This text is a survey of the general theory of stochastic processes, with a view towards random times and enlargements of filtrations. The first five chapters present standard materials, which were developed by the French probability school and which are usually written in French. The material presented in the last three chapters is less standard and takes into account some recent developments. AMS 2000 subject classifications: Primary 05C38, 15A15; secondary 05A15, 15A18. Keywords and phrases: General theory of stochastic processes, Enlargements of filtrations, Random times, Submartingales, Stopping times, Honest times, Pseudo-stopping times. Received August 2005.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic notions of the general theory . . . . . . . . . . . . . . . . . . . . 2.1 Stopping times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Progressive, Optional and Predictable σ-fields . . . . . . . . . . . 2.3 Classification of stopping times . . . . . . . . . . . . . . . . . . . 2.4 D´but theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . e 3 Section theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Projection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The optional and predictable projections . . . . . . . . . . . . . . 4.2 Increasing processes and projections . . . . . . . . . . . . . . . . 4.3 Random measures on (R+ × Ω) and the dual projections . . . . . 5 The Doob-Meyer decomposition and multiplicative decompositions . . 6 Multiplicative decompositions . . . . . . . . . . . . . . . . . . . . . . . 7 Some hidden martingales . . . . . . . . . . . . . . . . . . . . . . . . . 8 General random times, their associated σ-fields and Az´ma’s supere martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Arbitrary random times and some associated sigma fields . . . .
∗ This

346 347 347 347 348 351 353 355 357 357 360 362 371 372 375 381 381

is an original survey paper 345

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Az´ma’s supermartingales and dual projections associated with e random times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 The case of honest times . . . . . . . . . . . . . . . . . . . 8.2.2 The case of pseudo-stopping times . . . . . . . . . . . . . 8.3 Honest times and Strong Brownian Filtrations . . . . . . . . . . 9 The enlargements of filtrations . . . . . . . . . . . . . . . . . . . . . . 9.1 Initial enlargements of filtrations . . . . . . . . . . . . . . . . . . 9.2 Progressive enlargements of filtrations . . . . . . . . . . . . . . . ρ 9.2.1 A description of predictable and optional processes in (Gt ) L and Ft . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 The decomposition formula before ρ . . . . . . . . . . . . 9.2.3 The decomposition formula for honest times . . . . . . . . 9.3 The (H) hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Concluding remarks on enlargements of filtrations . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction

8.2

384 385 391 395 396 397 401 402 403 405 407 408 408

P.A. Meyer and C. Dellacherie have created the so called general theory of stochastic processes, which consists of a number of fundamental operations on either real valued stochastic processes indexed by [0, ∞), or random measures

on [0, ∞), relative to a given filtered probability space Ω, F , (Ft )t≥0 , P , where (Ft ) is a right continuous filtration of (F , P) complete sub-σ-fields of F . This theory was gradually created from results which originated from the study of Markov processes, and martingales and additive functionals associated with them. A guiding principle for Meyer and Dellacherie was to understand to which extent the Markov property could be avoided; in fact, they were able to get rid of the Markov property in a radical way. At this point, we would like to emphasize that, perhaps to the astonishment of some readers, stochastic calculus was not thought of as a basic “elementary” tool in 1972, when C. Dellacherie’s little book appeared. Thus it seemed interesting to view some important facts of the general theory in relation with stochastic calculus. The present essay falls into two parts: the first part, consisting of sections 2 to 5, is a review of the General Theory of Stochastic Processes and is fairly well known. The second part is a review of more recent results, and is much less so. Throughout this essay we try to illustrate as much as possible the results with examples. More precisely, the plan of the essay is as follows: • in Section 2, we recall the basic notions of the theory: stopping times, the optional and predictable σ-fields and processes,etc. • in Section 3, we present the fundamental Section theorems; • in Section 4, we present the fundamental Projection theorems;

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• in Section 5, we recall the Doob-Meyer decomposition of semimartingales; • in Section 6, we present a small theory of multiplicative decompositions of nonnegative local submartingales; • in Section 7, we highlight the role of certain “hidden” martingales in the general theory of stochastic processes; • in Section 8, we illustrate the theory with the study of arbitrary random times; • in Section 9, we study how the basic operations depend on the underlying filtration, which leads us in fact to some introduction of the theory of enlargement of filtrations; Acknowledgements I would like to thank an anonymous referee for his comments and suggestions which helped to improve the present text. 2. Basic notions of the general theory Throughout this essay, we assume we are given a filtered probability space Ω, F , (Ft )t≥0 , P that satisfies the usual conditions, that is (Ft )

is a right continuous filtration of (F , P) complete sub-σ-fields of F . A stochastic process is said to be c`dl`g if it almost surely has sample paths a a which are right continuous with left limits. A stochastic process is said to be c`gl`d if it almost surely has sample paths which are left continuous with right a a limits. 2.1. Stopping times Definition 2.1. A stopping time is a mapping T : Ω → R+ such that {T ≤ t} ∈ Ft for all t ≥ 0. To a given stopping time T , we associate the σ-field FT defined by: FT = {A ∈ F, A ∩ {T ≤ t} ∈ Ft f or all t ≥ 0} . We can also associate with T the σ-field FT − generated by F0 and sets of the form: A ∩ {T > t} , with A ∈ Ft and t ≥ 0. Proposition 2.2. Let T be a stopping time. Then T is measurable with respect to FT − and FT − ⊂ FT . Proposition 2.3. Let T be a stopping time. If A ∈ FT , then TA (ω) = T (ω) if ω ∈ A +∞ if ω ∈ A /

We recap here without proof some of the classical properties of stopping times.

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is also a stopping time. Proposition 2.4 ([26], Theorem 53, p.187). Let S and T be two stopping times. 1. For every A ∈ FS , the set A ∩ {S ≤ T } ∈ FT . 2. For every A ∈ FS , the set A ∩ {S < T } ∈ FT −. Proposition 2.5 ([26], Theorem 56, p.189). Let S and T be two stopping times such that S ≤ T . Then FS ⊂ FT . One of the most used properties of stopping times is the optional stopping theorem. Theorem 2.6 ([69], Theorem 3.2, p.69). Let (Mt ) be an (Ft ) uniformly integrable martingale and let T be a stopping time. Then, one has: E [M∞ | FT ] = MT and hence: E [M∞ ] = E [MT ] (2.2) One can naturally ask whether there exist some other random times (i.e. nonnegative random variables) such that (2.1) or (2.2) hold. We will answer these questions in subsequent sections. 2.2. Progressive, Optional and Predictable σ-fields Now, we shall define the three fundamental σ-algebras we always deal with in the theory of stochastic processes. Definition 2.7. A process X = (Xt )t≥0 is called (Ft ) progressive if for every t ≥ 0, the restriction of (t, ω) → Xt (ω) to [0, t] × Ω is B [0, t] ⊗ Ft measurable. A set A ∈ R+ × Ω is called progressive if the process 1A (t, ω) is progressive. The set of all progressive sets is a σ-algebra called the progressive σ-algebra, which we will denote M. Proposition 2.8 ([69], Proposition 4.9, p.44). If X is a (Ft ) progressive process and T is a (Ft ) stopping time, then XT 1{T ε} if Tn+1 < ∞.

Since X has left limits, Tn ↑ ∞. Now set: Y ≡ Zn 1[Tn ,Tn+1 [ . n≥0 Then |X − Y | ≤ ε and this completes the proof. Remark 2.17. We also have: O = σ {[0, T [ , T is a stopping time} . Remark 2.18. It is useful to note that for a random time T , we have that [T, ∞[ is in the optional sigma field if and only if T is a stopping time. A similar result holds for the predictable σ-algebra (see [26], [39] or [69]). Proposition 2.19 ([26], Theorem 67, p. 200). The predictable σ-algebra is generated by one of the following collections of random sets: 1. A × {0} where A ∈ F0 , and [0, T ] where T is a stopping time; 2. A × {0} where A ∈ F0 , and A × (s, t]where s < t, A ∈ Fs ; Now we give an easy result which is often used in martingale theory. Proposition 2.20. Let X = (Xt )t≥0 be an optional process. Then: 1. The jump process ∆X ≡ X − X− is optional; 2. X is predictable; 3. if moreover X is predictable, then ∆X is predictable.

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2.3. Classification of stopping times We shall now give finer results about stopping times. The notions developped here are very useful in the study of discontinuous semimartingales (see [39] for example). The proofs of the results presented here can be found in [24] or [26]. We first introduce the concept of predictable stopping times. Definition 2.21. A predictable time is a mapping T : Ω → R+ such that the stochastic interval [0, T [ is predictable. Every predictable time is a stopping time since [T, ∞[ ∈ P ⊂ O. Moreover, as [T ] = [0, T ] \ [0, T [, we deduce that [T ] ∈ P. We also have the following characterization of predictable times: Proposition 2.22 ([26], Theorem 71, p.204). A stopping time T is predictable if there exists a sequence of stopping times (Tn ) satisfying the following conditions: 1. (Tn ) is increasing with limit T . 2. we have {Tn < T } for all n on the set {T > 0}; The sequence (Tn ) is called an announcing sequence for T . Now we enumerate some important properties of predictable stopping times, which can be found in [24] p.54, or [26] p.205. Theorem 2.23. Let S be a predictable stopping time and T any stopping time. For all A ∈ FS− , the set A ∩ {S ≤ T } ∈ FT − . In particular, the sets {S ≤ T } and {S = T } are in FT − . Proposition 2.24. Let S and T be two predictable stopping times. Then the stopping times S ∧ T and S ∨ T are also predictable. Proposition 2.25. Let A ∈ FT − and T a predictable stopping time. Then the time TA is also predictable. Proposition 2.26. Let (Tn ) be an increasing sequence of predictable stopping times and T = limn Tn . Then T is predictable. We recall that a random set A is called evanescent if the set {ω : ∃ t ∈ R+ with (t, ω) ∈ A} is P−null. Definition 2.27. Let T be a stopping time. 1. We say that T is accessible if there exists a sequence (Tn ) of predictable stopping times such that: [T ] ⊂ (∪n [Tn ]) up to an evanescent set, or in other words, P [∪n {ω : Tn (ω) = T (ω) < ∞}] = 1

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2. We say that T is totally inaccessible if for all predictable stopping times S we have: [T ] ∩ [S] = ∅ up to an evanescent set, or in other words: P [{ω : T (ω) = S (ω) < ∞}] = 0. Remark 2.28. It is obvious that predictable stopping times are accessible and that the stopping times which are both accessible and totally inaccessible are almost surely infinite. Remark 2.29. There exist stopping times which are accessible but not predictable. Theorem 2.30 ([26] Theorem 81, p.215). Let T be a stopping time. There exists a unique (up to a P−null set) partition of the set {T < ∞} into two sets A and B which belong to FT − such that TA is accessible and TB is totally inaccessible. The stopping time TA is called the accessible part of T while TB is called the totally inaccessible part of T . Now let us examine a special case where the accessible times are predictable. For this, we need to define the concept of quasi-left continuous filtrations. Definition 2.31. The filtration (Ft ) is quasi-left continuous if FT = FT − for all predictable stopping times. Theorem 2.32 ([26] Theorem 83, p.217). The following assertions are equivalent: 1. The accessible stopping times are predictable; 2. The filtration (Ft ) is quasi-left continuous; 3. The filtration (Ft ) does not have any discontinuity time: FTn = F(lim Tn ) for all increasing sequences of stopping times (Tn ). Definition 2.33. A c`dl`g process X is called quasi-left continuous if ∆XT = 0, a a a.s. on the set {T < ∞} for every predictable time T . Definition 2.34. A random set A is called thin if it is of the form A = ∪ [Tn ], where (Tn ) is a sequence of stopping times; if moreover the sequence (Tn ) satisfies [Tn ] ∩ [Tm ] = ∅ for all n = m, it is called an exhausting sequence for A. Proposition 2.35. Let X be a c`dl`g adapted process. The following are equiva a alent: 1. X is quasi-left continuous;

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2. there exists a sequence of totally inaccessible stopping times that exhausts the jumps of X; 3. for any increasing sequence of stopping times (Tn ) with limit T , we have lim XTn = XT a.s. on the set {T < ∞}. 2.4. D´but theorems e In this section, we give a fundamental result for realizations of stopping times: the d´but theorem. Its proof is difficult and uses the same hard theory (capacities e theory) as the section theorems which we shall state in the next section. Definition 2.36. Let A be a subset of R+ × Ω. The d´but of A is the function e DA defined as: DA (ω) = inf {t ∈ R+ : (t, ω) ∈ A} , with DA (ω) = ∞ if this set is empty. It is a nice and difficult result that when the set A is progressive, then DA is a stopping time ([24], [26]): Theorem 2.37 ([24], Theorem 23, p. 51). Let A be a progressive set, then DA is a stopping time. Conversely, every stopping time is the d´but of a progressive (in fact optional) e set: indeed, it suffices to take A = [T, ∞[ or A = [T ]. The proof of the d´but theorem is an easy consequence of the following dife ficult result from measure theory: Theorem 2.38. If (E, E) is a locally compact space with a countable basis with its Borel σ-field and (Ω, F , P) is a complete probability space, for every set A ∈ E ⊗ F, the projection π (A) of A into Ω belongs to F . Proof of the d´but theorem. We apply Theorem 2.38 to the set At = A∩([0, t[ × Ω) e which belongs to B ([0, t[)⊗Ft . As a result, {DA ≤ t} = π (At ) belongs to Ft . We can define the n-d´but of a set A by e n DA (ω) = inf {t ∈ R+ : [0, t] ∩ A contains at least n points} ;

we can also define the ∞−d´but of A by: e n DA (ω) = inf {t ∈ R+ : [0, t] ∩ A contains infinitely many points} .

Theorem 2.39. The n-d´but of a progressive set A is a stopping time for e n = 1, 2, . . . , ∞.

1 Proof. The proof is easy once we know that DA (ω) is a stopping time. Indeed, n+1 by induction on n, we prove that DA (ω) which is the d´but of the progressive e n ∞ set An = A ∩ ]DA (ω) , ∞[. DA (ω) is also a stopping time as the d´but of the e progressive set ∩An .

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It is also possible to show that the penetration time T of a progressive set A, defined by: T (ω) = inf {t ∈ R+ : [0, t] ∩ A contains infinitely non countable many points} is a stopping time. We can naturally wonder if the d´but of a predictable set is a predictable e stopping time. One moment of reflexion shows that the answer is negative: every stopping time is the d´but of the predictable set ]T, ∞[ without being e predictable itself. However, we have: Proposition 2.40. Let DA be the d´but of a predictable set A. If [DA ] ⊂ A, e then DA is a predictable stopping time. Proof. If [DA ] ⊂ A, then [DA ] = A∩[0, DA ], is predictable since A is predictable and DA is a stopping time. Hence DA is predictable. One can deduce from there that: Proposition 2.41. Let A be a predictable set which is closed for the right e topology1 . Then its d´but DA is a predictable stopping time.

Now we are going to link the above mentioned notions to the jumps of some stochastic processes. We will follow [39], chapter I. Lemma 2.42. Any thin random set admits an exhausting sequence of stopping times. Proposition 2.43. If X is a c`dl`g adapted process, the random set U ≡ a a {∆X = 0} is thin; an exhausting sequence (Tn ) for this set is called a sequence that exhausts the jumps of X. Moreover, if X is predictable, the stopping times (Tn ) can be chosen predictable. Proof. Let Un ≡ (t, ω) : |Xt (ω) − Xt− (ω) > 2−n , for n an integer and set V0 = U0 and Vn = Un − Un−1 . The sets Vn are optional (resp. predictable if X is predictable) and are disjoint. Now, let us define the stopping times
1 Dn k+1 Dn

= =

k inf t > Dn : (t, ω) ∈ Vn ,

inf {t : (t, ω) ∈ Vn }

1 We recall that the right topology on the real line is a topology whose basis is given by intervals [s, t[.

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j so that Dn represents the j−th jump of X whose size in absolute value is between 2−n and 2−n+1 . Since X is c`dl`g, Vn does not have any accumulation point and a a k the stopping times Dn (k,n)∈N2 enumerate all the points in Vn . Moreover, from k Proposition 2.41, the stopping times Dn are predictable if X is predictable. k To complete the proof, it suffices to index the doubly indexed sequence Dn into a simple indexed sequence (Tn ).

In fact, we have the following characterization for predictable processes: Proposition 2.44. If X is c`dl`g adapted process, then X is predictable if and a a only if the following two conditions are satisfied: 1. For all totally inaccessible stopping times T , ∆XT = 0, a.s.on {T < ∞} 2. For every predictable stopping time T , XT 1{T 1 − t . g g g

1 µ−1 Γ (µ) 2

∞
R √ t 1−t

dyy 2µ−1 exp −

y2 2

.

(8.3)
(−ν)

Now, following Borodin and Salminen ([21], p. 70-71), if for −ν > 0, P0 denotes the law of a Bessel process of parameter −ν, starting from 0, then the law of Ly ≡ sup {t : Rt = y}, is given by: P0
(−ν)

(Ly ∈ dt) =

y −2ν 2−ν Γ (−ν) t−ν+1

exp −

y2 2t

dt.

Now, from the time reversal property for Bessel processes ([21] p.70, or [69]), we have: (−ν) P H0 ∈ dt = P0 (LRt ∈ dt) ; consequently, from (8.3), we have (recall µ = −ν): g Zt µ R2

R2µ =1− µ t 2 Γ (µ)

∞ 1−t

du

t exp − 2u

u1+µ

,

and the desired result is obtained by straightforward change of variables in the above integral. Remark 8.17. The previous proof can be applied mutatis mutandis to obtain: P [gµ (T ) > t | Ft ] = and At µ g (T )

1 2µ−1 Γ (µ) 1 2µ Γ (1 + µ)

∞ √Rt
T −t

dyy 2µ−1 exp − dLu µ. (T − u)

y2 2

;

t∧T 0

=

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Remark 8.18. It can be easily deduced from Proposition 8.16 that the dual g predictable projection At µ of 1(gµ ≤t) is: At µ = g 1 µ Γ (1 + µ) 2

t∧1 0

dLu µ. (1 − u) g Indeed, it is a consequence of Itˆ’s formula applied to Zt µ and the fact that o 2µ Nt ≡ Rt − Lt is a martingale and (dLt ) is carried by {t : Rt = 0}.

1 When µ = , Rt can be viewed as |Bt |, the absolute value of a standard 2 Brownian Motion. Thus, we recover as a particular case of our framework the celebrated example of the last zero before 1 of a standard Brownian Motion (see [42] p.124, or [81] for more references). Corollary 8.19. Let (Bt ) denote a standard Brownian Motion and let g ≡ sup {t ≤ 1 : Bt = 0} .

Then: P [g > t | Ft ] = and Ag = t Proof. It suffices to take µ ≡ 2 π
0

2 π

∞
|B √ t| 1−t

dy exp −

y2 2

,

t∧1

1 in Proposition 8.16. 2 Corollary 8.20. The variable 1 2µ Γ (1 + µ)
1 0

dLu √ . 1−u

dLu µ (1 − u)

is exponentially distributed with expectation 1; consequently, its law is independent of µ. Proof. The random time gµ is honest by definition (it is the end of a predictable g set). It also avoids stopping times since At µ is continuous (this can also be seen as a consequence of the strong Markov property for R and the fact that 0 is instantaneously reflecting). Thus the result of the corollary is a consequence of Remark 8.18 following Proposition 8.16 and Lemma 8.15. Given an honest time, it is not in general easy to compute its associated supermartingale Z L . Hence it is important (in view of the theory of progressive enlargements of filtrations) to dispose some characterizations of Az´ma’s supere martingales which also provide a method way to compute them explicitly. We will give two results in this direction, borrowed from [63] and [60].

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Let (Nt )t≥0 be a continuous local martingale such that N0 = 1, and limt→∞ Nt = 0. Let St = sups≤t Ns . We consider: g = = sup {t ≥ 0 : N t = S∞ }

sup {t ≥ 0 :

St − Nt = 0} .

(8.4)

Proposition 8.21 ([63]). Consider the supermartingale Zt ≡ P (g > t | Ft ) . 1. In our setting, the formula: Zt = Nt , t≥0 St

holds. 2. The Doob-Meyer additive decomposition of (Zt ) is: Zt = E [log S∞ | Ft ] − log (St ) . (8.5)

The above proposition gives a large family of examples. In fact, quite remarkably , every supermartingale associated with an honest time is of this form. More precisely: Theorem 8.22 ([63]). Let L be an honest time. Then, under the conditions (CA), there exists a continuous and nonnegative local martingale (Nt )t≥0 , with N0 = 1 and limt→∞ Nt = 0, such that: Zt = P (L > t | Ft ) = Nt . St

We shall now outline a nontrivial consequence of Theorem 8.22 here. In [7], the authors are interested in giving explicit examples of dual predictable projections of processes of the form 1L≤t , where L is an honest time. Indeed, these dual projections are natural examples of increasing injective processes (see [7] for more details and references). With Theorem 8.22, we have a complete characterization of such projections: Corollary 8.23. Assume the assumption (C) holds, and let (Ct ) be an increasing process. Then C is the dual predictable projection of 1g≤t , for some honest time g that avoids stopping times, if and only if there exists a continuous local martingale Nt in the class C0 such that Ct = log St . Now let us give some examples. Example 8.24. Let Nt ≡ Bt ,

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where (Bt )t≥0 is a Brownian Motion starting at 1, and stopped at T0 = inf {t : Bt = 0}. Let St ≡ sup Bs . s≤t Let g = sup {t : Bt = St } . Then P (g > t | Ft ) = Example 8.25. Let Nt ≡ exp 2νBt − 2ν 2 t , where (Bt ) is a standard Brownian Motion, and ν > 0. We have: St = exp sup 2ν (Bs − νs) , s≤t Bt . St

and g = sup t : (Bt − νt) = sup (Bs − νs) . s≥0 Consequently, P (g > t | Ft ) = exp 2ν (Bs − νs) − sup (Bs − νs) s≤t .

Example 8.26. Now, we consider (Rt ), a transient diffusion with values in [0, ∞), which has {0} as entrance boundary. Let s be a scale function for R, which we can choose such that: s (0) = −∞, and s (∞) = 0. s (Rt ) , t≥0 s (x) satisfies the required conditions of Proposition 8.21, and we have: Then, under the law Px , x > 0, the local martingale Nt = Px (g > t|Ft ) = where g = sup {t : Rt = It } , and It = inf Rs . s≤t s (Rt ) s (It )

Theorem 8.22 is a multiplicative characterization; now we shall give an additive one.

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Theorem 8.27 ([60]). Again, we assume that the conditions (CA) hold. Let (Xt ) be a submartingale of the class (Σc D) satisfying: lim Xt = 1. Let t→∞ L = sup {t : Xt = 0} . Then (Xt ) is related to the Az´ma’s supermartingale associated with L in the e following way: L Xt = 1 − Zt = P (L ≤ t|Ft ) . Consequently, if (Zt ) is a nonnegative supermartingale, with Z0 = 1, then, Z may be represented as P (L > t|Ft ), for some honest time L which avoids stopping times, if and only if (Xt ≡ 1 − Zt ) is a submartingale of the class (Σ), with the limit condition: lim Xt = 1. t→∞ Now, we give some fundamental examples: Example 8.28. First, consider (Bt ), the standard Brownian Motion, and let + T1 = inf {t ≥ 0 : Bt = 1) . Let σ = sup {t < T1 : Bt = 0}. Then Bt∧T1 satisfies the conditions of Theorem 8.27, and hence:
+ P (σ ≤ t|Ft ) = Bt∧T1 = t∧T1 0

1 1Bu >0 dBu + ℓt∧T1 , 2

where (ℓt ) is the local time of B at 0. This example plays an important role in the celebrated Williams’ path decomposition for the standard Brownian Motion on [0, T1 ]. One can also consider T±1 = inf {t ≥ 0 : |Bt | = 1) and τ = sup {t < T±1 : |Bt | = 0}. |Bt∧T±1 | satisfies the conditions of Theorem 8.27, and hence: t∧T±1 P (τ ≤ t|Ft ) = |Bt∧T±1 | =

sgn (Bu ) dBu + ℓt∧T±1 .
0

Example 8.29. Let (Yt ) be a real continuous recurrent diffusion process, with Y0 = 0. Then from the general theory of diffusion processes, there exists a unique continuous and strictly increasing function s, with s (0) = 0, limx→+∞ s (x) = +∞, limx→−∞ s (x) = −∞, such that s (Yt ) is a continuous local martingale. Let T1 ≡ inf {t ≥ 0 : Yt = 1) . Now, if we define Xt ≡ s (Yt∧T1 ) , s (1)
+

we easily note that X is a local submartingale of the class (Σc ) which satisfies the hypotheses of Theorem 8.27. Consequently, if we note σ = sup {t < T1 : Yt = 0} ,

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we have: P (σ ≤ t|Ft ) =

s (Yt∧T1 ) . s (1)

+

Example 8.30. Now let (Mt ) be a positive local martingale, such that: M0 = x, Mt ∧1 , x > 0 and limt→∞ Mt = 0. Then, Tanaka’s formula shows us that 1 − y for 0 ≤ y ≤ x, is a local submartingale of the class (Σc ) satisfying the assumptions of Theorem 8.27, and hence with g = sup {t : Mt = y} , we have: P (g > t|Ft ) = Mt 1 ∧1=1+ y y t 0

1(Mu 0, the local martingale (Mt = −s (Rt )) satisfies the conditions of the previous example and for 0 ≤ x ≤ y, we have: Px (gy > t|Ft ) = where Lt s(y) s (Rt ) 1 ∧1=1+ s (y) s (y)

t 0

1(Ru >y) d (s (Ru )) +

1 s(y) L , 2s (y) t

is the local time of s (R) at s (y), and where gy = sup {t : Rt = y} .

This last formula was the key point for deriving the distribution of gy in [67], Theorem 6.1, p.326. 8.2.2. The case of pseudo-stopping times In this paragraph, we give some characteristic properties and some examples of pseudo-stopping times. We do not assume here that condition (A) holds, but we assume that P [ρ = ∞] = 0. Theorem 8.32 ([59]). The following properties are equivalent: 1. ρ is a (Ft ) pseudo-stopping time, i.e (8.1) is satisfied; 2. Aρ ≡ 1, a.s ∞

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Remark 8.33. We shall give a more complete version of Theorem 8.32 in the section on progressive expansions of filtrations. Proof. We have: E [Mρ ] = E
0 ∞

Ms dAρ = E [M∞ Aρ ] . ∞ s

Hence, E [Mρ ] = E [M∞ ] ⇔ E [M∞ (Aρ − 1)] = 0, ∞ and the announced equivalence follows now easily. Remark 8.34. More generally, the approach adopted in the proof can be used to solve the equation E [Mρ ] = E [M∞ ] , where the random time ρ is fixed and where the unknown are martingales in H1 . For more details and resolutions of such equations, see [64].

ρ Corollary 8.35. Under the assumptions of Theorem 8.32, Zt = 1 − Aρ is a t ρ decreasing process. Furthermore, if ρ avoids stopping times, then (Zt ) is continuous.

Proof. The follows from the fact that µρ = E [Aρ |Ft ] = 1. t ∞ Remark 8.36. In fact, we shall see in next section, that under condition (C), ρ ρ is a pseudo-stopping time if and only if (Zt ) is a predictable decreasing process. For honest times, Az´ma proved that AL follows the standard exponential e law. For pseudo-stopping times, we have: ρ Proposition 8.37 ([59]). For simplicity, we shall write (Zu ) instead of (Zu ). Under condition (A), for all bounded (Ft ) martingales (Mt ), and all bounded Borel measurable functions f , one has: 1

E [Mρ f (Zρ )] = E [M0 ]
0 1

f (x) dx f (x) dx.
0

= E [Mρ ]

Consequently, Zρ follows the uniform law on (0, 1).

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Proof. Under our assumptions, we have E [Mρ f (Zρ )] = = = = = E
0 ∞ ∞ 0

Mu f (Zu ) dAρ u Mu f (1 − Aρ ) dAρ u u
∞ 0 1

E

E M∞ E M∞ E M∞

f (1 − Aρ ) dAρ u u

0 1

f (1 − x) dx f (x) dx .

0

Now, we give a systematic construction for pseudo-stopping times, generalizing D. Williams’s example. We assume we are given an honest time L and that conditions (CA) hold (the condition (A) holds with respect to L). Then the following holds:
L Proposition 8.38 ([59]). (i) IL = inf u≤L Zu is uniformly distributed on [0, 1]; ρ (ii) The supermartingale Zt = P [ρ > t | Ft ] associated with ρ is given by ρ L Zt = inf Zu . u≤t

As a consequence, ρ is a (Ft ) pseudo-stopping time.
L Proof. For simplicity, we write Zt for Zt . (i) Let

Tb = inf {t, then

Zt ≤ b} ,

0 < b < 1,

P [IL ≤ b] = P [Tb < L] = E [ZTb ] = b. (ii) Note that for every (Ft ) stopping time T , we have {T < ρ} = T < L where T = inf Consequently, we have ρ E [ZT ] = P [T < ρ] = P T < L = E [ZT ′ ] = E inf Zu , u≤T
′ ′ ′

t > T,

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394

which yields: ρ E ZT 1{T 0

which satisfies the usual assumptions. We first need the conditional laws of A∞ which were obtained under conditions (CA) in [6] and in a more general setting and by different methods in [61]. Proposition 9.8 ([61],[6]). Let G be a Borel bounded function. Define: MtG ≡ E (G (A∞ ) |Ft ) . Then, MtG = F (At ) − (F (At ) − G (At )) (1 − Zt ) , where F (x) = exp (x)
∞ x

dy exp (−y) G (y) .

Moreover, MtG has the following stochastic integral representation: MtG = E [G (A∞ )] + t 0

(F − G) (Au ) dµu .

Now, define, for G any Borel bounded function, g λt (G) ≡ MtG = F (At ) − (F (At ) − G (At )) (1 − Zt ) .

From Proposition 9.8, we also have: t λt (G)

= E [G (A∞ )] + ≡ E [G (A∞ )] +

0 t 0

(F − G) (As ) dµs ˙ λs (G) dµs .

Hence we have: λt (G) = λt (dx) G (x) ,

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with λt (dx) = (1 − Zt ) δAt (dx) + Zt exp (At ) 1(At ,∞) (x) exp (−x) dx, where δAt denotes the Dirac mass at At . Similarly, we have: ˙ λt (G) = with: ˙ λt (dx) G (x) ,

˙ λt (dx) = −δAt (dx) + exp (At ) 1(At ,∞) (x) exp (−x) dx. ˙ λt (dx) = λt (dx) ρ (x, t) , ρ (x, t) = 1 1 1{x>At } − 1{x=At } . Zt 1 − Zt (9.1) (9.2)

It then follows that: with

Now we can state our result about initial expansion with A∞ , which was first obtained by Jeulin ([42]), but the proof we shall present is borrowed from [60]. Theorem 9.9. Let L be an honest time. We assume, as usual, that the condiσ(A ) tions (CA) hold. Then, every (Ft ) local martingale M is an Ft ∞ semimartingale and decomposes as: t Mt = Mt +
0

1{L>s}

d M, µ Zs

s

t

−

0

1{L≤s}

d M, µ s , 1 − Zs

(9.3)

where Mt

t≥0

denotes an Ft

σ(A∞ )

local martingale.

Proof. We can first assume that M is an L2 martingale; the general case follows by localization. Let Λs be an Fs measurable set, and take t > s. Then, for any bounded test function G, we have: E (1Λs G (A∞ ) (Mt − Ms )) = E (1Λs (λt (G) Mt − λs (G) Ms )) = E (1Λs ( λ (G) , M t s t

(9.4) (9.5) (9.6) u t − λ (G) , M s )) u

= E 1Λs = E 1Λs s t

˙ λu (G) d M, µ

λu (dx) ρ (x, u) G (x) d M, µ d M, µ u ρ (A∞ , u) .

= E 1Λs s (9.7)

But from (9.2), we have: ρ (A∞ , t) = 1 1 1{A∞ >At } − 1{A∞ =At } . Zt 1 − Zt

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It now suffices to notice that (At ) is constant after L and L is the first time when A∞ = At , or in other words (for example, see [28] p. 134): 1{A∞ >At } = 1{L>t} , and 1{A∞ =At } = 1{L≤t} . Let us emphasize again that the method we have used here applies to many other situations, where the theorems of Jacod do not apply. Each time the differ˙ ent relationships we have just mentioned between the quantities: λt (G) , λt (G) , ˙ t (dx) , ρ (x, t) , hold, the above method and decomposition formula and λt (dx) , λ apply. Moreover, the condition (C) can be dropped and it is enough to have only a stochastic integral representation for λt (G) (see [63] for a discussion). In the case of enlargement with A∞ , everything is nice since every (Ft ) local martinσ(A ) gale M is an Ft ∞ semimartingale. Sometimes, an integrability condition is needed as is shown by the following example. Example 9.10 ([81], p.34). Let Z = 0 ϕ (s) dBs , for some ϕ ∈ L2 (R+ , ds). Recall that σ {Z} . Gt = ∩ε>0 Ft+ε We wish to address the following question: is (Bt ) a (Gt ) semimartingale? The above method applies step by step: it is easy to compute λt (dx), since t conditionally on Ft , Z is gaussian, with mean mt = 0 ϕ (s) dBs , and variance t 2 2 σt = 0 ϕ (s) ds. Consequently, the absolute continuity requirement (9.1) is satisfied with: x − ms ρ (x, t) = ϕ (s) . 2 σs But here, the arguments in the proof of Theorem 9.9 (replace M with B) do not always work since the quantities involved there (equations (9.4) to (9.7)) might be infinite; hence we have to impose an integrability condition. For example, if we assume that t |ϕ (s) | ds < ∞, σs 0 then (Bt ), is a (Gt ) semimartingale with canonical decomposition: t ∞

Bt = B0 + Bt +
0

ds

ϕ (s) 2 σs

∞ s

ϕ (u) dBu ,

where Bt is a (Gt ) Brownian Motion. As a particular case, we may take: Z = Bt0 , for some fixed t0 . The above formula then becomes: t∧t0 Bt = B0 + Bt +
0

ds

Bt0 − Bs , t0 − s

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where

Bt

is a (Gt ) Brownian Motion. In particular,

Bt

is independent

of G0 = σ {Bt0 }, so that conditionally on Bt0 = y, or equivalently, when (Bt , t ≤ t0 ) is considered under the bridge law Pt0 , its canonical decompox,y sition is: t y − Bs Bt = x + Bt + ds , t0 − s 0 where Bt , t ≤ t0 is now a Pt0 ; (Ft ) Brownian Motion. x,y Example 9.11. For more examples of initial enlargements using this method, see the forthcoming book [50]. 9.2. Progressive enlargements of filtrations The theory of progressive enlargements of filtrations was originally motivated by a paper of Millar [57] on random times and decomposition theorems. It was first independently developed by Barlow [13] and Yor [77], and further developed by Jeulin and Yor [43] and Jeulin [41, 42]. For further developments and details, the reader can also refer to [45] which is written in French or to [81, 50] or [68] chapter VI. for an English text. Let (Ω, F , (Ft ) , P) be a filtered probability space satisfying the usual assumptions, and for simplicity (and because it is always the case with practical examples), we shall assume that: F = F∞ = Ft .

t≥0

Again, we will have to distinguish two cases: the case of arbitrary random times and honest times. Let ρ be random time. We enlarge the initial filtration (Ft ) ρ with the process (ρ ∧ t)t≥0 , so that the new enlarged filtration (Ft )t≥0 is the smallest filtration (satisfying the usual assumptions) containing (Ft ) and making ρ o o ρ a stopping time (i.e. Ft = Kt+ , where Kt = Ft σ (ρ ∧ t)). Sometimes it is more convenient to introduce the larger filtration ρ Gt = {A ∈ F∞ : ∃At ∈ Ft , A ∩ {L > t} = At ∩ {L > t}} , ρ which coincides with Ft before ρ and which is constant after ρ and equal to F∞ ([28], p. 186). In the case of an honest time L, one can show that in fact (see [41]): L Ft = {A ∈ F∞ : ∃At , Bt ∈ Ft , A = (At ∩ {L > t}) ∪ (Bt ∩ {L ≤ t})} . ρ L In the sequel, we shall only consider the filtrations (Gt ) and Ft : the first one when we study arbitrary random times and the second one when we consider the special case of honest times.

A. Nikeghbali/The general theory of stochastic processes ρ L 9.2.1. A description of predictable and optional processes in (Gt ) and Ft

402

All the results we shall mention in what follows can be found in [43] (or in [42, 28]) and are particulary useful in mathematical finance ([30], [40]). Proposition 9.12. Let ρ be an arbitrary random time. The following hold: ρ 1. If H is a (Gt ) predictable process, then there exists a (Ft ) predictable process J such that Ht 1t≤ρ = Jt 1t≤ρ . ρ 2. If T is a (Gt ) stopping time, then there exists a (Ft ) stopping time S such that: T ∧ ρ = S ∧ ρ.

ρ 3. Let ξ ∈ L1 . Then a c`dl`g version of the martingale ξt = E [ξ|Gt ] is given a a by: 1 ξt = ρ 1ts ρ Zs−

(9.8)

We shall now give two applications of this decomposition. The first one is a refinement of Theorem 8.32, which brings a new insight to peudo-stopping times: Theorem 9.20. The following four properties are equivalent: 1. ρ is a (Ft ) pseudo-stopping time, i.e (8.1) is satisfied; 2. µρ ≡ 1, a.s t 3. Aρ ≡ 1, a.s ∞

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404

4. every (Ft ) local martingale (Mt ) satisfies ρ (Mt∧ρ )t≥0 is a local (Gt ) martingale.

If, furthermore, all (Ft ) martingales are continuous, then each of the preceding properties is equivalent to 5. ρ (Zt )t≥0 is a decreasing (Ft ) predictable process

Proof. (1) =⇒ (2) For every square integrable (Ft ) martingale (Mt ), we have E [Mρ ] = E
0 ∞

Ms dAρ = E [M∞ Aρ ] = E [M∞ µρ ] . ∞ ∞ s

Since EMρ = EM0 = EM∞ , we have E [M∞ ] = E [M∞ Aρ ] = E [M∞ µρ ] . ∞ ∞ ρ Consequently, µρ ≡ 1, a.s, hence µρ ≡ 1, a.s which is equivalent to: A∞ ≡ 1, ∞ t a.s. Hence, 2. and 3. are equivalent. (2) =⇒ (4) . This is a consequence of the decomposition formula (9.8). (4) =⇒ (1) It suffices to consider any H1 -martingale (Mt ), which, assuming ρ (4), satisfies: (Mt∧ρ )t≥0 is a martingale in the enlarged filtration (Gt ). Then as ρ a consequence of the optional stopping theorem applied in (Gt ) at time ρ, we get E [Mρ ] = E [M0 ] ,

hence ρ is a pseudo-stopping time. Finally, in the case where all (Ft ) martingales are continuous, we show: ρ a) (2) ⇒ (5) If ρ is a pseudo-stopping time, then Zt decomposes as ρ Zt = 1 − Aρ . t

As all (Ft ) martingales are continuous, optional processes are in fact predictable, ρ and so (Zt ) is a predictable decreasing process. ρ b) (5) ⇒ (2) Conversely, if (Zt ) is a predictable decreasing process, then from the uniqueness in the Doob-Meyer decomposition, the martingale part µρ t is constant, i.e. µρ ≡ 1, a.s. Thus, 2 is satisfied. t Now, we apply the progressive enlargements techniques to the study of the Burkholder-Davis-Gundy inequalities. More precisely, what remains of the Burkholder-Davis-Gundy inequalities when stopping times T are replaced by arbitrary random times ρ? The question of probabilistic inequalities at an arbitrary random time has been studied in depth by M. Yor (see [79], [81, 50] for details and references). For example, taking the special case of Brownian motion, it can easily be shown that there cannot exist a constant C such that: √ E [|Bρ |] ≤ CE [ ρ]

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405

for any random time ρ. For if it were the case, we could take ρ = 1A , for A ∈ F∞ , and we would obtain: E [|B1 | 1A ] ≤ CE [1A ] which is equivalent to: |B1 | ≤ C, a.s., which is absurd. Hence it is not obvious that the ”strict” BDG inequalities might hold for stopped local martingales at other random times than stopping times. However, we have the following positive result: Theorem 9.21 ([66]). Let p > 0. There exist two universal constants cp and Cp depending only on p, such that for any (Ft ) local martingale (Mt ), with M0 = 0, and any (Ft ) pseudo-stopping time ρ we have cp E (< M >ρ ) 2 ≤ E p ∗ Mρ

p

≤ Cp E (< M >ρ ) 2 .

p

Proof. It suffices, with the previous Theorem, to notice that in the enlarged ρ filtration (Gt ), (Mt∧ρ ) is a martingale and ρ is a stopping time in this filtration; then, we apply the classical BDG inequalities. Remark 9.22. The constants cp and Cp are the same as those obtained for martingales in the classical framework; in particular the asymptotics are the same (see [17]). Remark 9.23. It would be possible to show the above Theorem, just using the definition of pseudo-stopping times (as random times for which the optional stopping theorem holds); but the proof is much longer. 9.2.3. The decomposition formula for honest times One of the remarkable features of honest time (discovered by Barlow [13]) is the L fact that the pair of filtrations Ft , Ft satisfies the (H ′ ) hypothesis and every L (Ft ) local martingale X is an Ft semimartingale. More precisely: Theorem 9.24. An (Ft ) local martingale (Mt ), is a semimartingale in the L larger filtration Ft and decomposes as: t∧L Mt = Mt +
0

d M, Z L L Zs−

s

t

−

L

d M, Z L s , L 1 − Zs−

(9.9)

where Mt

t≥0

denotes a

L Ft , P local martingale.

ρ Remark 9.25. There are non honest times ρ such that the pair (Ft , Gt ) satisfies the (H ′ ) hypothesis: for example, the pseudo-stopping times of Proposition 8.38 enjoy this remarkable property (see [60]) (for other examples see [42]).

Proof. We shall give a proof under the conditions (CA), which are general enough for most of the applications. In this special case, it is an consequence

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of Theorem 9.9. Indeed, we saw in the course of the proof of Theorem 9.9 that (for ease of notations we drop the upper index L): 1{A∞ >At } = 1{L>t} , and 1{A∞ =At } = 1{L≤t} .
L Thus, by definition of Ft , we have: L Ft ⊂ Ft σ(A∞ )

.

Now, let M be an L2 bounded (Ft ) martingale; the general case follows by localization. From Theorem 9.9 t Mt = Mt +
0

1{L>s}

d M, µ Zs

s

t

−

0

1{L≤s}

d M, µ s , 1 − Zs

where Mt

t≥0

L denotes an Ft L2 martingale. Thus, Mt , which is equal to: t

Mt −

0

1{L>s}

d M, µ Zs

s

t

−

0

1{L≤s}

d M, µ s 1 − Zs

,

L L is Ft adapted, and hence it is an L2 bounded Ft martingale.

There are many applications of progressive enlargements of filtrations with honest times, but we do not have the place here to give them. At the end of this section, we shall give a list of applications and references. Nevertheless, we mention an extension of the BDG inequalities obtained by Yor: Proposition 9.26 (Yor [81], p. 57). Assume that (Ft ) is the filtration of a standard Brownian Motion and let L be an honest time. Then we have: √ E [|BL |] ≤ CE ΦL L , with ΦL = 1 + log 1 IL
1/2 L where IL = inf Zu , u