On Classical Soluions of Linear Sochasic Inegro-Differenial Equaions November 27, 204 arxiv:404.0345v2 [mah.pr] 26 Nov 204 James-Michael Leahy The Universiy of Edinburgh, E-mail: J.Leahy-2@sms.ed.ac.uk Remigijus Mikulevičius The Universiy of Souhern California, E-mail: mikulvcs@mah.usc.edu Absrac We prove he exisence of classical soluions o parabolic linear sochasic inegrodifferenial equaions wih adaped coefficiens using Feynman-Kac ransformaions, condiioning, and he inerlacing of space-inverses of sochasic flows associaed wih he equaions. The equaions are forward and he derivaion of exisence does no use he general heory of SPDEs. Uniqueness is proved in he class of classical soluions wih polynomial growh. Conens Inroducion 2 2 Ouline of main resuls 4 3 Proof of main heorems 3. Proof of uniqueness for Theorem 2.2 3.2 Small jump case 5 3.3 Adding free and zero-order erms 9 3.4 Adding uncorrelaed par Proof of Theorem 2.2) 24 3.5 Inerlacing a sequence of large jumps Proof of Theorem 2.5) 26 4 Appendix 29 4. Maringale and poin measure measure momen esimaes 29 4.2 Opional projecion 3 4.3 Esimaes of Hölder coninuous funcions 3 4.4 Sochasic Fubini hoerem 4 4.5 Iô-Wenzell formula 43
Inroducion 2 Inroducion Le Ω,F,P) be a complee filered probabiliy space andf 0 be a sub-sigma-algebra of F. We assume ha his probabiliy space suppors a sequence w ; ), 0, N, of independen one-dimensional Wiener processes and a Poisson random measure p d, dz) on R +,BR + ) wih inensiy measureπ dz)d, where,,π ) is a sigma-finie measure space. We also assume ha w ; ) and p d, dz) are independen off 0. Le F=F ) 0 be he sandard augmenaion of he filraion F ) 0, where for each 0, F =σ F 0, w s), p [0, s],γ) : s,γ ). For each real number T> 0, we ler T,O T, andp T be he F-progressive, F-opional, and F-predicable sigma-algebra onω [0, T], respecively. Denoe by q d, dz)= p d, dz) π dz)d he compensaed Poisson random measure. Le D, E, V be disjoin - measurable subses such ha D E V = andπv )<. Le 2, 2,π 2 ) be a sigma-finie measure space and D 2, E 2 2 be disjoin 2 -measurable subses such ha D 2 E 2 = 2. Fix an arbirary posiive real number T> 0 and inegers d, d 2. Leα 0, 2] and leτ T be a sopping ime. LeF τ be he sopping ime sigma-algebra associaed wihτ and leϕ :Ω R d R d 2 bef τ BR d )-measurable. We consider he sysem of sochasic inegro-differenial equaions on [0, T] R d given by du l = L ;l + +L 2;l )u + [,2] α)b i i u l + c l l u l + f l ) d+ N ;l ) u + g l dw ; I ;l,zu + h l z)) [ D z)q d, dz)+ E V z)p d, dz)], τ T, u l =ϕl, τ, l {,..., d 2 },.) where forφ C c R d ), k {, 2}, and l {,..., d 2 }, and L k;l N ;l φx) := {2} α) σk;i k; j x)σ x) i j φ l x)+ {2} α)σ k;i 2 + ρ k;l l x, z) φ l x+h k x, z)) φ l x) ) π k dz) D k + φ l x+h k x, z)) φ l x),2] α)h k;i x, z) i φ l x) ) π k dz) D k + {2} k) I l l d 2 +ρ 2;l l x, z))φ l x+h 2 x, z)) φl x) ) π 2 dz), E 2 φx) := {2} α)σ ;i I ;l,zφx) :=I l l d 2 +ρ ;l l x) i φ l x)+υ ;l l x)φ l x),, x, z))φ l x+h x, z)) φl x), x)υ k;l l x) φ l i x) H k x, z) α + ρ k x, z) 2) π k dz)+ H k x, z) α + ρ k x, z) ) π k dz)<. D k E k
Inroducion 3 The summaion convenion wih respec o repeaed indices i, j {,..., d }, l {,..., d 2 }, and N is used here and below. The d 2 d 2 dimensional ideniy marix is denoed by I d2. For a subse A of a larger se X, A denoes he{0, }-valued funcion aking he value on he se A and 0 on he complemen of A. We assume ha for each k {, 2}, σ k x)=σk;i ω, x)) i d,, b x)=b i ω, x)) i d, c x)=c l l ω, x)) l, l d 2, υ k x)=υk;l l ω, x)) l, l d 2,, f x)= f i ω, x)) i d 2, g x)=g i ω, x)) i d2,, are random fields onω [0, T] R d ha arer T BR d )-measurable. Moreover, for each k {, 2}, we assume ha H k x, z)=hk;i ω, x, z)) i d,ρ k x, z)=ρk;l l ω, x, z)) l, l d 2, are random fields onω [0, T] R d k ha arep T BR d ) k -measurable. Moreover, we assume ha h x, z)=h i ω, x, z)) i d 2, is a random field onω [0, T] R d ha isp T BR d )-measurable Sysems of linear sochasic inegro-differenial equaions appear in many conexs. They may be considered as exensions of boh firs-order symmeric hyperbolic sysems and linear fracional advecion-diffusion equaions. The equaion.) also arises in non-linear filering of semimaringales as he equaion for he unormalized filer of he signal see, e.g., [Gri76] and [GM]). Moreover,.) is inimaely relaed o linear ransformaions of inverse flows of jump SDEs and i is precisely his connecion ha we will exploi o obain soluions. There are various echniques available o derive he exisence and uniqueness of classical soluions of linear parabolic SPDEs and SIDEs. One approach is o develop a heory of weak soluions for he equaions e.g. variaional, mild soluion, or ec...) and hen sudy furher regulariy in classical funcion spaces via an embedding heorem. We refer he reader o [Par72, Par75, MP76, KR77, Tin77, Gyö82, Wal86, DP92, Kry99, CK0, P07, Hau05, R07, BvNVW08, HØU0, LM4a] for more informaion abou weak soluions of SPDEs driven by coninuous and disconinuous maringales and maringale measures. This approach is especially imporan in he non-degenerae seing where some smoohing occurs and has he obvious advanage ha i is broader in scope. Anoher approach is o regard he soluion as a funcion wih values in a probabiliy space and use he mehod deerminisic PDEs i.e. Schauder esimaes, see, e.g. [Mik00, MP09]). A hird approach is a direc one ha uses soluions of sochasic differenial equaions. The direc mehod allows o obain classical soluions in he enire Hölder scale while no resricing o ineger derivaive assumpions for he coefficiens and daa. In his paper, we derive he exisence of a classical soluions of.) wih regular coefficiens using a Feynman-Kac-ype ransformaion and he inerlacing of he space-inverse firs inegrals [KR8]) of a sochasic flow associaed wih he equaion. The consrucion of he soluion gives an insigh ino he srucure of he soluion as well. We prove ha he soluion of.) is unique in he class of classical soluions wih polynomial growh
Ouline of main resuls 4 i.e. weighed Hölder spaces). As an immediae corollary of our main resul, we obain he exisence and uniqueness of classical soluions of linear inegro-differenial equaions wih random coefficiens, since he coefficiensσ, H, a,ρ, and free erms g and h can be zero. Our work here direcly exends he mehod of characerisics for deerminisic firs-order parial differenial equaions and he well-known Feynman-Kac formula for deerminisic second-order PDEs. In he coninuous case i.e. H 0, H 2 0, h 0), he classical soluion of.) was consruced in [KR8, Kun8, Kun86, Roz90] see references herein as well) using he firs inegrals of he associaed backward SDE. This mehod was also used o obain classical soluions of.) in [DPMT07]. In he references above, he forward Liouville equaion for he firs inegrals of associaed sochasic flow was derived direcly. However, since he backward equaion involves a ime reversal, he coefficiens and inpu funcions are assumed o be non-random. The generalized soluions of.) wih d 2 =, non-random coefficiens, non-degenerae diffusion, and finie measuresπ =π 2 were discussed in [MB07]. In his paper, we give a direc derivaion of.) and all he equaions considered are forward, possibly degenerae, and he coefficiens and inpu funcions are adaped. For oher ineresing and relaed developmens, we refer he reader o [Pri2, ha3, Pri4]. This paper is organized as follows. In Secion 2, our noaion is se forh and he main resuls are saed. In Secion 3, he main heorem is proved and is divided ino a proof of uniqueness and exisence. In Secion 4, he appendix, auxiliary facs ha are used hroughou he paper are discussed. 2 Ouline of main resuls For each ineger n, le R n be he space of d-dimensional Euclidean poins x=x,..., x n ). For each x, denoe by x he Euclidean norm of x. Le R + denoe he se of non-negaive real-numbers. Le N be he se of naural numbers. Elemens of R d and R d 2 are undersood as column vecors and elemens of R 2d and R 2d 2 are undersood as marices of dimension d d and d 2 d 2, respecively. For each ineger n, he norm of an elemen x ofl 2 R n ), he space of square-summable R n -valued sequences, is denoed by x. For a opological space X,X) we denoe he Borel sigma-field on X bybx). For each i {,..., d }, le i = x i be he spaial derivaive operaor wih respec o x i and wrie i j = i j for each i, j {,..., d }. For a once differeniable funcion f = f..., f d ) : R d R d, we denoe he gradien of f by f= j f i ) i, j d. Similarly, for a once differeniable funcion f= f,..., f d ) : R d l 2 R d ), we denoe he gradien of f by f = j f i ) i, j d, and undersand i as a funcion from R d ol 2 R 2d ). For a muli-indexγ=γ,...,γ d ) {0,, 2,...,} d of lengh γ :=γ + +γ d, denoe by γ he operaor γ = γ γ d d, where 0 i is he ideniy operaor for all i {,..., d }. For each ineger d, we denoe by C c Rd ; R d ) he space of infiniely differeniable funcions wih compac suppor in R d. For a Banach space V wih norm V, domain Q of R d, and coninuous funcion f : Q
Ouline of main resuls 5 V, we define f 0;Q;V = sup f x) x Q and f x) f y) V [ f ] β;q;v = sup, β 0, ]. x,y Q,x y x y β V For each real numberβ R, we wrieβ=[β] +{β} +, and{β} + 0, ]. For a Banach space V wih norm V, real numberβ>0, and domain Q of R d, we denoe byc β Q; V) he Banach space of all bounded coninuous funcions f : Q V having finie norm f β;q;v := γ f 0;Q;V + [ γ f ] {β} + ;Q;V. γ [β] γ =[β] When Q=R d and V= R n or V=l 2 R n ) for any ineger n, we drop he subscrips Q and V from he norm β;q;v and wrie β. For a Banach space V and for eachβ>0, denoe byc β loc Rd ; V) he Fréche space of coninuous funcions f : R d V saisfying f C β Q; V) for all bounded domains Q R d. We call a funcion f : R d R d a C β loc Rd ; R d )-diffeomorphism if f is a homeomorphism and boh f and is inverse f are inc β loc Rd ; R d ). For a Fréche spaceχ, we denoe by D[0, T];χ) he space ofχ-valued càdlàg funcions on [0, T]. Unless oherwise specified, we endow D[0, T]; χ) wih he supremum seminorms. The noaion N= N,, ) is used o denoe a posiive consan depending only on he quaniies appearing in he parenheses. In a given conex, he same leer is ofen used o denoe differen consans depending on he same parameer. If we do no specify o which space he parameersω,, x, y, z and n belong, hen we meanω Ω, [0, T], x, y R d, z k, and n N. Le r x) := + x 2, x R d. Le us inroduce some regulariy condiions on he coefficiens and free erms. We consider hese assumpions for β > α and β > α. Assumpion 2. β). ) There is a consan N 0 > 0 such ha for each k {, 2} and all ω, Ω [0, T], r b 0 + b β + r σk 0 + σ k β N 0. Moreover, for each k {, 2} and all ω,, z) Ω [0, T] D k E k ), r Hk z) 0 K k z) and Hk z) β K k z) where K k, K k :Ω [0, T] D k E k ) R + arep T k -measurable funcions saisfying K k z)+ K D k z)+ K k z) α + K k z)2) π k dz)+ K k z) α + K k z)) π k dz) N 0, k E k for all ω,, z) Ω [0, T] D k E k ).
Ouline of main resuls 6 2) For each k {, 2}, here is a consanη k 0, ) such ha for all ω,, x, z) {ω,, x, z) Ω [0, T] R d D k E k ) : H kω, x, z) >ηk }, Id + H k x, z)) N 0. Assumpion 2.2 β). There is a consan N 0 > 0 such ha for each k {, 2} and all ω, ) Ω [0, T], c β+ υ k β+ r θ f β+ r θ g β N 0. Moreover, for each k {, 2} and all ω,, z) Ω [0, T] D k E k ), ρ k z) β l k z), r θ h z) β l k z), where l k :Ω [0, T] k R + arep T k -measurable funcion saisfying l k D z)+ l k z)2 π k dz)+ l k z)πk dz) N 0, k E k for all ω,, z) Ω [0, T] D k E k ). Remark 2.. I follows from Lemma 4.0 and Remark 4. ha if Assumpion 2. β) holds for some β> α, hen for allω,, and z D k E k, x H k x, z) := x+h k x, z) is a diffeomorphism. Le Assumpions 2. β) and 2.2 β) hold for some β > α and β > α. In our derivaion of a soluions of.), we firs obain soluions of equaions of a special form. Specifically, consider he sysem of SIDEs on [0, T] R d given by dû l = ) ) L ;l +L 2;l )û + ˆb i i u l + ĉ l l u l + fˆ l d+ N ;l û + g l dw ; + I ;l,zû + h l z)) [ D z)q d, dz)+ E z)p d, dz)], τ< T, û l =ϕl, τ, l {,..., d 2 }, 2.) where ˆb i x) := [,2] α)b i x)+ + k= ĉ l l x) :=c l l x)+ + 2 {2} α)σ k= 2,2] α) H k;i D k 2 k; j {2} α)σ 2 k= k= fˆ l x) := f l x)+σ ρ k;l l D k ; j x, z) ρ k;l l x) j g l x)+ k; j x, z) H k;i x) j υ k;l l x) x) j σ k;i x) H k; x, z), z) ) π k dz), H k; x, z), z))π k dz), h l x, z) h l H ; x, z), z) ) π dz). D
Ouline of main resuls 7 Le w 2; ), 0, N, be a sequence of independen one-dimensional Wiener processes. Le p 2 d, dz) be a Poisson random measure on [0, ) 2,B[0, ) 2 ) wih inensiy measureπ 2 dz)d. Exending he probabiliy space if necessary, we ake w 2 and p 2 d, dz) o be independen of w and p d, dz). Le Fˆ =σ w 2 s ), p 2 [0, s],γ) : s,γ 2) and F = F ) T be he sandard augmenaion of F ˆ F ) T. Denoe by q2 d, dz) = p 2 d, dz) π 2 dz)d he compensaed Poisson random measure. We associae wih he SIDE 2.), he F-adaped sochasic flow X = X x)=x τ, x),, x) [0, T] R d, generaed by he SDE dx = [,2] α)b X )d+ 2 k= 2 k= H k H k; D k H k H k; E k 2 k= {2} α)σ k; X )dw k; X, z), z)[p k d, dz),2] α)π k dz)d] X, z), z)p k d, dz), τ< T, X = x, τ, 2.2) and he F-adaped random fieldφ x)=φ τ, x),, x) [0, T] R d, solving he linear SDE given by dφ x)=c X x))φ x)+ f X x))) d+ + 2 k= + ρ k H k; k h H ; Φ x)=ϕx), τ. 2 k= υ k; X x))φ x)dw k; + g X x))dw ; X x), z), z)φ x)[ D kz)q k d, dz)+ E kz)p k d, dz)] X x), z), z)[ D z)q d, dz)+ E z)p d, dz)], τ< T, The coming heorem is our exisence, uniqueness, and represenaion heorem for 2.). Le us describe our soluion class. For eachβ 0, ), denoe byc β R d ) he linear space of all F-adaped random fields v=v x) such ha P-a.s. [τn,τ n+ )r λ n v D[0, T];C β R d, R d 2 )), where τ n ) n 0 is an increasing sequence of F-sopping imes wihτ 0 = 0 andτ n = T for sufficienly large n, and where for each n,λ n is a posiivef τn -measurable random variable.
Ouline of main resuls 8 Theorem 2.2. Le Assumpions 2. β) and 2.2 β) hold for some β> α and β>α. For each sopping imeτ T andf τ BR d )-measurable random fieldϕsuch ha for some β α, β β) andθ 0, P-a.s. r θ ϕ C β R d ), here exiss a unique soluion û=ûτ) of 2.) inc β R d ) and for all, x) [0, T] R d, P-a.s. û τ, x)=e [ Φ τ, X τ, x)) F ]. 2.3) Moreover, for eachǫ> 0 and p 2, [ ] E sup r θ θ ǫ û τ) p Fτ β N r θ ϕ p β + ), 2.4) T for a consan N= Nd, d 2, p, N 0, T,β,η,η 2,ǫ,θ,θ ). Using Iô s formula i is easy o check ha if m= and g x)=0, h x)=0, and ρ k x, z), for all ω,, x, z) Ω [[τ, T]] R d D k E k ), k {, 2}, hen Φ x)=ψ x)φx)+ψ x) Ψ s x) f sx s x))ds, where P-a.s. for all and x, Ψ x)=e [τ,τ ] cs X s x)) 2 k= 2 υ k; s e 2 k= ]τ,τ ] e 2 k= ]τ,τ ] X s x))υ k; s ]τ,τ ] X s x)) ) ds+ 2 υk; k= ]τ,τ ] s X s x))dw k; s D k ln +ρ k s H s k; X s x),z),z) ) ρ k s H s k; X s x),z),z) ) π k dz)ds k ln +ρ k s H k; s X s x),z),z) ) [ D k z)q k ds,dz)+ E k z)p k ds,dz)]. 2.5) The following corollary hen follows direcly from 2.3) and he 2.5). Corollary 2.3. Le m=and assume ha g x)=0, h x, z)=0, ρ k x, z), ω,, x, z) [[τ, T]] Rd D k E k ), k {, 2}. Moreover, le Assumpions 2. β) and 2.2 β) hold for some β > α and β > α. Le τ T be sopping ime andϕbe af τ BR d )-measurable random field such ha for some β α, β β) andθ 0, P-a.s. r θ ϕ C β R d ). ) If for all ω,, x) [[τ, T]] R d, f x) 0 andϕx) 0, hen he soluion û of.) saisfies û x) 0, P-a.s. for all, x) [0, T] R d. 2) If for all ω,, x, z) [[τ, T]] R d D k E k ), k {, 2},υ k x)=0, f x) 0, c x) 0, ϕx), andρ k x, z) 0, hen he soluion û of.) saisfies û x), P-a.s. for all, x) [0, T] R d.
Ouline of main resuls 9 Remark 2.4. SinceL 2 can be he zero operaor, boh Theorem 2.2 and Corollary 2.3 apply o fully degenerae equaions and parial differenial equaions wih random coefficiens. Now, le us discuss our exisence and uniqueness heorem for.). We consruc he soluion of u=uτ) of.) by inerlacing he soluions of 2.) along a sequence of large jump momens see Secion 3.5). By using an inerlacing procedure we are also able o drop he condiion of boundedness of I+ H x, z)) on he se ω,, x, z) {ω,, x, z) Ω [0, T] R d D E ) : H ω, x, z) >η k }. Also, in order o remove he erms in ˆb, ĉ, and fˆ ha appear in 2.), bu no in.), we subrac erms from he relevan coefficiens in he flow and he ransformaion. However, in order o do his, we need o impose sronger regulariy assumpions on some of he coefficiens and free erms. We will inroduce he parameersµ,µ 2,δ,δ 2 [0, α ], which essenially allows one o rade-off inegrabiliy in z 2 and regulariy in x of he coefficiens H k x, z),ρ k x, z), h k x, z). I is worh menioning ha he removal of erms and he inerlacing procedure are independen of each oher and ha i is due only o he weak assumpions on H andρ on he se V ha we do no have momen esimaes and a simple represenaion propery like 2.4) for he soluion of.). Neverheless, here is a represenaion of sors and we refer he reader o he proof of he coming heorem for an explici consrucion of he soluion. We inroduce he following assumpion for β> α, β>α, andδ,δ 2,µ,µ 2 [0, α]. 2 Assumpion 2.3 β,µ,µ 2,δ,δ 2 ). ) There is a consan N 0 {, 2} and all ω, ) Ω [0, T], > 0 such ha for each k r b 0 + b β + σ k β+ N 0. 2) For each k {, 2} and all ω, ) Ω [0, T], H k z) 0 K k z), H k z) β, z D k, r Hk z) 0 K k z), Hk z) β K k z), z Ek, ρ k, z) β l k z), z Dk, r θ h z) β l z), z D, where K k, K k, l k :Ω [0, T] D k E k ) R + arep T k -measurable funcions saisfying for all ω,, z) Ω [0, T] D k E k ), K k z)+ K k z)+l k z) N 0 and K k z) α + K k z)2 + l k z)2) π k dz)+ K k z) α + K k z)) π k dz) N 0. D k E k
Ouline of main resuls 0 3) For each k {, 2} and all ω, ) Ω [0, T], υ k β+ N 0, ifσ k 0, g β+ N 0, ifσ 0, γ H k z)) { β} + +δ k K k z), z Dk, if{ β} + +δ k, γ =[ β] γ H k z) 0 K k z), γ H k z)) { β} + +δ k K k z), z D k, if{ β} + +δ k >, γ =[ β] γ =[ β] γ ρ k z)) { β} + +µ k l k z), z Dk, if{ β} + +µ k, γ =[ β] γ ρ k z) 0 l k z), γ ρ k z)) { β} + +µ k l k z), z D k, if{ β} + +µ k >, γ =[ β] γ =[ β] γ h z)) { β} + +µ l z), z D, if{ β} + +µ, γ =[ β] γ h z) 0 l z), γ h z)) { β} + +µ l z), z D, if{ β} + +µ >, γ =[ β] γ =[ β] where K k, l k :Ω [0, T] D k R + arep T k -measurable funcions saisfying for all ω,, z) Ω [0, T] D k, K k z)+ l k z)+ K k z) α α δ k [0, α 2 ]δ k )+ K k z) 2 + l k z) α α µ k [0, α 2 ]µ k )+ l k z) 2) π k dz) N 0. D k 4) There is a consanη 2 0, ) such ha for all ω,, x, z) {ω,, x, z) Ω [0, T] R d 2 : H 2ω, x, z) >η2 }, Id + H 2 x, z)) N 0. Assumpion 2.4 β). ) There is a consan N 0 > 0 such ha for each k {, 2} and all ω, ) Ω [0, T], c β+ r θ f β N 0, υ k β N 0, ifσ k = 0, g β N 0, ifσ = 0, ρ k, z) β l k z), z Ek, r θ h z) β l z), z E, where for all ω, ) Ω [0, T], E k l k z)π k dz) N 0. 2) There exis processesξ,ζ :Ω [0, T] V R + ha arep T measurable saisfying r ξ z) H z) β + r ξ z) ρ z) β+ r ξ z) h z) β ζ z), for all ω,, z) Ω [0, T] V. We now sae our exisence and uniqueness heorem for.). Theorem 2.5. Le Assumpions 2.3 β,δ,δ 2,µ,µ 2 ) and 2.4 β) hold for some β> α, β>α, andδ,δ 2,µ,µ 2 [0, α ]. For each sopping imeτ T andf 2 τ BR d )-measurable random fieldϕsuch ha for someβ α, β β) andθ 0, P-a.s. r θ ϕ C β R d ), here exiss a unique soluion u=uτ) of.) inc β R d ).
Proof of main heorems 3 Proof of main heorems We will firs prove uniqueness of he soluion of 2.) in he classc β R d ). The exisence par of he proof of Theorem 2.2 is divided ino a series of seps. In he firs sep, by appealing o he represenaion heorem we derived for soluions of coninuous SPDEs in Theorem 2.4 in [LM4b], we use an inerlacing procedure and he srong limi heorem given in Theorem 2.3 in [LM4b] o show ha he space inverse of he flow generaed by a jump SDE i.e. he SDE 2.2) wihou he uncorrelaed noise) solves a degenerae linear SIDE. Then we linearly ransform he inverse flow of a jump SDE o obain soluions of degenerae linear SIDEs wih free and zero-order erms and an iniial condiion. In he las sep of he proof of Theorem 2.2, we inroduce an independen Wiener process and Poisson random measure as explained above, apply he resuls we know for fully degenerae equaions, and hen ake he opional projecion of he equaion. In he las secion, Secion 3.4, we prove Theorem 2.5 using an inerlacing procedure and removing he exra erms in ˆb, ĉ and f ˆ. The uniqueness of he soluion u of.) follows direcly from our consrucion. 3. Proof of uniqueness for Theorem 2.2 Proof of Uniqueness for Theorem 2.2. Fix a sopping imeτ T andf τ BR d )-measurable random fieldϕsuch ha for someβ α, β β) andθ 0, P-a.s. r θ ϕ C β R d ). In his secion we will drop he dependence of processes, x, and z when we feel i will no obscure he argumen. Le û τ) and û 2 τ) be soluions of 2.) inc β. I follows ha v := û τ) û 2 τ) solves and P-a.s. dv l = [L;l + +L 2;l )v + ˆb i iv l + ĉl l v l ]d+n ; v l dw; I ;l,zv [ D z)q d, dz)+ E z)p d, dz)], τ< T, v l = 0, τ, l {,..., d 2}, [τn,τ n+ )r λ n v D[0, T];C β R d, R d 2 )), where τ n ) n 0 is an increasing sequence of F-sopping imes wihτ 0 = 0 andτ n = T for sufficienly large n, and where for each n,λ n is a posiivef τn -measurable random variable. Clearly i suffices o akeτ =τ andλ 0 = 0. Thus, v x)=0 for all ω, ) [[τ 0,τ )). Assume ha for some n, P-a.s. for all and x, v τn x)=0. We will show ha P-a.s. for all and x, ṽ x) := v τn ) τ n+ x)=0. Applying Iô s formula, for each x, P-a.s. for all, we find d ṽ 2 = 2ṽ l L;l ṽ + N ṽ 2 + 2ṽ l bi iṽ l + ) 2ṽl cl l ṽ l d ) + 2ṽ l I;l,z ṽ + I ;l,zṽ 2 π dz) d D E + ) 2v l L2;l ṽ + 2ṽ l I2;l,zṽ d+2v l N ; ṽ l dw;
3. Proof of uniqueness for Theorem 2.2 2 + 2ṽ l I ;l,zṽ + I ;l,zṽ 2) q d, dz), τ n < τ n+, ṽ 2 = 0, τ n, l {,..., d 2 }, 3.) where forφ C c Rd ), k {, 2}, and l {,..., d 2 }, and L k;l φ := 2 σk;i σ k; j i j φ l +σ k; j j σ k;i i φ l +σ k;i υ k;l l i φ l +σ k; j j a k;l l φ l I k;l φ := ρ k;l l φ l H k ) ρ k;l l H ) k; )φ l π k dz) D k + φ l H k ) φ l +,2] α)f k;i i φ l) π k dz) D k + I l l d 2 +ρ k;l l )φ l H k ) φ l) π k dz). E k For eachωand, le Q = ṽ x) 2 r λ x)dx, R d whereλ=λ n + d + 2)/2 and d > d. Noe ha EQ r d x)dxe r λ n ṽ 0 <. R d I suffices o show ha sup T EQ = 0. To his end, we will muliply he equaion 3.) by he weigh r 2λ = r 2λ n+ r d, inegrae in x, and change he order of he inegrals in ime and space. Thus, we mus verify he assumpions of sochasic Fubini heorem hold see Corollary 4.3 and Remark 4.4 as well) wih he finie measureµdx)=r d x)dx on R d. Since b andσ k have linear growh anυ k and c are bounded, owing o Lemma 4.6, we easily obain ha here is a consan N= Nd, d 2, N 0,λ n ) such ha P-a.s for all, 2 2 r λ n ṽ r λ n 2 L k ṽ + r λ n N ṽ 2 r d dx N sup r λ n ṽ 2 β, and R d R d k= R d 4 r λ n ṽ 2 r λ n N ṽ 2 r d dx N sup 2 r λ n ṽ r λ n b i ṽ +2 r λ n ṽ r λ n cṽ ) r d T T r λ n ṽ 4 β, dx N sup T r λ n ṽ 2 β.
3. Proof of uniqueness for Theorem 2.2 3 For allφ C α loc Rd ) and all k,ω,, x, p, and z, r p φ H k ) φ+,2] α)f k;i i φ) = φ H k ) φ,2] α)h k;i i φ+,2] α)h k;i + F k;i ) i φ +p,2] α)h k;i + F k;i )r 2 xi φ+ k= +,2] α) rp H k ) r p rp H k ) r p φ H k ),2] α) φ) + ph k;i r 2 xi φ, 3.2) where φ := r p φ. By Taylor s formula, for allφ C α R d ) and all k,ω,, x, and z, we have φ H k ) φ,2] α)h k;i i φ r α φ α r H α 0. 3.3) Combining 3.2), 3.3), and he esimaes given in Lemma 4.0 ), for all k,ω,, x and z, we obain r α ρk H k; ) ρ k N ρ α r Hk α 0 and r λ n α ṽ H k ) ṽ+,2] α)f k;i i ṽ N r λ n ṽ α r Hk α 0 + r H ) 0[H k ] + r + H [α] 0 + [H] [α] +, 3.4) for some consan N= Nd,λ n, N 0,η,η 2 ). Therefore, P-a.s for all, 2 2 r λ n R d ṽ r λ n 2 I k ṽ + r λ I z ṽ 2 π dz) r d dx N sup r λ n ṽ 2 β, D E T and R d 2 r λ n ṽ r λ n 2 I k zṽ + r λ n I z ṽ 2) 2 r d dx N sup T r λ n ṽ 4 β, for some consan N= Nd, d 2,λ n, N 0,η,η 2 ). Le L 2 R d ) be he space of square-inegrable funcions f : R d R d 2 wih norm 0 and inner produc, ) 0. Moreover, le L 2 R d ;l 2 R d 2 )) be he space of square-inegrable funcions f : R d l 2 R d 2 ) wih norm 0. Wih he help of he above esimaes and Corollary 4.3, denoing v=r λ ṽ, P-a.s. for all, we have ) d v 2 0 2 v = l, L v ) 0 + N v 2 0 + 2 v,ī,z v ) 0 + Ī,z v 2 0 π dz) d D E + 2ṽ, b i iṽ + c l ṽ l ) 0 + 2ṽ, L 2 ṽ) ) 0 + 2ṽ,Ī 2,zṽ) 0 d+2v, N ; ṽ ) 0 dw ; ) + 2ṽ,Ī,zṽ ) 0 + Ī,zṽ 2 0 q d, dz), τ n < τ n+, v 2 0 = 0, τ n, l {,..., d 2 }, 3.5) where all coefficiens and operaors are defined as in 2.) wih he following changes:
3. Proof of uniqueness for Theorem 2.2 4 ) for each k {, 2},υ k is replaced wih 2) for each k {, 2},ρ k replaced wih k= D k ῡ k;l l :=υ k;l l 2 + {2} α)λσ k;i r xi δ l l; ρ k;l l :=ρ k;l l + rλ H k ) r λ I l l d 2 +ρ k;l l ); 3) c is replaced wih 2 c l l = c l l +λb i r 2 x i δ l l+ λ 2 k; 4 σ k;i σ j r xi x j k= 2 r λ + r λ H k; ) Im+ρ l l k H k; )),2] α)λr 2 x ih k;i H k; ) π k dz). Since for all k,ω and, r σk 0 + r σk β + υ k β N 0, for β> α and β>α, i is clear ha ῡ k α N. Moreover, since for all k,ω and, r Hk 0 + H k β K k and ρ β lk, applying he esimaes in Lemma 4.0) ), we ge ρ k α l k + K k +l k ) and c α N 0. We will now esimae he drif erms of 3.5) in erms of v 2 0. We wrie f g if f x) dx R d = gx) dx and f g if f x) dx gx) dx. Using he divergence heorem, for R d R d R d any v : R d R d 2,σ : R d R d andυ:r d R 2d 2 and all x, we ge σ i σ j v l v l i j 2 σi σ j ) i j v σ i σ j v l i vl j = σi i j σ j +σ i j σ j i ) v 2 σ i σ j v l i vl j, and 2σ i j σ j v l v l i σi j σ j ) i v 2 = σ i i j σ j +σ i j σ j i ) v 2, σ i v l υ l l v l i +σi v l υ l l v l i =σi v l υ l l sym v l i σi υ l l sym ) i v 2 = σ i i υl l sym +σ i υ l l sym ) v 2, whereυ l l sym = υ l l +υ ll )/2. Consequenly, for allω,, and x, we have and 2 v l L ;l v+ N v 2 2 divσ 2 i σ ; j j σ ;i ) v 2 ῡ ;l l sym v l v l divσ ; + ῡ v 2 N v 2 2 v l L 2);l v +ǫ) σ 2;i i v 2 + N v 2, for anyǫ> 0, where in he las esimae we have also used Young s inequaliy. By Lemma 4.0 2) and basic properies of he deerminan, here is a consan N= Nd, N 0,η,η 2 ) such ha for all k,ω,, x, and z, de H k; =dei d + F k ) F k N H k
3.2 Small jump case 5 and de H k; div F k F k 2 N H k 2. Thus, inegraing by pars, for allω,, and x, we ge 2 v l Ī ;l v+ Ī v 2 π dz) 2 ρ ;l l sym H ; )de H ; )π dz) v l v l D E D + de H ; +,2] α) D div F ) π dz) v 2 D E + E 2 ρ ;l l sym H ; ) v l v l + ρ H ; ) v 2) de H ; π dz) D E N K z) 2 + l z)k z)+l z) 2) π dz)+ K k z)+l k z) ) ) π dz) v 2. D E Analogously, for allω,, and x, we obain 2 v l Ī 2;l v +ǫ) v H 2 ) v 2 π 2 dz)+n v 2. D 2 E 2 Therefore, combining he above esimaes, P-a.s. for all, Q N 0 Q s ds+ M, 3.6) where M ) T is a càdlàg square-inegrable maringale. Taking he expecaion of 3.6) and applying Gronwall s lemma, we ge sup T EQ = 0, which implies ha P-a.s. for all and x, ṽ x)=0. This complees he proof. 3.2 Small jump case Se w ) = w ; ),,,π) =,,π ), pd, dz) = p d, dz), and qd, dz) = q d, dz). Leσ x)=σ i x)) i d, be al 2 R d )-valuedr T BR d )-measurable funcion defined onω [0, T] R d and H x, z)=h ix, z)) i d be ap T BR d ) -measurable funcion defined onω [0, T] R d. We inroduce he following assumpion for β > α. Assumpion 3. β). ) There is a consan N 0 > 0 such ha for all ω, ) Ω [0, T], Moreover, for all ω,, z) Ω [0, T], r b 0 + r σ 0 + b β + σ β N 0. r H z) 0 K z) and H z) β K z), where K :Ω [0, T] R + is ap T -measurable funcion saisfying K z)+ K z)+ K z) α + K z) 2) πdz) N 0, for all ω,, z) Ω [0, T].
3.2 Small jump case 6 2) There is a consanη 0, ) such ha for all ω,, x, z) {ω,, x, z) Ω [0, T] R d : H ω, x, z) >η}, I d + H x, z) ) N 0. Le Assumpion 3.β) hold for some β > α. Le τ T be a sopping ime. Consider he sysem of SIDEs on [0, T] R d given by dv x)= {2} α) ) σi x)σ j x) i j v x)+b i 2 x) iv x) d+ {2} α)σ i x) i v x)dw +,2] α) v x+h x, z)) v x)+f x, z) i v x))πdz)d + v x+h x, z)) v x)) [,2] α)qd, dz)+ [0,] α)pd, dz)], τ< T, where and v x)= x, τ, b i x) := [,2] α)b i x)+ {2} α)σ j x) j σ i x) F x, z) := H H x, z), z). 3.7) We associae wih 3.7), he sochasic flow Y = Y τ, x),, x) [0, T] R d, generaed by he SDE dy = [,2] α)b Y )d {2} α)σ Y )dw + F Y, z)[,2] z)qd, dz)+ [0,] z)pd, dz)], τ< T, 3.8) Y = x, τ. Owing o pars ) and 2) of Lemma 4.0, for eachω,, and z, he inverse of he mapping F x, z) := x+f x, z)= x H H x, z), z) is H x, z) := x+h x, z) and here is a consan N= Nd, N 0,β,η) such ha for allω,, x, y, and z, r F z) 0 NK z), F z) β K z), I d + F x, z)) N. Thus, by Theorem 2. in [LM4b], here is a modificaion of he soluion of 3.8), which we sill denoe by Y = Y τ, x), ha is ac β loc -diffeomorphism for anyβ [,β). Moreover, P-a.s. Y τ, ), Y τ, ) D[0, T];C β loc Rd ; R d )), and Y τ, ) coincides wih he inverse of Y τ, ) for all. The following proposiion shows ha he inverse flow Y τ) solves 3.7). Proposiion 3.. Le Assumpion 3.β) hold for someβ> α. For each sopping ime τ T andβ [ α,β), v x)=v τ, x)=y τ, x) solves 3.7) and for eachǫ> 0 and p 2, here is a consan N= Nd, p, N 0, T,β,η,ǫ) such ha [ ] [ ] E sup r +ǫ) v τ) p 0 + E sup r ǫ v τ) p β N. 3.9) T T
3.2 Small jump case 7 Proof. The esimae 3.9) is given in Theorem 2. in [LM4b] see also Remark 2.), so we only need o show ha Y τ, x) solves 3.7). Le δ n ) n be a sequence such haδ n 0,η) for all n andδ n 0 as n. I is clear ha here is a consan N= NN 0 ) such ha for all ω and, dv n) π{z : K z)>δ n }) N. 3.0) δ α n For each n, consider he sysem of SIDEs on [0, T] R d given by x)= {2} α) σi x)σ j x) i j v n) x)+b i 2 x) iv n) +,2] α) + {K >δ n }z) v n) x+h x, z)) v n) ) x) d x)+f i x, z) iv n) x) ) πdz)d {K >δ n }z) v n) x+h x, z)) v n) x) ) [,2] α)qd, dz)+ [0,] α)pd, dz)], and he sochasic flow Y n) dy n) + {2} α)σ i x) i v n) x)dw, τ< T, v n) x)= x, τ, 3.) = Y n) τ, x),, x) [0, T] R d, generaed by he SDE = [,2] α)b Y n) )d {2} α)σ + Y n) )dw {K >δ n }z)f Y n), z)[,2] α)qd, dz)+ [0,] α)pd, dz)], τ< T, Y n) x)= x, τ. 3.2) Since 3.0) holds, we can rewrie equaion 3.2) as ) dy n) = [,2] α)b Y n) )+,2] α) {K >δ n }z)f Y n), z)πdz) d 3.3) {2} α)σ Y n ))dw + {K >δ n }z)f Y, n) z)pd, dz), τ< T, and 3.) as dv n) x)= {2} α) 2 σi x)σ j ) x) x) i j v n) x)+b i x) jσ i d + {2} α)σ i x) i v n) x)dw +,2] α) {K >δ n }z)fx, i z)πdz) i v n) x)d + {K >δ n }z) v n) x+h x, z)) v n) x) ) pd, dz), τ< T. 3.4) We claim ha he soluion Y n) = Y n) x) of 3.3) can be wrien as he soluion of coninuous SDEs wih a finie number of jumps inerlaced. Indeed, for each n and sopping imeτ T,
3.2 Small jump case 8 consider he sochasic flow Ỹ n) dỹ n) = Ỹ n) = [ [,2] α)b Ỹ n) )+,2] α) {2} α)σ Ỹ n) = x, τ. Ỹ n) τ, x),, x) [0, T] R d, generaed by he SDE {K>δn }, z)f Ỹ n), z)πdz)]d )dw,τ < T, By Theorems 2. and 2.4 and Remark 2.2 in [LM4b], here is a modificaion of Ỹ n) = Ỹ n) τ, x), sill denoed Ỹ n) τ, x), ha is ac β loc-diffeomorphism. Furhermore, P-a.s. we have ha and ṽ n) = ṽ n) For each n, le τ, x)=ỹ n); dṽ n) x)= Ỹ n) τ, ), Ỹ n); τ, ) C[0, T];C β loc ) τ, x) solves he SPDE given by {2} α) 2 σi x)σ j + {2} α)σ i x) i v n) +,2] α) ṽ n) x)= x, τ. x) i j v n) x)dw x)+b i x) iv n) x) ) d {K>δn }, z)f i, z)πdz)d i v n) x), τ < T, A n) = {Ks >δ n }z)pds, dz), 0, ]0,] and define he sequence of sopping imes τ n) l ) l= recursively byτn) 0 =τand τ n) l+ = inf{ >τ n) l : A n) 0 } T. Fix some n. I is clear ha P-a.s. for all x and [0,τ n) ), Y n); τ, x)=ỹ n); τ, x)=ṽ n) τ, x) saisfies 3.4) up o, bu no including imeτ n). Moreover, P-a.s. for all x, and hence Consequenly, v n) for some l, v n) Y n) τ n) τ, x)=ỹ n) τ n τ, x)+ Y n); τ, x)= τ n) τ, x)=y n); τ, x)=y n); ṽ n) τ n) F τ n) Ỹ n) τ n) τ, x+h τ n) x), z)p{τn) τ, }, dz), x, z))p{τ n) }, dz). τ, x) solves 3.4) up o and including imeτ n) τ, x) solves 3.4) up o and including imeτ n) l. Assume ha. Clearly,
3.3 Adding free and zero-order erms 9 P-a.s. for all x and [τ n) l,τ n) l+ [τ n) l,τ n) l+ ), Y n); Moreover, P-a.s. for all x, which implies ha v n) ), Yn) x)=ỹ n) x)=ỹ n) τ n) l, Y n) τ n) l τ n) l, Y n) x))=ṽn) τ n) l x)), and hus P-a.s. for all x and τ n) l, Y n) x)). τ n) Yn τn l+, x)= ṽ n τ n l,τn l+, x+hτn l+, x, z))p{τn l+ }, dz), U τ, x)=y n); τ, x) solves 3.4) up o and including imeτ n l+. There- τ, x)=y n); τ, x) solves 3.4). I is easy o see ha for fore, by inducion, for each n, v n) allω,, and z, and hus r {K >δ n }z)f z) r F z) 0 + {K >δ n }z) F z) F z) β {K δ n }z)k z) dpd lim {K δn n }, z)k z) D 2 πdz)+dpd lim {K δn n }, z)k z)πdz)=0. E By virue of Theorem 2.3 in [LM4b], for eachǫ> 0, and p 2, we have [ ] [ ]) lim E sup r +ǫ) n Y n) τ) r +ǫ) Y τ) p 0 + E sup r ǫ Yn) τ) r ǫ Y τ) p β = 0, T T [ ] lim E sup r +ǫ) n Y n); τ) r +ǫ) Y τ) p 0 = 0 T and [ lim E sup r ǫ n T Yn); τ) r ǫ Y l ] τ) p β = 0. Then passing o he limi in boh sides of 3.) and making use of Assumpion 3.β), he esimae 3.4), and basic convergence properies of sochasic inegrals, we find ha v τ, x)=x τ, x) solves 3.7). 3.3 Adding free and zero-order erms Se w ) = w ; ),,,π) =,,π ), pd, dz) = p d, dz), and qd, dz)= p d, dz) π dz)d. Also, se D=D, E = E, and assume = D E. Leυ x)= υ l l ω, x)) l, l d 2, be al 2 R 2d 2 )-valuedr T BR d )-measurable funcion defined onω [0, T] R d andρ x, z)=ρ l l ω, x, z)) l, l d 2 be ap T BR d ) -measurable funcion defined onω [0, T] R d. We inroduce he following assumpions forβ> α and β>α.
3.3 Adding free and zero-order erms 20 Assumpion 3.2 β). ) There is a consan N 0 > 0 such ha for all ω, ) Ω [0, T], Moreover, for all ω,, z) Ω [0, T], r b 0 + r σ 0 + b β + σ β N 0. r H z) 0 K z) and H z) β K z), where K :Ω [0, T] R + is ap T -measurable funcion saisfying K z)+ K z)+ K z) α + K z) 2) πdz)+ K z) α + K z) ) πdz) N 0, for all ω,, z) Ω [0, T]. D 2) There is a consanη 0, ) such ha for all ω,, x, z) {ω,, x, z) Ω [0, T] R d : H ω, x, z) >η}, I d + H x, z) ) N 0. Assumpion 3.3 β). There is a consan N 0 > 0 such ha for all ω, ) Ω [0, T], Moreover, for all ω,, z) Ω [0, T], c β+ υ β+ r θ f β+ r θ g β N 0. ρ z) β+ r θ h z) β l z), where l :Ω [0, T] R + is ap T -measurable funcion saisfying l z)+ l z) 2 πdz)+ l z)πdz) N 0. ω,, z) Ω [0, T]. D Le Assumpions 3.2 β) and 3.3 β) hold for some β> α and β>α. Leτ T be a sopping ime andϕ :Ω R d R d 2 be af τ BR d )-measurable random field. Consider he sysem of SIDEs on [0, T] R d given by dv l = ) ) L l v + ˆb i i φ l +ĉ l l φ l + ˆf l d+ N l v + g l dw + I l,z v + h l z)) [ D z)qd, dz)+ E z)pd, dz)], τ< T, E v l =ϕl, τ, l {,..., d 2 }, 3.5) where forφ C c Rd ) and l {,..., d 2 }, L l φx) := {2} α) σi x)σ j x) i j φ l x)+ {2} α)σ i x)a l l x) φ l i x) 2 + ρ l l x, z) φ l x+h x, z)) φ l x) ) πdz) D k + φ l x+h x, z)) φ l x),2] α) i φ l x)h i x, z)) πdz) D k Nφ l l x) := {2} α)σ i x) i φ l x)+υ l l x)φ l x), I l,z φl x) := I d2 +ρ l l x, z))φ l x+h x, z)) φ l x), E
3.3 Adding free and zero-order erms 2 and where ˆb i x) := [,2]α)b i x)+ {2}α)σ j x) j σ i x) +,2] α)h i x, z) Hi H x, z), z) ) πdz), D ĉ l l x) :=c l l x)+ {2} α)σ j x) j υ l l x)+ ρ l l x, z) ρ l l H x, z), z) ) πdz), ˆf l l x) := f x)+ {2}α)σ j x) j g l x)+ D D h l x, z) h l H x, z), z) ) πdz). We associae wih 3.5) he sochasic flow X = X x)=x τ, x),, x) [0, T] R d, given by 3.8). LeΓ x)=γ τ, x),, x) [0, T] R d, be he soluion of he linear SDE given by dγ x)=c X x))γ x)+ f X x))) d+ υ X x))γ x)+g X x)) ) dw + ρ H X x), z), z)γ x)[ D z)qd, dz)+ E z)pd, dz)] + h H X x), z), z)[ D z)qd, dz)+ E z)pd, dz)], τ< T, Γ x)=0, τ. 3.6) LeΨ x)=ψ τ, x),, x) [0, T] R d, be he unique soluion of he linear SDE given by dψ x)=c X x))ψ x)d+υ X x))φ x)dw + ρ H X x), z), z)ψ x)[ D z)qd, dz)+ E z)pd, dz)], τ< T, Ψ x)=i d2, τ. In he following lemma, we obain p-h momen esimaes of he weighed Hölder norms ofγandψ. Lemma 3.2. Le Assumpions 3.2 β) and 3.3 β) hold for some β> α and β>α. For each sopping imeτ T andβ [0, β β), here exiss a D[0, T],C β loc Rd ))-modificaion ofγτ) andψτ), also denoed by Γτ) andψτ), respecively. Moreover, for eachǫ> 0 and p 2, here is a consan N= Nd, d 2, p, N 0, T,β,η,ǫ,θ) such ha [ [ E sup[ r θ+ǫ) Γ τ) p β ]+E sup r ǫ Ψ τ) p β ] N. 3.7) T T Proof. Leτ T be a fixed sopping ime andβ := β β. Esimaing 3.6) direcly and using he Burkholder-Davis-Gundy inequaliy, Lemma 4., he muliplicaive decomposiion h x, H X x), z), z)=r θ X x)) rθ H X x), z)) r θ X x)) h H X x), z), z) r θ H X x), z)),
3.3 Adding free and zero-order erms 22 Hölder s inequaliy, Lemma 4.0 ), Lemma 3.2 in [LM4b], and Gronwall s inequaliy, we ge ha for all x and y, [ ] E sup Γ x) p Nr θp x) T and [ ] E sup Γ x) Γ y) p Nr pθ x) r pθ y)) x y β )p, T where N= Nd, p, N 0, T,η,θ) is a posiive consan. Now, assume ha [β]. As in he proof of Theorem 3.4 in [Kun04], i follows hau = Γ τ, x) solves du = ) υ X )U + υ X ) X Γ + g X ) X dw + ρ H X, z), z)u [ D z)qd, dz)+ E z)pd, dz)] + ρ H X, z), z) [ H X )]Γ [ D z)qd, dz)+ E z)pd, dz)] + h x, H X, z), z) [ H X )]][ D z)qd, dz)+ E z)pd, dz)] + c X )U + c X ) X Γ + f X ) X ) d, τ< T, U = 0, τ. Recall ha by Lemma 4.6, a funcionφ : R d R n, n saisfies r θ φ β < if an only if r θ φ 0,..., r θ γ φ 0, γ [β], and [r θ γ φ] {β} + are finie. Esimaing as above and using Proposiion 3.4 in [LM4b], we obain ha for each p 2, here is a consan N= Nd, d 2, p, N 0, T,θ) such ha for all x and y, [ ] E sup Γ x) p r pθ x)n T and [ ] E sup Γ x) Γ y) p Nr pθ x) r pθ y)) x y β ) )p. T Using inducion, we ge ha for each p 2 and all muli-indicesγ wih 0 γ [β] and all x, E sup [ γ Γ x) p ] r pθ x)n, T and for all muli-indicesγ wih γ =[β] and all x, y, [ ] E sup γ Γ x) γ Γ y) p Nr pθ x) r pθ y)) x y β [β] )p, T for a consan N = Nd, d 2, p, N 0, T,β,η,θ). I is also clear ha for each p 2 and all muli-indicesγ wih 0 γ [β] and all x, [ E sup γ Ψ x) ] N, p T
3.3 Adding free and zero-order erms 23 and for all muli-indicesγ wih γ =[β] and all x, y, [ E sup γ Ψ x) γ Ψ y) ] N x y p β [β] )p. T We obain he exisence of a D[0, T],C β loc Rd ))-modificaion ofγτ) andψτ) using esimae 3.7) and Corollary 5.4 in [LM4b]. This complees he proof. Le Φ x)= Φ τ, x),, x) [0, T] R d, be he soluion of he linear SDE given by d Φ x)= c X x)) Φ x)+ f X x)) ) d+ υ X x)) Φ x)+g X x)) ) dw + ρ H X x), z), z) Φ x, y)[ D z)qd, dz)+ E z)pd, dz)] + h H X x), z), z)[ D z)qd, dz)+ E z)pd, dz)], τ< T, Φ x)=ϕx), τ. The following is a simple corollary of Lemma 3.2. Corollary 3.3. Le Assumpions 3.2 β) and 3.3 β) hold for some β> α and β>α. For each sopping imeτ T andf τ BR d )-measurable random fieldϕsuch ha for some β [0, β β), P-a.s.ϕ C β loc Rd ), here is a D[0, T];C β loc Rd, R d 2 ))-modificaion of Φτ), also denoed by Φτ), and P-a.s. for all, x) [0, T] R d, Φ τ, x)=ψ x)ϕx)+γ x). Moreover, if for someθ 0 andβ [0, β β), P-a.s. r θ ϕ C β R d ), hen for each ǫ> 0 and p 2, here is a consan N= Nd, d 2, p, N 0, T,θ,θ,β,ǫ) such ha [ ] E sup r θ θ ) ǫ Φ τ) p Fτ β N r θ ϕ p β + ). 3.8) T Now we are ready o sae our main resul concerning fully-degenerae SIDEs and heir connecion wih linear ransformaions of inverse flows of jump SDEs. Proposiion 3.4. Le Assumpions 3.2 β) and 3.3 β) hold for some β> α and β>α. For each sopping imeτ T andf τ BR d )-measurable random fieldϕsuch ha for some β α, β β) andθ 0, P-a.s. r θ ϕ C β R d ), we have ha P-a.s. Φτ, X τ)) D[0, T];C β loc Rd )) and v x)=v τ, x)= Φ τ, X τ, x)) solves 3.5). Moreover, for eachǫ> 0 and p 2, [ ] E sup r θ θ ) ǫ v τ) p Fτ β N r θ ϕ p β + ), 3.9) T for a consan N= Nd, d 2, p, N 0, T,β,η,ǫ,θ,θ ).
3.4 Adding uncorrelaed par Proof of Theorem 2.2) 24 Proof. Fix a sopping imeτ T and random fieldϕsuch ha for someβ α, β β) and θ 0, P-a.s. r θ R d ). By virue of Corollary ϕ Cβ 3.3 and Theorem 2. in [LM4b], P-a.s. Φτ, X τ)) D[0, T];C β loc Rd, R d 2 )). Then using he Io-Wenzell formula Proposiion 4.6) and following a simple calculaion, we obain ha v τ, x) := Φ τ, X τ, x)) solves 3.5). By Theorem 2. in [LM4b] and Corollary 3.3, for eachǫ> 0 and p 2, here exiss a consan N= Nd, p, N 0, T,β,η,ǫ) such ha E[sup r +ǫ) X τ) p β ]+E[sup r ǫ X τ) p β ] N. 3.20) T T Therefore applying Lemma 4.9 and Hölder s inequaly and using he esimaes 3.20) and 3.8), we obain 3.9), which complees he proof. 3.4 Adding uncorrelaed par Proof of Theorem 2.2) Proof of Theorem 2.2. Fix a sopping imeτ Tand random fieldϕsuch ha for some β α, β β) andθ 0, P-a.s. r θ ϕ C β R d ). Consider he sysem of SIDEs given by dṽ l = ) ) L ;l +L 2;l )ṽ + [,2] α)ˆb i i u l + ĉ l l u l x)+ fˆ l d+ N ;l ṽ + g l dw ; +N 2;l ṽ dw 2; + I ;l,zṽ + h l z)) [ D z)q d, dz)+ E p d, dz)] + I 2;l,zṽ [ D 2z)q 2 d, dz)+ E 2z)p 2 d, dz)] τ< T, 2 ṽ l =ϕl, τ, l {,..., d 2 }, where forφ C c Rd ) and l {,..., d 2 }, N 2;l φx) := {2} α)σ 2;i I 2;l,zφx) := I l l d 2 +ρ 2;l l x) i φ l x)+υ 2;l l x)φ l x),, x, z))φ l x+h 2 x, z)) φ l x). By Proposiion 3.4, P-a.s.Φτ, X τ)) D[0, T];C β loc Rd )) and ṽ τ, x)=φ τ, X τ, x)) solves 3.5). We wrie v x)=v τ, x). Moreover, for eachǫ> 0 and p 2, [ ] E sup r θ θ ) ǫ ṽ τ) p Fτ β N r θ ϕ p β + ), 3.2) T where N= Nd, d 2, p, N 0, T,β,η,η 2,ǫ,θ,θ ) is a posiive consan. Wihou loss of generaliy we will assume ha for allωand, r θ ϕ β N, since we can always muliply he equaion by indicaor funcion. For each n N {0}, le C n loc Rd ) be he separable
3.4 Adding uncorrelaed par Proof of Theorem 2.2) 25 Fréche space of n-imes coninuously differeniable funcions f : R d R d 2 endowed wih he counable se of semi-norms given by f n,in = sup γ f x), k N. 0 γ n x k Owing o Lemma 4.2, here is a he family of measures Eω du), ω, ) Ω [0, T] on D[0, T]; C [β] loc Rd )), corresponding oa=ṽ such ha for all bounded G :Ω [0, T] [0, T] D[0, T]; C [β] loc Rd )) R d 2 ha areo T B [0, T]) BD[0, T]; C [β] loc Rd ))) measurable, P-a.s. for all, we have E [G, ṽ)]= D[0,T];C [β ] loc R d ;R d 2 )) G, U)E du)=e [G, ṽ) F ], where he righ-hand-side is he càdlàg modificaion of he condiional expecaion. Se û x)=û τ, x)= E [ṽ τ, x)]= U x)e du). D[0,T];C [β ] loc R d ;R d 2 )) Leλ=θ θ )+ǫ. We claim ha for all muli-indicesγ wih γ [β], P-a.s. for all and x, γ [r λ x)û x)]= γ [r λ D[0,T];C [β ] x)u x)]e du)=e [ γ [r λ x)ṽ x)]]. R loc d ;R d 2 )) Indeed, since ] M = E [sup γ [r λ ṽs] 0, [0, T], s T is a F, P) maringale, we have [ [ E and hence P-a.s. for all, D[0,T];C [β ] loc R d ;R d 2 )) sup M ] 4E [ 2 M T 2] 4E T sup γ [r λ s T,x R d sup γ [r λ ṽ] 2 0 ]<, 3.22) T x)u sx)] E du)=e [sup Similarly, since E [ sup T r λ ṽ 2 β ] <, P-a.s. for each x and y, and hence, P-a.s. γ [r λ x)û x)] γ [r λ y)û y)] x y {β } + sup r λ T û β sup T T γ [r λ ṽ] 0 ]<. [ γ [r λ E x)ṽ x)] γ [r λ y)ṽ y)] x y {β } + E [ r λ ṽ β ], ] E [sup r λ ṽ β <. T ]
3.5 Inerlacing a sequence of large jumps Proof of Theorem 2.5) 26 Thus, P-a.s. r λ )ûτ) D[0, R d )) and 2.4) T];Cβ follows from 3.2) see he argumen 3.22)). For each l {,..., d 2 }, le A l x)=ϕl x)+ L ;l s +L 2;l s )û sx)+ [,2] α)ˆb i s x) iû l s x)+ĉl l s x)û l s x)+ fˆ s l x)) ds ]τ,τ ] + N ;l s û s x)+g l s x)) dw ; s ]τ,τ ] + I ;l s,zû s x)+h l sx, z) ) [ D z)q ds, dz)+ E z)p ds, dz)]. ]τ,τ ] By Theorem 2.2 in [Jac79], he represenaion propery holds for F, P), and hence every bounded F, P)- maringale issuing from zero can be represened as M = o s dw; s + e s z)q ds, dz), [0, T], ]0,] ]0,] where E o s 2 ds+e e s z) 2 π dz)ds<. ]0,T] ]0,T] Then for an arbirary F-sopping ime τ T and bounded F, P)- maringale, applying Iô s produc rule and aking he expecaion, we obain Eṽ τ τ, x) M τ = EA τ x) M τ. Since he opional projecion is unique, P-a.s. for all and x, û x)=a x). This complees he proof. 3.5 Inerlacing a sequence of large jumps Proof of Theorem 2.5) Proof of Theorem 2.5. Fix a sopping imeτ Tand random fieldϕsuch ha for some β α, β β) andθ 0, P-a.s. r θ ϕ C β R d ). For anyδ>0, we can rewrie.) as du l = L ;l +L 2;l )u + [,2] α) b i iu l + ) ) cl l u l + f l d+ N ;l u + g l dw ; Ī;l +,zu + h l z)) [ D z)q d, dz)+ E z)p d, dz)] + D E ) {K >δ} z)+ V z) ) I ;l,zu + h l z)) p d, dz), τ< T, u l =ϕl, τ, l {,..., d 2 }, 3.23) where forφ C c Rd ) and l {,..., d 2 }, L ;l φx) := {2} α) σ;i ; j x)σ x) i j φ l x)+ {2} α)σ k;i x)υ ;l l x) φ l i x) 2 + ρ ;l l x, z) φ l x+ H x, z)) φ l x) ) π dz) D + φ l x+ H x, z)) φ l x),2] α) H ;i x, z) i φ l x) ) π dz), D
3.5 Inerlacing a sequence of large jumps Proof of Theorem 2.5) 27 Ī,z φl x)=i l l d 2 + {K δ}z)ρ ;l l x, z))φ l x+ {K δ}z)h x, z)) φl x), H := {K δ}h, ρ := {K δ}ρ, h := {K δ}h, b i x) := bi D x),2] α)h ;i {K >δ} x, z)π dz), c l l x) := c l l x) ρ ;l l D {K >δ} x, z)π dz). For an arbirary sopping imeτ T andf τ BR d )-measurable random fieldϕ τ :Ω R d R d 2 saisfying for someθτ )>0, P-a.s. r θτ ) ϕ τ C β R d ), consider he sysem of SIDEs on [0, T] R d given by dv l = ) ) L ;l +L 2;l )v + [,2] α) b i i v l + c l l v l + f l d+ N ;l v + g l dw ; Ī;l +,zu + h l z)) [ D z)q d, dz)+ E z)p d, dz)], τ < T, v l =ϕ τ ;l, τ, l {,..., d 2 }. 3.24) Se H 2 = H 2 and ρ 2 =ρ 2. In order o invoke Theorem 2.2 and obain a unique soluion v = v τ, x)=v τ,ϕ τ, x) of 3.24), we will show ha for allωand, where r b 0 + b β + c β+ r θ f β N 0, 3.25) 2 b i x) := [,2]α) b i x) {2} α)σ 2 k= c l l x) := c l l x) 2 k= k=,2] α) H k;i D k 2 k= f l x) := f l x) σ k;l l ρ D k ; j {2} α)σ k;i x, z) ρ k;l l x) j g l x) k; j x, z) H k;i x) i υ k;l l x) x) j σ k;i x) k; H x, z), z) ) π k dz), k; H x, z), z) ) π k dz), D h l x, z) h l ; H x, z), z) ) π dz). Owing o Assumpion 2.3 β,δ,δ 2,µ,µ 2 ), we easily deduce ha here is a consan N= Nd, N 0, β) such ha for each k {, 2} and allωand, σ k; j j σ k; β+ σ k; j j a k; x) β+ σ ; j j g β N, ifα=2. Since H 0 δ, for any fixedη <, for all ω,, x, z) {ω,, x, z) Ω [0, T] R d D E ) : H ω, x, z) >η }, + H Id ω, x, z)) δ.
3.5 Inerlacing a sequence of large jumps Proof of Theorem 2.5) 28 Appealing o Assumpion 2.3 β,δ,δ 2,µ,µ 2 ) and applying Lemma 4.0, we obain ha here is a consan N= Nd, d 2, N 0 ) such ha for each k {, 2} and allω,, and z, and H k;i z) H k;i ρ k z) ρ k r θ h z) r θ h k; H z), z) β NK k z)+ K k z)) 2 + N 0,] { β} + +δ k ) K k z)k k z) δk + N,2] { β} + +δ k ) K k z)kk z)δk + K k z)2), H k; z), z) β Nl k z)k k z)+ K k z))+ N 0,] { β} + +µ k ) l k z)k k z) µk + N,2] { β} + +µ k ) l k z)kk z)µk + l k z) K k z)), ; H z), z) β Nl z)k z)+ K k z))+n 0,]{ β} + +µ ) l k z)k z)µ + N,2] { β} + +µ ) l k z)k z) µ + l k z) K z) ). Moreover, using Lemma 4.0, we find ha here is a consan N= Nd, d 2, N 0 ) such ha for each k {, 2}, and allω,, and z, r H k k; H z), z) 0 r Hk 0, [ H k;i k; H z), z)] β H k β. Combining he above esimaes and using Hölder s inequaliy and he inegrabiliy properies of l k z) and Kk z), we obain 3.25). Therefore, by Theorem 2.2, for each sopping ime τ T and andf τ BR d )-measurable random fieldϕ τ saisfying for someθτ )>0, P-a.s. r θτ ) ϕ τ C β R d ), here exiss a unique soluion v x)=v τ,ϕ τ, x) of 3.24) such ha [ ] E sup r θτ ) θ ǫ v τ ) p Fτ β N r θτ ) ϕ τ p β + ), 3.26) T where N= Nd, d 2, p, N 0, T,β,η,η 2,ǫ,θ,θτ )) is a posiive consan. Le A = D E ) {Ks>η }z)+ V z) ) p ds, dz), T. ]0,] Define a sequence of sopping imes τ n ) n 0 recursively byτ =τ and τ n+ = inf>τ n : A 0) T. We obain he exisence of a unique soluion u=uτ) of 3.23) inc β R d ) by inerlacing soluions of 3.24) along he sequence of sopping imes τ n ). For ω, ) [[0,τ )), we se u τ, x)=v τ,ϕ, x) and noe ha [ ] E sup r θ θ ǫ u τ) p Fτ β N r θ ϕ p β + ). τ For eachωand x, we se u τ x)=u τ x)+ D E ) {K >η }, z)+ V z) ) I,zu τ x)+h l τ x, z) ) p {τ }, dz).
Appendix 29 By virue of Lemma 4.9, here is a consan N= Nd, d 2,θ,θ,ζ τ z),β ) and hence where u τ H τ z) r ξ τ z)θ θ +ǫ+β ) β N r θ θ ǫ u l τ β, r λ u τ x) β N r θ θ ǫ u l τ β +ζ τ z), λ = ξ τ z)θ θ + +ǫ+β )) θ θ θ +ǫ). We hen proceed inducively, each ime making use of he esimae 3.26), o obain a unique soluion u=uτ) of 3.23), and hence.), inc β R d ). This complees he proof of Theorem 2.5. 4 Appendix 4. Maringale and poin measure measure momen esimaes Se,,π)=,,π ), pd, dz)= p d, dz), and qd, dz)=q d, dz). We will make use of he following momen esimaes o derive he esimaes ofγ andψ in Lemma 3.2. The noaion a p b is used o indicae ha he quaniy a is bounded above and below by a consan depending only on p imes b. Lemma 4.. Le h :Ω [0, T] R d bep T -measurable ) For each sopping imeτ T and p 2, [ p] E h s z)qds, dz) sup τ ]0,] 2) For each sopping imeτ T and p, E [ sup τ ) p ] h s z) pds, dz) ]0,] [ E p ]0,τ] + E [ E p ]0,τ] ]0,τ] [ + E ] h s z) p πdz)ds p/2 h s z) πdz)ds). 2 ]0,τ] ] h s z) p πdz)ds ) p ] h s z) πdz)ds, Proof. We will only prove par 2), since par ) is well-known see, e.g., [Kun04]) and i follows from 2) by he Burkholder-Davis-Gundy inequaliy. Assume ha h ω, z)>0 for allω, and z. Le A = h s z)pds, dz) and L = h s z)πdz)ds, T. ]0,] ]0,]
4. Maringale and poin measure measure momen esimaes 30 I suffices o prove he claim for p>, since he case p= is obvious. Fix an arbirary sopping imeτ T and p>. For allωand, we have A p [ = As + A s ) p A p ] s. Thus, using he inequaliy s b p a+b) p a p pa+b) p b p2 p [a p b+b p ], a, b 0, for allωand, we ge [ p2 p 2 and A p 0 A p ] A p s h s z)pds, dz)+ h s z) p pds, dz). ]0,] h ]0,] h s z) p pds, dz). Then since A is an increasing process, we have [ ] E h s z) p pds, dz) EAτ p p2p 2 E Aτ p L τ + h s z) p pds, dz). ]0,τ] ]0,τ] I is easy o see ha EL p τ = pe ]0,τ] Ls p dl s = pe ]0,τ] Applying Young s inequaliy, for allε>0, P-a.s., Ls p da s pe[lτ p A τ ]. A p τ L τ εa p τ+ p )p ε p p p L p τ and L p τ A τ εl p τ+ p )p ε p p p A p τ. Combining he above esimaes, for anyε 0, ), we have p ε p p p pε ) pp ) p EL p τ E ]0,τ] h s z) p pds, dz) EA p τ. and for anyε 2 0, p2 p 2 ) EAτ p p2 p 2 p2 p 2 ε 2 ) E which complees he proof. ]0,τ] h s z) p pds, dz)+ p )p L p ε p 2 p p τ,
4.2 Opional projecion 3 4.2 Opional projecion The following lemma concerning he opional projecion plays an inegral role in Secion 3.4 and he proof of Theorem 2.2. Lemma 4.2. cf. Theorem in [Mey76]) LeXbe a Polish space and D [0, T];X) be he space ofx-valued càdlàg rajecories wih he SkorokhodJ -opology. IfAis a random variable aking values in D [0, T];X), hen here exiss a family ofb[0, T]) F -measurable non-negaive measures E du), ω, ) Ω [0, T], on D [0, T];X) and a random-variable ζ saisfying P ζ< T)=0 such ha E D [0, T];X))= for <ζand E D[0, T];X))=0 for ζ. In addiion, E is càdlàg in he opology of weak convergence, E = E + for all [0, T], and for each coninuous and bounded funcional F on D [0, T];X), he process E F) is he càdlàg version of E[F A) F ]. If G :Ω [0, T] [0, T] D [0, T];X) R d 2 is bounded ando B [0, T]) B D [0, T];X))-measurable, hen G ω,, U)E du)=e G ) D[0,T];X) is he opional projecion of G A)=G ω,,a). Furhermore, if G= G ω,, U) is bounded andp B[0, T]) BD[0, T];X))-measurable, hen E G ) is he predicable projecion of G A)=G ω,,a). Proof. We follow he proof of Theorem in [Mey76]. Since D[0, T];X) is a Polish space, for each [0, T], here is family of probabiliy measures Ẽ ωdw),ω Ω, on D[0, T];X) such ha for each A BD[0, T];X)), Ẽ A) isf -measurable and P-a.s., P A A F )=Ẽ A). For eachω Ω, le I ω) be he se of all 0, T] such ha for each bounded coninuous funcion F on D[0, T];X), he funcion r Ẽω r F)= Fw)Ẽ r dw) D[0,T];X) has a righ-hand limi on [0, s) Q and a lef-hand limi on 0, s] Q for every raional s [0, T] Q. Leζ ω)=sup : Iω)) T. I is easy o see ha P ξ<t)=0. We se Ẽω = 0 ifξω)< T. The funcion Ẽ ω has lef-hand and righ-hand limis for all Q [0, T]. We define Eω=Ẽ ω + for each [0, T) he limi is aken along he raionals), and Eω T is he lef-hand limi a T along he raionals. The saemen follows by repeaing he proof of Theorem in [Mey76] in an obvious way. 4.3 Esimaes of Hölder coninuous funcions In he coming lemmas, we esablish some properies of weighed Hölder spaces ha are used Secion 3.5 and he proof of Theorem 2.5.