Random Times and Enlargements of Filtrations

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1 Random Times and Enlargemens of Filraions A hesis submied in fulfillmen of he requiremens for he degree of Docor of Philosophy Libo Li Faculy of Science School of Mahemaics and Saisics Universiy of Sydney April, 2012

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3 To My Family Believe in me, because I was made for chasing dreams. Saind - Illusion of Progress 3

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5 Acknowledgemen If I feel unhappy, I do mahemaics o become happy. If I am happy, I do mahemaics o keep happy. - Alfréd Rényi, Alhough he pah of research is largely a lonesome one, i would no have been possible wihou he generous suppor of numerous individuals o whom I am inellecually and personally indebed. However, before I ake his opporuniy o hank he many people who have made his hesis possible. I mus firs apologize o my grandparens o whom I bear he guil of no been able o fulfill my duies as a grandson. I mus graefully acknowledge my parens for providing me wih he opporuniy o pursue my selfish endeavours and o my moher for her quie, everyday suppor ha enabled me o focus on wriing my hesis. I mus reserve he bulk of my graiude for my friends Elise, Aliser, Ken, Laurence, Jon, Jaq, Bao and Huiming whose friendship, empahy and undersanding coninues o ranscend boh ime and geography. I mus hank my friends in Sydney for keeping me alive and sane. Thank you Belina, Sheryl and Emi for feeding me when I am hungry, for looking afer me when I am sick and for comforing me when I am down. Thank you Eienne, John and Jay for eneraining me when I am bored, for driving me when I am in need and for disracing me when I am working. I mus humbly hank all my eachers, boh pas and presen, whose knowledge and enhusiasm helped illuminae hese pages. Thank you in paricular o Trevor Sanon for providing me wih boh mahemaical simulus and refuge in a foreign land. Thank you o Ian Dous and Benjamin Goldys for heir immeasurable assisance and paience which considerably enriched my honours experience. Thank you o Pavel V. Gapeev and Monique Jeanblanc for drafing he join paper which manifesed ino Chaper 3 of his hesis. I is only wih heir insighs ha I was able o exend and complee he paper wih he help of my supervisor Marek Rukowski. Thank you Shiqi Song and again o Monique Jeanblanc for heir commens and inpus which helped grealy he developmen of Chaper 6. Finally, I am exremely graeful o Monique Jeanblanc and especially o my supervisor Marek Rukowski for all heir valuable ime and inpus which helped o fill he pages of his hesis. I is only wih heir paience and guidance ha I was able o complee my sudies. Thank you boh for helping me along his road of mahemaical research, a road which I hope o coninue for quie some imes o come. 5

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7 Conens Preface 9 1 Elemens from he General Theory of Sochasic Processes Sochasic Processes and Sopping Times Predicable and Opional Ses Projecion Theorems Dual Projecions and Increasing Processes Random Times and Relaed Processes Enlargemen of Filraion Preliminaries Iniial Enlargemen Progressive Enlargemen Hones Times Proofs of Semimaringale Decomposiion Resuls Sopped Processes: Arbirary Random Times Non-Sopped Processes: Hones Times Consrucing Random Times wih Given Survival Processes Inroducion Filering Example Azéma Supermaringale Condiional Disribuions Preliminary Resuls Consrucion Through a Change of Measure Case of a Brownian Filraion Applicaions o Valuaion of Credi Derivaives Defaulable Zero-Coupon Bonds Credi Defaul Swaps Random Times and Muliplicaive Sysems Inroducion Condiional Disribuions of Random Times

8 8 Random Times and Enlargemens of Filraions Characerisics of Random Times Inverse Problems Properies of Condiional Disribuions Exended Canonical Consrucions Random Time wih a Predeermined Generaor Random Time wih a Predeermined Condiional Disribuion Family of Random Times Muliplicaive Sysems Muliplicaive Sysems Associaed wih a Submaringale Predicable Muliplicaive Sysems Muliplicaive Approach o Random Times Muliplicaive Consrucion of a Random Time Non-Uniqueness of Condiional Disribuions Hones Times Non-Muliplicaive Approach o Condiional Disribuions Progressive Enlargemen of Filraion wih Pseudo-Hones Times Inroducion Random Times and Filraions Properies of Condiional Disribuions Enlargemens of Filraions Condiional Expecaions under Progressive Enlargemens Condiional Expecaions for Pseudo-Hones Times Condiional Expecaions for Pseudo-Iniial Times Properies of G-Local Maringales G-Local Maringales for Pseudo-Hones Times G-Local Maringales for Pseudo-Iniial Times Compensaors of he Indicaor Process Compensaor of H under Complee Separabiliy Compensaor of H for a Pseudo-Iniial Time Hypohesis H and Semimaringale Decomposiions Hypohesis H for he Progressive Enlargemen Hypohesis H for Pseudo-Iniial Times Applicaions o Financial Mahemaics Arbirage Free Markes Models Informaion Drif Sabiliy of Random Times under Min and Max Inroducion Hypohesis H under Min and Max Preliminary Resuls

9 L. Li Exended Canonical Consrucion Hypohesis H under Min and Max Preliminary Resuls General Case Pseudo-Sopping Time and Min and Max Admissibiliy of Generic Marke Models of Forward Swaps Raes Inroducion Admissible Families of Forward Swap Raes Linear Sysems Associaed wih Forward Swaps Graph Theory Terminology Inverse Problem for Deflaed Bonds Weak Admissibiliy of Forward Swaps T -Admissibiliy of Forward Swaps Admissible Marke Models of Forward Swap Raes Inverse Problem for Swap Annuiies Dynamics of Forward Swap Raes Appendix Proof of Proposiion Proof of Proposiion Marke Model of Forward CDS Spreads Inroducion Forward Credi Defaul Swaps Preliminaries Absrac Semimaringale Seups Seup A Seup B Seup C Seup D Marke Models for Forward CDS Spreads Model of One-Period CDS Spreads Model of LIBORs and One-Period CDS Spreads Model of One- and Two-Period CDS Spreads Model of One-Period and Co-Terminal CDS Spreads Generic Top-Down Approach Concluding Remarks

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11 Preface C es par la logique qu on démonre, c es par l inuiion qu on invene. 1 - Henri Poincaré, This hesis is based on series of research papers co-auhored by he candidae during his PhD sudies. They include Gapeev e al. [44], Li and Rukowski [76, 74, 75, 73] and Jeanblanc e al. [58]. Each chaper of his hesis is in fac self-conained wih he main heme revolving around he sudy of random imes, enlargemens of filraion and consrucion of marke models. Hisorical background The firs par of his hesis is devoed o he sudy of enlargemen of filraions. The sudy of enlargemens of filraions was moivaed by he following quesion: is he semimaringale propery invarian wih respec o a given enlargemen G of he base filraion F? In oher words, is any P, F-semimaringale a P, G-semimaringale where G is a fixed filraion such ha F G? I can be shown ha he answer o his quesion is negaive, in general. If he answer is posiive hen we say ha he hypohesis H holds beween F and G. If he hypohesis H holds, one is also ineresed in he semimaringale decomposiion of an F-semimaringale wih respec he enlarged filraion G. Tradiionally, he lieraure has focused on he iniial and progressive enlargemen of F wih a single finie valued random ime τ. These wo ypes of enlargemens has been sudied exensively in he works including, bu no limied o, Jeulin [62, 63, 64], Jeulin and Yor [65, 66], Jacod [50], Yor [107, 108] and Meyer [86]. In he sudy of iniial enlargemen, he sandard assumpion is he Jacod s crierion see [50], which assumes he condiional disribuion of he random ime τ is absoluely coninuous wih respec o a posiive measure on R +. Wihin his framework, i can shown ha he hypohesis H is indeed saisfied beween F and G and he G-semimaringale decomposiion of F-maringale can be derived explicily. On he oher hand, he sudy of progressive enlargemen has focused mainly on hones ime wih he main references been Jeulin [62, 63], Jeulin and Yor [66]. More recenly, we have he works of Gasbarra e al. [45], Jeanblanc and Le Cam [57], Jeanblanc and Song [59, 60] and El Karoui e al. [36]. In he works of Gasbarra e al. [45], Jeanblanc and Le Cam [57] and El Karoui e al. [36], he auhors sudied he Jacod s crierion also referred o as he densiy hypohesis in he conex of progressive enlargemen and have derived he G-compensaor of H = 1 τ, and he G-semimaringale decomposiion of F-maringales. The works of Jeanblanc and Song [59, 60] answer similar quesions ha are invesigaed in Chapers 3, 4 and 7 of his hesis, bu wih differen echniques and hey aim o describe a larger class of models for condiional disribuions, raher han focusing on he hypohesis HP see Definiion The second par of he hesis is devoed o he consrucion of Marke Models for forward swap raes and Credi Defaul Swaps CDS spreads. The marke model for forward LIBORs was firs examined in papers by Brace e al. [19] and Musiela and Rukowski [89]. Their approach was 1 I is by logic ha we prove, bu by inuiion ha we discover. 11

12 12 Random Times and Enlargemens of Filraions subsequenly exended by Jamshidian in [53, 54] o he marke model for co-erminal forward swap raes. Since hen, several papers on alernaive marke models for LIBORs and oher families of forward swap raes were published. To he bes of our knowledge, here is relaively scarce financial lieraure in regard eiher o he exisence or o mehods of consrucion of marke models for forward CDS spreads. This apparen gap is a bi surprising, especially when confroned wih he marke praciioners approach o credi defaul swapions, which hinges on a suiable varian of he Black formula. As background readings, he reader are direced o he works of Galluccio e al. [42], Pieersz and Regenmorel [95], Rukowski [98], or he monographs by Brace [18] or Musiela and Rukowski [90] and he references herein. Original conribuions We firs ouline he problems addressed in he hesis and he conribuions. The purpose of preliminary Chapers 1 and 2 is o give an overview of he classic resuls in he general heory of processes and enlargemens of filraions. For an exensive sudy of he general heory, he reader is referred o, e.g., Dellacherie [30] or He e al. [48]. The reader familiar wih he background maerial is hus advised o skip he firs wo chapers and move direcly o Chaper 3. Chapers 3 and 4, based on he papers Gapeev e al. [44] and Li and Rukowski [74], focus on he following main quesion: given a supermaringale G, which is assumed o be non-negaive and bounded by one, can we show ha here exiss a random ime such ha he Azéma supermaringale associaed wih his random ime is given by G? In he conex of random imes and heir applicaions o credi risk modeling, he above quesion was parially answered in a join paper by Gapeev e al. [44] and, using a differen mehod, in papers by Jeanblanc and Song in [59, 60] under he assumpion ha a supermaringale G is sricly posiive and saisfies cerain addiional coninuiy assumpions. Their research was coninued by Li and Rukowski in [74], who answered he above-menioned quesion in full generaliy wih he help of he concep of a muliplicaive sysem inroduced by Meyer [85]. Sadly, when one ook a closer look in he lieraure, i was menioned by Meyer on page 186 of [82] ha he original quesion was o some exen answered in a differen conex and using a differen mehod by Blumenhal and Geoor [16]. Wih he above in view, he main conribuions in Chapers 3 and 4 are: alernaive consrucions of a random ime wih a given in advanced Azéma supermaringale, he proof of he exisence of muliple random imes ha are consisen wih a given in advanced family of Azéma supermaringales, he proof of he uniqueness of condiional disribuion under hypohesis HP, a resul showing ha an hones ime is an F -measurable random ime saisfying he hypohesis HP. In Chapers 5 and 6, we focus on he heory of enlargemen of filraions by placing ourselves in he usual filered probabiliy space Ω, F, P wih he base filraion F = F 0. We work specifically under he progressive enlargemen of F wih τ. The aim of Chaper 5, derived from Li and Rukowski [75], is o sudy, in he conex of he heory of progressive enlargemen of filraion, he random imes consruced in Chaper 3 and 4. Similarly as in he classic paper of Jeulin [62] and he recen paper by Jeanblanc and Song [60], our goal is o examine he semimaringale decomposiion of F-maringales in he progressive enlargemen G of F. Unforunaely, one mus poin ou ha we were unable o show ha he hypohesis H is saisfied beween F and he progressive enlargemen G under only he hypohesis HP. The conribuions of his chaper o he curren lieraure can be summarised as follows: he semimaringale resul derived in his chaper exends he resul of Jeanblanc and Song [59] by obaining he form of he G-semimaringale decomposiion for a pseudo-hones ime wih a sricly posiive condiional disribuion, while removing coninuiy assumpions,

13 L. Li 13 we esablish he G-semimaringale decomposiion in he case where τ is consruced using he predicable muliplicaive sysem inroduced in Meyer [85]; i appears ha i resembles closely he classical G-semimaringale decomposiion for an hones ime and, in fac, i reduces o he classical decomposiion if we assume ha eiher Propery C or A holds see Definiion 1.5.4, inspired by Meyer [85], we inroduce he opional muliplicaive sysem and derive he G- semimaringale decomposiion when he random ime is consruced hrough he opional muliplicaive sysem, we show ha he hypohesis H is saisfied beween F and G if τ saisfies he exended densiy hypohesis see Definiion and derive he G-semimaringale decomposiion of F-maringales. In Chaper 6, we coninue working under he seing of progressive enlargemen and assume ha one is given wo random imes boh which saisfy eiher he hypohesis H or he hypohesis H. The quesion of ineres is herefore under which condiions are he hypohesis H or he hypohesis H sable under aking minimum and maximum. More specifically, we follow some preliminary works done in [58] o show under which condiions is he hypohesis H or hypohesis H saisfied beween F and he progressive enlargemen of F wih τ 1 τ 2 and/or τ 1 τ 2. The main conribuions of his chaper are he following resuls: a generalized version of he Norros lemma see Proposiion 3.1 in [43], he proof of sabiliy of he hypohesis H under minimum and maximum when he random imes are Cox-imes see Definiion 6.2.1, he proof of sabiliy of he hypohesis H under minimum and maximum for arbirary random imes. Diverging from he previous chapers, we focus in Chapers 7 and 8 on examining he necessary and sufficien condiions for he exisence of marke models for forward swap raes and CDS spreads which are applicable in he pricing of exoic insrumens such as credi defaul swapions. Given a family of forward swap raes, he aim of Chaper 7 is o firs re-examine and exend he works of Galluccio e al. [42] and Pieersz and van Regenmorel [95]. Chaper 8 is in he same vein as Chaper 7, bu we work under a defaulable framework and concenrae on exensions of CDS marke models presened in Brigo [22, 23] and exend he resuls o a semimaringale framework. This par of he hesis conribues o he exising lieraure in several respecs: we provide couner-examples o some resuls of Galluccio e al. [42], he resuls of Pieersz and van Regenmorel [95] on he posiiviy of bonds wihin a class of marke model are clarified, necessary and sufficien condiions for he exisence of a marke model consisen wih he given in advanced family of forward swaps and derive are given, he join dynamics of forward swap raes under a single probabiliy measure are derived, a sysemaic sudy of he marke models for CDS spreads is presened and he join dynamics of a family of CDS spreads are derived in each case. To conclude, one mus poin ou ha he derivaion of he join dynamics of forward swap rae or he CDS spreads does no jusify by iself ha he model is suiable for he arbirage pricing of credi derivaives. To show he viabiliy of he model, i would be sufficien o demonsrae ha he model arbirage-free in some sense, for insance, ha i can be suppored by an associaed arbirage-free model for defaul-free and defaulable zero-coupon bonds.

14 14 Random Times and Enlargemens of Filraions Overview of chapers Chaper 1. We inroduce, for he purpose of his hesis, he mos perinen resuls from he general heory of sochasic processes. Chaper 2. We give a brief overview of he heory of enlargemens of filraions. The basics of iniial and progressive enlargemen of a filraion wih a random ime are inroduced in Secions 2.2 and 2.3, respecively, while some classic resuls regarding he semimaringale decomposiion under he progressive enlargemen wih a random ime or an hones ime are presened in Secion 2.5. An original resul is Proposiion 2.2.1, which exends Jacod s crierion o he case where he F-condiional disribuion saisfies he random Lipschiz condiion. Chaper 3. We follow here he work of Gapeev e al. [44] o provide an explici consrucion of a random ime when he associaed Azéma semimaringale is given in advance. Our approach hinges on he use of a varian of Girsanov s heorem combined wih a judicious choice of he Radon-Nikodým densiy process. The proposed soluion is also parially moivaed by he classic example arising in he filering heory. Chaper 4. We presen he work by Li and Rukowski [74], which was moivaed by Gapeev e al. [44] and he recen resuls of Jeanblanc and Song [59, 60]. Our aim is o demonsrae, wih he help of muliplicaive sysems inroduced in Meyer [85], ha for any given posiive F-submaringale F such ha F = 1, here exiss a random ime τ on some exension of he filered probabiliy space such ha he Azéma submaringale associaed wih τ coincides wih F. Perinen properies of his consrucion are sudied and i is subsequenly exended o he case of several correlaed random imes wih he predeermined univariae condiional disribuions and an arbirary correlaion srucure given by a choice of a copula funcion. Chaper 5. This chaper derives from Li and Rukowski [75], where we deal wih various alernaive decomposiions of F-maringales wih respec o he filraion G which represens he enlargemen of a filraion F by a progressive flow of observaions of a random ime ha eiher belongs o he class of pseudo-hones imes or saisfies he exended densiy hypohesis. Several relaed resuls from he exising lieraure are essenially exended. We ouline wo poenial applicaions of our resuls o specific problems arising in Financial Mahemaics. Chaper 6. We move away from a single random ime and we presen some preliminary resuls obained by Jeanblanc, Li and Song [58]. The moivaion for his chaper derives from he well-known fac ha he minimum and maximum of wo sopping imes is again a sopping ime in he same filraion. Therefore, working under progressive enlargemen, he aim is o provide sufficien and/or necessary condiions for he sabiliy of he hypohesis H and H under minimum and maximum of wo random imes. Chaper 7. Following Li and Rukowski [76], we re-examine and exend cerain resuls from he papers by Galluccio e al. [42] and Pieersz and van Regenmorel [95]. We esablish several resuls providing alernaive necessary and sufficien condiions for admissibiliy of a family of forward swaps, ha is, he propery ha i is suppored by a family of bonds associaed wih he underlying enor srucure. We also derive he generic expression for he join dynamics of a family of forward swap raes under a single probabiliy measure and we show ha hese dynamics are uniquely deermined by a selecion of volailiy processes wih respec o he se of driving maringales. Chaper 8. Following Li and Rukowski [73], we provide he consrucion of several varians of marke models for forward CDS spreads, as firs presened by Brigo [23]. We compue explicily he join dynamics for some families of forward CDS spreads under a common probabiliy measure. We firs examine his problem for single-period CDS spreads under cerain simplifying assumpions. Subsequenly, we derive, wihou any resricions, he join dynamics under a common probabiliy measure for he family of one- and wo-period forward CDS spreads, and he family of one-period and co-erminal forward CDS spreads.

15 Chaper 1 Elemens from he General Theory of Sochasic Processes The purpose of his chaper is o inroduce he mos perinen resuls from he general heory of sochasic process for he purpose of his hesis. Mos resuls are presened here wihou proofs and hey can be found in Dellacherie [30] and He e al. [48]. I is assumed in his hesis ha he reader is familiar wih he heory of maringales and sochasic inegraion. 1.1 Sochasic Processes and Sopping Times This secion is devoed o basic properies of sopping imes and relaed resuls for sochasic processes. We assume hroughou ha we are given a probabiliy space Ω, F, P wih a filraion F, saisfying he usual condiions. Definiion A random ime τ is a map from Ω o R +. A random ime is a F-sopping ime if {τ } is an elemen of F for all 0. Definiion A random ime τ is said o avoid all F-sopping imes, if for all F-sopping ime T, we have Pτ = T = 0. Definiion A sopping ime T is predicable, if here exis a monoone increasing sequence of sopping imes T n n N such ha lim n T n = T Definiion Given a F-sopping ime T, he σ-algebras F T and F T are defined as follows F T = {A F A {T } F, 0} F T = F 0 σ{a {T > } where A F and 0} Proposiion Le T be a sopping ime. Then T F T F T. Proposiion If S and T are wo sopping imes hen 1. For all A F S, he se A {S T } F T 2. For all A F S, he se A {S < T } F T In paricular, if A = Ω hen {S T } belongs o F T and {S < T } belongs o F T Proposiion If S and T are wo sopping imes such ha S T hen F S F S is a sub σ-algebra of F T F T 15

16 16 Random Times and Enlargemens of Filraions Theorem Le T n n N be a monoone sequence of sopping imes such ha T = lim n T n. i If T n T hen F T = n F Tn. ii If T n T hen F T = n F Tn Corollary Le T n n N be a monoone sequence of sopping imes such ha lim n T n = T. i If a sequence T n n N is decreasing on he se {0 < T n < } for all n hen F T = n F Tn ii if a sequence T n n N is increasing on he se {0 < T n < } for all n hen F T = n F Tn Proof. The resul is an applicaion of Theorem and Proposiion The above heorems generalizes he definiion of F, where R +. Theorem Opional Sopping Theorem Le X be a righ-coninuous and uniformly inegrable maringale, hen for any sopping ime S and T such ha S T, E P XT F S = XS Theorem Predicable Sopping Theorem Le X be a righ-coninuous and uniformly inegrable maringale, hen for any predicable sopping imes S and T such ha S T, E P XT FT = EP X FT = XT. Corollary If he filraion F 0 is coninuous hen all F-maringales are coninuous. Proof. Noe ha T = is a predicable sopping ime. Hence by Theorem and coninuiy of filraion F = F = n F n. Therefore X is coninuous. X = E P X F = E P X F = X Theorem Doob-Meyer decomposiion Le X be a submaringale of class D. Then here exiss unique processes M and A such ha X = X 0 + M + A where M is a local maringale and A is a predicable process of finie variaion. Remark One adoped in Theorem he convenion ha M 0 = 0 and A 0 = 0.

17 L. Li Predicable and Opional Ses In his secion, we work on he space Ω R +, F BR +, P λ where λ is he Lebesgue measure on R +. The aim is o inroduce he noions of he predicable and opional σ-algebras on Ω R +. Definiion Given a filraion F, we define he following σ-algebras on Ω R + i he σ-algebra OF, which is generaed by all F-adaped càdlàg processes. ii he σ-algebra PF. which is generaed by all F-adaped càglàd processes. The σ-algebra OF is he F-opional σ-algebra and PF is he F-predicable σ-algebra. For breviy, we shall ofen wrie O for OF P for PF, if here is no confusion wih he filraion which we work on. Lemma The predicable σ-algebra is conained inside he opional σ-algebra. Proof. Suppose ha X is càglàd and F-adaped, so ha X P. We define X n = k=0 X k 2 n 1 [ k 2 n, k+1 2 n Then X n X poinwise and hus X is measurable wih respec o O because for each n N he process X n is measurable wih respec O. This shows ha P O. Definiion Le T and S be wo sopping imes, such ha S T. The se S, T included in Ω R + and given by he following is called a sochasic inerval S, T = {ω, Sω < T ω} The sochasic inervals S, T, S, T and S, T are defined similarly. The sochasic inerval T is also called he graph of T. Lemma i The predicable σ-algebra can also be characerised by he following PF = {A {0} : A F 0 } {A [s, : 0 < s <, s, Q +, A r<s F r }. ii The opional σ-algebra can also be characerised by he following equaliy O = σ{ S, S is a sopping ime} Theorem Opional Secion Theorem Le X and Y be wo opional predicable processes. If for every predicable sopping ime T, he random variables X T 1 {T < } and Y T 1 {T < } are inegrable and he equaliy E P XT 1 {T < } = EP YT 1 {T < } holds hen X = Y up o an evanescen se. 1.3 Projecion Theorems In his secion, we inroduce he noion of he opional and predicable projecion of a sochasic process. Theorem Le X be a measurable process, such ha for all sopping predicable ime T, X T 1 {T < } is inegrable. Then i There exiss a unique opional process o X such ha, for all sopping imes T, E P XT 1 {T < } F T = o X T 1 {T < }.

18 18 Random Times and Enlargemens of Filraions The process o X is called he F-opional projecion of X. ii There exiss a unique predicable process p X such ha, for all predicable sopping imes T, E P XT 1 {T < } F T = p X T 1 {T < }. The process p X is called he F-predicable projecion of X. Proposiion Le X be a measurable process and Y an opional predicable, resp. process. If he opional predicable, resp. projecion of X exiss, hen he opional predicable, resp. projecion of XY exiss and if given by o XY = o XY, p XY = p XY. Proof. For he opional case, invoking he propery of condiional expecaion, we obain E P XT Y T 1 {T < } FT = YT 1 {T < } E P XT 1 {T < } FT The proof for he predicable case is similar. = 1 {T < } Y T o X T. Proposiion Suppose ha X is a measurable process. If he opional and predicable projecions of X exis hen p o X = p X. Proof. The proof follows by direc compuaions: p o X T 1 {T < } = E o P X T 1 {T < } F T = E P EP XT 1 {T < } F T F T = E P XT 1 {T < } F T = p X1 {T < } where he hird equaliy holds by Theorem We conclude his secion by providing an ineresing characerisaion of he opional and predicable σ-algebras. Theorem Le IF be he σ-algebra generaed by he jump processes M where M ranges hrough all bounded F-maringales. Then OF = PF IF In paricular, if all F-maringales are coninuous hen OF = PF and every sopping ime is predicable. 1.4 Dual Projecions and Increasing Processes In his secion, we work on eiher he filered probabiliy space Ω, F, P wih he filraion F saisfying he usual condiions or he produc space Ω R +, F BR +, P λ. Le A be an non-adaped increasing process. We define a non-negaive measure µ A on he σ-algebra F BR + by seing µ A H := E P [0, 1 H ω, s da s ω, H F BR +.

19 L. Li 19 We say ha µ A is he measure generaed by he process A. The corresponding measure on he space of non-negaive, bounded processes is defined by µ A X := E P X s da s [0, where X is any non-negaive, bounded process. Definiion A measure µ on F BR + is said o be an opional measure if for any nonnegaive, bounded process X µx := E µ X = E µ o X =: µ o X where o X is he opional projecion of X. A measure µ is said o be a predicable measure if for any non-negaive, bounded process X where p X is he predicable projecion of X. µx := E µ X = E µ p X =: µ p X Theorem Le A be a non-adaped increasing process. Then he measure µ A generaed by A on F BR + is an opional predicable, resp. measure if and only if A is an adaped predicable, resp. process. Definiion If µ is a σ-finie measure on F BR + hen we define he measures µ o and µ p on he space of non-negaive processes by seing µ o X := µ o X, µ p X := µ p X Then µ o µ p, resp. is called he opional predicable, resp. projecion of µ. I is clear from definiion ha µ o µ p, resp. is an opional predicable, resp. measure. Remark The resricion of µ o he opional σ-algebra O is equal o µ o while he resricion of µ o P is equal o µ p. A measure ν on F BR + is opional predicable, resp. if and only if ν = ν o ν = ν p, resp.. Theorem Le µ A be a measure generaed by an increasing process A on F BR +. Then if A is a locally inegrable process. i he opional measure µ o A is generaed by a unique adaped increasing process, ii he predicable measure µ p A is generaed by a unique predicable increasing process, Lemma i If A is a locally inegrable increasing process hen here exiss a unique increasing opional process A o such ha, for any non-negaive bounded process X, E P [0, o X s da s = E P [0, X s da o s. The process A o is called he dual predicable projecion of A. ii If A is a locally inegrable increasing process hen here exiss a unique increasing predicable process A p such ha, for any non-negaive bounded process X, E P [0, p X s da s = E P [0, X s da p s. The process A p is called he dual predicable projecion of A.

20 20 Random Times and Enlargemens of Filraions Proof. We only presen he predicable case, since he proof for he opional case is similar. Le µ A be he measure generaed by A and µ p A be he predicable projecion of µ A. Theorem gives he exisence of a unique predicable, increasing process, which generaes µ p A ; we denoe his process by A p. Then, by definiion, E p P X s da s = µ A p X = µ p A X = E P X s da p s. [0, [0, Lemma i If A is a locally inegrable, increasing process and S and T are wo sopping imes such ha S T hen he following holds E o P X s da s F S = E P X s da o s F S. [S,T [S,T ii If A is a locally inegrable, increasing process hen he following holds E p P X s da s FS = E P X s da p s FS. [S,T [S,T Lemma Le A be a non-adaped process of inegrable variaion. Then he processes are uniformly inegrable maringales. o A A o and o A A p The following resuls shows how o calculae he jumps of he opional predicable dual projecions. Proposiion i Le S be any sopping ime. Then A o S1 {S< } = E P [ AS 1 {S< } F S ] ii Le S be any predicable sopping ime. Then A p S 1 {S< } = E P [ AS 1 {S< } F S ] 1.5 Random Times and Relaed Processes We will now apply he general heory of sochasic process o he sudy of finie random imes, which are given on he probabiliy space Ω, F, P endowed wih an arbirary filraion F = F 0 saisfying he usual condiions. Our goal is also o inroduce here he noaion for several perinen characerisics of a random ime τ defined on a filered probabiliy space Ω, F, F, P. Given a finie random ime τ and he non-adaped process H := 1 τ,, we focus on he dual opional predicable projecions of he process H. I is eviden ha he opional and predicable projecion depends on he choice of filraion. Thus hroughou he res of his hesis, if here is a need o disinguish beween projecions on differen filraions, hen we will adap he following noaion for he dual F-opional projecion and he dual F-predicable projecion of H = 1 τ, : Ā τ,f := H o,f, A τ,f := H p,f, [0,. Furhermore, he F-opional projecions are denoed by: F τ,f := o,f H, G τ,f := o,f 1 H, [0,. Since he random ime τ is finie, one can show ha G τ,f = 0 almos surely, and hus F τ,f F τ,f = 1. =

21 L. Li 21 Definiion Given a filraion F, he process G τ,f associaed wih he random ime τ. is ermed he Azéma supermaringale Proposiion The process G τ,f is generaed by Āτ,F or A τ,f, ha is G τ,f Āτ,F = E P Āτ,F F and G τ,f = E P A τ,f A τ,f F 1.1 for all [0,. Proof. Using Lemma 1.4.3, we can define he following F-maringale associaed wih he dual F- opional projecion m τ,f := F τ,f Āτ,F, [0,. Similarly, he F-maringale associaed wih he dual F-predicable projecion is given by m τ,f I should be sressed ha m τ,f are boh coninuous a infiniy. m τ,f = E P 1 Ā τ,f K and m τ,f he represenaion saed in m τ,f 0 := F τ,f A τ,f, [0,. and Āτ,F 0 are no necessarily zero and mτ,f and m τ,f Since F τ,f = 0, he processes m τ,f and m τ,f ake he form, = E P 1 A τ,f K for [0,. I is now easy o obain 0 A τ,f Definiion From Proposiion 1.5.1, we obain on [0, he following decomposiions: i he opional addiive decomposiion of he Azéma supermaringale associaed wih τ G τ,f Āτ,F = E P K Ā τ,f = M τ,f Āτ,F 1.2 τ,f Āτ,F where M := E P K. ii he predicable addiive decomposiion of he Azéma supermaringale associaed wih τ where M τ,f := E P A τ,f K. G τ,f = E P A τ,f K A τ,f = M τ,f A τ,f 1.3 Anoher process of ineres is V τ,f := o,f 1 1 τ,. I is known from [62] ha V τ,f = G τ,f + Âτ,F, V τ,f = G τ,f, V τ,f + = G τ,f. 1.4 The above equaliies will laer be useful in obaining an opional muliplicaive sysem associaed wih a random ime τ. For ease of presenaion, whenever here is no danger of confusion wih he random ime τ and he filraion F, we shall omi he superscrips τ,f on he dual projecions of H = 1 τ,. We hus denoe he dual opional and predicable projecions of H by Ā and A, respecively. The opional projecion of H will be denoed by F and he opional projecion of 1 0,τ by V. Therefore, equaions 1.2 and 1.3 can be rewrien as for he opional addiive decomposiion and G = M Ā, G = M A, for he predicable addiive decomposiion of G. Consequenly, he opional addiive decomposiion of F is given by F = 1 M + Ā,

22 22 Random Times and Enlargemens of Filraions and he predicable addiive decomposiion becomes F = 1 M + A, Anoher imporan process under consideraion will be he increasing process D, such ha 0 D 1 and D is F -measurable. I is defined by seing, for all R +, D := Pτ F = E P H F. 1.5 Noe ha we can recover he P, F-submaringale F associaed wih τ since F := Pτ F = E P D F Recall ha he supermaringale G = 1 F = Pτ > F is called he Azéma supermaringale of τ. Therefore, we find i naural o refer o he P, F-submaringale F as he Azéma submaringale of τ. Definiion The P, F-condiional disribuion of τ is he random field F u, u, R+ given by F u, := Pτ u F, u, R +. The P, F-condiional survival disribuion of τ is he random field G u, u, R+ given by G u, := Pτ > u F = 1 F u,, u, R +. Noe ha he following equaliies are valid, for all u, F u, = E P H u F = E P F u F 1.6 and hus, in paricular, he equaliy F = F, holds for all R +. I is also worh noing ha D = F, for all R +. I is obvious ha he random field F u, u, R+ provides more informaion abou he probabilisic properies of a random ime han he Azéma submaringale F. We will laer argue ha he knowledge of a process D also conveys more informaion abou a random ime han he Azéma submaringale F. This is inuiively clear, since when a random ime τ is no known hen F can always be recovered from D bu, in general, he converse implicaion does no hold. I appears ha, for a given Azéma submaringale F, one may find several increasing processes D such ha D is F -measurable and D generaes F, meaning ha he equaliy F = E P D F holds for all. In he following, we will someimes refer o he following wo assumpions, which are also frequenly used in he exising lieraure. Definiion i We say ha a random ime saisfies assumpion A whenever Pτ = S = 0 for all F sopping imes S, meaning ha τ avoids all F sopping imes. ii We say ha assumpion C is saisfied whenever all P, F-local maringales are coninuous. Le us conclude his secion wih an applicaion of Proposiion The following resul provides a sufficien condiion for he coninuiy of he dual opional predicable projecion of H. Lemma If a random ime τ avoids all F-sopping imes hen he F-dual opional projecion of H = 1 τ, is coninuous. Proof. By seing A = H = 1 {τ } in Proposiion 1.4.1, we obain since, by assumpion, Pτ = S = 0. E P H o S 1 {S< } = EP 1{τ=S} 1 {S< } Pτ = S = 0,

23 Chaper 2 Enlargemen of Filraion The heory of enlargemen of filraion has been sudied in he pas hiry years and is cenered around he wo following hypoheses: i The hypohesis H: any maringale in he smaller filraion is again a maringale in he larger filraion. ii The hypohesis H : any semimaringale in he smaller filraion is again a semimarinagale in he larger filraion. More specifically, he sudy of enlargemen of filraion usually assumes he exisence of a finie random ime τ on some filered probabiliy space Ω, F, P endowed wih a filraion F = F 0 saisfying he usual condiions. I is hen ypical o sudy he hypohesis H and he hypohesis H under he assumpion ha he larger filraion is obained by eiher he iniial or he progressive enlargemen wih he random ime τ see Definiion and Definiion The purpose of his chaper is o presen several known resuls from he lieraure of enlargemen of filraions. Noe ha he main focus of his hesis is on he progressive enlargemen wih a random ime, herefore, we shall only presen only a few resuls on he iniial enlargemen, while he res of he chaper is devoed o resuls on characerizaion of progressive enlargemen and semimaringale decomposiions. The main source of reference for his chaper are he works by Jacod [50], Jeulin [62, 63], Jeulin and Yor [65, 66] and Jeanblanc and Le Cam [57]. 2.1 Preliminaries We work on a probabiliy space Ω, F, P endowed wih a filraion F = F 0 saisfying he usual condiions and we assume ha τ is a random ime wih values in R + defined on his space. Definiion By an enlargemen of F associaed wih τ or, briefly, an enlargemen of F we mean any filraion K = K R+ in Ω, F, P, saisfying he usual condiions, and such ha: i he inclusion F K holds, meaning ha F K for all R +, and ii τ is a K-sopping ime. Definiion We say ha he hypohesis H is saisfied by filraions F and K under P if any P, F-maringale is also a P, K-maringale. We someimes wrie F K or say ha F is immersed in K. Lemma Assume ha F K. Then F K if and only if any of he following equivalen condiions holds: i for any R +, he σ-fields F and K are condiionally independen given F under P, ha is, for any bounded, F -measurable random variable ξ and any bounded, K -measurable random variable η we have E P ξη F = EP ξ F EP η F

24 24 Random Times and Enlargemens of Filraions ii for any R + and any u, he σ-fields F u and K are condiionally independen given F, iii for any R + and any bounded, F -measurable random variable ξ E P ξ K = EP ξ F, 2.2 iv For any R + and any bounded, K -measurable random variable η E P η F = EP η F. 2.3 v In paricular, if K is he progressive enlargemen of F see Definiion wih τ, hen for any R + P τ > F = P τ > F. 2.4 Definiion We say ha he hypohesis H is saisfied by filraions F and K under P if any P, F-semimaringale is also a P, K-semimaringale. The sudy of he hypohesis H is usually conduced in wo seps. Firs, one aims o esablished wheher he hypohesis H is saisfied by F and K under P. If he hypohesis H holds rue, hen he nex sep is o find ou wha is he P, K-semimaringale decomposiion of any P, F-local maringale. Therefore i is imporan o know characerisaion of he class of semimaringales given an arbirary filraion F. For his purpose, we quoe he Bicheler-Dellacherie heorem, which shall be useful in showing ha he hypohesis H holds beween F and K some enlargemens of F. Theorem A F-adaped process X is a F-semimaringale if and only if for any sequence of simple F-predicable process converging uniformly o zero, he sochasic inegral wih respec o X converges o zero in measure. The nex proposiion shows ha if he hypohesis H is saisfied beween F and K, hen he hypohesis H is also saisfied beween F and K +. he righ coninuous modificaion of K. Proposiion If he hypohesis H is saisfied beween F = F 0 and K = K 0 hen he hypohesis H is also saisfied beween F = F 0 and K + = K + 0. Proof. Suppose M F is a bounded F-maringale he righ coninuous version. The hypohesis H is saisfied beween F and K wih he K-semimaringale decomposiion of M F is given by M F = M K +A K, where M K is a K-local maringale and A K a K-adaped locally bounded variaion process. Le T n n N be a common localisaion sequence for he M K and A K, hen for every n N, he process M K T n = M F T n A K T n is a K-maringale. For every n N, using Proposiion 2.44 from [48], we define he K + -maringale M n := lim Ms T K s n = M T F n lim A K s T s n, wih he second equaliy holds by righ coninuiy of M F. I is also no hard o see ha for every n N, we have M n = M n T n. One can hen define a K + -local maringale by seing T 0 = 0 M K+ := M i 1 Ti 1,T i = M F i=1 i=1 lim A K s T s i 1 Ti 1,T i. This ells us ha K + -semimaringale decomposiion of any bounded F-maringale M F is given by M F K+ = M + ÂK+, where he K + -adaped locally bounded variaion process ÂK+ is given by Wih his we conclude he proof. Â K+ := i=1 lim A K s T s i 1 Ti 1,T i.

25 L. Li 25 Remark One should be careful ha regularisaion is done afer he process is sopped. In general, one can no inerchange sopping and limi from he righ as he processes are no assumed o be cádlág. In he following secions, we inroduce he wo forms of enlargemens, which were sudied in he exising lieraure, ha is, he iniial and he progressive enlargemen of F wih a random ime τ. In addiion, we shall also menion he class of hones imes. To conclude his chaper, we give an overview of known resuls concerning semimaringale decomposiion of F-maringales in he respecive enlargemen of F. 2.2 Iniial Enlargemen Definiion The iniial enlargemen of F is he filraion F τ = F τ R+ given by he equaliy F τ = s> στ F s for all R +. Lemma Suppose ha X is an inegrable F τ -adaped process. Then X akes he form where he map X s,q is B R + F q measurable. X = lim q X τ,q, Proof. By Corollary 2.4 in [97], for an inegrable process X, X = E P X τ F = lim E P X στ Fq q Since for q >, we have E P X στ Fq Fq στ, herefore, here exiss an B R + F q measurable map s, ω X s,qω such ha and his concludes he proof lim E P X στ Fq = lim Xτ,q q q Lemma Suppose X is a bounded F τ -predicable process zero a = 0, hen for any > 0, he process X has he following represenaion X = X τ,, where X s, B R + PF. Proof. The F τ -predicable σ-algebra PF τ is generaed by P F τ = {A {0} : A F τ 0 } {A ]s, ] : 0 < s <, s, Q +, A F r τ } r<s Since he σ-algebra r<s F r τ is conained in στ F s. Therefore, any r<s F r τ random variable A akes he form A τ,s, where A u,s is B R + F s measurable. measurable We need only o prove he claim of he lemma for he generaors of PF τ resriced o 0,. The generaor hus akes he form A τ,s 1 ]s,]. I is obvious ha 1 ]s,] is a lef-coninuous process in. I remains o observe ha for a fixed u 0 he map A u,s is F s -measurable. Hence A u,s 1 ]s,] is B R + PF measurable and i is sufficien o ake X u, := A u,s 1 ]s,]. Theorem Jacod s Crierion If he F-condiional disribuion of a random ime τ is absoluely coninuous wih respec o an σ-finie measure on B R +, hen hypohesis H is saisfied beween F and F τ. Jacod s crierion was firs formulaed for he iniial enlargemen F τ only. However, in general, one can formulae Jacod s crierion as an assumpion on he F-condiional disribuion of τ and can sudy any enlargemen of F.

26 26 Random Times and Enlargemens of Filraions Definiion A pair of τ, P is said o saisfy he densiy hypohesis if here exiss a posiive random field m s, and a σ-finie measure η on B R + such ha for a fixed s 0, he process m s, s is a P, F-maringales and he F-condiional disribuion Fu, τ admis he following represenaion Fu, τ = m s, dη s, u. [0,u] Semimaringale decomposiion of F-maringale under he densiy hypohesis has been examined by several papers including, Jeanblanc and Le Cam [57] and El Karoui e al. [35]. Theorem Suppose ha X is a P, F-local maringale and τ saisfies he densiy hypohesis. Then he process X, which is given by 1 u=τ X = X + X, m u, s, m u,s is a P, F τ -local maringale. 0,] Remark Jeanblanc and Le Cam [57] have shown ha if he random ime saisfies densiy hypohesis hen hypohesis H is also saisfied beween F and he progressive enlargemen of F wih τ. They also compued explicily see Theorem he semimaringale decomposiion of any P, F-maringale in he progressive enlargemen of F. In he following, we provide a simple exension of Jacod s crierion. Proposiion Given a random ime τ, if he F-condiional disribuion of τ saisfied he random Lipschiz condiion, hen he hypohesis H is saisfied beween F and he iniial enlargemen of F wih τ. Proof. Given an F-semimaringale X, we will prove he claim by conradicion. If X is no a F τ semimaringale, hen by Theorem here exiss 0, ɛ > 0 and a sequence of simple F τ - predicable process {ξ n } n N converging uniformly o zero, such ha inf n N E P 1 ξ n X ɛ. Firsly, for large enough n, he process ξ n s s 0 is bounded and F τ -predicable. The non-decreasing process F u u 0 is random Lipschiz, ha is, here exiss some posiive inegrable process K, such ha F u F s K η u η s where η is a σ-finie measure on BR +. Then i is easy o see ha F u F s K η u K η s F u F s. Then by he Kunia-Waanabe inequaliy, we obain 1/2 1/2 1 ξ n u X F du 1 ξ n u X 2 K dη u df u, [0, [ [0, [ [0, [ 1/2 1 ξ n u X K dη u F, F 0, 1/2 [0, [ By aking he expecaion and using he Cauchy-Schwarz inequaliy, we obain E P 1 ξ n X E P 1 ξ n u X K dη u E P F, F 0, [0, [ Pτ > 0 E Q 1 ξ n u X dη u [0, [

27 L. Li 27 where dq = K dp. The erm E Q 1 ξ n u X in he righ-hand side converges o zero, since he semimaringale propery is invarian under an absoluely change of measure and X is an F-semimaringale. 2.3 Progressive Enlargemen The iniial enlargemen does no seem o be well suied for he analysis of a random ime since i posulaes ha στ F τ 0, meaning ha all he informaion abou τ is already available a ime 0. I appears ha he following noion of he progressive enlargemen is more suiable for formulaing and solving problems associaed wih an addiional informaion conveyed by observaions of occurrence of a random ime τ. Definiion The progressive enlargemen of F is he minimal enlargemen, ha is, he smalles filraion F τ = F τ R+, saisfying he usual condiions, such ha F F τ and τ is a F τ -sopping ime. More explicily, F τ = s> στ F for all R +. We shall ofen denoe he progressive enlargemen of F wih a random ime τ by G when here is no confusion beween random imes. Le H be he filraion generaed by he process H = 1 {τ }. I is clear ha F τ F H F τ. In fac, he inclusion F τ K necessarily holds for any enlargemen of F. In he res of he hesis, we will mainly work wih he progressive enlargemen F τ. However, we would like firs o clarify he relaionships beween various filraions encounered in he exising lieraure. One can find a commen by Meyer [86], where he essenially inroduces he progressive enlargemen as in he form given in Definiion We firs make his commen explici and find i convenien o inroduce he following definiion, which hinges on a naural modificaion of F which is inroduced in he nex secion, where we sudy he class of hones imes. Definiion The family F τ = F τ R+ is defined by seing, for every 0, F τ = {A F τ Ã F and Âτ, F τ such ha A = Ã {τ > } Âτ, {τ }}. We noe ha, for all R +, F τ {τ > } = F {τ > }, F τ {τ } = F τ {τ }. 2.5 I is easily seen ha he σ-field F τ is uniquely characerized by condiions 2.5. The nex resul shows ha he family F τ coincides in fac wih he progressive enlargemen F τ. Lemma If τ is any random ime hen F τ = F τ. Proof. Recall ha F τ = s> στ s F s and F τ = s> στ F s see Definiion and Definiion To show ha F τ = F τ, i suffices o check ha condiions 2.5 are saisfied by F τ. The following relaionship for all R + is immediae, F {τ > } F τ {τ > } F τ {τ > } = F {τ > }. This shows ha F τ {τ > } = F {τ > }, while on he oher hand, F τ {τ } = s> στ s F s {τ } = s> στ F s {τ } = F τ {τ } since στ s {τ } = στ {τ } for every s >. Definiion We say ha an enlargemen K is admissible before τ if he equaliy K {τ > } = F {τ > } holds for every R +.