PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION

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1 PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION Anna Aksami, Monique Jeanblanc, Marek Rukowski To cie his version: Anna Aksami, Monique Jeanblanc, Marek Rukowski. PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION. 25. <hal > HAL Id: hal hps://hal.archives-ouveres.fr/hal Submied on 2 Jan 26 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION Anna Aksami Mahemaical Insiue, Universiy of Oxford, Oxford OX2 6GG, Unied Kingdom Monique Jeanblanc Laboraoire de Mahémaiques e Modélisaion d Évry LaMME, Universié d Évry-Val-d Essonne, UMR CNRS Évry Cedex, France Marek Rukowski School of Mahemaics and Saisics Universiy of Sydney Sydney, NSW 26, Ausralia December 25 Absrac We sudy problems relaed o he predicable represenaion propery for a progressive enlargemen G of a reference filraion F hrough observaion of a finie random ime τ. We focus on cases where he avoidance propery and/or he coninuiy propery for F-maringales do no hold and he reference filraion is generaed by a Poisson process. Our goal is o find ou wheher he predicable represenaion propery PRP, which is known o hold in he Poisson filraion, remains valid for a progressively enlarged filraion G wih respec o a judicious choice of G-maringales. Keywords: predicable represenaion propery, Poisson process, random ime, progressive enlargemen Mahemaics Subjecs Classificaion 2: 6H99 Corresponding auhor: Monique Jeanblanc, posal address: Laboraoire de Mahémaiques e Modélisaion d Évry, Universié d Évry-Val-d Essonne, 925 Évry Cedex, France, phone: , monique.jeanblanc@univ-evry.fr.

3 2 A. Aksami, M. Jeanblanc and M. Rukowski Inroducion We sudy various problems associaed wih a progressive enlargemen G = G R+ of a reference filraion F = F R+ hrough observaion of he occurrence of a finie random ime τ. We focus on he cases where he so-called avoidance propery and/or he coninuiy propery for F-maringales do no hold. Under he assumpions ha F is he Brownian filraion, he probabiliy disribuion of τ is coninuous, and he filraion F is immersed in is progressive enlargemen G, i was shown by Kusuoka [2] ha any G-maringale can be decomposed as a sochasic inegral wih respec o he Brownian moion and a sochasic inegral wih respec o he compensaed maringale of he indicaor process H := {τ }. We are ineresed in various exensions of Kusuoka s resul, in paricular, o he case where he filraion F is generaed by a Poisson process N. The main goal is o examine wheher he predicable represenaion propery PRP, which is well known o hold for he Poisson process and is naural filraion, is also valid in he progressively enlarged filraion G wih respec o judiciously chosen family of G-maringales. In conras o he classical versions of he predicable represenaion propery in a Brownian filraion, we aemp here o derive explici expressions for he inegrands, raher han o esablish he exisence and uniqueness of inegrands. As a main inpu, we posulae he knowledge of he Azéma supermaringale of τ wih respec o he filraion generaed by a Poisson process. The paper is srucured as follows. In Secion 2, we inroduce he se-up and noaion. We also review briefly he classic resuls regarding he G-semimaringale decomposiion of F-maringales sopped a τ. I is worh noing ha no general resul on a G-semimaringale decomposiion afer τ of an F-maringale is available in he exising lieraure. Moreover, mos papers in his area are devoed o eiher he case of hones imes see, for insance, Jeulin and Yor [8, 9] or he case where he densiy hypohesis holds see Jeanblanc and Le Cam [2]. In Secion 3., we examine he case of a general filraion F. We posulae ha he immersion propery holds beween F and G and hus he Azéma supermaringale of a random ime τ is a decreasing process, which is also assumed o be F-predicable. We derive several alernaive inegral represenaions for a G-maringale sopped a τ see Proposiions 3.2 and 3.3. In Secion 3.2, we mainain he assumpion ha he immersion propery holds, bu we no longer posulae ha he Azéma supermaringale of τ is F-predicable. Under he assumpion ha he reference filraion is generaed by he Poisson process, we obain an explici inegral represenaion for paricular G-maringales sopped a τ in erms of he compensaed maringale of he Poisson process and he compensaed maringale of he indicaor process of τ see Proposiion 3.4. In Secion 4, he immersion propery is relaxed, bu we sill assume ha he filraion is generaed by he Poisson process. Since we work wih he Poisson filraion, all F-maringales are necessarily processes of finie variaion and hus any F-maringale is manifesly a G-semi-maringale; in oher words, he so-called hypohesis H is saisfied. We examine he validiy of he predicable represenaion propery for he Poisson filraion wihou making any addiional assumpions, excep for he fac ha we sill consider G-maringales sopped a ime τ. We also aemp o idenify cases where he mulipliciy [6, 7] of he enlarged filraion wih respec o he class of all G-maringales sopped a τ equals wo. The main resul of his work, Theorem 4., offers a general represenaion formula for any G-maringale sopped a τ in erms of he compensaed maringale of he Poisson process, he compensaed maringale of he indicaor process of a random ime and an addiional G-maringale, which can be seen as a correcion erm, which presence is due o he possibiliy of an overlap of jumps of he Poisson process and a single jump he indicaor process H associaed wih a random ime τ. We also show ha his addiional erm has a naural inerpreaion as an opional sochasic inegral as opposed o he usual predicable sochasic inegrals. I is worh noing ha he Azéma supermaringale of a random ime does no uniquely deermine all he properies of τ wih respec o a filraion F see Jeanblanc and Song [3, 4] and Li and Rukowski [2, 22]. In our conex, due o only a parial informaion abou he random ime τ and possible join jumps of he Poisson process N and he indicaor process H, more explici compuaions seem o be ou of reach.

4 Progressive Enlargemens of a Poisson Filraion 3 2 Preliminaries We firs inroduce he noaion for an absrac se-up in which a reference filraion F is progressively enlargedhrough observaionsof a randomime. Le Ω,G,F,P be a probabiliyspace where F is an arbirary filraion saisfying he usual condiions and such ha F G. Le τ be a random ime, ha is, a sricly posiive, finie random variable on Ω,G,F,P. The indicaor process H := [τ, [ is a raw increasing process. We denoe by G he progressive enlargemen of he filraion F wih he random ime τ, ha is, he smalles righ-coninuous and P-compleed filraion G such he inclusion F G holds and he process H is G-adaped, so ha τ is a G-sopping ime. In oher words, he progressive enlargemen is he smalles filraion saisfying he usual condiions ha renders τ a sopping ime. For a filraion K = F or K = G, we denoe by B p,k resp., B o,k he dual K-predicable resp., he dual K-opional projecion of a process B of finie variaion, whereas p,k U resp., o,k U sands for he K-predicable resp., K-opional projecion of a process U. The K-predicable covariaion process of wo semi-maringales X and Y is denoed by X,Y K. Recall ha X,Y K is defined as he dual K-predicable projecion of he covariaion process [X,Y], ha is, X,Y K := [X,Y] p,k. The càdlàg, bounded F-supermaringale Z given by Z := Pτ > F = o,f [,τ[ is called he Azéma supermaringale see Azéma [4] of he random ime τ. Noe ha since τ is assumed o be finie, we have ha Z := lim Z =. The process Z admis a unique Doob- Meyer decomposiion Z = µ A p where µ is an F-maringale and A p is an F-predicable, increasing process. Specifically, A p is he dual F-predicable projecion of he process H, ha is, A p = H p,f, whereas he posiive F-maringale µ is given by he equaliy µ = EA p F. I is well known see, e.g., Jeulin [7, p. 64] ha for any bounded, G-predicable process U, he process by convenion, we denoe s = ]s,] τ U τ H U s da p s 2. is a uniformly inegrable G-maringale. In paricular, he process M τ given by M τ := H τ is a uniformly inegrable G-maringale. Also, i is worh noing ha U τ H τ U s da p s = da p s 2.2 U s dm τ s. We shall also use he opional decomposiion Z = m A o where A o := H o,f is he dual F-opional projecion of H and he posiive F-maringale m is given by m := EA o F. Since τ is sricly posiive, we have ha µ = m =. In he special case where τ avoids all F-sopping imes, ha is, Pτ = σ = for any F-sopping ime σ, he equaliy A p = A o holds and hus also µ = m. Alhough mos exising resuls regarding he G-semimaringale decomposiion of an F-maringale are formulaed in erms of he Azéma supermaringale Z, i is also possible o use for his purpose he supermaringale Z, which is given by Z := Pτ F = o,f [,τ]. Noe ha he Azéma supermaringale Z is a càdlàg process, bu he supermaringale Z fails o be càdlàg, in general. I is also worh menioning see Jeulin and Yor [8, p. 79] ha Z = m H o,f = m A o and hus Z = Z + m. I is known ha he processes Z and Z = Z do no vanish before τ. Specifically, he random ses { Z = } and {Z = } = { Z = } are disjoin from ],τ], so ha he se {Z = } is disjoin

5 4 A. Aksami, M. Jeanblanc and M. Rukowski from ],τ[. Moreover, he ses {Z = }, { Z = } and {Z = } = { Z = } are known o have he same débu, which is he F-predicable sopping ime R given by R := inf{ > : Z = } = inf{ > : = } = inf{ > : = } where he middle equaliy holds since Z is a non-negaive, càdlàg supermaringale and he las one is obvious. For he reader s convenience, we firs recall some resuls on progressive enlargemen of a generic filraion F and an arbirary random ime τ. As was already menioned, no general resul furnishing a G-semimaringale decomposiion afer τ of an F-maringale is available in he exising lieraure, alhough such resuls were esablished for paricular classes of random imes, such as: he hones imes see Jeulin and Yor [9], he random imes saisfying he densiy hypohesis see Jeanblanc and Le Cam [2], as well as for some classes of random imes obained hrough various exensions of he so-called Cox consrucion of a random ime see, e.g., [9, 3, 4, 2, 22, 25]. In his work, we will focus on decomposiion resuls for F-maringales sopped a τ and hus we firs quoe some classical resuls regarding his case. For par i in Proposiion 2., he reader is referred o Jeulin [7, Proposiion 4.6]; he second par is borrowed from Jeulin and Yor[8, Théorème, pp ]. Proposiion 2.. i For any F-local maringale X, he process X := X τ is a G-local maringale sopped a τ. ii For any F-local maringale X, he process X := X τ τ τ {Zs <} where J := H X τ p,f, is a G-local maringale sopped a τ. d X,m F s 2.3 d X,µ F Z s +d J s s By comparing pars i and ii in Proposiion 2., we obain he following well known resul, which will be used in he proof of he main resul of his noe see Theorem 4.. Corollary 2.. The following equaliy holds for any F-local maringale X τ d J s = τ 2.4 d X,m µ F s. 2.5 Proof. For he sake of compleeness, we provide he proof of he corollary. We noe ha he processes τ d X,m F s and τ {Zs <} d X,µ F Z s +d J s s are G-predicable. Hence equaions 2.3 and 2.4 yield wo Doob-Meyer decomposiions of he special G-semi-maringale X τ. The uniqueness of he Doob-Meyer decomposiion leads o he equaliy X = X, and hus also o = = = τ {Zs <} τ τ d X,µ F Z s +d J s s d X,µ m F s +d J s d X,µ m F Z s +d J s s τ τ {Zs =} d X,m F s d X,µ F s +d J s where he las equaliy follows from Lemme 4b in Jeulin and Yor [8]. We conclude ha 2.5 is valid.

6 Progressive Enlargemens of a Poisson Filraion 5 3 The PRP under he Immersion Hypohesis Le us inroduce he following noaion for he class of G-maringales sudied in his work MG,τ := { Y h := Eh τ G : h H o F,τ } where H o F,τ := {h : h is an F-opional process and E h τ < }. Noe ha MG,τ is in fac he se of all G-maringales sopped a τ. We denoe by H p F,τ he se of all processes h from H o F,τ ha are F-predicable, raher han merely F-opional. Our main quesion reads as follows: under which assumpions any process Y h from he class MG, τ admis an inegral represenaion wih respec o some fundamenal G-maringales? Of course, a par of he problem is a judicious specificaion of he fundamenal G-maringales, which will serve as inegraors in he represenaion resuls. Our goal is o characerize he dynamics ofheprocessesy h MG,τandoobainsufficiencondiionsforhePRPoholdinhefilraion G wih respec o he G-maringale M τ given by equaion 2.2 and some auxiliary F- or G- maringales. Of course, he choice of hese maringales depend on he se-up a hand. Somewha surprisingly, i is no easy o obain he dynamics of Y h by direc compuaions in a general seup and hus we will firs focus on he case when he immersion propery holds. Recall ha he immersion propery beween F and G, which is also known as he hypohesis H, means ha any F- local maringale is a G-local maringale. Specifically, in Secion 3. we will work under Assumpion 3. and we will assume ha F is an arbirary filraion. Nex, in Secion 3.2, we will examine some examples of represenaion heorems when he immersion holds for he filraion generaed by a Poisson process and he Azéma supermaringale is eiher an F-predicable process, or merely an F-opional process. In Secion 3., we will work under he following assumpion. Assumpion 3.. We posulae ha he process h belongs o H o F,τ, he random ime τ is such ha he filraion F is immersed in is progressive enlargemen G, and he Azéma supermaringale Z is decreasing and F-predicable, so ha Z = A p. If he filraion F is immersed in G, hen Z is a decreasing process. I is also known see Lemma 6.3 in Nikeghbali [24] ha if τ avoids F-sopping imes respecively, if all F-maringales areconinuous, hen he processa o = A p isconinuousrespecively, heprocessa o isf-predicable and hus A o = A p. Proposiion 3.. If eiher i he process h belongs o H p F,τ or ii Assumpion 3. holds, hen he G-maringale Y h saisfies where we denoe Y h = {τ } h τ + {<τ} E Z X h := E h s da p s F = H h τ + H X h Z 3. h s da p s F = µ h and µ h sands for he following uniformly inegrable F-maringale µ h := E h s da p s 3.2 h s da p s F. 3.3 Proof. Under assumpion i, equaliy 3.2 was esablished in Ellio e al. [8]. They firs show ha see Lemma 3. in [8] for an F-opional process h we have Y h = {τ } h τ + {<τ} Z E h τ {<τ} F. 3.4

7 6 A. Aksami, M. Jeanblanc and M. Rukowski Nex, in Secion 3.4, hey show ha he properies of he dual F-predicable projecion imply ha E h τ {<τ} F = E h s da p s F. 3.5 for any process h H p F,τ. By combining 3.4 wih 3.5, we obain 3.. In he second par of he proof, we suppose ha Assumpion 3. is saisfied. We claim ha H o,f = o,f H = A p. Indeed, under hypohesis H, τ is a pseudo-sopping ime and hus, by Theorem in [25], he equaliy H o,f = o,f H holds. Moreover, by our assumpion Z = o,f H = o,f H = A p, so ha he equaliy o,f H = A p is valid as well. Using he properies ofhe dual F-opionalprojecion and he equaliy H o,f = A p, we obain, for any process h H o F,τ, E h τ {<τ} F = E h s dhs o,f F = E h s da p s F. 3.6 By combining he las equaliy wih 3.4, we conclude once again ha 3. holds. Remark 3.. Le us observe ha if Z = A p wihou assuming ha he immersion propery holds, hen he equaliy H p,f = p,f H is valid. Indeed, i is well known ha he equaliy p,f o,f X = p,f X holds for any bounded measurable process see propery.26 on p. 4 in Jacod []. From he equaliy Z = A p, we obain o,f H = A p, and hus p,f o,f H = p,f A p = A p = H p,f where he second equaliy is obvious, since A p is an F-predicable process. We conclude ha under Assumpion 3., all four projecions of H are idenical as classes of equivalences of measurable sochasic processes, ha is, o,f H = H o,f = H p,f = p,f H. I is well known ha he immersion propery implies ha Pτ > F = Pτ > F for all R +. Therefore, under immersion, he Azéma supermaringale Z is an F-adaped, decreasing process and, obviously, he F-maringale µ h is also a G-maringale. Le us menion ha if Z is an F-predicable, decreasing process, hen i is no necessarily rue ha he immersion propery holds. In fac, i was shown by Nikeghbali and Yor [25] see Theorem herein ha, if all F-maringales are coninuous, hen he propery ha Z is an F-predicable, decreasing process is equivalen o he propery ha τ is an F-pseudo-sopping ime he laer propery is weaker ha he immersion propery beween F and G. Finally, i was recenly shown by Aksami and Li [3] ha Z is càglàd and decreasing if and only if τ is an F-pseudo-sopping ime. 3. The PRP for he Enlargemen of a General Filraion In his subsecion, we work wih a general righ-coninuous and P-compleed filraion F and we search for an inegral represenaion of Y h in erms of he G-maringales M τ and µ h, which are given by equaions 2.2 and 3.3, respecively. We sar by noing ha for any càdlàg F-adaped process U, he jump U = U U process is a hin process, i.e., here exiss a sequence of F- sopping imes S n such ha { U } n [S n ] see Definiion 7.39 in []. As a consequence, if Z is a finie variaion process hen X h s Z s dzs = <s X h s Zs, R +, 3.7 Z s for an arbirary process h from he class H o F,τ and he associaed F-maringale µ h given by equaion 3.3 and he process X h given by 3.2. We are in a posiion o prove he following resul yielding an inegral represenaion for any process Y h from MG,τ.

8 Progressive Enlargemens of a Poisson Filraion 7 Proposiion 3.2. If Assumpion 3. holds, hen Y h MG,τ admis he following represenaion dy h = h Xh dm τ + H dµ h Z Z. 3.8 Proof. Since Z = A p, i is clear ha dz = da p. If we denoe Y = Y h and X = X h, hen he Iô formula yields X dy = h dh + H d X X dh H Z Z = h X X X dh + H d H Z Z and hus, since µ =, X d Z = dx +X Z 2 dz + Z 2 Z + + X. Z Z From equaion 3.2, we have dx = dµ h h da p and hus, using dz = da p, which leads o where dx X Z 2 dz = dµ h h X da p, dy = X dl := H = H = H h X dm τ + H dµ h Z +dl 3.9 Z 2 X Z X Z Z +X Z dz Z X Z X + X H Z Z dh = X Z X Z dm τ H where he penulimae equaliy follows from 3.7 and he las one holds since dz = da p. The assered equaliy 3.8 now easily follows from 3.9. I is no obvious ha he firs erm in he righ-hand side of 3.8 is a G-maringale, since he inegrand is no necessarily G-predicable. However, his is indeed rue, since Y h is a G- maringale, he inegrand in he second erm in righ-hand side of 3.8 is G-predicable and, due o he immersion propery, he F-maringale µ h is also a G-maringale. Remark 3.2. Assume ha Z is decreasing, bu no necessarily F-predicable, so ha Z = µ A p, where he F-maringale µ is of finie variaion for an explici example, see Lemma 3.2 below. A sligh modificaion of he proof of Proposiion 3.2 yields dy h = h Xh dm τ + H dµ h Z Z H X h dµ. 3. Z I is unclear, however, wheher he firs and he las erms in he righ-hand side of 3. are G-maringales, even hough i is obvious ha heir sum is a G-maringale. Inheproofofhenex resul, wewillneedhe followingelemenarylemmain whichweimplicily assume ha he inegrals are well defined.

9 8 A. Aksami, M. Jeanblanc and M. Rukowski Lemma 3.. Le Assumpion 3. be valid. If V is a process of finie variaion µ h s dv s =, R +, 3. hen h s Xh s dv s = h s Xh s dv s, R +. Z s Z s Proof. We wrie Y = Y h and X = X h. Noing ha and using 3., we obain h X dv = h X Z = X = µ h h A p = µ h +h Z X dv Z h X X + µ h +h Z X Z Z = h Z X +h Z dv Z = Z h X dv, Z dv which is he desired equaliy. By combining Proposiion 3.2 wih Lemma 3., we obain he following resul yielding an alernaive represenaion for maringales from he class MG, τ. Corollary 3.. Le Assumpion 3. be valid. If hen Y h MG,τ admis he following represenaion dy h = A p µ h s dmτ s =, R +, 3.2 h Xh dm τ + H dµ h Z. 3.3 Proof. From Proposiion 3.2, we know ha 3.8 is valid. If, in addiion, condiion 3.2 is saisfied hen, using Lemma 3., we obain h Xh dm τ = Z h Xh dm τ = Z Z A p h Xh dm τ since M τ is a process of finie variaion and Z = + Z = A p. Hence represenaions 3.8 and 3.3 of Y h are equivalen under he presen assumpions. Remark 3.3. In view of 2.2, o esablish 3.2, i suffices o show ha µ h s da p s = = µ h s dh s, R In fac, he firs equaliy is rue if he filraion F is quasi-lef-coninuous and he second one is saisfied under he avoidance propery. I is also obvious ha boh equaliies are saisfied when all F-maringales are coninuous.

10 Progressive Enlargemens of a Poisson Filraion 9 Le us now assume, in addiion, ha here exiss a d-dimensional F-maringale M, which has he predicable represenaion propery wih respec o he filraion F. The following resul, which is an immediae consequence of Proposiion 3.2, shows ha he d + -dimensional G-maringale M τ,m generaes any G-maringale sopped a ime τ of he form Eh τ G for some process h H o F,τ. Proposiion 3.3. Le Assumpion 3. be valid. If an F-maringale M has he PRP wih respec o F, hen Y h MG,τ admis he following represenaion dy h = h Xh dm τ + H φ h dm 3.5 Z for some F-predicable process φ h. If, in addiion, condiion 3.2 holds hen also dy h = A p h Xh dm τ + H φ h dm. 3.6 Remark 3.4. Le us noe ha his framework was sudied by Kusuoka [2] under he addiional assumpion ha he filraion F is generaed by a Brownian moion. Noe ha if F is a Brownian filraion, hen all F-maringales in paricular, he maringale µ h defined by 3.3 are coninuous, so ha condiion 3.2 is rivially saisfied and hus Proposiion 3.3 is valid when M = W is a Brownian moion. Remark 3.5. I is also worh sressing ha Proposiion 3.3 is no covered by he recen resuls of Jeanblanc and Song [5], where he auhors prove ha if he hypohesis H holds beween F and G and he PRP for he filraion F is valid wih respec o an F-maringale M, hen he following condiions are equivalen: i M τ F τ and he immersion propery beween F and G holds, ii he PRP wih respec o M and M τ holds and he equaliy G τ = G τ is saisfied. For an example of he se-up when he immersion propery beween F and G holds and he propery ha M τ is F τ -measurable is no valid, see he case of he filraion F generaed by a Poisson process, which is examined in Secion The PRP for he Enlargemen of he Poisson Filraion Le N be a sandard Poisson process wih he sequence of jump imes denoed as T n n= and he consan inensiy λ. We ake F o be he filraion generaed by he Poisson process N and we assume, as usual, ha F G. We now denoe by M := N λ he compensaed Poisson process, which is an F-maringale. From he well known predicable represenaion propery of he compensaed Poisson process see, for insance, Proposiion in [6], he F-maringales µ and µ h admi he inegral represenaions µ = + for some F-predicable processes φ and φ h. φ s dm s, µ h = µ h + φ h s dm s 3.7 Remark 3.6. Observe ha condiion 3.2 is saisfied when he filraion F is generaed by a Poisson process N and he random ime τ is independen of F, so ha he immersion propery holds and he Azéma supermaringale Z is a decreasing, deerminisic funcion hence an F-predicable process. Indeed, from 3.7, we deduce ha Moreover, N s dz s = { µ h } { M > } = { N > }. 3.8 <s N s Z s = = Recall ha F τ is he σ-field generaed by F-predicable processes sopped a τ. N s dh s, R +, 3.9

11 A. Aksami, M. Jeanblanc and M. Rukowski where he second equaliy holds since he jumps of Z occur a deerminisic imes. The las equaliy follows from he fac ha N sdh s is non-negaive and ha, due o he independence of N and τ, we have E N s dh s = E N τ = P N = dpτ =. In he remaining par of his secion, we work under he following posulae. Assumpion 3.2. The probabiliy space Ω, G, F, P suppors a random variable Θ wih he uni exponenial disribuion and such ha Θ is independen of he filraion F generaed by he Poisson process N. The random ime τ is given hrough he Cox consrucion, ha is, by he formula τ = inf{ R + : Λ Θ} 3.2 where Λ is an F-adaped, increasing process such ha Λ = and Λ := lim Λ =. Under Assumpion 3.2, he immersion propery holds for he filraions F and G. Furhermore, he Azéma supermaringale Z equals Z = Pτ > F = e Λ and hus i is a decreasing and an F-adaped bu no necessarily F-predicable, process. I is also worh noing ha i may happen, for insance, ha { H > } { N > } see equaion 3.2, so ha he validiy of he second equaliy in 3.4 is no ensured, in general Predicable Azéma s Supermaringale Le us firs consider he siuaion where he process Λ in Assumpion 3.2 is F-predicable. Then we have he following corollary o Proposiion 3.3, which covers, in paricular, he case where τ is independen of N see Remark 3.6. Corollary 3.2. Le Assumpion 3.2 be valid wih an F-predicable process Λ. Then for any process Y h MG,τ represenaions 3.5 and 3.6 hold wih M = N λ. Proof. Under he presen assumpions, he immersion propery holds and he Azéma supermaringale Z is decreasing and F-predicable, so ha Z = A p. I hus suffices o show ha condiion 3.2 holds for he F-maringale µ h. Noe ha as Poisson filraion is quasi-lef-coninuous, he maringale µ h can only jump a oally inaccessible imes. Since Z is predicable, he processes µ h and Z canno jump ogeher and hus he firs equaliy in 3.4 holds. I remains o show ha he second equaliy in 3.4 holds for all R +. We observe ha E N s dh s = P N τ = Θ = θdpθ θ = where he las equaliy holds since, for any fixed θ, he random ime τ is F-predicable and he jump imes of he Poissonprocess are F-oally inaccessible. Since N s dh s is non-negaiveand 3.8 is valid, we conclude ha he equaliy µh s dh s = is saisfied for all R +. The saemen now follows from Proposiion Non-Predicable Azéma Supermaringale We coninue he sudy of he Poisson filraion and he Cox consrucion of a random ime, bu we no longer assume ha he Azéma supermaringale of τ is F-predicable. Hence condiion 3.2 is no saisfied, in general, and hus Corollary 3.2 no longer applies. Despie he fac ha we will sill posulae he immersion propery beween F and G, i seems o us ha a general represenaion resul is raher hard o esablish. Therefore, we pospone an aemp o derive a general resul o he foregoing secion, where we will work wihou posulaing he immersion propery. Our immediae goal is merely o show ha explici represenaions are sill available when he Azéma supermaringale Z is no F-predicable, a leas for some paricular maringales from he

12 Progressive Enlargemens of a Poisson Filraion class MG, τ. In conras o he preceding subsecion, hese represenaions are derived by means of direc compuaions, raher han hrough an applicaion of he predicable represenaion propery of he compensaed Poisson process. Hence we will be able o compue explicily he inegrand φ h arising in suiable varians of represenaion 3.5. For he sake of concreeness, we also specialize Assumpion 3.2 by posulaing ha Λ = N. Assumpion 3.3. The probabiliy space Ω, G, F, P suppors a random variable Θ wih he uni exponenial disribuion and such ha Θ is independen of he filraion F generaed by he Poisson process N. The random ime τ is given by he formula τ = inf{ R + : N Θ}. 3.2 Under Assumpion 3.3, he Azéma supermaringale equals Z = e N for all R + and hus i is decreasing, bu no F-predicable. As in he preceding subsecion, he filraion F is immersed in G and hus he compensaed Poisson F-maringale M is also a G-maringale. I is crucial o observe ha he inclusion [τ] n [T n ] holds, meaning ha a jump of he process H may only occur when he Poisson process N has a jump, ha is, { H > } { N > }. We firs compue explicily he Doob-Meyer decomposiion of Z and we show ha he compensaor of H is coninuous. Lemma 3.2. Le Z = e N where N is he Poisson process. Then he following asserions hold: i Z admis he Doob-Meyer decomposiion Z = µ A p where and γ = e > ; µ = γe Ns dm s, A p = γλe Ns ds 3.22 ii he process M τ := H γλ τ is a G-maringale and hus he random ime τ given by 3.2 is a oally inaccessible G-sopping ime. Proof. The proof of pari is elemenary, since i relies on he sandard Sieljes inegraion and hus is is omied. For he second par, he G-maringale propery of he process M τ is a consequence of 2.2, and he fac ha τ is a oally inaccessible follows from he coninuiy of he compensaor A p of H. Recall ha he process X h is defined by 3.2 and noe ha, due o he form of A p, { M τ > } = { H > } { N > } = { M > } I is also worh sressing ha here he equaliy M τ = M τ is no saisfied see Remark 3.5. The following resul shows ha when F is he Poisson filraion, he immersion propery holds, bu he Azéma supermaringale Z is no F-predicable, hen an exension of Proposiion 3.3 is sill feasible in some circumsances. Proposiion 3.4. Le Assumpion 3.3 be valid. Consider he G-maringale Y h MG,τ where he process h H p F,τ is given by h = hn for some Borel funcion h : R R. Then Y h admis he following represenaion or, equivalenly, dy h = h hn + dm τ + H hn + hn dm 3.24 dy h = h hn dm τ + H hn + hn τ d M 3.25 where he processes M τ := H γλ τ and and he funcion h is given by 3.28 M τ := M M τ are orhogonal G-maringales

13 2 A. Aksami, M. Jeanblanc and M. Rukowski Proof. We wrie X = X h and Y = Y h. In view of Lemma 3.2 and equaion 3., we have Y = h s dh s + H E Z Using he independence of incremens of he Poisson process N, we obain where X = E Equaions yield γλh s e Ns ds F γλhn s e Ns ds F = hn e N = hn Z 3.27 hx := E Y = and hus, since { H > } { N > }, γλhn s +xe Ns ds h s dh s + H hn dy = h hn dh + H d hn H hn = h hn dh + H hn + hn dn hn + hn dh = h hn + dh + H hn + hn dn. Consequenly, dy = h hn + dm τ + H hn + hn dm + H λγ hn hn + d+ H λ hn + hn d. To complee he proof, i suffices o show ha he following equaliy is saisfied for all x To esablish 3.29, we firs observe ha hx+ = E γhx+ γ hx+ hx = γλhn s +x+e Ns ds = ee γλhn s +x+e Ns+ ds. If we denoe by T he momen of he firs jump of N, hen we obain hx = E T hn s +xe Ns ds = E = γhx+e γλhn s +x+e Ns+ ds = γhx+e hx+ = γhx+ γ hx+. γλhx ds +E γλhn s +xe Ns ds T We conclude ha 3.29 holds and hus he proof of 3.24 is compleed. Represenaion 3.25 is an easy consequence of equaion 3.24 and hus we omi he deails. Le us finally observe ha he orhogonaliy of M τ and M τ follows from he fac ha M τ and N are pure jump maringales wih jumps of size. Remark 3.7. Proposiion 3.4 will be revisied in Secion 4.2 see Example 4., where we will re-derive represenaion 3.25 using he general represenaion formula.

14 Progressive Enlargemens of a Poisson Filraion 3 4 The PRP Beyond he Immersion Hypohesis In his secion, we work wih he filraion F generaed by a Poisson process N, bu we no longer posulae ha he immersion propery beween F and G holds. Recall ha he equaliy Z = µ A p is he Doob-Meyer decomposiion of he Azéma supermaringale Z associaed wih a random ime τ. I is worh noing ha he Azéma supermaringale of any random ime τ is necessarily a process of finie variaion when he filraion F is generaed by a Poisson process. We argue ha he main difficuly in esablishing he PRP for a progressive enlargemen is due o he fac ha he jumps of F-maringales may overlap wih he jump of he process H. I appears ha even when he filraion F is generaed by a Poisson process N, he validiy of he PRP for he progressive enlargemen of F wih a random ime τ is a challenging problem for he par afer τ if no addiional assumpions are made. Since he inclusion { H > } { N > } may fail o hold, in general, i is hard o conrol a possible overlap of jumps of processes N and H when he only informaion abou he F-condiional disribuion of he random ime τ is is Azéma supermaringale Z. Neverheless, in he main resul of his secion Theorem 4. we offer a general represenaion formula for a G-maringale sopped a τ. I is fair o acknowledge ha we need o inroduce for his purpose an addiional maringale o compensae for a poenial mismach of jumps of H and N. Subsequenly, we illusrae our general resul by considering some special cases. We conclude his noe by emphasizing he role of he opional sochasic inegral in our general represenaion resul and by obaining in Corollary 4. an equivalen represenaion of Y h in erms of he opional and predicable sochasic inegrals. Remark 4.. From par i in Proposiion 2., we deduce ha he compensaed maringale M := N λ sopped a τ admis he following semimaringale decomposiion wih respec o G M = M τ τ d M,m F s. 4. Noe ha he compensaed Poisson process M = N λ is an F-adaped hence also G-adaped process of finie variaion, so ha i is manifesly a G-semimaringale for any choice of a random ime τ. Consequenly, due o he PRP of he compensaed Poisson process wih respec o is naural filraion, no addiional assumpions regarding he random ime τ are needed o ensure ha he hypohesis H is saisfied, ha is, any F-maringale is always a G-semimaringale. 4. Main Resul Weareinaposiionoprovehemainresulofhisnoe. IshouldbesressedhaheG-maringales M τ and M in he saemen of he following Theorem 4. are universal, in he sense ha hey do no depend on he choice of he process h H p F,τ see equaions 2.2 and 4.. By conras, he G-maringale M h, which is defined by 4.3, is clearly dependen on h. In he nex subsecion, we presen some special cases of represenaion esablished in Theorem 4. in which he process M h does no appear, despie he fac ha hey are valid for any process h H o F,τ. In he saemen of Theorem 4., we will use he following lemma, which exends a resul quoed in Jeulin [7] see Remark 4.5 herein or equaion 2. wih U =. Noe ha equaion 2.2 can be obained as a special case of 4.2 by seing ξ = κ =. Lemma 4.. Le he process B be given by he formula B = ξh where ξ is an inegrable and G τ -measurable random variable. Then he process M, which is given by he equaliy M = B H s db p,f s, 4.2 is a purely disconinuous G-maringale sopped a τ. Moreover, he dual F-predicable projecion of B saisfies B p,f = κ sda p s where κ is an F-predicable process such ha he equaliy κ τ = Eξ F τ holds.

15 4 A. Aksami, M. Jeanblanc and M. Rukowski Proof. Le he process B be given by B = ξh, where he inegrable random variable ξ is G τ - measurable, and le B p,f be is dual F-predicable projecion. On he one hand, we have, for any u, On he oher hand, we obain EB u B G = Eξ {u τ>} G = {τ>} Z Eξ {u τ>} F u H s E = {τ>} Z EB p,f u B p,f F. db p,f s We define he F-predicable process Λ by seing Λ = u τ E {τ>} u τ G = E {τ>} db p,f s G u τ = {τ>} E {τ>} db p,f s F. Z u dbs p,f F = E {τ>u} u = E Z u = E Z u Λ u Λ + = E Z u Λ u Λ db p,f s. Then we obain db p,f s + {u τ>} τ db p,f = E B p,f u B p,f F u v s + u u Λ s Λ da p s db p,f F Λ s Λ dz s F where he las equaliy is a consequence of he following elemenary compuaion Z u Λ u Λ u Λ s Λ dz s = Z u Λ u Λ +Λ Z u Z = Z u Λ u Λ Z Λ u Z u +Λ Z + u dλ s = B p,f u B p,f. dbs p,f F F s da p v u Λ s dz s Tha complees he proof of he firs saemen in Lemma 4.. For he second saemen, we noe ha, from he definiion of he σ-algebra F τ, he equaliy Eξ F τ = κ τ holds for some F- predicable process κ. Hence for any bounded, F-predicable process X, we obain noe ha X τ is F τ -measurable E X s dbs p,f = E = EX τ κ τ = E since A p = H p,f. We conclude ha B p,f X s db s = E ξx s dh s = EξX τ = E X τ Eξ F τ X s κ s dh s = E X s κ s da p s, = κ sda p s for all R +. The following heorem esablishes he inegral represenaion wih predicable inegrands for an arbirary G-maringale Y h associaed wih a process h from he class H p F,τ. Theorem 4.. If he process h belongs o he class H p F,τ, hen he G-maringale Y h sopped a τ admis he following predicable represenaion dy h = A p h Xh dm τ + H φ h +φ φ X h d M + +φ d M h

16 Progressive Enlargemens of a Poisson Filraion 5 where he F-predicable processes φ and φ h are given by 3.7 and he G-maringale M h equals M h := Bh H s where B h := ξ h H and he G τ -measurable random variable ξ h is given by db h,p,f s 4.3 ξ h := µ τ X h τ Z τ µ h τ. 4.4 Proof. As usual, we wrie X = X h and Y = Y h. By proceeding as in he proof of Proposiion 3.2 and recalling ha dz = dµ da p, we obain dy = h X dm τ + H dµ h Z X X Z X dµ H H. Z Z Recall ha Z = µ A p and X = µ h h A p. Moreover, he F-maringales µ and µh saisfy 3.7, so ha { µ } { N > } and { µ h } { N > }. Since he process A p is F-predicable and he jump imes of N are F-oally inaccessible, we obain Therefore, X Z and hus also X Z = X X Z Z µ A p = µh Ap =. = µ h X µ h X A p +φ A p Z = µ h X µ µ + h X A p da p +φ A p. Consequenly, using he definiion of M τ, we obain dy = h X A p + A p Le us se + H dm τ + ξ h +φ dh dµ h X dµ µ +φ µ h + µ +φ X µ U := X s dµ s µ h so ha τ U = ξ h. By applying Corollary 2. o he F-local maringale U, we obain and hus where H = H d ξ h H p,f dy = A p = H d U,m µ F X d µ,m F X d µ,µ F d µh,m F +d µh,µ F. h X dm τ + d M h +φ + H d K 4.5 +φ d K := +φ dµ h Z d µh,m F µ µ h +d µh,µ F + H +φ X +φ dµ +d µ,m F + µ 2 d µ,µ F = +φ φ h dm φ h d M,m F d[µ,µh ] +d µ h,µ F X Z 2 +φ φ dm φ d M,m F d[µ,µ] + µ,µ F

17 6 A. Aksami, M. Jeanblanc and M. Rukowski since, in view of 3.7, we have d µ µ h = d[µ,µh ] and d µ 2 = d[µ,µ]. To obain he assered formula from equaion 4.5, i remains o show ha d K = φ h X h φ d M where M is given by 4.2. Using again 3.7, we ge so ha finally d[µ,µ h ] d µ h,µ F = φ h φ dm, d[µ,µ] d µ,µ F = φ 2 dm, d K = +φ φ h Z dm φ h d M,m F φh φ dm X +φ φ dµ φ d M,m F φ 2 dm = Z 2 φ h X h φ d M, which is he desired equaliy. Hence he proof of he proposiion is compleed. Remark 4.2. Le us se H = N sdh s and σ = inf{ : H = } so ha H = {σ } and he jump imes of he process H H are disjoin from he sequence T n. The G-adaped, increasing process H sopped a τ admis a G-predicable compensaor and hus here exiss an F-predicable, increasing process Λ such ha he process M := H Λ τ is a G-maringale sopped a τ. Since { µ h } { N > } and { µ } { N > }, i is clear ha and hus µ h τ µ τ X h τ Z τ dh = d M h = db H µ h τ µ τ X h τ Z τ dh = φ h φ db h,p,f = φ h X h φ dm. X h dh Unforunaely, an explici formula for he G-compensaor Λ is no available, in general. However, his argumen formally gives he PRP for he riple M τ, M,M of G-maringales, meaning ha any process Y h MG,τ can be represened as follows dy h = A p h Xh dm τ + H +φ φ h φ X h I is worh sressing ha none of he processes M τ, M,M depends on h. d M +dm. 4.2 Varians of he Predicable Represenaion Formula We presen here some special cases of he inegral represenaion for he process Y h, which can be deduced from Theorem 4.. i Suppose firs ha τ is independen of he naural filraion F of a Poisson process N. Then N H = and hus also µ h τ = µ τ =. Then he random variable ξ h given by 4.4 saisfies ξ h =, so ha B h = for any process h H o F,τ. Since µ =, we have φ = and hus for any process Y h MG,τ, we obain dy h = A p h Xh dm τ + H φ h Z d M. 4.6 Due o he posulaed independence of τ and F, he Azéma supermaringale Z is a decreasing deerminisic funcion, so ha i is F-predicable. Recall ha his se-up was also covered by

18 Progressive Enlargemens of a Poisson Filraion 7 Corollary 3.2. As expeced, represenaion 4.6 coincides wih equaion 3.6, since under he presen assumpions he equaliy M = M holds. ii Le us now assume ha a random ime τ avoids all F-sopping imes, ha is, Pτ = σ = for any F-sopping ime σ. Then N H = and hus B h = for any process h H o F,τ. Moreover, he process A p in he Doob-Meyer decomposiion of Z is known o be coninuous. Therefore, any process Y h MG,τ saisfies dy h = h Xh dm τ + H φ h X h φ d M φ Hence he mulipliciy of he filraion G wih respec o maringales sopped a τ equals wo, where by he mulipliciy of he filraion G, we mean here he minimal number of muually orhogonal maringales needed o represen all maringales sopped a τ as sochasic inegrals. For he concep of mulipliciy of a filraion, see Davis and Varaiya [7] and he survey paper by Davis [6] and he references herein. iii Under he assumpion ha he graph of he random ime τ is included in n [T n ], ha is, when { H > } { N > }, we obain and hus db h = µ h X h τ τ µ τ dh = φ h X h φ dh Z τ d M h = db h H db h,p,f = φ h X h φ dm τ. Consequenly, any process Y h MG,τ admis he following represenaion dy h = A p h Xh dm τ + H φ h X h φ d M τ 4.8 +φ where he G-maringales M τ and M τ := M M τ are orhogonal. We hus see ha he mulipliciy of he filraion G wih respec o maringales sopped a τ is equal o wo. Example 4.. To illusrae par iii, we will re-examine he se-up inroduced in Secion Recall ha i was assumed in Secion 3.2.2ha h = hn,z = e N and he immersion propery holds, so ha he equaliy M = M holds. From 3.22 and he equaliy see 3.27 µ h = Xh + h s da p s = hn Z γλhn s Z s ds we deduce ha he processes φ and φ h appearing in 3.7 are given by φ = γe N = γ, φ h = γ hn + hn where h is given by By subsiuing hese processes ino 4.8, we obain recall ha, from Lemma 3.2, we have A p = dy h = h Xh dm τ + H γ hn + hn +γ hn d M M τ γ = h hn dm τ + H hn + hn d where M τ := M M τ = M M τ =: M τ. This resul coincides wih represenaion 3.25, which was previously esablished in Proposiion 3.4 by means of more direc compuaions. M τ

19 8 A. Aksami, M. Jeanblanc and M. Rukowski 4.3 Opional Represenaion Formula for he Poisson Filraion In his final subsecion, we make use of he opional sochasic inegral, in order o derive an inegral represenaion for any process Y h in erms of he G-maringales M τ and M see Corollary 4.. In essence, he idea of an opional sochasic inegral is o exend he noion of he Iô sochasic inegral from predicable o opional inegrands by ensuring ha he inegral of an opional inegrand wih respec o a local maringale is uniquely defined and follows a local maringale. I is worh noing ha he opional sochasic inegral was used in he recen paper of Aksami e al. [2] in he conex of arbirage properies of a financial model endowed wih a progressively enlarged filraion. Several alernaive approaches o he opional sochasic inegral were proposed in he lieraure see, e.g., [23, 26, 27]. We follow here he exposiion presened in Chaper III of Jacod [] where he opional sochasic inegral is inroduced as a special case of a sochasic inegral wih respec o a random measure. Le G be an arbirary filraion saisfying he usual condiions and le X be an arbirary G-local maringale null a ime. The ineger-valued random measure µ X on Ω R + R associaed wih he jumps of X is given by he following expression see Example 3.22 in [] µ X d,dx := s> { Xs }δ s, Xsd,dx. 4.9 We denoe by ν X he dual G-predicable projecion of he random measure µ X ; he exisence of ν X is esablished in Theorem 3.5 in []. Le W : Ω R + R R and V : Ω R + R R be some mappings. For any mapping W G loc µx resp., V Hloc µx, we denoe by W µ X ν X resp., V µ X he sochasic inegral of he firs resp., second kind wih respec o a random measure µ X ν X resp., µ X, which is given by Definiion 3.63 resp., Definiion 3.73 in []. For he definiions of he spaces G loc µx and Hloc µx of inegrands, he reader is referred o pages 98 and in [], respecively. We will need he propery ha, by definiion, he inegrals W µ X ν X and V µ X belong o he space M d loc of purely disconinuous G-local maringales. To be a bi more specific, for any process V Hloc µx, he inegral V µ X is defined as a unique process from M d loc such ha where D := { X }. V µ X = V, X D, R +, 4. For he definiion and properies of he opional sochasic inegral wih respec o a local maringale X, which is denoed hereafer as K X for any process K belonging o he space o L loc X of opional inegrands, he reader is referred o Jacod [] see pages 6 8 herein. Le us only menion here ha, for any process K o L loc X, he opional sochasic inegral K X is he unique local maringale such ha K X c = p,f K X c, K X = K M p,f K M, 4. where, as usual, X c sands for he coninuous maringale par of X. Noe ha when K is an G-predicable process, hen he condiions above reduce o he following condiions K X c = K X c, K X = K M, which are known o uniquely characerize he classic concep of predicable sochasic inegral see Definiion 2.46 in []. The nex resul, which is merely a resaemen of Theorem 3.84 from Jacod [], furnishes a link beween he opional sochasic inegral wih respec o a G-local maringale X and sochasic inegrals wih respec o he associaed random measures µ X ν X and µ X. Theorem 4.2. i Le X be a G-local maringale null a ime and le µ X be he associaed random measure given by 4.9. Then he se of opional sochasic inegrals { K X : K o L loc X}

20 Progressive Enlargemens of a Poisson Filraion 9 coincides wih he following se { U X c +W µ X ν X +V µ X : U L locx c, W G locµ X, V H locµ X }. ii Le U L loc Xc, W G loc µx, V H loc µx and le he process K be given by he following expression K = U Dc J c+ X W, X +V, X D where J := {ν X,R > }. Then K belongs o he space o L loc X of opional inegrands and hus he opional sochasic inegral K X is well defined. Moreover, he following equaliy holds K X = U X c +W µ X ν X +V µ X. In our applicaion of Theorem 4.2, we consider he progressive enlargemen of he Poisson filraion and we focus on he G-maringale M h given by equaion 4.3 or, equivalenly, by he following expressions M h = H s ξh H κ h s Z dap s = ξh κ h τ H s κ h s dmτ s 4.2 where he random variable ξ h is given by 4.4 and κ h is an F-predicable process such ha he equaliy κ h τ = Eξ h F τ holds. I is clear from 4.3 and 4.2 ha he process M ξ := ξ h κ h τh is a purely disconinuous G-maringale. Our goal is o derive he opional inegral represenaion of M ξ wih respec o M τ. Recall ha we denoe by Z = µ A p he Doob-Meyer decomposiion of he Azéma supermaringale Z of τ. Le S n be a sequence of F-predicable sopping imes exhausing he jumps of he F-predicable, increasing process A p. Since M τ = H H A p, i is clear ha he random measure µ τ := µ Mτ associaed wih he jumps of he G-maringaleM τ equals µ τ d,dx = δ τ, d,dx+ {Sn τ}δ Sn, Z Sn A p d,dx. Sn n Hence he dual G-predicable projecion ν τ of µ τ is given by he following expression ν τ d,dx = δ dx H da p + n {Sn τ}δ Sn, Z Sn A p Sn d,dx. Le us firs consider he special case where he process A p is assumed o be coninuous. Lemma 4.2. Assume ha he process A p is coninuous, so ha he G-maringale M τ is quasilef-coninuous. Then M ξ = L h M τ where L h = M ξ. Proof. To alleviae noaion, we wrie ξ = ξ h and κ = κ h in he proof. The asserion follows immediaelyfromtheorem4.2appliedox = M τ, asξ κ τ H = V µ τ wihvs,x = ξ κ τ H s since µ τ d,dx = δ τ, d,dx when A p is coninuous. The propery ha V H loc µx is a consequence of he he equaliy κ τ = Eξ F τ. Alernaively, one can esablish he lemma direcly. To his end, we observe ha he maringale M ξ M τ has a null coninuous maringale par and he jump process given by ξ κ τ H p,g ξ κ τ H = ξ κτ H since τ is a oally inaccessible G-sopping ime. The process M ξ is a purely disconinuous G- maringale wih he same jump process and hus we conclude ha M ξ = M ξ M τ. In he nex resul, he coninuiy assumpion for A p is relaxed.

21 2 A. Aksami, M. Jeanblanc and M. Rukowski Lemma 4.3. We have M ξ = L h M τ where he process L h equals, for all R +, L h = Mτ Z τ + A p τ M τ Z τ + A p τ ξh κ h τ H { M τ }. 4.3 Proof. As before, we denoe ξ = ξ h and κ = κ h. Le us se Vs,x := xz τ + A p τ Z τ + A p τ ξ κ τ H s. Then ξ κ τ H = V µ τ, since V µ τ = Vτ,H + n V S n, Ap S n {Sn τ} {Sn } = ξ κ τ H. Z Sn The assered equaliy 4.3 can now be deduced from Theorem 4.2. Once again, one can check ha V Hloc µx since κ τ = Eξ F τ. The final resul of his noe shows ha he opional and predicable sochasic inegrals can be coupled in order o obain an inegral represenaion of any G-maringale sopped a τ in erms of wo universal maringales: he G-maringale M τ associaed wih τ and he G-maringale par M of he compensaed F-maringale M of he Poisson process N. To esablish Corollary 4., i suffices o combine Theorem 4. wih Lemma 4.3. Corollary 4.. For any process Y h MG,τ, he following inegral represenaion is valid Y h = K h M τ + H φ h φ X Z +φ Z M where he process K h equals K h = A p h X +L h where he process L h is given by Opional Decomposiion Propery Represenaions 2.3 and 2.4 can be ermed predicable decomposiions. I appears ha a paricular opional decomposiion is also available see 4.4. Recall ha Z := Pτ F. Le he F-sopping ime R be given by 2 R := R { ZR=<Z R } = inf{ > : Z =, > }. For any process X, we denoe J := H X R p,f where H := [ R, [. The following opional decomposiion resul was esablished by Aksami [] see Theorem 7. herein. Proposiion 4.. For any F-local maringale X, he process X given by X := X τ is a G-local maringale sopped a ime τ. τ 2 For any even C and any random ime σ, we se σ C := σ C + {Ω\C}. Z s d[x,m] s + J τ 4.4

22 Progressive Enlargemens of a Poisson Filraion 2 Remark 4.3. Afer compleing his paper, we learn abou he innovaive work by Choulli e al. [5] where he auhors inroduced he following G-maringale M τ τ da o s := H Z s and esablished several decomposiions of a process Y h MG,τ based, in paricular, on represenaions of some G-maringalesas Lebesgue-Sieljes inegrals wih respec o M τ and wih opional inegrands. Acknowledgemens. The research of Anna Aksami was suppored by he Chaire Marchés en Muaion Fédéraion Bancaire Française and by he European Research Council under he European Union s Sevenh Framework ProgrammeFP7/27-23/ ERC gran agreemen no The research of Monique Jeanblanc was suppored by he Chaire Marchés en Muaion Fédéraion Bancaire Française. The research of Marek Rukowski was suppored by he Chaire Marchés en Muaion Fédéraion Bancaire Française and he Ausralian Research Council s Discovery Projecs funding scheme DP2895. References [] Aksami, A.: Random imes, enlargemen of filraion and arbirages. PhD hesis, Universié d Evry Val d Essonne, 24. [2] Aksami A., Choulli T., Deng J., and Jeanblanc M.: On arbirage sabiliy up o random horizons and afer hones imes. Working paper, 22. [3] Aksami A. and Li, L.: Pseudo-sopping imes and he immersion propery. Working paper, 24 arxiv: [4] Azéma, J.: Quelques applicaions de la héorie générale des processus I. Inveniones Mahemaicae 8 972, [5] Choulli, T., Daveloose, C. and Vanmaele, M. : Hedging moraliy risk and opional maringale represenaion heorem for enlarged filraion. Working paper arxiv: [6] Davis, M.H.A.: Maringale represenaion and all ha. In: Advances in Conrol, Communicaion Neworks, and Transporaion Sysems, Birkhäuser Boson, 25, pp [7] Davis, M.H.A. and Varaiya, P.: On he mulipliciy of an increasing family of sigma-fields. Annals of Probabiliy 2 974, [8] Ellio, R.J., Jeanblanc, M. and Yor, M.: On models of defaul risk. Mahemaical Finance 2, [9] Guo, X. and Zeng, Y.: Inensiy process and compensaor: A new filraion expansion approach and he JeulinYor heorem. Annals of Applied Probabiliy 8 28, [] He S., Wang, J. and Yan, J.: Semimaringale Theory and Sochasic Calculus. Science Press and CRS Press, Beijing and Boca Raon, 992. [] Jacod, J.: Calcul sochasique e problèmes de maringales. Lecure Noes in Mahemaics 74, Springer, Berlin Heidelberg New York, 979. [2] Jeanblanc, M. and Le Cam, Y.: Progressive enlargemen of filraion wih iniial imes. Sochasic Processes and heir Applicaions 9 29, [3] Jeanblanc, M. and Song, S.: An explici model of defaul ime wih given survival probabiliy. Sochasic Processes and heir Applicaions 2 2,