DOOB S MAXIMAL IDENTITY, MULTIPLICATIVE DECOMPOSITIONS AND ENLARGEMENTS OF FILTRATIONS

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1 Illinois Journal of Mahemaics Volume 5, Number 4, Winer 26, Pages S DOOB S MAXIMAL IDENTITY, MULTIPLICATIVE DECOMPOSITIONS AND ENLARGEMENTS OF FILTRATIONS ASHKAN NIKEGHBALI AND MARC YOR In he memory of J.L. Doob Absrac. In he heory of progressive enlargemens of filraions, he supermaringale Z = Pg > F associaed wih an hones ime g, and is addiive Doob-Meyer decomposiion, play an essenial role. In his paper, we propose an alernaive approach, using a muliplicaive represenaion for he supermaringale Z, based on Doob s maximal ideniy. We hus give new examples of progressive enlargemens. Moreover, we give, in our seing, a proof of he decomposiion formula for maringales, using iniial enlargemen echniques, and use i o obain some pah decomposiions given he maximum or minimum of some processes. 1. Inroducion Le Ω, F, F, P be a filered probabiliy space saisfying he usual hypoheses righ coninuous and complee. Given he end L of an F predicable se Γ, i.e., L = sup { :, ω Γ}, hese imes are also referred o as hones imes, M. Barlow [4] and Jeulin and Yor [1] have shown ha he supermaringale Z L = P L > F, chosen o be càdlàg, plays an essenial role in he enlargemen formulae wih respec o L, i.e., in expressing a general F maringale M as a semimaringale in F L, he smalles filraion which conains F, and makes Received May 18, 25; received in final form November 18, Mahemaics Subjec Classificaion. Primary 5C38, 15A15. Secondary 5A15, 15A c 26 Universiy of Illinois

2 792 ASHKAN NIKEGHBALI AND MARC YOR L a sopping ime. This enlargemen formula is 1.1 M = M + L d M, Z s Z s + L d M, 1 Z s 1 Z s, where M denoes an F L, P local maringale. Hence i is imporan o have an explici formula for Z L. In he lieraure abou progressive enlargemens of filraions, no many examples are fully developed see, e.g., [27], [9] or [8]; indeed, he compuaion of Z L is someimes difficul. Moreover, he examples are developed essenially in he Brownian seing, where, as we shall see, Z L is coninuous; no examples of disconinuous Z L s are known. In his paper, we firs consider a special family of hones imes g, and hen prove ha his family is generic in he sense ha every hones ime is in fac of his form under some reasonable assumpions. More precisely, we consider he following class of local maringales. Definiion 1.1. We say ha an F local maringale N belongs o he class C, if i is sricly posiive, wih no posiive jumps, and saisfies lim N =. Remark 1.2. Le N be a local maringale of class C. Then sup N s, s is supremum process, is coninuous. This propery is essenial in our paper. Hence, mos of he resuls we shall sae remain valid for posiive local maringales, which go o zero a infiniy, and whose suprema are coninuous. We associae wih a local maringale of class C he supermaringale N / and he random ime g defined as g sup { : N = S } = sup { : N = }. In Secion 2, we prove ha he associaed supermaringale Z saisfies 1.2 Z P g > F = N, and hen give he decomposiion formula 1.1 in erms of he local maringale N. This will provide us wih some new and explici examples of supermaringales Z which are disconinuous. We also esablish a relaionship beween he muliplicaive represenaion 1.2 and he Doob-Meyer addiive decomposiion of Z. In Secion 3, we sudy he problem of he iniial enlargemen of F wih he variable S, and hen give a new proof of 1.1.

3 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS 793 In Secion 4, we show ha he formula 1.2 is in fac very general. More precisely, for any end of a predicable se L, under he following assumpions CA he supermaringale Z L = PL > F may be represened in he form 1.2: C All F -maringales are c oninuous e.g., he Brownian filraion. A L avoids every F -sopping ime T, i.e., P [L = T ] =, In Secion 5, we give some new examples of enlargemens of filraions. Moreover, as an illusraion of our approach and he mehod of enlargemens of filraions, we recover and complee some known resuls of D. Williams [24] abou pah decomposiions of some diffusion processes, given heir minima. We add a new conribuion o hese pah decomposiions by inroducing a new family of random imes, as defined in [18] and called pseudo-sopping imes, which generalize he fundamenal noion of sopping imes, inroduced by J.L. Doob. We ake his opporuniy o quoe wo passages, resp., in he appendix of Meyer s book 1966, and in Dellacherie-Meyer s book, volume I [6, p. 184]: Les emps d arrê on éé uilisés, sans définiion formelle, depuis le débu de la héorie des processus. La noion apparaî ou à fai clairemen pour la première fois chez Doob en Il a sans doue fallu auan de génie aux créaeurs du calcul différeniel pour explicier la noion si simple de dérivée, qu à leurs successeurs pour faire ou le rese. L invenion des emps d arrê par Doob es ou à fai comparable. 2. A muliplicaive represenaion formula 2.1. Doob s maximal ideniy. Le N be a local maringale which belongs o he class C, wih N = x. Le = sup s N s. We consider 2.1 g = sup { : N = S } = sup { : N = }. To esablish our main proposiion, we shall need he following varian of Doob s maximal inequaliy, which we call Doob s maximal ideniy: Lemma 2.1 Doob s maximal ideniy. For any a >, we have: 1 x 2.2 P S > a = 1. a Hence, x/s is a uniform random variable on, 1. 2 For any sopping ime T we have 2.3 P S T NT > a F T = 1, a

4 794 ASHKAN NIKEGHBALI AND MARC YOR where is also a uniform random variable on, 1, indepen- Hence N T /S T den of F T. S T = sup N u. u T Proof. Formula 2.3 is a consequence of 2.2 when applied o he maringale N T +u u and he filraion F T +u u. Formula 2.2 iself is obvious when a x, and for a > x, i is obained by applying Doob s opional sopping heorem o he local maringale N Ta, where T a = inf{u : N u > a}. The nex proposiion gives an explici formula for Z Pg > F, in erms of he local maringale N. Wihou loss of generaliy, we assume from now on ha x = 1. Indeed, if N = x, we can consider he local maringale N /x which sars a 1. Proposiion In our seing, he formula Z = N,, holds. 2 The Doob-Meyer addiive decomposiion of Z is 2.4 Z = E [log S F ] log. Proof. We firs noe ha {g > } = { u > : S u = N u } = { u > : N u } { } = sup N u. u Hence, from 2.3, we ge Pg > F = N. To esablish 2.4, we represen N / by Io s formula o obain Z = S s dn s N s S s 2 ds s. Now, we remark ha he measure ds s is suppored by he se {s : Z s = 1}; hence Z = 1 + N = dn s S s 1 S s ds s, 1 S s dn s log.

5 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS 795 From he uniqueness of he Doob-Meyer decomposiion, log is he predicable increasing par of Z whils 1 S s dn s is is maringale par. As Z is of class D, 1 le. As Z =, log S = which proves 2. S s dn s is a uniformly inegrable maringale. Now, 1 S s dn s and hus 1 S s dn s = E [log S F ], Remark 2.3. I is well known, and i follows from 2.4, ha he maringale in 2.5 is in fac in BMO. Corollary 2.4. following hold: Assuming ha all F maringales are coninuous, he 1 log is he dual predicable projecion of 1 {g }, i.e., for any posiive predicable process k s, ds s E k g = E k s. S s 2 The random ime g is hones and avoids any F sopping ime T, i.e., P [g = T ] =. Proof. Under our assumpions, he predicable and opional sigma algebras are equal. Thus, i suffices o prove ha g avoids sopping imes, he oher asserions being obvious. Since log is he dual predicable projecion of 1 {g } and is coninuous, for any F sopping ime T, Thus we ge P g = T =. E [ 1 {g=t } ] = E [ log S T ] =. We can now wrie he formula 1.1 in erms of he maringale N. Proposiion 2.5. Le X be a local F maringale. Then, X has he following decomposiion as a semimaringale in F g : X = X + g d X, N s N s where X is an F g local maringale. g d X, N s S N s Proof. This is a consequence of formula 1.1 and Proposiion 2.2. We shall now give a relaionship beween and E[log S F ]. For his, we shall need he following easy exension of Skorokhod s reflecion lemma see [12, p. 72]:

6 796 ASHKAN NIKEGHBALI AND MARC YOR Lemma 2.6. Le y be a real-valued càdlàg funcion on [,, such ha y has no negaive jumps, and y =. Then, here exiss a unique pair z, a of funcions on [, such ha: 1 z=y+a, 2 z is posiive, càdlàg, and has no negaive jumps, 3 a is increasing, coninuous, vanishing a zero and he corresponding measure da s is suppored by {s : zs = }. The funcion a is moreover given by Proposiion 2.7. we have or, equivalenly, Wih Proof. By 2.4, we can wrie From Lemma 2.6, we deduce ha a = sup ys. s µ = E [log S F ], log = sup µ s 1 µ 1, s = exp µ 1. 1 Z = 1 µ + log. log = sup µ s 1. s 2.2. Some hidden Azéma-Yor maringales. We shall now associae wih he wo dimensional process log, Z a family of maringales reminiscen of he Azéma-Yor maringales see, e.g., [3] which we shall now discuss. In fac, once again, we have o inroduce a slighly generalized version of wha are usually called Azéma-Yor maringales. Indeed, hese maringales were originally defined for coninuous local maringales see [21, Chaper VI], while we would like o define hem for local maringales wihou posiive jumps. This exension can be obained using he following balayage argumen: Lemma 2.8. Le Y = M + A be a special semimaringale, where M is a càdlàg local maringale, and A a coninuous increasing process. Se H = { : Y = }, and define g sup{s < : Y s = }. Then, for any locally bounded predicable process k, k g is predicable and 2.6 k g Y = k Y + k gs dy s.

7 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS 797 Proof. The proof is he same as he proof for coninuous semimaringales. The reader can refer o [5, p. 144] for even more general versions of he balayage formula. Now, we can sae he following generalizaion of he classical Azéma-Yor maringales: Proposiion 2.9. Le N be a local maringale such ha is supremum process is coninuous his is he case if N is in he class C. Le f be a locally bounded Borel funcion and define F x = x dyfy. Then, X F f N is a local maringale and 2.7 F f N = f S s dn s + F S. Proof. In 2.6, ake k f, and Y N. Then, we have f S g N = Bu S g =, hence F f N = f S gs d S s N s. In conclusion, for any locally bounded funcion f, F f N = is a local maringale. f S s dn s + F S. f S s dn s + F S Remark 2.1. Alhough very simple, hese maringales played an essenial role in he resoluion by Azéma and Yor of Skorokhod s embedding problem see [21, Chaper VI] for more deails and references. Remark In [15], a special case of Proposiion 2.9, for specrally negaive Lévy maringales is obained by differen means. Now, we associae wih he wo dimensional process log, Z, a canonical family of local maringales which are in fac of he form 2.7. Proposiion Le f be a locally bounded Borel funcion, and le F x = x dyfy. 1 The following processes are local maringales: 2.8 F log f log 1 Z,.

8 798 ASHKAN NIKEGHBALI AND MARC YOR 2 Denoing Kx = F x 1 and kx = fx 1, he local maringales in 2.8 are seen o be equal o 2.9 K µ k µ µ µ,. Proof. 1 The fac ha 2.8 defines a local maringale may be seen as an applicaion of Io s lemma when f is regular, followed by a monoone class argumen. 2 Formula 2.9 is obained by a rivial change of variables, and he fac ha 1 Z = µ µ, which was derived in Proposiion 2.7. Similar formulas are derived in [17] from differen consid- Remark eraions. 3. Iniial expansion wih S and enlargemen formulae In his secion, we shall deal wih he quesion of iniial enlargemen of he filraion F wih he variable S. This problem canno be deal wih using he powerful enlargemen heorem of Jacod see [9], bu can be reaed by a careful combinaion of differen proposiions in [8]. However, we shall give a simple proof which can also be adaped o deal wih some oher siuaions. Evenually, we will use our resul abou he iniial expansion of F wih he variable S o recover formula 1.1. Le us define he new filraion F σs ε> F +ε σ S, which saisfies he usual assumpions. The new informaion σs is brough in a he origin of ime and g is a sopping ime for his larger filraion. More precisely, we have: Lemma We have The following hold: g = inf { : N = S }, and hence g is an F σs sopping ime. 2 Consequenly, F σs. F g Proof. 1 The measure d is suppored by he se { : N = }. As g = sup{ : N = }, he process does no grow afer g, which also saisfies g = inf { : = S } ; hence g is an F σs sopping ime. 2 This is obvious.

9 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS 799 Now we inroduce some sandard erminology. Definiion 3.2. We shall say ha he pair of filraions F, F σs saisfies he H hypohesis if every F semimaringale is an F σs semimaringale. We shall now show ha he pair of filraions F, F σs saisfies he H hypohesis and give he decomposiion of an F local maringale in F σs. For his, we need o know he condiional law of S given F. Proposiion 3.3. For any Borel bounded or posiive funcion f, we have E f S F = f 1 N N/ N dxf x = f 1 N + N dx f x x 2. Proof. The proof is based on Lemma 2.1; in he following, U is a random variable which follows he sandard uniform law and which is independen of F. E f S F = E f F = E f 1 {S } F + E f S 1 {S< } F = f P F + E f S 1 {S< } F = f P U N N F + E f 1 N U {U< } F = f 1 N N/ N + dxf. x A sraighforward change of variable in he las inegral also gives E f S F = f 1 N + N dy f y y 2. One may now ask if EfS F is of he form 2.7. The answer o his quesion is posiive. Indeed, E f S F = f 1 N + N dy f y y 2 = dy f y y 2 N dy f y y 2 f S. Hence, E f S F = H 1 + H h N,

10 8 ASHKAN NIKEGHBALI AND MARC YOR wih and h x = h f x x H x = x dy f y y 2 x f x x dy f y y 2, = x dy f y f x. y2 Moreover, again from formula 2.7, we have he following represenaion of EfS F as a sochasic inegral: 3.2 E f S F = E f S + h S s dn s. Le us sum up hese resuls, inroducing some noaions: λ f E f S F = f 1 N N and 3.4 λ f = E f S + where 3.5 λs f = h f S s. λ s f dn s, dx f x x 2, Moreover, here exis wo families of random measures λ dx and λ dx, wih λ dx = 1 N 3.6 dx δ S dx + N 1 {x>s} x 2, 3.7 λ dx = 1 dx δ S dx + 1 {x>s} x 2, such ha λ f = λ dx f x, λ f = λ dx f x. Thus here is an absolue coninuiy relaionship beween λ dx and λ dx; more precisely, 3.1 λ dx = λ dx ρ x,, wih 3.11 ρ x, = 1 N 1 {S=x} + 1 N 1 {S<x}. Now, we can sae he main heorem of his secion.

11 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS 81 Theorem 3.4. Le N be a local maringale in he class C recall ha N = 1. Then, he pair of filraions F, F σs saisfies he H hypohesis and every F local maringale X is an F σs semimaringale wih canonical decomposiion X = X + 1 {g>s} d X, N s N s where X is an F σs local maringale. 1 {g s} d X, N s S N s, Remark 3.5. The following proof is modelled on he argumens found in [27], alhough our framework is more general since we do no assume ha our filraion has he predicable represenaion propery wih respec o some maringale nor ha all maringales are coninuous. Proof. We can assume ha X is in H 1 ; he general case follows by localizaion. Le Λ s be an F s measurable se, and ake > s. Then, for any bounded es funcion f, λ f is a bounded maringale, hence in BMO, and we have E 1 Λs f A X X s = E 1 Λs λ f X λ s f X s = E 1 Λs λ f, X λ f, X s = E 1 Λs λ u f d X, N u s = E 1 Λs λ u dx ρ x, u f x d X, N u s = E 1 Λs s d X, N u ρ S, u. Bu from 3.11, we have ρ S, = 1 1 {S=S N } {S<S N }. I now suffices o noe from Lemma 3.1 ha is consan afer g and g is he firs ime when S =, or, in oher words, This complees he proof. 1 {S >} = 1 {g>}, and 1 {S =} = 1 {g }. Theorem 3.4 yields a new proof of he decomposiion formula in he progressive enlargemen case. More precisely, we have: Corollary 3.6. The pair of filraions F, F g saisfies he H hypohesis. Moreover, every F local maringale X decomposes as X = X + 1 {g>s} d X, N s N s 1 {g s} d X, N s S N s,

12 82 ASHKAN NIKEGHBALI AND MARC YOR where X is an F g local maringale. Proof. Le X be an F maringale which is in H 1 ; he general case follows by localizaion. From Theorem 3.4, X = X + 1 {g>s} d X, N s N s 1 {g s} d X, N s S N s, where X denoes an F σs maringale. Thus, X, which is equal o d X, N s d X, N s X 1 {g>s} 1 {g s},, N s S N s is F g adaped recall ha F g F σs, and hence is an F g maringale. 4. A muliplicaive characerizaion of Z Usually, in he lieraure abou progressive enlargemens of filraions, i is assumed ha he condiions CA are saisfied. Now, we shall prove ha under his assumpion he supermaringale Z L = PL > F, associaed wih an hones ime, can be represened as N /, where N is a posiive local maringale. More precisely, we have he following resul: Theorem 4.1. Le L be an hones ime. Then, under he condiions CA, here exiss a coninuous and nonnegaive local maringale N, wih N = 1 and lim N =, such ha Z = P L > F = N. Proof. Under he condiions CA, Z is coninuous and can be wrien as see [1] or [5] for deails Z = M A, where M and A are coninuous, Z = 1 and da is suppored by { : Z = 1}. Then, for < T inf{ : Z = }, we have log Z = and hence 4.1 log Z = dm s Z s 1 2 dm s 1 Z s 2 d M s Z 2 s Also, from Skorokhod s reflecion lemma, we have u dm s 4.2 A = sup 1 u u Z s 2 A, d M s Zs 2 + A. d M s Z 2 s.

13 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS 83 Now, combining 4.1 and 4.2, we obain where Z = N, dm s N = exp 1 Z s 2 is a local maringale saring from 1, and u = sup exp u u = exp sup u = exp A. dm s Z s 1 2 dm s Z s 1 2 u u d M s Zs 2 d M s Zs 2 d M s Z 2 s Finally, we noe ha, since Z T =, lim T N =, which allows us o define N for all. Corollary 4.2. The supermaringale Z = PL > F admis he following addiive and muliplicaive represenaions: Z = N, Z = M A. Moreover, hese wo represenaions are relaed as follows: dm s N = exp 1 d M s Z s 2 Zs 2, = exp A ; and dn s M = 1 + = E log S F, S s A = log. Proof. The resul is a consequence of Proposiion 2.2 and Theorem 4.1. Now, as a consequence of Theorem 4.1, we can recover he enlargemen formulae and he fac ha he pair of filraions F, F L saisfies he H hypohesis: Corollary 4.3. Le L be an hones ime. Then under he condiions CA, he pair of filraions F, F L saisfies he H hypohesis and every

14 84 ASHKAN NIKEGHBALI AND MARC YOR F local maringale X is an F L semimaringale wih canonical decomposiion L X = X d X, Z s d X, 1 Z s + +, Z s L 1 Z s where X denoes an F L local maringale. Proof. The resul is a combinaion of Theorem 4.1 and Corollary 3.6. Remark 4.4. We hus see ha under he assumpions CA he iniial enlargemen of filraions wih A amouns o enlarging iniially he filraion wih S, he erminal value of he supremum process of a coninuous local maringale in C. We shall now ouline anoher nonrivial consequence of Theorem 4.1. In [2], he auhors give explici examples of dual predicable projecions of processes of he form 1 g, where g is an hones ime. Indeed, hese dual projecions are naural examples of increasing injecive processes see [2] for more deails and references. Wih Theorem 4.1, we have a complee characerizaion of such projecions: Corollary 4.5. Assume he assumpion C holds, and le C be an increasing process. Then C is he dual predicable projecion of 1 g, for some hones ime g ha avoids sopping imes, if and only if here exiss a coninuous local maringale N in he class C such ha C = log. These resuls can be naurally exended o he case where he supermaringale Z has only negaive jumps; we considered he special of he hypohesis CA because of is pracical imporance. We only give here he exension of Theorem 4.1; he corollaries are easily deduced. Proposiion 4.6. Le L be an hones ime ha avoids sopping imes. Assume ha Z L has no posiive jumps. Then, here exiss a local maringale N, in he class C, wih N = 1, such ha Z L = Z = P L > F = N. Proof. We use he same noaions as in he proof of Theorem 4.1. < T inf{ : Z = }, we have dms log Z = 1 d M c s Z s 2 Zs 2 + log 1 + Z s Z s + A. Z s Z s <s For

15 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS 85 Now, from Lemma 2.6, A = sup s + <s dms Z s 1 2 Combining he las wo equaliies, we obain where N = exp dms Z s 1 2 d M c s Zs 2 log 1 + Z s Z s. Z s Z s Z = N, d M c s Zs Z s exp Z s. Z s Z s <s 5. Examples and applicaions In his secion, we look a some specific local maringales N, and use he iniial enlargemen formula wih S o ge he pah decomposiions given he maxima or minima of some sochasic processes. Our aim here is o illusrae how echniques from enlargemen of filraions can be applied. To have a complee descripion for he pah decomposiions, we associae wih g a random ime, called pseudo-sopping ime, which occurs before g. Evenually, we give some explici examples of supermaringales Z wih jumps Pseudo-sopping imes. In [18], we have proposed he following generalizaion of sopping imes: Definiion 5.1. Le ρ : Ω, F R + be a random ime; ρ is called a pseudo-sopping ime if for every bounded F maringale we have E M ρ = E M. David Williams [25] gave he firs example of such a random ime, and he following sysemaic consrucion is esablished in [18]: Proposiion 5.2. CA, is a pseudo-sopping ime, wih Le L be an hones ime. Then, under he condiions { ρ sup < L : Z L } = inf u L ZL u, Z ρ P ρ > F = inf u ZL u, and Z ρ ρ follows he uniform disribuion on, 1.

16 86 ASHKAN NIKEGHBALI AND MARC YOR The following propery, also proved in [18], is essenial in sudying pah decomposiions: Proposiion 5.3. Le ρ be a pseudo-sopping ime and le M be an F local maringale. Then M ρ is an F ρ local maringale. In our seing, Proposiion 5.2 gives: Proposiion 5.4. Define he nonincreasing process r by Then, { ρ sup < g : N u r inf. u S u N = inf u g N u S u is a pseudo-sopping ime, and r ρ follows he uniform disribuion on, Pah decomposiions given he maxima or he minima of a diffusion. Now, we shall apply he echniques of enlargemens of filraions o esablish some pah decomposiions resuls. Some of he following resuls have been proved by David Williams in [24], using differen mehods. Jeulin has also given a proof based on enlargemens echniques in he case of ransien diffusions see [8]. Here, we complee he resuls of David Williams by inroducing he pseudo-sopping imes ρ defined in Proposiion 5.4, and we give some ineresing examples The killed Brownian Moion. Le N B, where B is a Brownian Moion saring a 1, and sopped a T = inf{ : B = }. Le sup B s. s Le g = sup { : B = } and { ρ = sup < g : B B u } }. = inf u g S u = S g is disribued as he reciprocal of a From Doob s maximal ideniy, S T uniform disribuion, 1, i.e., i has he densiy 1 [1, x1/x 2. Proposiion 5.5. Le B be a Brownian Moion saring a 1 and sopped when i firs his. Then: B ρ /S ρ follows he uniform law on, 1, and condiionally on B ρ /S ρ = r, B is a Brownian Moion up o he firs ime when B = r.

17 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS 87 B is an F g and F σs T semimaringale wih canonical decomposiion 5.1 B = B + g ds T B s g ds S T B s, where B is an F σs T Brownian Moion, sopped a T and independen of S T. Consequenly, we have he following pah decomposiion: Condiionally on S T = m we have: 1 The process B ; g is a Bessel process of dimension 3, sared from 1, considered up o T m, he firs ime when i his m. 2 The process S g B g+ ; T g is an F g+ hree dimensional Bessel process, sared from, considered up o T m, he firs ime when i his m, and is independen of B ; g. Proof. The resuls concerning he decomposiion unil ρ are consequences of he resuls of Subsecion 5.1. The decomposiion formula is a consequence of Theorem 3.4. Since B is an F σs T bracke, i follows from Lévy s heorem ha i is an F σs T Moreover, i is independen of F σs T local maringale, wih T as is Brownian Moion. = σs T. Now, condiionally on S T = m, wih T m = inf{ : B = m}, B saisfies he following sochasic differenial equaion: B = B Tm ds +. B s Hence i is a hree dimensional Bessel process up o T m. I also follows from he decomposiion formula ha B g+ = B g+ + g ds B s This equaion can also be wrien as S g B g+ = Bg+ B g + T g T g ds S g B g+s. ds S g B g+s. Now, B g+ B g is an F g+ Brownian Moion, saring from, and is independen of F g. Taking β B g+ B g, which is also an F g+ Brownian Moion, saring from, independen of F g, he process ξ S g B saisfies he sochasic differenial equaion ξ = β + T g ds ξ s ; hence i is a hree dimensional Bessel process, sared a, and considered up o T m, and condiionally on S g, is independen of F g.

18 88 ASHKAN NIKEGHBALI AND MARC YOR Some recurren diffusions. The previous example can be generalized o a wider class of recurren diffusions X, saisfying he sochasic differenial equaion 5.2 X = x + B + b X s ds, x >, where B is he sandard Brownian Moion, and b is a Borel inegrable funcion such ha exisence and uniqueness holds for equaion 5.2 for example, b bounded or Lipschiz coninuous. The infiniesimal generaor L of his diffusion is L = 1 d 2 2 dx 2 + b x d dx. Le T inf{ : X =, and denoe by s he scale funcion of X, which is sricly increasing and which vanishes a zero, i.e., z s z = exp 2 b y dy, where Hence, y b y = b u du. N s X T s x is a coninuous local maringale belonging o he class C. If denoes he supremum process of N and X he supremum process of X, we have = s X T. s x Now, le and g = sup { < T : X = X }, { ρ = sup < g : X X = inf u g Proposiion 5.6. Le X be a diffusion process saisfying equaion 5.2. Then: X ρ /X ρ follows he uniform law on, 1, and condiionally on X ρ /X ρ = r, X, ρ is a diffusion process, up o he firs ime when X = rx, wih he same infiniesimal generaor as X. X u X u }.

19 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS X is an F g and F σx T semimaringale wih canonical decomposiion X = B + + g b X u du s X u T s X u du g s X u s X T s Xu du, where B is an F σx T Brownian Moion, sopped a T and independen of X T. Consequenly, we have he following pah decomposiion: Condiionally on X T = m we have: 1 The process X ; g is a diffusion process sared from x >, considered up o T m, he firs ime when i his m, wih infiniesimal generaor 1 d 2 2 dx 2 + b x + s x d s x dx. 2 The process X g+ ; T g is an F g+ diffusion process, sared from m, considered up o T, he firs ime when i his, and is independen of X ; g; is infiniesimal generaor is given by 1 2 dx 2 + d 2 b x + s x d s x s m dx. 3 X T follows he same law as s 1 1/U, where U follows he uniform law on, 1. Proof. The proof is exacly he same as he proof of Proposiion 5.5, so we will no reproduce i here Geomeric Brownian Moion wih negaive drif. Le N exp 2νB 2ν 2, where B is a sandard Brownian Moion, and ν >. Wih he noaion of Theorem 3.4, we have = exp sup 2ν B s νs s, and { } g = sup : B ν = sup B s νs s. Before saing our proposiion, le us menion ha we could have worked wih more general coninuous exponenial local maringales, bu we preferred o keep he discussion as simple as possible he proof for more general cases is exacly he same.

20 81 ASHKAN NIKEGHBALI AND MARC YOR Proposiion 5.7. have: Wih he assumpions and noaions used above, we 1 The variable sup s B s νs follows he exponenial law of parameer 2ν. 2 Every local maringale X is an F σs semimaringale and decomposes as X = X + 2ν X, B g 2ν g N s S N s d X, B s, where X is an F σs local maringale. 3 Condiionally on S = m, he process B ν; g is a Brownian Moion wih drif +ν up o he firs hiing ime of is maximum m/2ν. Proof. From Doob s maximal equaliy, expsup s g 2νB s 2ν 2 s 1 follows he uniform law and hence sup s B s νs follows he exponenial law of parameer 2ν. The decomposiion formula is a consequence of Theorem 3.4 and he fac ha dn = 2νN db. To show 3, i suffices o noice ha B ν is equal o B +ν in he filraion F σs of S., wih B an F σs Brownian Moion which is independen General ransien diffusions. Now, we consider R, a ransien diffusion wih values in [,, which has {} as enrance boundary. Le s be a scale funcion for R, which we can choose such ha s =, and s =. Then, under he law P x, for any x >, he local maringale N = sr /sx, saisfies he condiions of Theorem 3.4, and we have where and P x g > F = s R s I, g = sup { : R = I }, I = inf s R s. We hus recover resuls of Jeulin [8, Proposiion 6.29, p. 112] by oher means. Jeulin used his formula and gave a quick proof of a heorem of David Williams [24], using iniial enlargemen of filraions argumens. Our proof follows he same lines, and so we refer o he book of Jeulin for he argumen and insead describe an ineresing example, he hree dimensional Bessel process.

21 DOOB S MAXIMAL IDENTITY AND EXPANSIONS OF FILTRATIONS 811 Proposiion 5.8. Le R be a hree dimensional Bessel process saring from 1, and se, as above, I = inf s R s, and g = sup{ : R = I }. Define ρ by { } I I u ρ = sup < g : = inf. R u g Then: 1 The variable I ρ /R ρ follows he uniform law on, 1 and, condiionally on I ρ = rr ρ, R, T r is a hree dimensional Bessel process saring from 1, up o he firs ime T r when I = rr. 2 I I g follows he uniform law on, 1. 3 Condiionally on I = r, he process R, g is a Brownian Moion saring from 1 and sopped when i firs his r. Proof. There exiss β, a Brownian Moion, such ha ds R = 1 + β +. R s 1 follows easily from he resuls of Subsecion 5.1. Now, from Io s formula, i follows ha 1 dβ s = 1 R Rs 2 ; hence, i is a local maringale. In F σi, g β g = β ds, R s where β is an F σi Brownian Moion independen of I. Hence, R g decomposes as R g = β in F σi, and his complees he proof of 3. 2 is an immediae consequence of Doob s maximal ideniy. Remark 5.9. The previous mehod applies o any ransien diffusion R, wih values in,, and which saisfies R = x + B + R u duc R u, where c : R + R allows uniqueness in law for his equaion. These diffusions were sudied in [23] o obain an exension of Piman s heorem see also [27].

22 812 ASHKAN NIKEGHBALI AND MARC YOR 5.3. Some examples of Z wih jumps. We shall conclude his paper by giving some explici examples of disconinuous Z s. Le X be a Poisson process wih parameer c and le N = X c. N is a maringale in he naural filraion F of X. Every local maringale Y in his filraion may be wrien as E f Y = Y + k s dn s, where k is an F predicable process. Now, for f : R + R + a locally bounded and Borel funcion, le = exp f s dx s + c 1 exp f s ds. E f is an F local maringale which can be represened as E f = 1 + If fsds =, hen lim E f =. E f s exp f s 1 dn s. Proposiion 5.1. Le f be a nonnegaive locally bounded and Borel funcion on R +, such ha lim E f =. Define { } g = sup : E f = E f, where E f = sup Es f. s Then: 1 sup s fsdx s +c 1 exp fsds is disribued as a random variable wih he exponenial law wih parameer 1. 2 The supermaringale Z g associaed wih g is given by P g > F = E f E f. 3 Every F local maringale Y = k sdn s is a semimaringale in he filraion F σef, wih canonical decomposiion Y = Ỹ + c c g where Ỹ is an F σef g k s exp f s 1 ds E f s k s exp f s 1 E f ds, Es f local maringale.

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24 814 ASHKAN NIKEGHBALI AND MARC YOR [21] D. Revuz and M. Yor, Coninuous maringales and Brownian moion, Grundlehren der Mahemaischen Wissenschafen, vol. 293, Springer-Verlag, Berlin, MR h:65 [22] L. C. G. Rogers and D. Williams, Diffusions, Markov processes, and maringales. Vol. 2, Wiley Series in Probabiliy and Mahemaical Saisics: Probabiliy and Mahemaical Saisics, John Wiley & Sons Inc., New York, MR k:6117 [23] Y. Saisho and H. Tanemura, Piman ype heorem for one-dimensional diffusion processes, Tokyo J. Mah , MR b:654 [24] D. Williams, Pah decomposiion and coninuiy of local ime for one-dimensional diffusions. I, Proc. London Mah. Soc , MR #3373 [25], A non-sopping ime wih he opional-sopping propery, Bull. London Mah. Soc , MR f:676 [26] C. Yoeurp, Théorème de Girsanov généralisé, e grossissemen d une filraion, in: Grossissemens de filraions: exemples e applicaions, Lecure Noes in Mahemaics, vol. 1118, Springer-Verlag, Berlin, 1985, pp [27] M. Yor, Some aspecs of Brownian moion. Par II, Lecures in Mahemaics ETH Zürich, Birkhäuser Verlag, Basel, MR e:614 Ashkan Nikeghbali, Laboraoire de Probabiliés e Modèles Aléaoires, Universié Pierre e Marie Curie, and CNRS UMR 7599, 175 rue du Chevalere, F-7513 Paris, France Curren address: ETH Zurich, Deparmen Mahemaik, HG G16, Ramisrasse 11, 892 Zurich Swizerland address: ashkan.nikeghbali@mah.ehz.ch Marc Yor, Laboraoire de Probabiliés e Modèles Aléaoires, Universié Pierre e Marie Curie, and CNRS UMR 7599, 175 rue du Chevalere F-7513 Paris, France address: deaproba@proba.jussieu.fr