CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS

Transcription

1 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS Aa Aksami, Tahir Choulli, Moique Jeablac To cie his versio: Aa Aksami, Tahir Choulli, Moique Jeablac. CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS. 26. <hal-36246> HAL Id: hal hps://hal.archives-ouveres.fr/hal Submied o 6 May 26 HAL is a muli-discipliary ope access archive for he deposi ad dissemiaio of scieific research documes, wheher hey are published or o. The documes may come from eachig ad research isiuios i Frace or abroad, or from public or privae research ceers. L archive ouvere pluridiscipliaire HAL, es desiée au dépô e à la diffusio de documes scieifiques de iveau recherche, publiés ou o, émaa des éablissemes d eseigeme e de recherche fraçais ou éragers, des laboraoires publics ou privés.

2 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS ANNA AKSAMIT, TAHIR CHOULLI AND MONIQUE JEANBLANC arxiv:65.395v mah.pr 2 May 26 Absrac. The paper gahers ogeher ideas relaed o hi radom ime, i.e., radom ime whose graph is coaied i a hi se. The cocep aurally complees he sudies of radom imes ad progressive elargeme of filraios. We develop classificaio ad ( )- decomposiio of radom imes, which is aalogous o he decomposiio of a soppig ime io oally iaccessible ad accessible pars, ad we show applicaios o he hypohesis (H ), hoes imes ad iformaioal drif via eropy.. Iroducio The paper develops ideas relaed o hi radom imes. The cocep aurally fis ad complees he sudies of radom imes ad progressive elargeme of filraios. A radom ime defied o a filered probabiliy space (Ω,G,F,P) is simply a radom variable wih values i,. I he lieraure of elargeme of filraio, i is commo o assume ha a give radom ime τ avoids all F-soppig imes, i.e., P(τ = T < ) = for ay F-soppig ime T. Here, we ake a closer look a his codiio ad make i a sarig poi o defie wo classes of radom imes. We come up wih sric radom imes which basically saisfy avoidace codiio ad hi radom imes which saisfy couer propery, i.e., hey are fully buil of F-soppig imes. The oio of hi radom ime was meioed, bu o developed, for he firs ime i Dellacherie ad Meyer uder he ame variable aléaoire arlequie referrig o he cosume of he Harlequi which is made of paches of differe colors. We begi, i Secio 2, wih defiig ad sudyig ( )-decomposiio of a radom ime io sric ad hi pars. The ( )-decomposiio is cogrue wih he decomposiio of a soppig ime io oally iaccessible ad accessible pars. Our sudy srogly relies o he oio of dual opioal projecio, we make use of oher processes liked o geeral heory of sochasic processes, i paricular, o elargeme of filraio heory. The mai resul, saed i Theorem 2.4, says ha ay radom ime ca be ( )-decomposed ad gives sric ad hi radom ime characerisaios i erms of is dual opioal projecio. Dae: May 3, 26. Key words ad phrases. hi radom imes, avoidace of soppig ime, dual opioal projecio, progressive elargeme of filraio, hypohesis (H ), hoes imes, eropy. AA ad MJ wish o ackowledge he geerous fiacial suppors of Chaire Marchés e Muaio, Fédéraio Bacaire Fraçaise. AA wishes o ackowledge he suppor of he Europea Research Coucil uder he Europea Uio s Seveh Framework Programme (FP7/27-23)/ ERC gra agreeme o The research of TC is suppored by he Naural Scieces ad Egieerig Research Coucil of Caada (NSERC), hrough Gra RGPIN 4987.

3 2 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC I Secio 3 we relae o progressive elargeme of filraio ad he hypohesis (H ). For a radom ime τ, F τ is he filraio F progressively elarged wih τ, i.e., (.) F τ := s>(f s σ(τ s)) for ay. The hypohesis (H )holds for(f,f τ ) ifay F-marigaleisaF τ -semimarigale. The mai resul here is Theorem 3. where we esablish he hypohesis (H ) for hi radom imes. I exeds previous resuls by Jeuli 7 which deal wih couably valued radom imes. Isead of couably may real values we chose couably may F-soppig imes which are already capured i referece filraio F. We may see Theorem 3. as a aleraive direcio of developme of he resul by Jacod (see 7, Theorem 3,2 ad 2) o he hypohesis (H ) i iiial elargeme wih aomic σ-field o he direcio based o desiy hypohesis i progressive seig (iiial imes from 5). Hoes imes are impora ad a well sudied class of radom imes; roughly speakig hey are las passage imes ad we refer o hem i Secio 4. Adopig he oio of jumpig filraio from Jacod ad Skorokhod 4 we show i Theorem 4.8 ha such a filraio ca oly suppor hoes imes which are hi. Tha icludes compoud Poisso process filraio. I4 he lik bewee jumpig filraio ad fiie variaio marigales is esablished; problems relaed o purely discoiuous marigale filraios are reaed i Haig 2. I Secio 4 we also exploi wo examples of hi hoes imes: las passage ime a a barrier a of a Compoud Poisso process ad a example based o Browia moio local ime approximaio. Some auxiliary resuls o hoes imes are colleced i Appedix B. The addiioal iformaio carried by elarged filraio ad is measureme was sudied by several auhors. Already i Meyer 2 ad Yor 25, he quesio o sabiliy of marigale spaces wih respec o iiial elargeme wih aomic σ-field was asked. From more rece sudies, geeralizig ad applyig previous resuls i differe coexs, we refer he reader o 3, 4, 5, 6. Here we defie he eropy of a hi radom ime by (5.2) ad we prove i Theorem 5.2 ha is fiieess is eough for sabiliy of some marigale spaces i progressive seig. Theorem 5.2 reveals ha he oio of he eropy of hi radom ime is a correc oe ad is a aswer o he quesio asked i 2 abou addiioal kowledge associaed wih a pariio ad disclosed i progressive maer: U problème voisi, mais plus iéressa peu-êre, cosise à mesurer le bouleverseme produi, sur u sysème probabilise, o pas e força des coaissaces à l isa, mais e les força progressiveme das le sysème. I Secio 6 we collec i Theorems 6.2 ad 6.3 he resuls cocerig he hypohesis (H ) ad eropy for more geeral progressive elargeme of filraio. I corary o previous resuls, addig several members of a pariio a he same ime is allowed i his case. For he reader s coveiece we provide a Eglish raslaio: A similar problem, bu perhaps of more ieres, cosiss i measurig he resulig perurbaio, i a probabilisic sysem, o by requirig kowledge a he isa, bu by addig hem progressively o he sysem.

4 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 3 For ay càdlàg process X we will deoe by X he lef-coiuous versio of X, by X he jump of X ad by X he limi lim X if i exiss. The process X is said o be icreasig if for almos all ω i saisfies X (ω) X s (ω) for all s. The radom variable is said o be posiive if i has values i, ). 2. The ( )-decomposiio Le (Ω,G,F,P) be a filered probabiliy space, where F := (F ) deoes a filraio saisfyig he usual codiios, ad such ha F G. Cosider a radom ime τ, i.e., a radom variable wih values i,. Noe ha a radom ime τ is o ecessarily F -measurable. For a radom ime τ we deoe by τ is graph. The followig defiiio coais he leadig idea of he paper. I discrimiaes wo classes of radom imes usig a crierio based o F-soppig imes. Defiiio 2.. A radom ime τ is called (a) a sric radom ime if τ T = for ay F-soppig ime T, i.e., if i avoids all F-soppig imes. (b) a hi radom ime if is graph τ is coaied i a hi se, i.e., if here exiss a sequece of F-soppig imes (T ) = wih disjoi graphs such ha τ T. We say ha such a sequece (T ) exhauss he hi radom ime τ or ha (T ) is a exhausig sequece of he hi radom ime τ. Noe ha a hi radom ime τ is buil of F-soppig imes, i.e., τ = C + T C where (T ) is a exhausig sequece for τ, ad (2.) C := {τ = } ad C := {τ = T < } for. We deoe by z he càdlàg F-marigale wih ermial value P(C F ), amely (2.2) z := P(C F ). Le us also remark ha a exhausig sequece (T ) of a hi radom ime is o uique. The sraighforward observaio ha he wo classes of radom imes have rivial iersecio is saed i he followig lemma. Lemma 2.2. A radom ime τ belogs o he class of sric radom imes ad o he class of hi radom imes if ad oly if τ =. The mai cocep of his secio, he ( )-decomposiio, is preseed i he ex defiiio. I is followed by he resul saig he exisece of such a decomposiio for ay radom ime ad some equivale characerisaios of wo classes of radom imes. Defiiio 2.3. Cosider a radom ime τ. A pair of radom imes (τ,τ 2 ) is called a ( )-decomposiio of τ if τ is a sric radom ime, τ 2 is a hi radom ime, ad τ = τ τ 2 τ τ 2 =.

5 4 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC Before preseig Theorem 2.4 le us recall, followig7, some useful processes associaed wih τ. For he process A := τ,, we deoe by A p is F-dual predicable projecio ad by A o is F-dual opioal projecio (see Appedix A). By he abuse of laguage, A o is also called he dual opioal projecio of he radom ime τ. We also defie wo F- supermarigales Z ad Z as opioal projecios of process A ad A respecively, i.e., Z := o,τ = P(τ > F ) ad Z := o,τ = P(τ F ). Sice he dual opioal projecio A o will play a crucial role i he paper, we recall wo equaliies where i appears: (2.3) A o = m Z ad A o = Z Z, where m is a BMO F-marigale. Furhermore, Z = Z + m. Theorem 2.4. (a) Ay radom ime τ has a ( )-decomposiio (τ,τ 2 ) which is uique o he se {τ < }. (b) A radom ime is a sric radom ime if ad oly if is dual opioal projecio is a coiuous process. (c) A radom ime is a hi radom ime if ad oly if is dual opioal projecio is a pure jump process. Proof. (a) I is eough o ake τ ad τ 2 of he followig form τ = τ { A o τ =} ad τ 2 = τ { A o τ >}, where τ C is he resricio of he radom ime τ o he se C, defied as τ C = τ C + C c. Properies ofdual opioalprojecio esure haτ adτ 2 saisfy herequired codiios. More precisely, he ime τ is a sric radom ime as, for ay F-soppig ime T, P(τ = T < ) = E {τ=t} { A o τ =} (T< ) = E =. ad he ime τ 2 is a hi radom ime as {u=t} { A o u =}da o u τ 2 = τ { A o > } = τ T T, where he sequece (T ) exhauss he jumps of he càdlàg process A o, i.e., { A o > } = T. (b) Le T be a F-soppig ime. Sice E( A o T {T< } ) = P(τ = T < ) ad A o is a icreasig process we deduce ha P(τ = T < ) = if ad oly if A o T {T< } = P-a.s.. Sice { A o > } is a opioal se, opioal secio heorem 3, Theorem 4.7 implies ha { A o > } is exhaused by disjoi graphs of F-soppig imes. Thus, we coclude

6 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 5 ha τ is a sric radom ime if ad oly if A o is coiuous. (c) For (T ) a sequece of F-soppig imes wih disjoi graphs, we have P(τ = T < ) = E( A o T {T< }). Sice E(A o ) = P(τ < ), by defiiio of he dual opioal projecio ad usig he fac ha A o is a icreasig process, we coclude ha he sequece (T ) saisfies he codiio P(τ = T < ) = P(τ < ) if ad oly if i saisfies he codiio E(A o ) = E( Ao T {T< }). I oher words, τ is a hi radom ime if ad oly if A o is a pure jump process. For i {,2}, correspodig o he wo ( )-pars of a radom ime, i.e., τ ad τ 2, we defie A i := τi,. The A i,p ad A i,o are respecively he F-dual predicable projecio ad he F-dual opioal projecio of A i. Le us deoe by Z i ad Z i he supermarigales associaed wih τ i. The, he followig relaios hold. Lemma 2.5. Le τ be a radom ime ad (τ,τ 2 ) is ( )-decomposiio. The, he supermarigales Z ad Z ca be decomposed i erms of he supermarigales Z, Z 2 ad Z, Z 2 as: Z = Z +Z 2 ad Z = Z + Z 2. Proof. The resul follows from he propery ha τ τ 2 =. The ex resul gives he supermarigales Z ad Z of a hi radom ime ad heir decomposiios io F-marigale m ad fiie variaio process A o i erms of he exhausig sequece of τ ad of he associaed F-marigales defied i (2.2). Lemma 2.6. Le τ be a hi radom ime wih exhausig sequece (T ) ad (z ) be he family of F-marigales associaed wih τ hrough (2.2). The (a) z > ad z > a.s. o C for each, (b) Z τ > a.s. o {τ < }, (c) Z = { T}z, Z = {<T}z, Ao = { T}z T ad m = z T. Proof. (a) Defie he F-soppig ime (2.4) R := if{ : z = }. As z is a posiive càdlàg marigale, by 23, Proposiio (3.4) p.7, i vaishes o R,. Sice z is bouded z exiss ad: {R < } = {if z = } = {z = }. Moreover, he equaliy = E(z {z =}) = E( C {z =}) implies ha C {z = } =, so as well C {if z = } =. We obai ha z > ad z > a.s. o C. (b) We have Z τ {τ< } = C Z T ad, o {T < }, we have Z T = P(τ T F T ) P(τ = T F T ) = z T.

7 6 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC From par (a), his implies ha Z τ > a.s. o {τ < }. We omi he proof of (c) as i is sraighforward. The followig resul describes how, afer a hi radom ime, he codiioal expecaios wih respec o elemes of F τ ca be expressed i erms of he codiioal expecaios wih respec o elemes of F. Lemma 2.7. Le τ be a hi radom ime wih exhausig sequece (T ) ad (z ) be he family of F-marigales associaed wih τ hrough (2.2). The, for ay G-measurable iegrable radom variable X ad s we have Proof. Noe ha EX F τ EX C F {s T} C = {s T} C. z F τ = u>f u σ(c {T s}, s u, N). Thus, by Moooe Class Theorem, for each G F τ here exiss F F such ha (2.5) G {T s} C = F {T s} C. The, we have o show ha E X {s T} C z Fτ = {s T} C E X {s T} C F. For ay G F τ, we choose F F saisfyig (2.5), ad we obai E X {s T} C G z = E X{s T} C F E C F = E {s T} F E C F EX C F = E {s T} C F EX C F = E {s T} C G EX C F which eds he proof. We ed his subsecio wih a remark o he ( )-decomposiio of a radom ime τ as a F τ -soppig ime, where F τ is he filraio F progressively elarged wih τ as i (.). Remark 2.8. We ca also decompose he radom ime τ 2 io accessible ad oally iaccessible pars. The, we cosider a decomposiio of τ oo hree pars as: τ = τ { A o τ =}, τ i 2 = τ { A o τ >, A p τ=} ad τ a 2 = τ { A o τ >, A p τ>}. The τ τ i 2 is a Fτ -oally iaccessible par ad τ a 2 is a Fτ -accessible par of he F τ - soppig ime τ. These ypes of resuls were already show i 7, p.65 ad 9. We oe ha τ is a F τ -predicable soppig ime if ad oly if τ is a F-predicable soppig ime. Moreover, a filraio F τ such ha τ a 2 he accessible hi par of τ is o a F-soppig ime is o quasi-lef coiuous. The las observaio provides a sysemic way o cosruc examples of o quasi-lef coiuous filraios.

8 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 7 3. The hypohesis (H ) We exploi here he hypohesis (H ) i progressive elargeme i he coecio o he ( )-decomposiio of a radom ime. Le us firs recall ha hypohesis (H ) holds for (F,F τ ) if ay F-marigale is a F τ -semimarigale. Firs, i subsecio 3., we examie he case of hi radom imes. The, i subsecio 3.2, we work wih geeral radom imes. 3.. Thi radom ime. Before formulaig he resul of his subsecio we mus recall a vial resul by Jacod (see 7, Theorem 3,2 ad 2) o he hypohesis (H ) i iiial elargeme wih aomic σ-field. Le F C deoe he iiial elargeme of he filraio F wih he aomic σ-field C := σ(c, ) wih C defied i (2.), i.e., (3.) F C := s>f s σ(c, ). I his case of elargeme, Jacod s resul says ha he hypohesis (H ) holds for (F,F C ) ad he decomposiio of ay F-marigale X as a F C -semimarigale is (3.2) X = X + C z s d X,z s, where X is a F C -local marigale ad z are give i (2.2). Theorem 3.. Le τ be a hi radom ime. The F F τ F C ad he hypohesis (H ) is saisfied for (F,F τ ). Moreover, for each F τ -predicable ad bouded process G ad each F-local marigale Y he iegral X := G Y is a F τ -semimarigale wih caoical decomposiio (3.3) X = X + τ where X is a F τ -local marigale. d X,m s + Z s C {s>t} zs d X,z s Proof. The firs par follows from Jacod s resul (3.2) ad Sricker s Theorem 22, Theorem 4, Chaper II sice F τ F C. Le H be a F τ -predicable bouded process. The, 7, Lemma (4,4) implies ha H = { τ} J + {τ<} K (τ) where J is a F-predicable bouded process ad K : R + Ω R + R is P B(R + )- measurable ad bouded. Noe ha, sice { τ} {Z > }, J ca be chose o saisfy J = J {Z >}. Sice τ is a hi radom ime, we ca rewrie he process H as H = J { τ} + {T<}K (T ) C wih C = {τ = T }. Noe ha each process K := {T<}K (T ) is F-predicable ad bouded ad, sice C {z > }, K ca be chose o saisfy K = K {z >}.

9 8 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC Le X be a H F-marigale. The sochasic iegrals J X ad K X are well defied ad each of hem is H F-marigale. For each ad for each bouded F-marigale N, by iegraio by pars, we have ha (3.4) E C N = Ez,N = E z,n. Sice N E( C N ) is a liear form, by he dualiy (H,BMO) implies ha (3.4) holds for ay H F-marigale N. Similarly, by 2, Proposiio.32, for ay H F-marigale N, he process N,m exiss ad we have E(N τ ) = E(N,m ) = E( N,m ) where m is give i (2.3). Therefore τ E H s dx s =E J s dx s + E C KsdX s =E J s d m,x s + E Ksd z,x s The, sice for ay predicable fiie variaio process V, E h s dv s = E deduce E H s dx s = E + τ =E Z s {Zs >}J s d m,x s Z s zs E zs {z s >}Ksd z,x s Z s J s d m,x s + E C z s K sd z,x s. p h s dv s, we For ay H F-marigale Y ad F τ -predicable process G he asserio of he heorem follows as for ay s ad F Fs τ he process H = (s, F is clearly F τ -predicable. To ed he proof we recall ha ay local marigale is locally i H. Remark , Lemma (4,) is a special case of Theorem 3. where he radom ime wih couably may values is cosidered. I correspods o he siuaio of hi radom ime whose graph is icluded i couable uio of cosa secios, i.e, τ wih = {(ω, ) : ω Ω}. Our proof is similar o he proof of 7, Lemma (4,). We ed his secio wih a resul likig processes i F τ ad F C. I ca be used as a aleraive approach o show he decomposiio i Theorem 3. usig (3.2). I is relaed o he ideas i 8. Proposiio 3.3. Le τ be a hi radom ime ad X be a process such ha X = τ, X. The (a) The process X is a F C -(super-, sub-) marigale if ad oly if he process X is a F τ -(super-, sub-) marigale. (b) Le ϑ be a F C -soppig ime. The ϑ τ is a F τ -soppig ime.

10 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 9 (c) The process X is a F C -local marigale if ad oly if he process X is a F τ -local marigale. Proof. (a) Noe ha he filraios F τ ad F C are equal afer τ, i.e., for each ad for each se G F C here exiss a se F F τ such ha (3.5) {τ } G = {τ } F. To show (3.5), by Moooe Class Theorem, i is eough o cosider G = C ad o ake F = C {τ } which belogs o F τ as C Fτ τ by 3, Corollary 3.5. Tha implies ha he process τ, X is F τ -adaped if ad oly if i is F C -adaped. The equivalece of (super-, sub-) marigale propery comes from (3.5). (b) For each we have {ϑ τ } = {ϑ } {τ } F τ by (3.5). (c) We combie he wo previous pois Geeral radom ime. I his secio we work wih he ( )-decomposiio (τ,τ 2 ) of he radom ime τ. We defie hree elarged filraios F τ := (F τ ), F τ 2 := (F τ 2 ) ad F τ,τ 2 := (F τ,τ 2 ) as Obviously, F F τ i F τ,τ 2 for i =,2. F τ i : = s>f s σ(τ i s) for i =,2 F τ,τ 2 : = s>f s σ(τ s) σ(τ 2 s). Theorem 3.4. Le τ be a radom ime ad (τ,τ 2 ) is ( )-decomposiio. The, he hypohesis (H ) is saisfied for (F,F τ ) if ad oly if he hypohesis (H ) is saisfied for (F,F τ ). Proof. I he firs sep, we show he followig iclusios of filraios for i =,2: F F τ i F τ,τ 2 = F τ. Le A o be a F-dual opioal projecio of τ. Noe ha τ, = τ, { A o τ =} ad τ2, = τ, { A o τ >}, hus, sice A o τ Fτ τ, he processes τ, ad τ2, are F τ -adaped which implies ha F τ,τ 2 F τ. O he oher had we have τ, + τ2, = τ, which implies ha F τ,τ 2 F τ. I he secod sep, oe ha if a F-marigale is a F τ -semimarigale, by Sricker s Theorem 22, Theorem 4, Ch II, p. 53, i is as well a F τ -semimarigale. Thus he ecessary codiio follows. Sice τ 2 is a hi radom ime, he previous sep ad Theorem 3. imply ha he hypohesis (H ) is saisfied for (F τ,f τ ). Thus he sufficie codiio follows. I he ex proposiio we see ha τ ad τ 2 are i some sese orhogoal (i erms of semimarigaledecomposiioadassociaedsupermarigales, whichisdueoτ τ 2 = ).

11 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC Proposiio 3.5. The F-supermarigale Z 2 of a hi radom ime τ 2 coicides wih he F τ -supermarigale Z 2,Fτ of τ 2, i.e., P(τ 2 > F ) = P(τ 2 > F τ ). Proof. Le T be a F-soppig ime. For each A F τ, here exiss B F such ha A {τ = T} = B {τ = T}, so P(τ = T F ) = P(τ = T F τ ) which eds he proof. 4. Thi hoes imes Here we resric our aeio o a special class of radom imes, amely o hoes imes. We recall is defiiio below (see 7, p. 73) ad some aleraive characerizaios i Appedix B. Defiiio 4.. A radom ime τ is a F-hoes ime if for every > here exiss a F -measurable radom variable τ such ha τ = τ o {τ < }. The, i is always possible o choose τ such ha τ. We refer o he Appedix B for furher helpful resuls o hoes imes. 4.. Fudameal properies. Le us sar wih some characerisaio ad properies of (hi) hoes imes. Proposiio 4.2. (a) Le τ be a hoes ime ad deoe by (τ,τ 2 ) is ( )-decomposiio. The, he imes τ ad τ 2 are hoes imes. (b) A radom ime τ is a sric hoes ime if ad oly if Z τ = a.s. o {τ < }. (c) Le τ be a hoes ime wih ( )-decomposiio (τ,τ 2 ). The, Z τ = o {τ = τ < } ad Z τ < o {τ = τ 2 < }. Proof. (a) O he se {τ < }, τ is equal o γ, he ed of he opioal se Γ (Theorem B.2). The, as {τ < } {τ < }, o he se {τ < }, oe has τ = γ, so τ is a hoes ime. Same argume for τ 2. (b) Assume ha τ is a sric hoes ime. The, he hoes ime propery preseed i Theorem B.2 (c) implies ha Z τ = ad he sric ime propery implies, by Theorem 2.4 (b), he coiuiy of A o. Therefore, he equaliy Z = Z + A o leads o equaliy Z τ = a.s. o {τ < }. Assume ow ha Z τ = o he se {τ < }. The, o {τ < } we have = Z τ Z τ, so Z τ = ad τ is a hoes ime. Furhermore, as A o τ = Z τ Z τ =, for each F-soppig ime T we have P(τ = T < ) = E( {τ=t} { A o τ =} (T< ) ) = E( {u=t} { A o u =}da o u ) =. So τ is a sric radom ime. (c) From he hoes ime propery of τ ad Lemma 2.5, o he se {τ < } = Z τ = Z τ + Z 2 τ.

12 O he se {τ = τ < }, we have CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS = Z τ + Z 2 τ = Z 2 τ, where he secod equaliy comes from (c) i Theorem B.2. Now le us compue Z 2 τ Z 2 τ = Z 2 τ A 2,o τ = Z 2 τ =, where we have used he sric radom ime propery of τ, i.e., { A 2,o > } = T (wih (T ) beig a exhausig sequece of τ 2 ) ad P(τ = T < ) =. Fially, o {τ = τ < } O he se {τ = τ 2 < }, Z τ = Z τ +Z 2 τ =. Z τ = Z τ 2 +Z 2 τ 2 Z 2 τ 2 <, where he las iequaliy is due o Lemma 2.6 (b). Remark 4.3. We would like o remark ha he codiio ha Z τ < for a hoes ime τ which, by Proposiio 4.2 (c), is equivale o he codiio ha τ is a hi hoes ime is a esseial assumpio i. Lemma 4.4. Le τ be a hi hoes ime ad τ be associaed wih τ as i Defiiio 4.. The, for each : (a) o {T = τ } = {T = τ } we have z = Z, A o = z T ad m = z z T ; (b) o {T < } we have z = {τ=t }( Z ) ad z = {τ=t }( Z ); i paricular Z = {τ=t <}( Z ) ad Z = {τ=t <}( Z ). Proof. (a) Usig properies of τ we deduce ha {T=τ }z = P(T = τ,τ = T < F ) = P(τ,T = τ = τ F ) = P(τ,T = τ F ) = {T=τ }( Z ) where he firs equaliy is due o τ, he hird oe follows by τ = τ o {τ } ad he las oe is rue sice T ad τ are wo F -measurable radom variable ad {T = τ } = {T = τ < } {T = τ = } = { {T = τ } {τ < } } { {T = } {τ = } }. The dual opioal projecio of a hi radom ime saisfies {T=τ }A o = k {T=τ,T k }z k T k = {T=τ }z T, where he secod equaliy is due o he fac ha for k we have {T=τ,T k }z k T k = {T=τ }E( {τ=tk } T k )

13 2 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC = {T=τ =T k }E( {τ=tk } T k ) = sice T ad T k have disjoi graphs ad τ is a hoes ime. Combiig he wo previous pois, we coclude ha m = Z A o = z z T o he se {T = τ }. (b) Agai usig properies of radom variable τ we derive {T<}z = P(τ = T = τ < F ) = {T=τ <}( Z ), {T<}z = P(τ = T = τ < F ) = {T=τ <}( Z ). The, Lemma 2.6 (c) complees he proof. For progressive elargeme wih a hoes ime, he hypohesis (H ) is saisfied (o oly for F-local marigales sopped a τ), ad he followig decomposiio is give i 7, Theorem (5,). Le M be a F-local marigale. The, here exiss a F τ -local marigale M such ha: τ (4.) M = M + d M,m s {s>τ} d M,m s. Z s Z s Remark 4.5. For a hi hoes ime τ, he wo decomposiio formulas, firs give i Theorem 3. ad secod give i (4.), coicide. I is eough o show ha {s>τ} Z s d X, m s = C {s>t} zs d X,z s. This is a simple cosequece of he se iclusio {τ < s} {τ = T } {T = τ s s} ad Lemma 4.4 (a): {s>τ} Z s d X, m s = = = {s>τ} {τ=t} {s>τ} {τ=t} zs C d X, m s Z s {s>t} zs d X,z s d X,z s Jumpig filraio. I his secio we develop he relaioship bewee jumpig filraio ad hi hoes imes. Le us firs recall he defiiio of a jumpig filraio ad he mai resul obaied i Jacod ad Skorokhod 4. Defiiio 4.6. A filraio F is called a jumpig filraio if here exiss a localizig sequece (θ ), i.e., a sequece of soppig imes icreasig a.s. o, wih θ = ad such ha for all ad > he σ-fields F ad F θ coicide up o ull ses o {θ < θ + }. The sequece (θ ) is he called a jumpig sequece. There exiss a impora aleraive characerizaio of jumpig filraio i erms of marigale s variaio (4, Theorem ). Theorem 4.7. The wo followig codiios are equivale: (a) a filraio F is a jumpig filraio; (b) all marigales i he filraio F are a.s. of locally fiie variaio.

14 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 3 I he remaiig par of his subsecio we ivesigae relaioship bewee jumpig filraioadhoesimes. Weshowhaheredoesoexissrichoesimeiajumpig filraio ad ha here exiss a sric hoes ime i a filraio which admis a o-cosa coiuous marigale (i paricular such a filraio is o a jumpig filraio). Theorem 4.8. The followig asserios hold. (a) If F is a jumpig filraio, he all F-hoes imes are hi. (b) If all F-hoes imes are hi, he all o-cosa F-local marigales are purely discoiuous. Proof. (a) Le τ be a hoes ime. The, ake he same process α as i he proof of Proposiio B., i.e., α is a icreasig, càdlàg, adaped process such ha α = τ o {τ } ad τ = sup{ : α = }. Le us defie he pariio (C ) = such ha C = {θ τ < θ } for ad C = {τ = } wih (θ ) beig a jumpig sequece for he jumpig filraio F. O each C wih we have τ = T := if{ θ : = α θ }. From he jumpig filraio propery, we kow ha α θ is F θ -measurable so each T is a soppig ime ad τ = T which shows ha he hoes ime τ is a hi radom ime. (b) The proof by coradicio is based o 23, Exercise (.26) p.235. Assume ha M is a o-cosa coiuous F-local marigale wih M =. Defie he F-soppig ime S = if{ > : M = }. The, defie he F-hoes ime τ := sup{ S : M = }. Sice M is coiuous, τ is o equal o ifiiy wih sricly posiive probabiliy. We ow show ha τ is a F-sric hoes ime. Le us deoe Z(ω) := { : M (ω) = }. The se Z(ω) is closed ad Z c (ω) is he uio of couably may ope iervals. We call G(ω) he se of lef eds of hese ope iervals. I wha follows we show ha for ay F-soppig ime T we have P(T G) =. Defie he F-soppig ime D T := if{ > T : M = } ad oe ha {T G} = {M T = } {T < D T } F T. Assume P(T G) = p >. The he process Y = {T G} M T+ { DT T} is a (F T+ ) -marigale. Ideed for s we have E(Y F T+s ) = {T G} sg(m T+ )E(M T+ { DT T} F T+s ) = {T G} sg(m T+ ) MT+s {s DT T} E( {s DT T} {>DT T}E(M T+ F DT ) F T+s )

15 4 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC =Y s {T G} sg(m T+ )E( {s DT T} {>DT T}M DT F T+s ) =Y s where we have used he marigale propery of M ad M DT =. Moreover Y = ad here exiss ε > such ha P(M T =,D T T > ε) p 2 >. Sice Y ε = {MT =} {DT T ε} M T+ε ad P(Y ε > ) >, we have E(Y ε ) > = Y. So, P(T G) =. Fially, as τ G a.s. we coclude ha τ is a sric hoes ime. Fially we give wo examples of sric hoes imes origiaig from purely discoiuous semimarigales of ifiie variaio. I he firs Example 4.9, we sudy he case of Azéma s marigale (see 22, IV.8 p ). I he secod Example 4., we recall he example 2. from 8 o Maximum of dowwards drifig specrally egaive Lévy processes wih pahs of ifiie variaio. Example 4.9. Le B be a Browia moio ad F is aural filraio. Defie he process The process g := sup{s : B s = }. µ := sg(b ) g is a marigale wih respec o he filraio G := (F g+) ad is called he Azéma marigale. The, he radom ime τ := sup{ : µ = } is clearly a G-hoes ime. Noe ha τ = τ B := sup{ : B = } ad τ B is a F- sric hoes ime (see i 2, Table α ), p.32 ha τ B has coiuous F-dual opioal projecio). Thus, sice G F, τ is a G-sric hoes ime. Example 4.. Le X be a Lévy process wih characerisic riple (α,σ 2 =,ν) saisfyig ν((, )) =, α+ xν(dx) < ad x ν(dx) =. The, ρ = sup{ : X = X } wih X = sup s X s is a sric hoes ime as show i 8, Secio Examples of hi hoes imes Compoud Poisso process: las passage ime a a barrier a. Le us cosider he filraio F geeraed by a Compoud Poisso Process (CCP) X, defied as N X = Y, = where N is a Poisso process wih parameer η ad sequece of jump imes (θ ) =, ad (Y ) = are i.i.d. sricly posiive iegrable radom variables, idepede from N, wih cumulaive disribuio fucio F. We will sudy a hi hoes ime i he filraio of X which is o a soppig ime. I a progressive elargeme framework, i order o sudy

16 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 5 he F τ -semimarigale decomposiio of F-marigales before τ, oe eeds o compue he marigale m = Z +A o. Therefore, we shall prese he compuaios of A o (hece m). Defie a radom ime τ as (4.2) τ := sup{ : µ X a} wih a > ad a cosa µ. From ow o we assume ha µ > ηe(y ). Uder his codiio, he radom ime τ is fiie a.s.. Sice τ is a las passage ime, i is a hoes ime i he filraio F. Furhermore, sice he process µ X has oly egaive jumps, oe has µτ X τ = a. The radom ime τ is hi as we shall see below. Lemma 4.. The hoes ime τ is a hi radom ime wih exhausig sequece (T ) give by (4.3) T := if{ > : µ X = a} ad T := if{ > T : µ X = a} for >. Each T is a predicable soppig ime. Furhermore Z τ <. Proof. Radom ime τ is hoes sice o {τ < } i is equal o τ := {s : µs X s a}. The ses (C ) = wih C = {τ = T } form a pariio of Ω. The, τ = = T C. Thus τ is a hi radom ime Noe ha τ is o a F-soppig ime as C / F T for ay. To show ha each T is predicable, le us defie he soppig imes J d ad J u as J d = if{ > : µ X = a,µ X < a} J u = if{ > : µ X > a,µ X = a}. Firs observe ha J d θ ad J u θ. For each, we have P(J d = θ ) = as P(J d = θ ) = E P J d = θ F θ = E {Jd >θ }P µθ X θ = a F θ E P θ θ = a µθ +X θ F θ =, µ so we coclude ha J d = a.s. For each, we have P(J u = θ ) = as P(J u = θ ) = E P J u = θ F θ σ(y ) = E {Ju>θ }P µθ X θ = a F θ σ(y ) E P θ θ = a µθ +X θ +Y F θ σ(y ) µ =, so we coclude ha J u = a.s.. Now, for each, we simply defie a aoucig sequece (T,m ) m for T as T,m = if{ > T : µ X a m }

17 6 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC wih T =. We see ha J d = ad J u = a.s. esure ha each sequece (T,m ) m is ideed a aoucig sequece of T. Le us remark ha i fac he radom ime τ (defied i (4.2)) ca be see as he ed of he opioal se Γ = T as τ(ω) = sup{ : (ω,) Γ}. Proposiio 4.2. The supermarigales Z ad Z associaed wih he hoes ime τ are give by (4.4) Z = Ψ(µ X a) {µ X a} + {µ X<a}, (4.5) Z = Ψ(µ X a) {µ X>a} + {µ X a}, where Ψ(x) is he rui probabiliy associaed wih he process µ X, i.e., for every x (4.6) Ψ(x) := P( x < ) wih x := if{ : x+µ X < }. The fucio Ψ saisfies he followig properies: (a) for x < we have Ψ(x) = ; (b) he fucio Ψ is coiuous ad decreasig o (, ); (c) for x =, we have Ψ() = ηe(y ) <. µ I paricular, Z τ = where κ = µ +κ ηe(y ). The supermarigale Z admis he decomposiio Z = m A o where m = ( Ψ()) ( T) +Ψ(µ X a) {µ X a} + {µ X<a} A o = ( Ψ()) { T}. The F-dual opioal projecio ad he F-dual predicable projecio of τ, are equal, i.e. A o = A p. Proof. The form of Z follows from he saioary ad idepede icremes propery of µ X P(τ > F) = P(if s (µs X s) < a F ) = P(if s (µ(s ) (X s X )) < a µ+x F ) = Ψ(µ X a) {µ X a} + {µ X<a}. Le us ow compue he dual opioal projecio A o of he process τ,. For ay bouded opioal process X we have EX τ = E {τ=t}x T = E X T E {τ=t} F T which implies ha A o = P(τ = T F T ) T,.

18 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 7 To compue P(τ = T F T ) le us defie, for ay x ad T, he soppig ime S T x = if{ > T : x+µ X < } ad oice ha P(τ = T F T ) = E(S T a = F T ) = E(S = ) = Ψ(). The, A o = ( Ψ()) T,. This is also he dual predicable projecio as, by previous lemma, T are predicable soppig imes. The marigale m = Z + A o equals he m = E ( Ψ()) {T< } F = ( Ψ()) { T} +Ψ(µ X a) {µ X a} + {µ X<a}. Fially, from he geeral relaio Z = Z + A o, we coclude he form of Z Browia moio: local ime approximaio. We give a example relaed o a approximaio resul for he local ime. Le B be a Browia moio. For ε >, defie a double sequece of soppig imes by U ε =, V ε = We cosider he radom ime U ε = if{ V ε : B = ε}, V ε = if{ Uε : B = }. (4.7) τ ε := sup{v ε : V ε T } wih T = if{ : B = }. From he defiiio we easily see ha i is a hoes hi radom ime. Le us iroduce he processes X ε, Y ε ad J ε X ε := sup{s T : B s = ε} Y ε := sup{s T : B s = } ad he fucio ζ J ε := {X ε >Y ε } ζ(x) := P x (T < T ) = x, for x,. The supermarigale Z ε associaed wih τ ε is equal o Z ε = J ε ζ(b T )+( J ε )ζ(ε) = J ε ( B T )+( J ε )( ε) = J ε B T ( J ε )ε. Le us defie he process D ε = max{ : V ε } which is equal o he umber of dowcrossigs of Browia moio from level ε o level before ime. By iegraio by pars we obai B J ε +ε( Jε ) = Js ε db s + B s djs ε +ε( Jε )

19 8 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC The dual opioal projecio of τ ε equals = J ε s db s +εd ε +ε. A o,ε = ( { τ ε }) p = εd ε T +ε ad we easily see ha i is a pure jump process wih he propery { A o,ε > } =,T = V ε. We ca ierpre he sequece τ ε wih ε goig o zero as a approximaio of he sric hoes ime τ give by (4.8) τ := sup{ < T : B = }, as τ ε τ P a.s. (by ime reversal a τ). The supermarigale Z associaed wih τ equals Z = T {Bs>}dB s 2 L T ad, by 23, Chaper VI Theorem (.) ad he fac ha E( T ) <, we have he followig covergece for dual opioal projecios lim E sup εd T ε ε 2 L T =. I order o sudy he relaioship bewee progressive elargemes wih τ ε, le us recall he defiiio of weak covergece of σ-fields (see Defiiio i 9 ad refereces herei). Defiiio 4.3. A sequece of σ-fields G coverges weakly o a σ-field G if ad oly if for all G G, E( B G ) coverges i probabiliy o G. We wrie G w G. Lemma 4.4. Le F be a progressive elargeme of he filraio F wih radom ime τ / defied i (4.7) ad F be he progressive elargeme of he filraio F wih radom ime τ defied i (4.8). The, for each, he sequece of σ-fields (F ) coverges weakly o F. Proof. We have o check ha for each F F, P(F F ) coverges i probabiliy o F. We limi our aeio o he ses F belogig o he geeraor of F. If F F, he codiio is obviously saisfied. If F = {τ s} for s, usig Proposiio B.3 ad he hoesy of τ /, we have E( {τ s} F ) = {τ s}e( / {τ s} F ) E( {τ s} F ) = {τ / s} P(τ / s F ) {τ s} a.s. where he covergece comes from τ / τ a.s.

20 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 9 5. Eropy of a hi radom ime The addiioal iformaio carried by elarged filraio ad is measureme was sudied by several auhors. Already i Meyer 2 ad Yor 25, he quesio o sabiliy of marigale spaces wih respec o iiial elargeme wih aomic σ-field was asked. Here we complee previous sudies by givig a simple coecio bewee progressive elargeme wih hi radom ime ad codiioal eropy of a pariio associaed o his ime. I he case of iiial elargeme wih a pariio C := (C ), he addiioal kowledge is measured by eropy, amely H(C) := P(C )logp(c ). I he case of progressive elargeme wih a hi radom ime τ, we sugges he measureme of addiioal kowledge by he eropy of a hi radom ime defied hrough: (5.) H(τ) := E C logz T, where C ad z are defied i (2.) ad (2.2). Le us remark ha he codiio H(τ) < is weaker ha he codiio H(C) <. To sae he mai resul of his secio i Theorem 5.2, which cosiss of a geeralisaio of 25, Theorem 2 we eed o defie more geeral objec, amely (5.2) H γ (τ) := E C log γ γ >. z T Remark 5.. (a) If τ is a F-soppig ime he H γ (τ) =. (b) If for ay he se C is i F T, he we do o gai ay addiioal iformaio sice C logz T = C log C =. (c) As oed i Secio 2, he exhausig sequece (T ) of a hi radom ime is o uique. However H γ (τ) is ivaria uder differe decomposiios of τ sice for F-soppig imes T, T ad T 2 such ha T T 2 = ad {τ = T} = {τ = T } {τ = T 2 } we have {τ=t} log γ P(τ = T F T ) = {τ=t }log γ P(τ = T F T )+ {τ=t2 }log γ P(τ = T F T ) = {τ=t }log γ P(τ = T F T )+ {τ=t2 }log γ P(τ = T 2 F T2 ). The eropy of hi radom ime reveals o be a adequae oio o rea he sabiliy of marigale spaces wih respec o progressive elargeme of filraio (wih hi radom ime). I his secio we work uder sadig assumpio ha (C) all F-marigales are coiuous The we iroduce some relaed oaios. For ay p, ), we deoe H p ad S p he Baach spaces cosisig respecively of coiuous local marigales ad coiuous semimarigales equipped wih he followig orms: (a) a coiuous F-local marigale X belogs o H p if X H p := X /2 L p < ;

21 2 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC (b) a coiuous F-semimarigale X, wih caoical decomposiio X = M + V, belogs o S p if X S p := M /2 L p + dv L p <. We are ready o sae he mai resul of his secio. I is a geeralisaio of 25, Theorem 2 ad he proof here is based o he origial proof. Theorem 5.2. Le τ be a hi radom ime wih a exhausig sequece (T ) ad family of F-marigales (z ) saisfyig z = C. Assume (C) ad le r, ), p,γ > saisfy = +. The he followig codiios are equivale: r p γ (a) for eachf-localmarigaley adeachf τ -predicableprocessg, he F τ -semimarigale X := G Y saisfies: (b) H γ/2 (τ) <. X S r (F τ ) C p,r X /2 L p; I paricular, if he codiios (a) ad (b) are saisfied, he H p (F) is coiuously embedded i S r (F τ ). Proof. By Theorem 3., uder assumpio (C), each X of he form X = G Y where Y is a F-local marigale ad G is a F τ -predicable process, has he decomposiio (5.3) X = X + X,Y b + X,Y a where Y b ad Y a are F τ -local marigales give by Y b := τ Z s d m s ad Y a := C dẑ T zs s wih X, m ad ẑ he F τ -local marigale pars from Doob-Meyer decomposiio of correspodig F τ -semimarigales X, m ad z. Sice r < p ad coiuiy of X i always holds ha X /2 L r = X /2 L r = X /2 L p. Thus, showig (a) (b) is equivale o showig ha (5.4) d X,Y b + X,Y a L r C p,r X /2 L p holds for ay X. By 25, Lemma 2 ad he fac ha sochasic iervals,τ ad T, C for N are pairwise disjoi, iequaliy (5.4) holds for ay adequae X if ad oly if Firsly we show ha E Y b γ/2 < ad E Y a γ/2 <. (5.5) Y b /2 L γ < γ >. By 25, Remark 5. 2) p.23, sice, for γ > 2, x γ/2 is moderae Orlicz fucio, we have τ /2 /2 L s C + log Zsd m 2 γ U L γ

22 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 2 where U is radom variable wih uiform disribuio o,. Noe ha ( logx)γ/2 < ad hus, by usig he fac ha L µ L γ for < µ < γ <, he (5.5) holds. Showig ha E Y a γ/2 < if ad oly if H γ/2 (τ) < for γ = 2 is a simpler special case which will be useful aferwards ad we sar wih i. By properies of dual predicable projecio, we have ha E( Y a ) = E C T (z ) 2d z = E d z T z. Cosider he fucio f : R + R + defied as f(x) = x xlogx for x > ad f() =. The, Iô s formula for z implies C = z z logz logz sdz s 2 d z s. We deduce, by akig codiioal expecaio wih respec o F, ha (5.6) E d z s F = 2z log. z Fially E( Y a ) = z s = 2 E d z T z z s E z T logz T = 2H(τ) <. IorderocompleeheproofiremaisoshowhaH γ (τ) < ifadolyife Y a γ <, firs oice ha Lemma C. implies ha i is equivale o provig H γ (τ) < if ad oly if E log I γ <. To his ed, oe ha, by Lemma C.2, we have E log I γ = = E E C log I γ FT z z T T E C log γ zt β ( β) 2dβ. Deoig by z = C zt, akig ay ε (,) ad defiig f(x) = x x (,), we furher obai E log I γ z = E z E {z>ε} C +C 2 E C +C 2 H γ (τ). z f(ε) ε( ε) + 2 ε {z ε} log γ z log γ β log ε ( β) 2dβ γ +E f(z) {z ε} z( ε) 2 log β γdβ for Thus we coclude ha H γ (τ) < if ad olyif E log I γ < ad he proofis complee.

23 22 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC 6. Geeral progressive elargeme wih a pariio I his secio we cosider more geeral progressive elargeme of filraio wih give a pariio (C m ) m of Ω, which, i corary o previous resuls, allows for addig several members of a pariio a he same ime. Le τ be a hi radom ime wih exhausig sequece (T ) ad ξ be a discree radom variable wih values i N. Sice for each m N we ca fid uique decomposiio (,k) N N such ha m = 2 (2k ), by rearragig ad reumberig erms i (C m ) m, we ca obai double sequece (C k ),k of ses such ha he members (C k ) k are added a ime T. More precisely, for double sequece (C k ),k ad a sequece of F-soppig imes (T ) wih disjoi graphs, we defie ξ ad τ by ξ :=,k 2 (2k ) C k ad τ := T k Ck. Noe ha {τ = T < } = {T < } k Ck ad σ(ξ {τ=t}) = σ((c k ) k). Aalogously o (2.2) we defie (6.) z,k := P(C k F ). Described above siuaio is capured by he progressively elarged filraio F ξ,τ := (F ξ,τ ) defied hrough (6.2) F ξ,τ := s>f s σ ( ξ {u τ} : u s ). Similarly o Lemma 2.7 we obai: Lemma 6.. For ay G-measurable iegrable radom variable X ad s we have E X F ξ,τ = E X C k F {s T} C k {s T} C k. Theorem 3. ca be easily exeded o he case of F ξ,τ as saed below. Theorem 6.2. The hypohesis (H ) is saisfied for (F,F ξ,τ ). Moreover, for each F ξ,τ - predicable ad bouded process G ad each F-local marigale Y he iegral X := G Y is a F ξ,τ -semimarigale wih caoical decomposiio (6.3) X = X + τ where X is a F ξ,τ -local marigale. d X,m s + Z s,k C k z,k {s>t} z,k s d X,z,k s Proof. The proof is almos ideical o he origial proof of Theorem 3.. We jus give he form of F ξ,τ -predicable process H by H = { τ} J + {τ<} K (ξ,τ) wherej isaf-predicableboudedprocessadk : R + Ω N R + RisP 2 N B(R + )- measurable. Noe ha, sice { τ} {Z > }, J ca be chose o saisfy J =

24 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 23 J {Z >}. Sice τ is a hi radom ime, we ca rewrie he process H as H = J { τ} +,k {T<}K (2 (2k ),T ) C k. Noe ha each process K,k := {T<}K (2 (2k ),T ) is F-predicable ad bouded = K,k {z,k >}. ad, sice C k {z,k > }, K,k ca be chose o saisfy K,k Likewise, Theorem 5.2 ca be geeralized o he case of F ξ,τ. To his ed le us defie (6.4) H γ (ξ,τ) := E C k log γ γ >. z,k,k T The, he followig resul follows by he same proof as Theorem 5.2. Theorem 6.3. Le (C) k,k be a F -measurable pariio, τ be a hi radom ime wih exhausive sequece (T ) ad F ξ,τ be give by (6.2). Assume (C) ad le r, ), p,γ > saisfy = +. The he followig codiios are equivale: r p γ (a) for eachf-localmarigaley adeachf ξ,τ -predicableprocessg, he F ξ,τ -semimarigale X := G Y saisfies: (b) H γ/2 (ξ,τ) <. X S r (F ξ,τ ) C p,r X /2 L p; I paricular, if he codiios (a) ad (b) are saisfied, he H p (F) is coiuously embedded i S r (F ξ,τ ). Appedix A. Projecios We collec here he defiiios of he key ools we have used alog he paper. Projecios ad dual projecios oo he referece filraio F play a impora role i he heory of elargeme of filraios. Firs we recall he defiiio of opioal ad predicable projecios, see 3, Theorems 5. ad 5.2 ad 6, p Defiiio A.. Le X be a measurable bouded (or posiive) process. The opioal projecio of X is he uique opioal process o X such ha for every soppig ime T we have E X T {T< } F T = o X T {T< } a.s.. The predicable projecio of X is he uique predicable process p X such ha for every predicable soppig ime T we have E X T {T< } F T = p X T {T< } a.s.. For defiiio of dual opioal projecio ad dual predicable projecio see 6, p.265, 22, Chaper 3 Secio 5,, Chaper 6 Paragraph 73 p.48, 3, Secios 5.8, 5.9. We poi ou ha he coveio we use here allows a jump a.

25 24 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC Defiiio A.2. (a) Le V be a càdlàg pre-locally iegrable variaio process (o ecessary adaped). The dual opioal projecio of V is he uique opioal process V o such ha for every opioal process H we have E H s dv s = E, ) H s dvs o, ) (b) Le V be a càdlàg locally iegrable variaio process (o ecessary adaped). The dual predicable projecio of V is he uique predicable process V p such ha for every predicable process H we have E H s dv s = E H s dvs p., ), ) Appedix B. Hoes imes I his Appedix we gaher complemeary resuls o hoes imes. They complee or slighly exed he exisig resuls maily from 7. We have made use of hem alog he paper. Proposiio B.. (a) A radom ime τ is a F-hoes ime if ad oly if for every > here exiss a F -measurable radom variable τ such ha τ = τ o {τ < }. (b) A radom ime τ is a F-hoes ime if ad oly if for every > here exiss a F -measurable radom variable τ such ha τ = τ o {τ }. Proof. Sufficiecy of boh codiios is sraighforward. Usig he oaio from Defiiio 4. we iroduce he process α as α = sup r Q,r< τ r. This defiiio implies ha α is a icreasig, lef-coiuous, adaped process such ha α = τ o {τ < } hus he ecessary codiio i (a) is prove. Le us deoe by α he righ-coiuous versio of α, i.e., α = α+. The, α is a icreasig, càdlàg, adaped process such ha α = τ o {τ } ad τ = sup{ : α = } hus he ecessary codiio i (b) is proved. The ex heorem gives some characerisaios of a hoes ime. We prove equivalece of he codiio (f). We also refer he reader o Azéma 7, Theorem.4 ad Proposiio.2, p o compare wih predicable case. Theorem B.2. Le τ be a radom ime. The, he followig codiios are equivale: (a) τ is a hoes ime; (b) here exiss a opioal se Γ such ha τ(ω) = sup{ : (ω,) Γ} o {τ < }; (c) Z τ = a.s. o {τ < }; (d) τ = sup{ : Z = } a.s. o {τ < }; (e) P τ, is geeraed by P, ad,τ, where P τ is he predicable σ-field liked o F τ ; (f) A o = Ao τ a.s. o he se {τ < }. Proof. The equivalece amog codiios (a), (b), (c), (d) ad (e) is saed i Theorem (5,) from Jeuli 7..

26 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 25 To prove ha (b) (f) suppose ha τ is a ed of he opioal se Γ o {τ < }. Wihou loss of geeraliy we may assume ha Γ is lef-closed, i.e., for each ω if x ր x ad x Γ(ω) he x Γ(ω), sice lef-closure of a opioal se is opioal. Thus oe has τ Γ. Defie he suppor of he measure da o, i.e., S(dA o ) := {(ω,) : ε > A o (ω) > Ao ε (ω)}, wih he exesio of A o such ha A o = for. Noe ha S(dAo ) is lef-closed ad opioal. Moreover τ S(dA o ) sice P(τ < ) = E(A o ) = E S(dA o )(s)da o s = P τ S(dA o ), τ <., ) Le Λ be ay lef-closed, opioal se coaiig τ. The E Λ (s)da o s = E Λ (s)da s = P(τ < ), ), ) = E(A o ) = E S(dA o )(s)da o s,, ) where he firs ad he hird equaliy comes from Defiiio A.2, he secod is due o τ Λ. Hece S(dA o ) Λ. Therefore we coclude ha S(dA o ) is he smalles lef-closed opioal se coaiig τ ad we deduce ha τ S(dA o ) Γ. Sice, by assumpio, τ is a ed of Γ o {τ < }, we obai ha, o he se {τ < }, A o = Ao τ for. To fiish he proof, we show implicaio (f) (b). I is sraighforward o see ha τ is he ed of he suppor of da o which is a opioal se. For a hoes ime, o he se { τ}, he projecio o F τ ca be expressed i erms of projecio o F as i he followig proposiio. I is kow ha, o he se { τ}, Z >. Proposiio B.3. Le τ be a hoes ime. The, for ay G-measurable iegrable radom variable X ad s we have EX F τ E X {τ s} F {τ s} = {τ s}. Pτ s F Proof. Noe ha for each G F τ here exiss F F such ha G {τ s} = F {τ s} as, by Moooe Class Theorem, i is eough o check i for G F for which i is obviously saisfied ad for G = {τ B} where B is a Borel se for which, by hoes ime propery, we have {τ B} {τ s} = {τ s B} {τ s} wih {τ s B} F s F. The, we have o show ha E X {τ s} P(τ s F ) F τ = {τ s} E(X {τ s} F ). For ay G F τ, we choose F F such ha G {τ s} = F {τ s}, ad we obai E X {τ s} G P(τ s F ) = E X {τ s} F P(τ s F ) = E {τ s} F E(X {τ s} F )

27 26 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC = E {τ s} G E(X {τ s} F ). which eds he proof. Appedix C. Eropy of hi radom ime Here we prese geeralisaios of wo auxiliary lemmas, Lemma 3 ad Lemma 4, from 25. They serve o prove Theorem 5.2. Lemma C.. For all γ > here exis c γ ad C γ such ha (C.) c γ E Y a γ E log γ C γ E Y a γ I where I is defied as (C.2) I := C I where I := if T z Proof. Sep. We firs prove he firs iequaliy i (C.). We have ha { T} C E Y a Y a F τ = { T} C E = { T} C z = { T} C z = 2 { T} C log z E C E (zs s F )2d z d z s F z s (z s) 2d z s F τ 2 C log I where he secod equaliy follows from Lemma 2.7, he hird oe from dual predicable projecio properies ad he four oe from (5.6). Therefore, for each µ (, we deduce ha µ F µ { T} C E Y a Y a τ { T} C E Y a Y a F τ 2 µ C log µ. (C.3) I Cosequely, he required iequaliy for γ (, follows by E( Y a γ ) = γ F E C E Y a Y a T τt γ F E sup { T} C E Y a Y a τ 2 γ E C log γ I = 2 γ E log γ I

28 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 27 ad for γ > follows by γ E( Y a γ ) γγ E sup { T} C E Y a Y a F τ (2γ) γ E C log γ I = (2γ) γ E log γ, I where he firs iequaliy is due o Burkholder-Gudy iequaliy for ermial value of icreasig process ad supremum of he associaed poeial ( p.88) ad he secod iequaliy is due o (C.3) for µ =. Sep 2. We ow prove he secod iequaliy i (C.). Le µ (, ad p > ad cosider F τ -marigale M := E( Y a µ Fτ ). Firsly we will show ha log µ (C.4) C µ +supm. I By Iô s lemma ad decomposiio (6.3) applied o z we obai ha o { T } C log z = dẑs + 2 z s (z s) 2d z s. Nex, by akig codiioal expecaio w.r. F τ ad usig iequaliy x+y µ x µ + y µ for µ (,, we deduce ha o { T } C log µ µ E dẑ z zs s F τ + 2 µ µe (zs s F τ )2d z µ/2 µ C E (zs s F τ )2d z +E (zs s F τ )2d z C E Y a µ/2 F τ +E Y a µ F τ where i he secod iequaliy we have used Burkholder-Davis-Gudy iequaliy for coiuous local marigales (see 24, IV-42, p.93) applied o F-local marigale N F for ay F F where N T := T+ dẑs ad F := (F T ) T wih FT := FT+. τ z s Fially, usig iequaliy x+x 2 2x 2 +/4o x = Y a µ/2 we coclude ha o{ T } C log µ C +E Y a µ z F τ ad by akig supremum over ad summig over he iequaliy (C.4) follows. I order o prove geeral case, relyig o (C.4) ad he iequaliy (+x) p 2 p (+x p ), we obai log µp Cµ +sup p I 2p M p.

29 28 A. AKSAMIT, T. CHOULLI AND M. JEANBLANC The, by akig expecaios ad applyig Doob s maximal iequaliy o M we obai E log µp C pµ2 p +E(supM p ) I Cµ2 + p p p p (p ) pe( Y a µp ). Tha complees he proof sice ay γ = µp > ca be obaied wih µ (, ad p >. Lemma C.2. Le, for each, Q be a absoluely coiuous measure give by dq = C. dp z The, for each F T -measurable radom variable β wih values i radom ierval (,zt ), we have Q (I < β F T ) = β zt β zt where I is defied i (C.2). Or, equivalely Q (I dβ F T ) = z T z T ( β) 2 {<β<z T }. Proof. For ay F T -measurable radom variable β wih values i radom ierval (,z T ) we defie a F-soppig ime T β by T β := if{ T : z < β}. The we compue P({I < β} C F T ) = P({T β < } C F T ) = E P (P({T β < } C F T β ) F T ) = E P ( {T β < } z F T β T ) = βp({t β < } F T ) = βp({i < β} C F T )+βp({i < β} C c F T ). Sice {I < β} C c = Cc we deduce ha ( β)p({i < β} C F T ) = β( z T ) ad herefore, by Bayes rule, we have which complees he proof. Q (I < β F T ) = P({I < b} C F T ) z T = β zt β zt Refereces Aa Aksami, Tahir Choulli, Ju Deg, ad Moique Jeablac. No-Arbirage uder a Class of Hoes Times. hp://arxiv.org/abs/3.42, Aa Aksami ad Moique Jeablac. Elargemes of filraios wih fiace i view. Spriger, Jürge Amediger, Peer Imkeller, ad Mari Schweizer. Addiioal logarihmic uiliy of a isider. Sochasic processes ad heir applicaios, 75(2): , 998.

30 CLASSIFICATION OF RANDOM TIMES AND APPLICATIONS 29 4 Sefa Akircher, Seffe Dereich, ad Peer Imkeller. The Shao iformaio of filraios ad he addiioal logarihmic uiliy of isiders. The Aals of Probabiliy, 34(2): , Sefa Akircher, Seffe Dereich, ad Peer Imkeller. Elargeme of filraios ad coiuous Girsaov-ype embeddigs. I Sémiaire de probabiliés XL, pages Spriger, Sefa Akircher ad Peer Imkeller. Fiacial markes wih asymmeric iformaio: iformaio drif, addiioal uiliy ad eropy. Proceedigs of he 6h Risumeika Ieraioal Symposium, pages 2, Jacques Azéma. Quelques applicaios de la héorie géérale des processus. I. Iveioes mahemaicae, 8(3): , Giorgia Callegaro, Moique Jeablac, ad Behaz Zargari. Carhagiia elargeme of filraios. ESAIM: Probabiliy ad Saisics, 7:55 566, Delia Coculescu. From he decomposiios of a soppig ime o risk premium decomposiios. arxiv prepri arxiv:92.432, 29. Claude Dellacherie ad Paul-Adré Meyer. A propos du ravail de Yor sur le grossisseme des ribus. I Sémiaire de Probabiliés XII, pages Spriger, 978. Claude Dellacherie ad Paul-Adré Meyer. Probabiliés e poeiel: Chapires 5 à 8. Théorie des marigales. Herma, Ja Haig. O filraios relaed o purely discoiuous marigales. I Sémiaire de Probabiliés XXXVI, pages Spriger, Sheg-wu He, Jia-gag Wag, ad Jia-a Ya. Semimarigale heory ad sochasic calculus. Beijig: Sciece Press. Boca Rao, FL: CRC Press Ic., Jea Jacod ad Aaoli V. Skorohod. Jumpig filraios ad marigales wih fiie variaio. I Sémiaire de Probabiliés XXVIII, pages Spriger, Moique Jeablac ad Ya Le Cam. Progressive elargeme of filraios wih iiial imes. Sochasic Processes ad heir Applicaios, 9(8): , Moique Jeablac, Marc Yor, ad Marc Chesey. Mahemaical mehods for fiacial markes. Spriger, Thierry Jeuli. Semi-marigales e grossisseme d ue filraio. Number Spriger, Cosaios Kardaras. O he characerisaio of hoes imes ha avoid all soppig imes. Sochasic Processes ad heir applicaios, (24): , Youes Kchia ad Philip Proer. O progressive filraio expasio wih a process, applicaios o isider radig. arxiv prepri arxiv: , Roger Masuy ad Marc Yor. Radom imes ad elargemes of filraios i a Browia seig. Number 873. Spriger Berli, Paul-Adré Meyer. Sur u héorème de J. Jacod. I Sémiaire de Probabiliés XII, pages Spriger, Philip Proer. Sochasic Iegraio ad Differeial Equaios: Versio 2., volume 2. Spriger Verlag, 24.