On some universal sigma finite measures and some extensions of Doob's optional stopping theorem
aa r X i v : . [ m a t h . P R ] J un ON SOME UNIVERSAL σ -FINITE MEASURES AND SOMEEXTENSIONS OF DOOB’S OPTIONAL STOPPING THEOREM JOSEPH NAJNUDEL AND ASHKAN NIKEGHBALI
Dedicated to Marc Yor for his 60th birthday
Abstract.
In this paper, we associate, to any submartingale of class (Σ), defined on afiltered probability space (Ω , F , P , ( F t ) t ≥ ), which satisfies some technical conditions, a σ -finite measure Q on (Ω , F ), such that for all t ≥
0, and for all events Λ t ∈ F t : Q [Λ t , g ≤ t ] = E P [ Λ t X t ]where g is the last hitting time of zero of the process X . This measure Q has already beendefined in several particular cases, some of them are involved in the study of Brownianpenalisation, and others are related with problems in mathematical finance. More precisely,the existence of Q in the general case solves a problem stated by D. Madan, B. Roynette andM. Yor, in a paper studying the link between Black-Scholes formula and last passage timesof certain submartingales. Moreover, the equality defining Q remains true if one replacesthe fixed time t by any bounded stopping time. This generalization can be viewed as anextension of Doob’s optional stopping theorem. Introduction
This work finds its origin in a recent paper by Madan, Roynete and Yor [7] and a set oflectures by Yor [4] where the authors are able to represent the price of a European put optionin terms of the probability distribution of some last passage time. More precisely, they provethat if ( M t ) t ≥ is a continuous nonnegative local martingale defined on a filtered probabilityspace (Ω , F , ( F t ) t ≥ , P ) satisfying the usual assumptions, and such that lim t →∞ M t = 0, then( K − M t ) + = K P ( g K ≤ t |F t ) (1.1)where K ≥ g K = sup { t ≥ M t = K } . Formula (1.1) tells that it isenough to know the terminal value of the submartingale ( K − M t ) + and its last zero g K toreconstruct it. Yet a nicer interpretation of (1.1) is suggested in [4] and [7]: there exists ameasure Q , a random time g , such that the submartingale X t = ( K − M t ) + satisfies Q [ F t g ≤ t ] = E [ F t X t ] , (1.2)for any t ≥ F t -measurable random variable F t . Indeed it easilyfollows from (1.1) that in this case Q = K. P and g = g K . It is also clear that if a stochasticprocess X satisfies (1.2), then it is a submartingale. The problem of finding the class ofsubmartingales which satisfy (1.2) is posed in [4] and [7] and is the main motivation of thispaper: Problem 1 ( [4] and [7] ): for which submartingales X can we find a σ -finite measure Q Date : November 1, 2018. nd the end of an optional set g such that Q [ F t g ≤ t ] = E [ F t X t ]? (1.3)Identity (1.3) is reminiscent of the stopping theorem for uniformly integrable martingales.Indeed, if M is a c`adl`ag, uniformly integrable martingale, then it can be represented as M t = E [ M ∞ |F t ], and hence the terminal value of M , i.e. M ∞ , is enough to obtain themartingale M . But we also note that if we write g = sup { t ≥ M t = 0 } , then M t = E [ M ∞ g ≤ t |F t ] , since E [ M ∞ g>t |F t ] = 0. Thus (1.3) holds for M , where the measure Q is the signed measure Q = M ∞ . P . Consequently, the stopping theorem can also be interpreted as the existence ofa (signed) measure and the end of an optional set from which one can recover the uniformlyintegrable martingale M . But (1.3) does not admit a straightforward generalization tomartingales which are not uniformly integrable: indeed, such a measure Q would be realvalued and infinite. We hence propose the following problem: Problem 2: given a continuous martingale M , can we find two σ -finite measures Q (+) and Q ( − ) , such that for all t ≥ F t -measurable variables F t : (cid:0) Q (+) − Q ( − ) (cid:1) [ F t g ≤ t ] = E [ F t M t ] , (1.4)with g = sup { t ≥ M t = 0 } ?Identities (1.3) and (1.4) can hence be interpreted as an extension of Doob’s optional stoppingtheorem for fixed times t .It is also noticed in [7] that other instances of formula (1.2) have already been discovered:for example, in [3], Az´ema and Yor proved that for any continuous and uniformly martingale M , (1.3) holds for X t = | M t | , Q = | M ∞ | . P and g = sup { t ≥ M t = 0 } , or equivalently | M t | = E [ | M ∞ | g ≤ t |F t ] . Here again the measure Q is finite. Recently, other particular cases where the measure Q is not finite were obtained by Najnudel, Roynette and Yor in their study of Brownianpenalisation (see [8]). For example, they prove the existence of the measure Q when X t = | W t | is the absolute value of the standard Brownian Motion. In this case, the measure Q isnot finite but only σ -finite and is singular with respect to the Wiener measure: it satisfies Q ( g = ∞ ) = 0, where g = sup { t ≥ W t = 0 } .In the special case where the submartingale X is of class ( D ), Problem 1 was recentlysolved in [5] in relation with the study of the draw-down process.. In this case, the measure Q is finite. The relevant family of submartingales is the class (Σ): Definition 1.1 (([9, 11])) . Let (Ω , F , ( F t ) t ≥ , P ) be a filtered probability space. A nonneg-ative (local) submartingale ( X t ) t ≥ is of class (Σ), if it can be decomposed as X t = N t + A t where ( N t ) t ≥ and ( A t ) t ≥ are ( F t ) t ≥ -adapted processes satisfying the following assumptions: • ( N t ) t ≥ is a c`adl`ag (local) martingale; • ( A t ) t ≥ is a continuous increasing process, with A = 0; • The measure ( dA t ) is carried by the set { t ≥ , X t = 0 } . In fact, as mentioned in [5], the solution is essentially contained and somehow hidden in [1]. he definition of the class (Σ) goes back to Yor ([11]) when X is continuous and someof its main properties which we shall use frequently in this paper were studied in [9]. It isshown in [1] and [5] that if X is of class (Σ) and of class ( D ), then it satisfies (1.2) with g = sup { t ≥ X t = 0 } and Q = X ∞ . P , or equivalently X t = E [ X ∞ g ≤ t |F t ] . Now, what happens if X is of class (Σ) but satisfies A ∞ = ∞ almost surely? This is the casefor example if one works on the space C ( R + , R ) of continuous functions endowed with thefiltration ( F t ) t ≥ generated by the coordinate process ( Y t ) t ≥ and with the Wiener measure W , and if X t = | Y t | (here the increasing process is the local time which is known to beinfinite almost surely at infinity). In this case, as it was already mentioned, the existenceof the measure Q , which is singular with respect to W , was established in [8]. But thegeneral situation when X is of class (Σ) and satisfies A ∞ = ∞ almost surely is neverthelessmore subtle: one will need to make some non-standard assumptions on the filtration ( F t ) t ≥ .Indeed, let us assume that in the filtration ( F t ) t ≥ , F contains all the P -negligible sets (i.e.the filtration is complete), and that under P , A ∞ = ∞ almost surely, and then g = ∞ a.s.(e.g. X is the absolute value of a Brownian motion). For all t ≥
0, the event { g > t } hasprobability one (under P ) and then, is in F and, a fortiori, in F t . If one assumes that Q exists, one has Q [ g > t, g ≤ t ] = E P [ g>t X t ] , and then: E P [ A t ] ≤ E P [ X t ] = 0which is absurd.The goal of this paper is to show that under some technical conditions on the filtration( F t ) t ≥ , Problem 1 can be solved for all submartingales of the class (Σ). The measure Q isconstructed explicitly. Since for continuous martingales, M + and M − are of class (Σ), weshall be able to solve Problem 2 and hence interpret our results as an extension of Doob’soptional stopping theorem. Our approach is based on martingale techniques only and weare hence able to obtain the measure Q for a wide range of processes which can possiblyjump, thus including the generalized Az´ema submartingales in the filtration of the zeros ofBessel processes of dimension in (0 ,
2) and the draw-down process X t = S t − M t where M is a martingale with no positive jumps and S t = sup u ≤ t M u . In particular, the existenceof Q does not require any scaling or Markov property for X . More precisely, the paper isorganized as follows: • in Section 2, we state our technical condition on the filtration ( F t ) t ≥ and then stateand prove our main theorem about the existence and the uniqueness of the measure Q for submartingales of the class (Σ). We then deduce the solution to Problem 2, henceinterpreting (1.3) and (1.4) together as an extension of Doob’s optional stoppingtheorem. We also give the image of the measure Q by the functional A ∞ ; • in Section 3, we give several examples of such a measure Q in classical and lessclassical settings. . Description of the σ -finite measure The main result of this paper states that the existence and the uniqueness of Q essentiallyholds for all submartingales ( X t ) t ≥ of class (Σ). However, before stating this result, we needto introduce some technicalities on filtrations.2.1. A measure extension theorem.
Let us first introduce the following definition:
Definition 2.1.
Let (Ω , F , ( F t ) t ≥ ) be a filtered measurable space, such that F is the σ -algebra generated by F t , t ≥ F = W t ≥ F t . We shall say that the property (P) holds ifand only if F t = G t + , where ( G t ) t ≥ is a filtration, enjoying the following conditions: • For all t ≥ G t is generated by a countable number of sets; • For all t ≥
0, there exists a Polish space Ω t , and a surjective map π t from Ω to Ω t ,such that G t is the σ -algebra of the inverse images, by π t , of Borel sets in Ω t , andsuch that for all B ∈ G t , ω ∈ Ω, π t ( ω ) ∈ π t ( B ) implies ω ∈ B ; • If ( ω n ) n ≥ is a sequence of elements of Ω, such that for all N ≥ N \ n =0 A n ( ω n ) = ∅ , where A n ( ω n ) is the intersection of the sets in G n containing ω n , then: ∞ \ n =0 A n ( ω n ) = ∅ . Note that if (Ω , F , ( F t ) t ≥ ) satisfies the property (P), then ( F t ) t ≥ is right-continuous. Thetechnical definition 2.1 is needed in order to state the following lemma, which is fundamentalin the construction of the measure Q . In particular, it provides sufficient conditions underwhich a finite measure b P defined on each F t by b P = M t . P , where M t is a (true) positivemartingale, can be extended to a measure on F = W t ≥ F t . Lemma 2.2.
Let (Ω , F , ( F t ) t ≥ ) be a filtered probability space satisfying the property (P),and let, for t ≥ , Q t be a finite measure on (Ω , F t ) , such that for all t ≥ s ≥ , Q s is therestriction of Q t to F s . Then, there exists a unique measure Q on (Ω , F ) such that for all t ≥ , its restriction to F t is equal to Q t .Proof. The uniqueness is a direct application of the monotone class theorem, then, let usprove the existence. One can obviousely suppose that Q t is a probability measure for all t ≥ Q (Ω), if it is different from zero). We first define, for t ≥ e Q t as therestriction of Q t to G t . For t > s ≥
0, and for all events Λ s ∈ G s , one has: e Q t (Λ s ) = Q t (Λ s ) = Q s (Λ s ) = e Q s (Λ s ) , hence, e Q s is the restriction of e Q t to G s . Now, for t ≥
0, the map π t is, by assumption,measurable from (Ω , G t ) to (Ω t , B (Ω t )) (where B (Ω t ) is the Borel σ -algebra of Ω t ). Let ¯ Q t be the image of e Q t by π t . By Theorem 1.1.6 of [10], for 0 ≤ s ≤ t , there exists a conditionalprobability distribution of ¯ Q t given the σ -algebra π t ( G s ), generated by the images, by π t , ofthe sets in G s . Note that this σ -algebra is included in B (Ω t ). Indeed, if B ∈ G s , there exists A ∈ B (Ω t ) such that B = π − t ( A ), and then π t ( B ) = π t ◦ π − t ( A ), which is equal to A bysurjectivity of π t . Now, the existence of the conditional probability distribution described bove means that one can find a family ( Q ω ) ω ∈ Ω t of probability measures on (Ω t , B (Ω t )) suchthat: • For each B ∈ B (Ω t ), ω → Q ω ( B ) is π t ( G s )-measurable; • For every A ∈ π t ( G s ), B ∈ B (Ω t ):¯ Q t ( A ∩ B ) = Z A Q ω ( B ) ¯ Q t ( dω ) . Now, for all ω ∈ Ω, let us define the map R ω from G t to R + , by: R ω [ B ] = Q π t ( ω ) [ π t ( B )] . The map R ω is a probability measure on (Ω , G t ). Indeed, R ω [Ω] = Q π t ( ω ) [Ω t ] = 1 , since π t is surjective. Moreover, let ( B k ) k ≥ be a family of disjoint sets in G t , and B theirunion. By assumption, there exists ( ˜ B k ) k ≥ in B (Ω t ) such that B k = π − t ( ˜ B k ). Since π t is surjective, π t ( B k ) = ˜ B k . Moreover, the sets ( ˜ B k ) k ≥ are pairwise disjoint. Indeed, if x ∈ ˜ B k ∩ ˜ B l for k > l ≥
1, then, by surjectivity, there exists y ∈ Ω, such that x = π t ( y ),which implies y ∈ π − t ( ˜ B k ) ∩ π − t ( ˜ B l ), and then y ∈ B k ∩ B l , which is impossible. Therefore: R ω [ B ] = Q π t ( ω ) [ ˜ B ]= X k ≥ Q π t ( ω ) [ ˜ B k ]= X k ≥ R ω [ B k ]Hence, R ω is a probability measure. Moreover, for each B ∈ G t , the map ω → R ω ( B )is the composition of the measurable maps ω → π t ( ω ) from (Ω , G s ) to (Ω t , π t ( G s )), and ω ′ → Q ω ′ [ π t ( B )] from (Ω t , π t ( G s )) to ( R + , B ( R + )), and hence, it is G s -measurable. Themeasurability of π t follows from the fact that for A ∈ G s , the inverse image of π t ( A ) by π t is exactly A (by an assumption given in the definition of the property (P)). Moreover, forevery A ∈ G s , B ∈ G t : e Q t ( A ∩ B ) = e Q t [( π − t ◦ π t ( A )) ∩ ( π − t ◦ π t ( B ))]= e Q t [ π − t ( π t ( A ) ∩ π t ( B ))]= ¯ Q t [ π t ( A ) ∩ π t ( B )]= Z Ω t ω ∈ π t ( A ) Q ω ( π t ( B )) ¯ Q t ( dω )= Z Ω π t ( ω ) ∈ π t ( A ) Q π t ( ω ) ( π t ( B )) e Q t ( dω )= Z A R ω ( B ) e Q t ( dω )Finally, we have found a conditional probability distribution of e Q t with respect to G s . Since G s is countably generated, this conditional probability distribution is regular, by Theorem1.1.8 in [10]. One can then apply Theorem 1.1.9, again in [10], and since F is the σ -algebragenerated by G t , t ≥
0, one obtains a probability distribution Q on (Ω , F ), such that for all ntegers n ≥
0, the restriction of Q to G n is e Q n . Now, for t ≥
0, let Λ t be an event in F t .One has, for n > t , integer: Q [Λ t ] = e Q n [Λ t ] = Q n [Λ t ] = Q t [Λ t ] , which implies that Q satisfies the assumptions of Lemma 2.2. (cid:3) Now we give important examples of filtered measurable spaces (Ω , F , ( F t ) t ≥ ) which satisfythe property (P). Corollary 2.3.
Let Ω be C ( R + , R d ) , the space of continuous functions from R + to R d , or D ( R + , R d ) , the space of c`adl`ag functions from R + to R d (for some d ≥ ). For t ≥ , define F t = G t + , where ( G t ) t ≥ is the natural filtration of the canonical process Y , and F = W t ≥ F t .Then (Ω , F , ( F t ) t ≥ ) satisfies property (P).Proof. Let us prove this result for c`adl`ag functions (for continuous functions, the result issimilar and is proved in [10]). For all t ≥ G t is generated by the variables Y rt , for r ,rational, in [0 , t , the set of c`adl`ag functions from [0 , t ] to R d , and for π t , the restriction to the interval[0 , t ]. The space Ω t is Polish if one endows it with the Skorokhod metric, moreover, its Borel σ -algebra is equal to the σ -algebra generated by the coordinates, a result from which oneeasily deduces the properties of π t which need to be satisfied. The third property is easy tocheck: let us suppose that ( ω n ) n ≥ is a sequence of elements of Ω, such that for all N ≥ N \ n =0 A n ( ω n ) = ∅ , where A n ( ω n ) is the intersection of the sets in G n containing ω n . Here, A n ( ω n ) is the set offunctions ω ′ which coincide with ω n on [0 , n ]. Moreover, for n ≤ n ′ , integers, the intersectionof A n ( ω n ) and A n ′ ( ω n ′ ) is not empty, and then ω n and ω n ′ coincide on [0 , n ]. Therefore, thereexists a c`adl`ag function ω which coincides with ω n on [0 , n ], for all n , which implies: ∞ \ n =0 A n ( ω n ) = ∅ . (cid:3) The main theorem.
We can now state the main result of the paper:
Theorem 2.4.
Let ( X t ) t ≥ be a (true) submartingale of the class (Σ) (in particular X t isintegrable for all t ≥ ), defined on a filtered probability space (Ω , F , P , ( F t ) t ≥ ) which satisfiesthe property (P). In particular, ( F t ) t ≥ is right-continuous and F is the σ -algebra generatedby F t , t ≥ . Then, there exists a unique σ -finite measure Q , defined on (Ω , F , P ) such thatfor g := sup { t ≥ , X t = 0 } : • Q [ g = ∞ ] = 0 ; • For all t ≥ , and for all F t -measurable, bounded random variables F t , Q [ F t g ≤ t ] = E P [ F t X t ] . emark . If g < ∞ , then A ∞ = A g < ∞ . Hence, the first condition satisfied by Q impliesthat: Q [ A ∞ = ∞ ] = 0 . In other words, A ∞ is finite a.s. under Q . Proof.
Let f be a Borel function from R + to R + , bounded and integrable, and let, for x ≥ G ( x ) := Z ∞ x f ( y ) dy. By [9] (Theorem 2.1), one immediately checks that the process( M ft := G ( A t ) + f ( A t ) X t ) t ≥ , where A is the increasing process of X , is a nonnegative local martingale. Moreover, for all t ≥
0, if N is the martingale part of X and T t is the set containing all the stopping timesbounded by t , then the family ( N T ) T ∈T t is uniformly integrable (it is included in the set ofconditional expectations of N t , by stopping theorem), and ( A T ) T ∈T t is bounded by A t ( A isincreasing), which is integrable (it has the same expectation as X t − X ). Hence, ( X T ) T ∈T t is uniformly integrable, which implies, since f and G are uniformly bounded, that ( M fT ) T ∈T t is also uniformly integrable. Hence, M f is a true martingale. Therefore, by Lemma 2.2, itis possible to construct a finite measure P f on (Ω , F , P ), uniquely determined by: P f [Λ s ] = E P [ Λ s M fs ]for all s ≥ s ∈ F s . Let us now prove that: P f [ A ∞ = ∞ ] = 0 . Indeed, for u ≥
0, let us consider, as in [9], the right-continuous inverse of A : τ u := inf { t ≥ , A t > u } . It is easy to check that for t, u ≥
0, the event { τ u ≤ t } is equivalent to {∀ t ′ > t, A t ′ > u } ,which implies that τ u is a stopping time (recall that ( F t ) t ≥ is right-continuous). Moreover,if τ u < ∞ , then A τ u = u and X τ u = 0. Indeed, for all t > τ u , A t > u , and for all t < τ u , A t ≤ u , which implies the first equality by continuity of A , for 0 < τ u < ∞ (if τ u = 0 then u = 0 and the equality is also true). Moreover, if X τ u >
0, by right-continuity of X , thereexists a.s. ǫ > X > τ u , τ u + ǫ ], which implies that A is constanton this interval, and then A τ u = A τ u + ǫ > u , which is a contradiction. Now, for all t, u ≥ P f [ A t > u ] = E P [( G ( A t ) + f ( A t ) X t ) A t >u ] ≤ E P [( G ( A t ) + f ( A t ) X t ) τ u ≤ t ]= E P [( G ( A τ u ∧ t ) + f ( A τ u ∧ t ) X τ u ∧ t ) τ u ≤ t ]by applying stopping theorem to the stopping time τ u ∧ t . Therefore: P f [ A t > u ] ≤ E P [( G ( A τ u ) + f ( A τ u ) X τ u ) τ u ≤ t ]= G ( u ) P [ τ u ≤ t ] . By taking the increasing limit for t going to infinity, one deduces: P f [ ∃ t ≥ , A t > u ] ≤ G ( u ) P [ τ u < ∞ ] . his implies: P f [ A ∞ > u ] ≤ G ( u ) , and by taking u → ∞ , P f [ A ∞ = ∞ ] = 0 . Let us now suppose that f ( x ) > x ≥
0, and that
G/f is uniformly bounded on R + (for example, one can take f ( x ) = e − x ). Since P f [ A ∞ = ∞ ] = 0 and f ( A ∞ ) >
0, one candefine a measure Q f by the following equality: Q f [Λ] = P f (cid:20) Λ f ( A ∞ ) (cid:21) for all events Λ ∈ F . This measure is σ -finite, since for all ǫ > Q f [ f ( A ∞ ) ≥ ǫ ] ≤ ǫ P f (1) < ∞ . Now, for t ≥
0, and F t , bounded, F t -measurable: Q f [ F t g ≤ t ] = P f (cid:20) F t f ( A t ) g ≤ t (cid:21) = P f (cid:20) F t f ( A t ) d t = ∞ (cid:21) since A ∞ = A t on the event { g ≤ t } , which is equivalent to { d t = ∞} , where d t = inf { v >t, X v = 0 } . Let us now introduce the filtration ( H t ) t ≥ , where for all t ≥ H t is the σ -algebra generated by F t and all the Q -negligible sets of F , where Q := P + P f . From [6], p.183, ( H t ) t ≥ satisfies the usual assumptions and consequently, from the D´ebut theorem, d t is an ( H t ) t ≥ -stopping time. From [6], Theorem 59, p. 193, there exists an ( F t ) t ≥ -stoppingtime e d t such that d t = e d t , Q -a.s. One deduces: P f (cid:20) F t f ( A t ) d t ≤ u (cid:21) = P f (cid:20) F t f ( A t ) e d t ≤ u (cid:21) = E P (cid:20) F t f ( A t ) M fu e d t ≤ u (cid:21) . By applying stopping theorem to ˜ d t ∧ u , one obtains: E P [ M fu |F e d t ∧ u ] = M f e d t ∧ u . Hence: P f (cid:20) F t f ( A t ) d t ≤ u (cid:21) = E P (cid:20) F t f ( A t ) M f e d t e d t ≤ u (cid:21) = E P (cid:20) F t f ( A t ) M fd t d t ≤ u (cid:21) = E P (cid:20) F t G ( A t ) f ( A t ) d t ≤ u (cid:21) By taking u going to infinity, one obtains: P f (cid:20) F t f ( A t ) d t < ∞ (cid:21) = E P (cid:20) F t G ( A t ) f ( A t ) d t < ∞ (cid:21) oreover, P f (cid:20) F t f ( A t ) (cid:21) = E P (cid:20) F t G ( A t ) f ( A t ) + F t X t (cid:21) Therefore, P f (cid:20) F t f ( A t ) d t = ∞ (cid:21) = E P [ F t X t ] + E P (cid:20) F t G ( A t ) f ( A t ) d t = ∞ (cid:21) = E P [ F t X t ] + E P (cid:20) F t G ( A ∞ ) f ( A ∞ ) d t = ∞ (cid:21) and then: Q f [ F t g ≤ t ] = E P [ F t X t ] + E P (cid:20) F t G ( A ∞ ) f ( A ∞ ) g ≤ t (cid:21) Now, let us define the measure: P f := G ( A ∞ ) . P . and the unique measure P f such that for all t ≥
0, its restriction to F t has density: N ft := G ( A t ) − E P [ G ( A ∞ ) |F t ] + f ( A t ) X t with respect to P (note that N ft ≥ P -a.s.). It is easy to check that the measures P f and P f + P f have the same restriction to F t , and by monotone class theorem, they are equal.Under P f and P f , the measure of the event { A ∞ = ∞} is zero, since these two measuresare dominated by P f . Then, one can define the σ -finite measures: Q f := 1 f ( A ∞ ) . P f and Q f := 1 f ( A ∞ ) . P f . The measure Q f is the sum of Q f and Q f . Now, we have: Q f [ F t g ≤ t ] = E P (cid:20) F t G ( A ∞ ) f ( A ∞ ) g ≤ t (cid:21) , by using directly the definition of Q f . Moreover, let us recall that: Q f [ F t g ≤ t ] = E P [ F t X t ] + E P (cid:20) F t G ( A ∞ ) f ( A ∞ ) g ≤ t (cid:21) . In particular, since
G/f is assumed to be uniformly bounded: Q f [ F t g ≤ t ] < ∞ , his implies that the following equalities are meaningful, and then satisfied, since Q f = Q f + Q f : Q f [ F t g ≤ t ] = Q f [ F t g ≤ t ] − Q f [ F t g ≤ t ]= (cid:18) E P [ F t X t ] + E P (cid:20) F t G ( A ∞ ) f ( A ∞ ) g ≤ t (cid:21)(cid:19) − E P (cid:20) F t G ( A ∞ ) f ( A ∞ ) g ≤ t (cid:21) = E P [ F t X t ]Hence, the measure Q f satisfies the second property given in Theorem 2.4. By applying thisproperty to F t = f ( A t ) (which is bounded, since f is supposed to be bounded) and by usingthe fact that A t = A ∞ on { g ≤ t } , one deduces: P f [ g ≤ t ] = E P [ f ( A t ) X t ]and then (by using the fact that for all t ≥ N ft has an expectation equal to the total massof P f ): P f [ g > t ] = E P [ G ( A t ) − G ( A ∞ )] . Since G ( A t ) − G ( A ∞ ) ≤ G (0) tends P -a.s. to zero when t goes to infinity, one obtains: P f [ g = ∞ ] = 0 , and Q f [ g = ∞ ] = 0since Q f is absolutely continuous with respect to P f . Therefore, the measure Q exists: letus now prove its uniqueness (which implies, in particular, that Q f is in fact independent ofthe choice of f ). If Q ′ and Q ′′ satisfy the conditions defining Q , one has, for all t ≥ t ∈ F t : Q ′ [Λ t , g ≤ t ] = Q ′′ [Λ t , g ≤ t ]Now let u > t ≥
0, and let Λ u be in F u . One can check that: Q ′ [Λ u , g ≤ t ] = Q ′ [Λ t , d t > u, g ≤ u ]If H ′ t is the σ -algebra generated by F t and the ( Q ′ + Q ′′ )-negligible sets of F , then d t isa stopping time with respect to the right-continuous filtration ( H ′ t ) t ≥ . Hence the eventΛ ′ u := Λ t ∩ { d t > u } is in H ′ u , and then, there exists an event Λ ′′ u ∈ F u such that Q ′ [(Λ ′′ u \ Λ ′ u ) ∪ (Λ ′ u \ Λ ′′ u )] = 0and Q ′′ [(Λ ′′ u \ Λ ′ u ) ∪ (Λ ′ u \ Λ ′′ u )] = 0 . ne then deduces that: Q ′ [Λ u , g ≤ t ] = Q ′ [Λ ′ u , g ≤ u ]= Q ′ [Λ ′′ u , g ≤ u ]= Q ′′ [Λ ′′ u , g ≤ u ]= Q ′′ [Λ ′ u , g ≤ u ]= Q ′′ [Λ u , g ≤ t ] . By the monotone class theorem, applied to the restrictions of Q ′ and Q ′′ to the set { g ≤ t } ,one has for all Λ ∈ F : Q ′ [Λ , g ≤ t ] = Q ′′ [Λ , g ≤ t ] . By taking the increasing limit for t going to infinity, Q ′ [Λ , g < ∞ ] = Q ′′ [Λ , g < ∞ ] . Now, by assumption: Q ′ [ g = ∞ ] = Q ′′ [ g = ∞ ] = 0 , which implies: Q ′ [Λ] = Q ′′ [Λ] . This completes the proof of Theorem 2.4. (cid:3)
A careful look at the proof of Theorem 2.4 shows that the result is valid if t is replaced bya bounded stopping time T . Moreover, for submartingales of the class (Σ) which are also ofclass ( D ), we can take a filtration ( F t ) which satisfies the usual assumptions. More precisely,the following result holds: Corollary 2.6.
Let ( X t ) t ≥ be a submartingale of the class (Σ) . (1) If the filtered probability space (Ω , F , P , ( F t ) t ≥ ) satisfies the property (P), then thereexists a unique σ -finite measure Q , defined on (Ω , F , P ) such that for g := sup { t ≥ , X t = 0 } : • Q [ g = ∞ ] = 0 : • For any bounded stopping time T , and for all F T -measurable, bounded randomvariables F T , Q [ F T g ≤ T ] = E P [ F T X T ] . (2) if the X is of class ( D ) and the filtered probability space (Ω , F , P , ( F t ) t ≥ ) satisfiesthe usual assumptions or the property (P), then for any stopping time TX T = E [ X ∞ g ≤ T |F T ] , where as usual g := sup { t ≥ , X t = 0 } .Remark . Part (2) of Corollary 2.6, under the usual assumptions, is proved in [5].Let us note that, in the proof of Theorem 2.4, if f does not vanish, is bounded and if G/f is also bounded then the finite measure P f has density f ( A ∞ ) with respect to Q . Now, onecan prove that, in fact, these conditions on f are not needed. More precisely, one has thefollowing: roposition 2.8. Let f be an integrable function from R + to R + . Then, there exists aunique finite (positive) measure M f such that: M f [ F t ] = E P [ F t N ft ] for all t ≥ , and for all bounded, F t -measurable functionals F t , where the process ( N ft ) t ≥ is given by: N ft := G ( A t ) − E P [ G ( A ∞ ) |F t ] + f ( A t ) X t for G ( x ) := Z ∞ x f ( y ) dy. In particular, ( N ft ) t ≥ is a nonnegative martingale. Moreover, the measure M f is absolutelycontinuous with respect to Q , with density f ( A ∞ ) .Proof. In the proof of Theorem 2.4, we have shown this result if f is strictly positive,bounded, and if G/f is also bounded (recall that G ( x ) is the integral of f between x andinfinity). One can now prove Proposition 2.8 for any measurable, bounded, nonnegativefunctions f with compact support. Indeed, if f is such a function, one can find f and f ,bounded, strictly positive, integrable, such that, with obvious notation, G /f and G /f arebounded, and f = f − f (for example, one can take f ( x ) := f ( x ) + e − x and f ( x ) := e − x ).One has, for all t ≥
0, and for all bounded, F t -measurable random variables F t : M f [ F t ] = E P [ F t N f t ] , and M f [ F t ] = E P [ F t N f t ] . Now N f is the difference of N f and N f , and then, it is a (nonnegative) martingale. Hence,there exists a unique finite measure M such that: M [ F t ] = E P [ F t N ft ] . Therefore M f exists, is unique, and since M f and M f + M f coincide on F t for all t ≥ M f [ F t ] = M f [ F t ] − M f [ F t ](this equality is meaningful because all the measures involved here are finite). Since Propo-sition 2.8 is satisfied for f and f : M f [ F t ] = Q [ F t f ( A ∞ )] − Q [ F t f ( A ∞ )] , which implies M f [ F t ] = Q [ F t f ( A ∞ )] . By monotone class theorem, f satisfies Proposition 2.8. Now, let us only suppose that f is nonnegative and integrable. There exists nonnegative, measurable, bounded functions( f k ) k ≥ with compact support, such that: f = X k ≥ f k . With obvious notation, one has: G = X k ≥ G k , nd then, for all t ≥ G ( A t ) = X k ≥ G k ( A t )and E P [ G ( A ∞ ) |F t ] = X k ≥ E P [ G k ( A ∞ ) |F t ] , P -a.s., where the two sums are uniformly bounded by G (0). This boundedness implies thatone can substract the second sum from the first, and obtain: G ( A t ) − E P [ G ( A ∞ ) |F t ] = X k ≥ ( G k ( A t ) − E P [ G k ( A ∞ ) |F t ])almost surely. Moreover: f ( A t ) X t = X k ≥ f k ( A t ) X t , and then, P -a.s.: N ft = X k ≥ N f k t . We know that M f k is well-defined for all k ≥
1, hence, one can consider the measure: M := X k ≥ M f k . Now, for t ≥ F t , bounded, F t -measurable: M [ F t ] = X k ≥ M f k [ F t ]= X k ≥ E P [ F t N f k t ]= E P [ F t N ft ] . Hence, the measure M f is well-defined, unique by monotone class theorem, and is equal to M . Now, one has, for all k ≥ M f k = f k ( A ∞ ) . Q . Since M f is the sum of the measures M f k , M f = "X k ≥ f k ( A ∞ ) . Q = f ( A ∞ ) . Q which completes the proof of Proposition 2.8. (cid:3) Another question which is quite natural to ask is the following: since Q [ A ∞ = ∞ ] = 0,what is the image of Q by the functional A ∞ (in other words, what is the ”distribution of A ∞ under Q ”)? This question can be solved in any case: roposition 2.9. Let ( X t ) t ≥ be a submartingale of class (Σ) , which satisfies all the as-sumptions of Theorem 2.4. Then, if ( A t ) t ≥ is the increasing process of ( X t ) t ≥ , the imageby the functional A ∞ of the measure Q defined in Theorem 2.4, is a measure on R + , equal tothe sum of E P [ X ] times Dirac measure at zero, and another measure, absolutely continuouswith respect to Lebesgue measure, with density P [ A ∞ > u ] at any u ∈ R + . In particular, if A ∞ = ∞ , P -almost surely, then this image measure is E P [ X ] δ + R + λ , where λ is Lebesguemeasure on R + , and δ is Dirac measure at zero.Proof. Let f be an integrable function from R + to R + . By taking the notation of Proposition2.8, one has: M f = f ( A ∞ ) . Q . Therefore, Q [ f ( A ∞ )] is the total mass of M f , and then, the expectation of: N f = G (0) − E P [ G ( A ∞ ) |F ] + f (0) X By applying this result to f = [0 ,u ] , one deduces, for any u ≥ Q [ A ∞ ≤ u ] = u − E P [( u − A ∞ ) + ] + f (0) E P [ X ]= E P [ A ∞ ∧ u ] + f (0) E P [ X ]= Z u P [ A ∞ > v ] dv + f (0) E P [ X ] . which implies Proposition 2.9. (cid:3) Remark . When X is also of class ( D ), P [ A ∞ > v ] is computed in [9], Theorem 4.1.2.3. An extension of Doob’s optional stopping theorem.
We shall now see how The-orem 2.4 and Corollary 2.6 can be interpreted as an extension of Doob’s optional theoremto continuous martingales which are not necessarily uniformly integrable on the one hand,and to the larger class of processes of the class (Σ).Let M be a continuous martingale; then M + and M − are both of class (Σ). If g = sup { t ≥ M t = 0 } , then under the assumptions of Theorem 2.4, there exist two σ -finite measures Q (+) and Q ( − ) such that • Q ( ± ) [ g = ∞ ] = 0: • For all t ≥
0, and for all F t -measurable, bounded random variables F t , Q ( ± ) [ F t g ≤ t ] = E P (cid:2) F t M ± t (cid:3) . Now since M = M + − M − , we deduce from Theorem 2.4 and Corollary 2.6 the followingsolution to Problem 2: Proposition 2.11.
Let M be a continuous martingale defined on a filtered probability space (Ω , F , P , ( F t ) t ≥ ) which satisfies the property (P). Then there exist two σ -finite measures Q (+) and Q ( − ) , suchthat for any bounded stopping time T and any bounded F T -measurable variable F T , (cid:0) Q (+) − Q ( − ) (cid:1) [ F T g ≤ T ] = E [ F T M T ] , with g = sup { t ≥ M t = 0 } . The measures Q (+) and Q ( − ) are obtained by applyingTheorem 2.4 to the submartingales M + and M − . emark . If the martingale M is uniformly integrable, then following Corollary 2.6, onecan also work with a filtration satisfying the usual assumptions and take any stopping time T , not necessarily bounded. Consequently, Proposition 2.11 can be viewed as an extension ofDoob’s optional stopping theorem: the terminal value of the martingale M has to be replacedby (cid:0) Q (+) − Q ( − ) (cid:1) which is a signed measure when restricted to the sets g ≤ t . Theorem 2.4and Corollary 2.6 can in turn be interpreted as an extension of the stopping theorem to thelarger class of submartingales of the class (Σ).3. Some examples
Now, let us study in more details several consequences of Theorem 2.4.3.1.
The case of a the absolute value, or the positive part, of a martingale.
Wesuppose that X t = M + t , X t = M − t or X t = | M t | , where ( M t ) t ≥ is a continuous martingale.In this case, X is a submartingale of class (Σ), and its increasing process is half of the localtime of M in the two first cases, and the local time of M in the third case. Therefore,one can apply Theorem 2.4. In particular, if ( X t ) t ≥ is a strictly positive martingale, thenit is a submartingale of class (Σ), with increasing process identically equal to zero. Onededuces that for any nonnegative, integrable function f , N ft = f (0) X t , which implies thatfor all t ≥
0, the restriction of M f to F t has density f (0) X t with respect to P . Hence, since f ( A ∞ ) = f (0), the restriction of Q to F t has density X t with respect to P . In particular, Q is a finite mesure, and X does not vanish under Q , i.e. Q [ ∃ t ≥ , X t = 0] = 0 . The case of the draw-down of a martingale.
Let ( M t ) t ≥ be a c`adl`ag martingale,starting at zero, without positive jumps. This assumption implies that its supremum S t := sup s ≤ t M s is a.s. continuous with respect to t . The process( X t := S t − M t ) t ≥ is then a submartingale of class (Σ) with martingale part − M and increasing process S . Oneobtains, for all t ≥ F t bounded, F t -measurable: Q [ F t g ≤ t ] = E P [ F t ( S t − M t )]where, in this case, g is the last time when M reaches its overall supremum.3.3. The uniformly integrable case.
Let us suppose that, in Theorem 2.4, the family ofvariables ( X t ) t ≥ is uniformly integrable. In this case, ( E P [ X t ]) t ≥ , and then ( E P [ A t ]) t ≥ areuniformly bounded. By monotone convergence, A ∞ is integrable, and in particular finitea.s. Since ( A t ) t ≥ and ( X t ) t ≥ are uniformly integrable, ( N t ) t ≥ is a uniformly integrablemartingale, which implies that there exists N ∞ such that for all t ≥ N t = E [ N ∞ |F t ] and N t tends a.s. to N ∞ for t going to infinity. One deduces that X t tends a.s. to X ∞ := N ∞ + A ∞ .Now, for all nonnegative, bounded, integrable functions f , the martingale N f is uniformlyinegrable. Moreover, if f is continuous, G ( A t ) + X t f ( A t ) tends a.s. to G ( A ∞ ) + X ∞ f ( A ∞ )when t → ∞ , and the uniformly integrable martingale ( E [ G ( A ∞ ) |F t ]) t ≥ tends a.s. to ( A ∞ ). Therefore, the terminal value of N f is X ∞ f ( A ∞ ), which implies that M f hasdensity X ∞ f ( A ∞ ) with respect to P , and finally: Q = X ∞ . P . This case was essentially obtained by Az´ema, Meyer and Yor in [1] and in [5] in relation withproblems from mathematical finance. The particular case where X t = | M t | , where ( M t ) t ≥ is a continuous uniformly integrable martingale, starting at zero, and for which the measure Q has density | M ∞ | with respect to P , was studied in [2], [3].3.4. The case where A ∞ is infinite almost surely. In this case, for any nonnegative,integrable function f , one has: N ft = G ( A t ) + f ( A t ) X t . Moreover, if X = 0 a.s., then the image of Q by A ∞ is simply Lebesgue measure. There areseveral interesting examples of this particular case.1) It X t = M + t , X t = M − t or X t = | M t | , where M is a continuous martingale, then weare in the case: A ∞ = ∞ iff the total local time of M is a.s. infinite, or, equivalently,iff the overall supremum of | M | is a.s. infinite. This condition is satisfied, in particular,if M is a Brownian motion. More precisely, let us suppose that Ω = C ( R + , R ), ( F t ) t ≥ is the smallest right-continuous filtration, containing the natural filtration of the canonicalprocess ( Y t ) t ≥ , and P is Wiener measure. If X t = | Y t | , X t = Y + t or X t = Y − t , the σ -finitemeasure Q described in Theorem 2.4 was already studied in [8], Chapter 1. This measuresatisfies a slightly more general result than what is written in Theorem 2.4. Indeed, in theirmonograph, Najnudel, Roynette and Yor prove that there exists a unique σ -finite measure W on Ω such that for all t ≥
0, for all bounded, F t -measurable functionals F t , and for all a ∈ R : W [ F t g a ≤ t ] = P [ F t | Y t − a | ] , W [ g a = ∞ ] = 0where g a := sup { t ≥ , Y t = a } . Moreover W can be decomposed (in unique way) as the sum of two σ -finite measures W + and W − , such that: W + [ F t g a ≤ t ] = P [ F t ( Y t − a ) + ] , W − [ F t g a ≤ t ] = P [ F t ( Y t − a ) − ] , W + [ E − ] = W − [ E + ] = 0where E − is the set of trajectories which do not tend to + ∞ , and E + is the set of trajectorieswhich do not tend to −∞ . With these definitions, the measure Q is equal to W + if X t = Y + t , W − if X t = Y − t and W if X t = | Y t | .2) Let ( M t ) t ≥ be a c`adl`ag martingale, starting at zero, without positive jumps. Theprocess ( X t := S t − M t ) t ≥ is a submartingale of class (Σ) with martingale part − M and increasing process S , and onehas A ∞ = ∞ a.s., iff the overall supremum of M is a.s. infinite. A particular case where thiscondition holds is, again, when M is a Brownian motion. More precisely, if one takes thesame filtered probability space as in the previous example, and if X t = (sup s ≤ t Y s ) − Y t , then he σ -finite measure exists and is in fact equal to W − . Note that the image of this measureby X is equal to the image of W by the absolute value.3) Another interesting example is studied in Chapter 3 of [8]. Let us take the same filteredmeasurable space as in the previous examples, endowed with a probability measure P underwhich the canonical process ( Y t ) t ≥ is a recurrent, homogeneous diffusion with values in R + ,starting at zero, and such that zero is an instantaneously reflecting barrier. We suppose thatthe infinitesimal generator G of Y satisfies (for x ≥ G f ( x ) = ddm ddS f ( x )where S is a continuous, strictly increasing function such that S (0) = 0 and S ( ∞ ) = ∞ ,and m is the speed measure, satisfying m ( { } ) = 0. There exists a jointly continuous family( L yt ) t,y ≥ of local times of Y , satisfying: Z t h ( Y s ) ds = Z ∞ h ( y ) L yt m ( dy )for all borelian functions h from R + to R + . If we define the process ( X t ) t ≥ by: X t = S ( Y t )then ( X t − L t ) t ≥ is a ( F t ) t ≥ -martingale. Hence, if F is the σ -algebra generated by ( F t ) t ≥ ,the assumptions of Theorem 2.4 are satisfied, and L ∞ is infinite, since the diffusion Y isrecurrent. The σ -finite measure Q is given by the formula: Q = Z ∞ dl Q l , where Q l is the law of a process ( Z lt ) t ≥ , defined in the following way: let τ l be the inverselocal time at l (and level zero) of a diffusion R , which has a law equal to the distributionof Y under P , and let ( ˜ R u ) u ≥ be an homogeneous diffusion, independent of R , starting atzero, never hitting zero again, and such that for 0 ≤ u < v , x, y > P [ ˜ R v ∈ dy | ˜ R u = x ] = S ( y ) S ( x ) P [ R v ∈ dy, ∀ w ∈ [ u, v ] , R w > | R u = x ](intuitively, the law of ( ˜ R u ) u ≥ is the law of ( R u ) u ≥ , conditioned not to vanish), then Z l satisfies Z lt = R t for t ≤ τ l , and Z lτ l + u = ˜ R u for u ≥
0. Theorem 2.4 applies, in particular, if Y is a Bessel process of dimension d ∈ (0 , d = 2(1 − α ) (which imples 0 < α < X t = ( Y t ) α = ( Y t ) − d . In this case, the process ( ˜ R u ) u ≥ , involved in an essential way in the construction of Q , isa Bessel process of dimension 4 − d = 2(1 + α ). For d = 1 ( α = 1 / X t = Y t ) t ≥ is theabsolute value of a Brownian motion, and ˜ R is a Bessel process of dimension 3.4) Let Ω be the space of continous functions from R + to R , ( H t ) t ≥ the smallest right-continous filtration, containing the natural filtration of the canonical process ( Y t ) t ≥ , H the -algebra generated by ( H t ) t ≥ and P the probability measure under which ( Y t ) t ≥ is a Besselprocess of dimension d := 2(1 − α ) for 0 < α <
1. For t ≥
0, let us take the notation: g ( t ) := sup { u ≤ t, Y u = 0 } and let ( F t ) t ≥ be the filtration of the zeros of Y , i.e. F t = H g ( t ) . One defines the σ -algebra F as the σ -algebra generated by ( F t ) t ≥ , i.e. by the zeros of Y . Now, the process( X t := ( t − g ( t )) α ) t ≥ is a ( F t ) t ≥ -submartingale of class (Σ), and its increasing process ( A t ) t ≥ is given by: A t = 12 α Γ(1 + α ) L t ( Y )where L t ( Y ) is the local time of Y at zero, defined as the increasing process of the submartin-gale ( Y αt ) t ≥ , which is of class (Σ) (see [9], and the previous example). Since Y is recurrent, A ∞ = ∞ a.s. Now, let R be the σ -finite measure on (Ω , ( H t ) t ≥ , H ) which is equal to themeasure Q of example 3). Because of this example, one has, for all bounded, H t -measurablefunctions F t : R [ F t g ≤ t ] = E P [ F t Y αt ]where g is the last zero of Y , equal to the last zero of X . Now, if F t is F t -measurable, thenone obtains R [ F t g ≤ t ] = E P [ F t E P [ Y αt |F t ]]which implies: R [ F t g ≤ t ] = 2 α Γ(1 + α ) E P [ F t X t ]Therefore, the measure Q satisfying the conditions given in Theorem 2.4 is the restrictionof the measure ˜ Q := 12 α Γ(1 + α ) R to the σ -algebra F (generated by the zeros of Y ). Moreover, the image of Q by X is: S := 12 α Γ(1 + α ) Z ∞ dl S l where S l is the law of a process ( V lt ) t ≥ , defined in the following way: let τ l be the inverselocal time at l (and level zero) of a diffusion R , with the same law as Y under P , and let γ ( t ) be the last zero of R before time t , for all t ≥ V l satisfies V lt = ( t − g ( t )) α for t ≤ τ l , and V lτ l + u = u α for u ≥
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