzr) Ss r)dzr) + E St r)dzr) s s = St r) Ss r)) z drνdz) + St r)z drνdz) s s s = St s) I) Sr)z drνdz) + Sr)z drνdz). By 1) ad th Lbsgu domiatd covrgc thorm It, s) providd t s ad s is i a boudd itrval. 3. Càdlàg proprty This sctio is cocrd with th xistc of a càdlàg modificatio of th solutio X to 1). For som tchical raso w will d to assum that th smigroup S is aalytic o. Without ay loss of grality w may assum that S is xpotially stabl. Th blogs to th rsolvt st of th grator A. Lt ρ, ρ >, b th domai of A) ρ quippd with th orm z ρ := A) ρ z. If ρ <, th ρ is th dual spac to ρ whr th duality is giv by th idtificatio =. Th proof of th thorm blow is postpod to Sctio 6. Thorm 3.1. Lt X b th solutio to 1), whr A is th grator of a xpotially stabl aalytic smigroup S o a ilbrt spac. Lt Z b a Lévy procss takig valus i a ilbrt spac U = ρ for a crtai 14

15 ρ < 1/ ad havig rprstatio 11). Assum that th Lévy masur ν of Z satisfis ν ρ \ ) = ad that ) z ρ + z 4 ε νdz) < for a crtai ε >. Th X has a càdlàg modificatio i ad E Xt) q <, T <, q [1, 4). 13) t T Usig th stadard localisatio procdur w obtai th followig. Corollary 3.. Assum that thr ar ρ < 1/ ad ε > such that Z taks valus i ρ, th Lévy masur ν of Z satisfis ν ρ \ ) =, ad z 4 ενdz) <, R >. { z ρ R} Th X has a càdlàg modificatio i Diagoal cas Lt us ow cosidr th diagoal cas of 1). Clarly w ca assum that = l, ν = µ whr ach µ is a Lévy masur of a o dimsioal Lévy procss Z, ) is th caoical basis of l, A = γ, ad γ >, N. W assum that ach Z has Lévy Khichi rprstatio Z t) = zπ ds, dz) + z π ds, dz), 14) { z >1} { z 1} whr π is a Poisso radom masur with itsity masur µ satisfyig R z 1µ dz) <. Corollary 3.3. I th diagoal cas assum that thr ar ρ < 1/ ad ε > such that ) z γ ρ + zγ 4 4ε µ dz ) <. R 15

16 Th X has a càdlàg modificatio i = l ad 13) holds. If ad R { z γ ρ R} th X has a càdlàg modificatio i l. ) z γ ρ 1µ dz ) < z 4 γ 4ε µ dz ) <, R >, 15) I th followig xampl w show that solutio of th liar quatio ca b càdlàg i although th ois procss dos ot liv i. Exampl 3.4. Lt Z = σ L, N, whr L ar idpdt ad idtically distributd Lévy procsss of typ 14), ad σ ) is a squc of strictly positiv umbrs. Assum that th Lévy masur µ of L has fiit momts up to ordr 4. Th th Lévy masur µ of Z = σl quals µ /σ ). Cosqutly, if thr is a ε > such that [ γ ε 1 σ + γ ε σ 4 ] <, th by Corollary 3.3, X has a càdlàg modificatio i l. I particular, if γ α, σ κ, th X has a càdlàg modificatio i l providd that α > 1 κ ad κ > 1/4. Assum ow that ach L is a Poisso procss with itsity 1. Th µ = δ 1 ad σ L has a Lévy masur µ = δ σ. Thrfor Z = σ L ) livs i l if ad oly if R x 1µ k dx ) = σ 1 <. c, if σ κ, th Z taks valus i l if ad oly if κ > 1/. Summig up, if 1/4 < κ < 1/ ad α > 1 κ th Z dos ot tak valus i l but th solutio X has a càdlàg modificatio i l. Rmark 3.5. It turs out that our rsult is ot applicabl to th cas of α-stabl ois cosidrd by Liu ad Zhai [17]; s discussio i Sctio 1. Namly, still i th diagoal cas, assum that Z = Z ) whr Z = σ L, 16

17 σ > ad L ar idpdt ral-valud symmtric α-stabl procsss for a fixd α, ). Not that th Lévy masur µ of σ L is giv by µ ) = µσ 1 ), whr µ is th Lévy masur of th symmtric α-stabl procss. c µ has th dsity Cσ/ x α 1+α. Thrfor th coditio 15) has th form γ ρ R σ α zγ 4 4ε dz z 1+α = R4 α σ α 4 α γ 4ε+4 α)ρ <, R >. Thrfor 15) ad th fact that γ + imply that σα <. Th last iquality is howvr if ad oly if coditio udr which Z taks valus i l ; s [1, ]. 4. Wakly càdlàg proprty i diagoal cas I th prst sctio w ar cocrd with th wakly càdlàg proprty dalig oly with th diagoal cas. Rsults hr ar maily of gativ typ. Assum that A is gativ dfiit slf-adjoit with a compact rsolvt, ) is th orthoormal basis of igvctors of A, γ ) is th corrspodig squc of igvalus, ad Z = Z, whr Z ) ar idpdt ral-valud Lévy procsss with th Lévy Khichi rprstatio 14). Th whr Xt) = X t), dx = γ X dt + dz, X ) =. Not that ach procss X is càdlàg. Takig ito accout Thorm.3 th followig rsult provids, i particular, a cssary coditio for th xistc of th wakly càdlàg modificatio of X. Th proof is i th spirit of [17]. Propositio 4.1. If for ach r >, µ dy) =, {r y } 17

18 th for vry T >, P t [,T ] ) Xt) < =. I particular th procss X dos ot hav a wakly càdlàg modificatio. Proof Not that for ach, c Xt ) Xt). t [,T ] t [,T ] Z t)) = X t)) 4 Xt), t [,T ] t [,T ] t [,T ] whr Z t) := Z t) Z t ). Takig ito accout th obvious stimat w obtai whr 4 t [,T ] X t) 4 Sic ζ ar idpdt, ) P Xt) < t [,T ] t [,T ] ζ 4 t [,T ] X t) Xt), t [,T ] ζ := Z t)), N. t [,T ] P ζ < r Z t)), ) = lim P ζ r ). Lt π b th Poisso radom masur corrspodig to Z. Th P ζ r ) = P π [, T ] {y: y > r}) = ) = xp { T µ {y: r < y }}. Thrfor th dsird coclusio follows from th stimat ) { } P Xt) < lim xp T µ {y: r < y }. 16) r t [,T ] As a dirct cosquc of 16) w obtai th followig rsult. 18

19 Corollary 4.. Lt Z t) = σ L b t), t, whr L ar idpdt idtically distributd Lévy procsss of typ 14), ad σ, b >. If th Lévy masur µ of L has a uboudd port, ad thr is a T > such that ) th σ or b. P t [,T ] Xt) < >, Th followig rsult shows that th assumptio σ is i gral ot sufficit for th xistc of a wakly càdlàg modificatio of X. Propositio 4.3. Lt L b idpdt idtically distributd Lévy procsss, of typ 14), ach with th Lévy masur µ. Assum that µ has a uboudd port. Th thr is a squc σ such that ) P Xt) < =, T >, t [,T ] rgardlss th squc γ ). Proof Takig ito accout 16) it is ough to fid a squc σ such that µ{y: σ 1 r < y } =, r >. Lt ψx) = µ{y: y > x}, x >. By th assumptio ψ is a dcrasig fuctio from, + ) to, + ). Thus th rsult follows from th lmma blow. Lmma 4.4. Assum that ψ:, + ), + ) is a dcrasig fuctio. Th thr is a squc a ) such that a ad ψra ) =, r >. Proof Lt N k b a squc such that N k+1 1 =N k ψk ) 1. 19

20 Lt a = k for [N k, N k+1 1]. Th for ay m N, ψma ) = k=1 k=m N k+1 1 =N k N k+1 1 ψmk) k=m N k+1 1 =N k =N k ψk ) =. ψ 5. Cylidrical càdlàg proprty i diagoal cas This sctio dals with diagoal cas. assum that = l ad X = X ) whr As i th prvious sctio w dx = γ X dt + dz, X ) =, = 1,,..., 17) γ, N, ar strictly positiv ral umbrs, ad Z, N, ar idpdt ral-valud Lévy procsss with th Lévy Khichi dcompositio 14). Giv a squc β ) of strictly positiv umbrs, st { } lβ := x ): x ) l := x β β <. Sic Z ) ar idpdt, it is kow, s [19], that Z is a Lévy procss i lβ if ad oly if = R β x 1µ dx ) [ β β 1,β 1 ) x µ dx ) + µ R \ β 1, β 1 ) )] <. Thrfor, th assumptio that ach µ has a fiit scod momt surs that Z is a Lévy procss i a suitably chos wightd l -spac. W assum that for ay, th Lévy masur µ of Z has fiit momts up to ordr 4. Writ m j ) := x j µ dx), j = 1,..., 4. 18) R

21 Not that sic th Lévy masurs µ hav fiit first momts, th procsss Z hav trajctoris with boudd variatio. Subtractig th drifts w may assum that Z t) = x π ds, dx), 19) R whr π is th Poisso jump masur of Z ad π is th compsatd masur. I this way ach Z is a squar itgrabl martigal. Obviously, ad X t) = EX t) = m ) I this sctio w assum that m ) γ γt s) dz s), ) γt s) ds m ) γ. + m ) 1) <, 1) γ which is mor that is dd to guarat that th procss X := X ) rstrictd to ay fiit tim itrval [, T ], is a squar itgrabl radom lmt i l satisfyig E Xt) l <. t T Takig modificatios w ca also assum that th ral-valud procsss X ar càdlàg. Not that X solvs th liar quatio 1) with a diagoal liar oprator Ax ) = γ x ), ad with Z = Z ). Th mai rsults of th prst sctio ar th thorms blow. Thir proofs ar howvr postpod to Sctios 8 ad 9, rspctivly. Th first rsult covrs th cas whr Z ) ar idtically distributd with th Lévy masur havig fiit momts up to ordr 4. This particular cas rquirs γ +. By [] th solutio X has o càdlàg modificatio ulss µ =, Morovr, if µ has a uboudd port, th by Propositio 4.3, X dos ot hav v a wakly càdlàg modificatio. 1

22 Thorm 5.1. Assum th stimat 1), ad that thr is a ε, 1) such that: m1 ) + m ) + m 4 ) + γ ε 1)/ m ) + γ ε 1 m 4 ) ) <. ) Th X is cylidrical càdlàg. Morovr, T <, q [1, 4), E Xt), z l q <. 3) z l : z l 1 t [,T ] I additio for ay z l, th sris X t)z = Xt), z l covrgs i L q uiformly i t o ach compact itrval. Th proofs of th difficult first parts of th thorm ar postpod to th followig sctios. r w sktch th proof of th fial o. Lt Pr,m z) k = z k if m k ad othrwis. For ay z l ad for ay δ > thr is a δ N such that t [,T ] k= Pr,m z l δ z l, m > δ. Morovr, if Pr,m z, th m q E X k t)z k = E Pr,m z q Xt), Pr,m z q Pr,m z t [,T ] Pr,m z q E Xt), v l q. v l : v l 1 t [,T ] Cosqutly, th stimat i 3) guarats that for ay z l, for all T > ad ε > thr is a ε,t such that for all ε,t m, m q E X k t)z k ε. t [,T ] k=

23 5.1. lβ -cylidrical càdlàg proprty Rcall that lβ is a wightd l -spac. Th lβ ) l1/β, whr 1/β = 1/β ). Our xt rsult is cocrd with lβ -cylidrical càdlàg proprty ad i particular it covrs th cas whr Z = σ L, N, whr L ar idtically distributd with th Lévy masur havig fiit momts up to ordr 4, ad σ. Thorm 5.. Assum that lim m1 ) m 4 ) + γ 1 ) =. 4) Lt morovr β ) b a squc of positiv umbrs tdig to + such that β m ) + β ) m 1 ) <, γ γ ad for a crtai ε >, β m 1 ) γ <, β m 1 )) ε +ε <, [ β [ β m )) ε 1+ε + γ 1 m 4 )) 1 γ 1 ) 1 ε ] <, ) 1 ε + γ 1 ) 1/ ) + m 4 )) 1/1 1 1+ε ] <. Th for ay z lβ ) = l1/β, th procss Xt), z l, t, is wll-dfid ad has a càdlàg modificatio. Morovr, T <, q [1, 4), E Xt), z l q <. z l1/β : z l 1 1/β t [,T ] 5.. Commts o localisatio It is of itrst to xtd th rsults to th cas wh th Lévy masurs do ot hav fiit momts, lik for α-stabl procsss. owvr th localisatio ida mployd i Sctio 3 i th study of th càdlàg proprty dos ot lad to ay itrstig applicatios. I fact: assum that Z livs i th wightd spac lβ, whr β tds to. Lt B β, R) b th ball i lβ of radius R, ad lt ν R b th rstrictio of th Lévy masur ν of Z to B β, R). Th ν R = µ,r, whr µ,r is th rstrictio of µ to [ R/β, R/β ]. Lt m j,r ) b th momt of ordr j of µ,r. Th from Thorm 5.1 w hav th followig. 3

24 Corollary 5.3. If for ach R >, m,r ) γ + m ) 1,R) <, γ ad thr is a ε, 1) such that: m1,r ) + m,r ) + m 4,R ) + γ ε 1)/ m,r ) + γ ε 1 m 4,R ) ) <. Th X is cylidrical càdlàg. Ufortuatly, if Z = σ L ad L ar idpdt α-stabl, th th rmum apparig i th formulatio of th corollary abov is fiit if ad oly if σα <, which holds if ad oly if Z taks valus i l ; s Rmark 3.5. Thrfor th last qustio formulatd at th d th itroductio is op. 6. Proof of Thorm 3.1 Sic U z ρνdz) < ad Z is giv 11), w s that Zt) = Z t) + a + m)t, whr Z t) = zνdz)ds, a := zνdz)ds ρ. ρ { z ρ >1} Sic [, + ) t St s)a + m)ds is cotiuous, w may assum that a + m =. Thus Xt) = St s)dzs) = St s)z πds, dz). W will us th followig fact < ρ 1 < ρ < +, Sic Sh) I) z h U C ρ1,ρ := t ρ ρ 1 St) Lρ1, ρ ) <. 5) <t ASs)z ds 4 h A) 1 ρ Ss) L,) ds z ρ,

25 as a cosquc of 5) w hav Sh) I) z C,ρ h ρ z ρ, ρ >, h <, z ρ. 6) For t h < t < t + h T w hav E Xt + h) Xt) Xt) Xt h) = E ξ 1 + ξ + ξ 3 η 1 + η, whr ξ 1 := ξ := ξ 3 := η 1 := η := h +h t h St + h s) St s)) dzs), St + h s) St s)) dzs), St + h s)dzs), St s) St h s)) dzs), St s)dzs). Thrfor, takig ito accout th iquality w obtai ξ 1, ξ η 1, η 1 ξ1 η 1 + ξ η ) E Xt + h) Xt) Xt) Xt h) 3E [ ξ 1 η 1 + ξ η ] + E ξ1 E η +E ξ E η 1 + E ξ 3 E η 1 + E ξ 3 E η. Assum that κ > is such that κ + ρ < 1/. By 5) ad 6), w hav h E ξ 1 = St + h s) St s)) z νdz)ds h = Sh) I) St s)z νdz)ds h = Sh) I) St s)z νdz)ds 5

26 h κ Similarly ad C,κh κ C κ,ρ h C,κC ρ,κ 1 κ + ρ) T 1 κ+ρ) z ρνdz) =: Cκ)h κ z ρνdz). E ξ Cκ)hκ t s) κ+ρ) ds z ρ νdz) z ρνdz) E η 1 = E St h s + h) St h s)) dzs) Cκ)h κ z ρνdz). Nxt +h h E ξ 3 = St + h s)z νdz)ds = Ss)z νdz)ds t h z ρνdz) Ss) L ρ,)ds C ρ, 1 ρ h1 ρ z ρνdz) =: Bρ)h 1 ρ z ρνdz), ad usig th sam argumts w obtai E η Bρ)h1 ρ z ρνdz). Summig up, for ay κ > such that κ + ρ) < 1 thr is a costat C idpdt of h such that 3 ) E ξ i + η i C h 1 ρ + h κ) z ρνdz). 7) i=1 i=1 W procd to th calculatio of th trms E ξ 1 η 1 ad E ξ η. To do this lt k ) b a orthoormal basis of. By Propositio.4, E ξ 1 η 1 = k,l E ξ 1, k η 1, l = J 1 + J + J 3, 6

27 whr J 1 = h St + h s) St s)) z, j dsνdz) j,k h St s) St h s)) z, k dsνdz) h = St + h s) St s)) z dsνdz) h St s) St h s)) z dsνdz) h = Sh) I) St s)z dsνdz) h Sh) I) St h s)z dsνdz) ) h 4κ C ρ,κ z ν dz) C1)h 4κ h t s) ρ+κ) ds z ν dz) ). h t h s) ρ+κ) ds I th stimat abov, κ > is such that κ + ρ < 1/ ad C1) dpds o T, κ ad ρ. Not that J quals [ h St + h s) St s)) z, St s) St h s)) z dsνdz) is lss tha or qual to [ h Sh) I) St s)z Sh) I) St h s)z dsνdz) ) C)h 4κ z ν dz). Fially J 3 quals h St + h s) St s)) z St s) St h s)) z dsνdz) 7 ] ]

28 is lss tha or qual to h h 4ζ z 4 ε νdz)c ε,ζ t s) ζ ε) t h s) ζ ε) ds C3)h 4ζ z 4 ε νdz), whr ζ > is such that ζ ε < 1/4. To stimat E ξ η w us similar calculatios. Namly whr I 1 = ad [ I = ad I 3 = E ξ η = I 1 + I + I 3, St + h s) St s)) z dsνdz) St s)z dsνdz) ) z ρ νdz) C ρ,κc ρ,h κ+1 ρ 1 t t s) ρ+κ) ds 1 ρ ) z ρ νdz) C ρ,κc ρ,h κ+1 ρ+1 ρ+κ) ρ 1 ρ + κ) ) C4) z ρ νdz) h 4ρ ] St + h s) St s)) z, St s)z dsνdz) ) ) z ρ νdz) C ρ,κc ρ, t s) ρ+κ) t s) ρ ds h κ ) z ρ νdz) C ρ,κc 1 ρ, 1 ρ κ) hκ+1 ρ κ) ) C5) z ρ νdz) h 4ρ St + h s) St s)) z St s)z dsνdz) 8

29 z 4 ε z 4 ε C6) νdz) Ss) L s T ε,)cε,δ t s) δ ε) dsh δ νdz) Ss) L s T ε,)cε,δ 1 1 δ ε) hδ+1 δ ε) z 4 ε νdz)h1+ε. Summig up thr ar costats B ad ε > such that E [ ξ i η i Bh 1+ ε z 4 ε νdz) + i=1 z ρ νdz) ) ] Usig th stimat abov ad 7) w ca fid costats δ > ad R such that E Xt + h) Xt) Xt) Xt h) Rh1+ δ, ad th dsird coclusio follows form Corollary., ad th Bichtlr Jacod typ stimats from [18]. 7. Auxiliary rsult for th proofs of Thorms 5.1 ad 5. I this sctio L t) := R x π ds, dx), = 1,,..., ad λ ) is a squc of strictly positiv ral umbrs. Latr λ = γ z) 1/ε, whr z ) is a fixd squc ad ε >. Obviously L ar idpdt Lévy procsss, ad m j ) = x j ν d), N, j = 1,, 3, 4, whr ν is th Lévy masur of L. Cosidr a squc z ) of ral umbrs. If R z m )λ 1 <, 8). 9

30 th th ral-valud procss Y t) := λt s) d L s)z, 9) is wll dfid, as th sris o th right had sid covrgs i L Ω, F, P), ad E Y t)) <, T <. t T Th followig lmma will play a crucial rol i th proof of Thorms 5.1 ad 5.. Lmma 7.1. Lt ε, 1). Assum that 8) holds ad that th quatity Az, λ, ε) quals ) zm )λ ε 1)/ + z 4 m ) λ ε 1 + m 4 )λ ε 1 + λ ε ) ) is fiit. Th for all t h t t + h <. Proof St Th whr E Y t + h) Y t)) Y t) Y t h)) 6Az, λ, ε)h 1+ε. at, h) := E Y t + h) Y t)) Y t) Y t h)). I 1 := I := I 3 := I 4 := I 5 := at, h) = E I 1 + I + I 3 ) I 4 + I 5 ), +h t h h λt+h s) d L s) z, λ k t+h s) λt s)) d L s) z, λ t+h s) λt s)) d L s) z, λt s) d L s) z, λ t s) λ s)) d L s) z. 3

31 W hav E I 1I 4 + I 5 ) = E I 1 E I 4 + I 5 ) = E I 1 ) E I 4 + E I5, ad E I 1 I + I 3 )I 4 + I 5 ) =, E I + I 3 ) I 4 + I 5 ) = E I I 4 +E I E I 5 +4 E I I 4 E I 3 I 5 +E I 3E I 4 +E I 3I 5. Thus at, h) = ) E I1 E I 4 + E I5 + E I I4 + E I E I5 +4 E I I 4 E I 3 I 5 + E I3E I4 + E I3I 5. Giv δ, 1] dfi Cδ, z) := z m )λ δ 1, Dδ, z) := z 4 m4 ) + m ) ) λ δ. 3) Blow th last iquality i ach stimat follows from th followig lmtary iqualitis ad x >, δ, 1], 1 x x δ, 31) u λ 1 λ x ) x δ for all u, λ >, x >, ad δ, 1]. W hav u λ δ 1, 3) E I 1 = = zm ) +h t λt+h s) ds z m ) λ 1 λ h ) Cδ, z)h δ, E I 4 = zm ) λkt s) ds = 31 z m ) λ 1 λ h ) Cδ, z)h δ,

32 ad E I = = Cδ, z)h δ, zm ) λ t+h s) λt s)) ds z m ) λ 1 λ h ) 1 λh) E I 3 = = zm ) h λ t+h s) λt s)) ds z m ) λ λ h λt) 1 λh) z m ) λ 1 λ h ) 1 Cδ, z)hδ, Clarly E I I 4 E I 5 = = zm ) h λ t s) λ s)) ds z m ) λ 1 λ ) ) 1 λh) z m ) λ k 1 λ h ) 1 Cδ, z)hδ. = z m ) λt s) λt+h s) λt s)) ds, ad E I 3 I 5.What is lft is to fid good stimats for th trms E II 4 ad E I3I 5. W hav E II 4 quals E z k λ k t+h s) ) λ kt s) d L k s)) z j λjt s) d L j s) k=1 E k,k,j,j = λ k t+h s) λ kt s) ) d L k s) 3 j=1 )

33 whr J 1 := k j ad J 3 := = J 1 + J + J 3, = k j λ k t+h s) λ k t s)) d L k s) λjt s) d L j s) λ j t s) d L j s)z k z k z j z j zkz j E λ k t+h s) ) ) λ kt s) d L k s)) λjt s) d L j s) ) E zkz j E λ k t+h s) ) λ kt s) d L k s) λjt s) d L j s)), J := zkz j E λ k t+h s) ) λ kt s) d L k s) λkt s) d L k s) k j λ j t+h s) ) λ jt s) Lj ds) λjt s) d L j s) = zkz j E λ k t+h s) ) λ kt s) d L k s) λkt s) d L k s) k j E W hav J 1 = k j = k j λ j t+h s) λ jt s) ) d L j s) λ jt s) d L j s), z 4 E λ t+h s) λt s)) d L s)) λt s) d L s)). z kz j m k)m j) λ k t+h s) λ kt s) ) ds z k z j m k)m j) 4λ k λ j 1 λ k h ) 1 λ k h ) 1 λ jh ) 1 Cδ, z) h δ, 33 λ jt s) ds

34 ad J = zkz j m k)m j) λ k t+h s) ) λ kt s) λkt s) ds k j λ j t+h s) ) λ jt s) λjt s) ds = zk z j m k)m j) ) 1 λ k h 1 ) λ jh 1 ) λ kh 1 ) λ jh 4λ k λ j k j Cδ, z) h δ. Fially, by Propositio.5, J 3 = J 3,1 + J 3, + J 3,3, whr J 3,1 = = zm 4 ) λ t+h s) λt s)) ) λt s) ds zm 4 ) ) 1 λ h ) 1 λ h 4λ Dδ, z)h δ, J 3, = = zm 4 4 ) λ t+h s) λt s)) λ t s) ds z 4 m 4 ) 4λ 1 λ h ) 1 λ h ), ad J 3,3 = = zm 4 ) λ t+h s) λt s)) ds zm 4 ) ) 1 λ h ) 1 λ h 4λ 1 Dδ, z)hδ. λt s) ds Trm J 3, will b stimatd latr. It is th most difficult trm to valuat as i th domiator w hav λ i th first powr. 34

35 whr W procd to th stimatio of E I 3I 5. W hav h E I3I 5 = E z k λ k t+h s) ) λ kt s) d L k s) k=1 ) h z j λ j t s) ) λ j s) d L j s) j=1 = U 1 + U + U 3, U 1 := k j E z kz j E h h λ k t+h s) ) ) λ kt s) d L k s) λ j t s) ) ) λ j s) d L j s), ) ad U := k j h h W hav U 3 := U 1 = k j z kz j E h λ k t+h s) λ kt s) ) d L k s) λ k t s) λ k s) ) d L k s)e λ j t s) λ j s) ) d L j s), zk 4 E k=1 h h h λ j t+h s) λ jt s) ) d L j s) λ k t+h s) ) ) λ kt s) d L k s) λ k t s) ) ) λ k s) d L k s). z k z j m k)m j) 4λ k λ j λ k t+h) λ kt ) λ k ) 1 ) λ jt λ j) ) λ j ) 1 ) 35

36 k j z k z j m k)m j) 4λ k λ j 1 λ k h ) 1 λ j h ) 1 4 Cδ, z) h δ, U = k j z k z j m k)m j) 4λ k λ j λ k t+h) λ kt ) λ kt λ k) ) λ jt+h) λ jt ) λ jt λ j) ) λ k) 1 ) λ j) 1 ) k j 1 Cδ, z) h δ. z k z j m k)m j) 4λ k λ j 1 λ k h ) 1 λ kh ) 1 λ jh ) 1 λ jh ) Fially, by Propositio.5, U 3 = U 3,1 + U 3, + U 3,3, whr U 3,1 quals h zkm 4 k) k=1 = λ k t+h s) ) λ kt s) λkt s) ) ) λ k s) ds zk 4m k) λ k t+h) ) λ kt λ k t ) λ k) λ k ) 1 ) k=1 4λ k 1 Dδ, z)hδ, U 3, = = zkm 4 4 k) k=1 k=1 k=1 h λ k t+h s) λ kt s) ) λ k t s) λ k s) ) ds z 4 k m 4k) 4λ k λ k t+h) λ kt ) λ k t λ k) ) 4λ k ) 1 ) z 4 k m 4k) 4λ k 1 λ k h ) 1 λ k h ), ad U 3,3 quals h zkm 4 k) λ k t+h s) ) λ kt s) ds k=1 36

37 = h λ k t s) λ k s) ) ds zk 4m k) λ k t+h) ) λ kt λ k t ) λ k) λ k ) 1 ) k=1 4λ k 1 4 Dδ, z)hδ. So far w hav ot stimatd th trms U 3, ad J 3,. owvr w ot that R := z 4 m 4 ) λ 1 λ h ) 1 4λh), domiats U 3, + J 3,. Summig up, w obtai th followig stimat δ, 1] ad t h t t + h T, E Y t + h) Y t)) Y t) Y t h)) ) 6 zm )λ δ 1 + z 4 m4 ) + m ) ) λ δ h δ + R. Put δ = ε + 1)/. I ordr to stimat R ot that z R = m 4 4 )λ δ 1 λh ) λ λ δ 1 ) z 4 m 4 )λ ε Sic for all x, y > ad λ >, w hav 1 4λh ) λ δ 1 λh ) 1 4λh ). λ> λ δ λ δ λ 1 ε 1 λx) 1 4λy) 4 1+ε)/ λ 1 ε λ 1+ε)/ xy) 1+ε)/ R = 1+ε xy) 1+ε)/ 1+ε)/ x + y 1+ε, zm 4 4 )λ ε h 1+ε, which givs th dsird stimat. 37

38 8. Proof of Thorm 5.1 Rmark 8.1. I th proof w caot apply dirctly Lmma 7.1 puttig λ = γ. Idd th quatity Az, γ, ε) apparig i th formulatio of Lmma 7.1 domiats zm 4 4 )γ. ε To apply th Chtsov critrio w d to fid a ε > such that Az, γ, ε) <. z l 1 Thus i particular w would d zm 4 4 )γ ε <. z l 1 This coditio is vr satisfid if th squcs {m 4 )} ad {m )} ar costat ad by cosquc lim γ =, which corrspods to th cas whr Z ) ar idtically distributd. Rcall that th procsss L k wr dfid at th bgiig of th prvious sctio. Lt z l. St Th z whr λ := γ z /ε, N. 33) γt s) dz s) z Y z)t) := rz)t) := z z z 1+/ε l λt s) d L s), λt s) m 1 )ds λt s) dl s) = Y z)t)+rz)t), z 1+/ε m 1 ) + z m 1 )γ 1 m 1 ) + z l 38 z m 1 )λ 1 ) m 1 ) γ ) 1/.

39 Thrfor, thaks to 1) ad ), z l : z l R t [,T ] rz)t) <, R <. 34) Obviously 8) holds. Lt us dot by M th rmum apparig i ). Th th quatity Az, λ, ε) apparig i Lmma 7.1, is domiatd by ) z l M + M z z 1 ε)/ε Thrfor +M + M ) z 4 + z 4 z + z 4 z ) 1 ε) ε. Az, λ, ε) <, R <. z l : z l R By Lmma 7.1, ad Corollary., for ay q [1, 4), ad R >, z l : z l R E t,s [,T ] Y z)t) Y z)s) q C 1 E Y z)t ) q + C, 35) whr C 1 ad C ar costat. By th Bichtlr Jacod iquality for Poisso itgrals i ifiit dimsios s [18]) it follows that thr is a costat C dpdig oly o T, such that Thrfor, E Y z)t ) 4 C m 4 )z 4 λ 1 + C m )z λ 1 ). 36) E Y z)t ) 4 <. 37) z l : z l R Combiig 34) 37) w obtai 3). Th càdlàg proprty follows from th càdlàg proprty of all procsss X k as th sris covrgs uiformly i t [, T ]; s th argumts blow th formulatio of th thorm. 9. Proof of Thorm 5. As i th prvious sctio, λ ) is giv by 33). Usig th argumts from th prvious sctio w s that th proof will b complt as soo as 39

40 w show that thr is a squc of strictly positiv umbrs β ), icrasig to +, such that for ay R >, Az, λ, ε) + ) m 4 )zλ m )zλ 1 < z l 1/β : z l 1/β R ad z l 1/β : z l 1/β R { z 1+/ε m 1 ) + z m 1 )γ 1 ) } + zm )λ 1 <. Lt us dot by MR, β) th scod rmum abov. For ay squc β ), w hav Nxt MR, β) R 1+ ε +R ad fially β m 1 ) γ Az, λ, ε) R 4 +R 4 1 ε 4+ +R ε β ε m 1 ) β m 4 )γ ε 1 β ) 1/ + R ε 1 m 4 )z 4 λ 1 + R 4 +R 4+ ε β m ) γ ε 1 β m )γ + R +ε 1 ε 4+ + R ε + β m ) γ ε 1 m ) + m 4 ) ), β m 4 )γ 1 β 1 ε + m )z λ 1 + R 4 ε 1 ε β 1+ε m ). β m ) ) + R β m ) ) β m ) γ m 4 ) + R 4+ 4 ε β + ε m ). By dirct calculatios ad assumptio 4) w arriv at th statmt of th thorm. 4

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