Conditioned stochastic dierential equations: theory, examples and application to nance

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1 Stochastic Processes and their Applications 1 (22) Conditioned stochastic dierential equations: theory, examples and application to nance Fabrice Baudoin a;b; a Laboratoire de Probabilites et Modeles aleatoires, Universite Paris 6, 175 rue du Chevaleret, F-7513 Paris, France b CREST, Laboratoire de Finance-Assurance, 15 Bd Gabriel Peri, Malako, France Received 3 May 21; received in revised form 7 February 22; accepted 21 February 22 Abstract We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic dierential equations that we call conditioned stochastic dierential equations. The link with the theory of initial enlargement of ltration is made and after a general presentation several examples are studied: the conditioning of a standard Brownian motion (and more generally of a Markov diusion) by its value at a given date, the conditioning of a geometric Brownian motion with negative drift by its quadratic variation and nally the conditioning of a standard Brownian motion by its rst hitting time of a given level. As an application, we introduce the notion of weak information on a complete market, and we give a quantitative value to this weak information. c 22 Elsevier Science B.V. All rights reserved. MSC: 6H1; 6J65; 6J6; 91A44; 91B16; 91B28 Keywords: Brownian bridge; Conditioning; Initial enlargement of ltration; Exponential generalization of Pitman s 2M-X theorem; Filtering; Portfolio optimization 1. Introduction In this paper, we present a natural generalization of the Brownian bridges. More precisely, (1) A constant horizon time T (; + ]. Correspondence address: Laboratoire de Probabilites et Modeles aleatoires, Universite Paris 6, 175 rue du Chevaleret, F-7513 Paris, France. address: fabrice.baudoin@ensae.fr, symplectik@aol.com (F. Baudoin) /2/$ - see front matter c 22 Elsevier Science B.V. All rights reserved. PII: S (2)19-6

2 11 F. Baudoin / Stochastic Processes and their Applications 1 (22) (2) A functional Y on the Wiener space measurable with respect to events which can occur before time T and valued in some Polish space S. (3) A probability measure on B(S). We construct a probability P on the Wiener space which satises (1) P and P coincide on the events independent of Y. (2) The law of Y under P is precisely. The dynamics of the coordinate process under P denes a stochastic dierential equation that we call a conditioned stochastic dierential equation (in abbreviate CSDE). Hence, under some pathwise uniqueness conditions, this CSDE denes on a general ltered probability space a process with law P. The present paper is organized as follows. In Section 2, we show how to construct the probability P and we give explicitly the associated CSDE. Then we show the link between the classical theory of initial enlargement of ltration (see Amendinger et al., 1998; Follmer and Imkeller, 1993; Jacod, 1985; Jeulin, 198; Jeulin and Yor, 1985) and the theory of CSDEs. First, the decomposition of the coordinate process X in its natural ltration enlarged by Y does not depend on the probability and then we can recover the CSDE satised by X from this decomposition by a simple ltering formula. Moreover, we show how our probability P is related to the martingale preserving measure considered in Amendinger et al. (1998). To conclude the section, we show how Malliavin calculus and Fourier analysis can be used in the computations when S = R N and the functional Y is dierentiable in Malliavin s sense. In Section 3, we provide several examples. (1) First, we consider the important case where the functional Y is the value at a given date of a given Markov diusion. In the particular case of the Brownian motion, this corresponding CSDE is studied in detail and many particular examples are given. In the general case, we study the CSDE by two dierent methods: Malliavin calculus and Burgers equation. We then deduce the set of diusions which have the same bridges as a given diusion, recovering a result of Fitzsimmons (1998). We also deduce a probabilistic representation of fundamental solutions of Burgers type partial dierential equation. Roughly speaking, this representation, can be seen as a Hopf Cole transformation on the Wiener path space of the classical Feynman Kac formula. (2) As a second example, we study the case where the conditioned functional is the quadratic variation of a geometric Brownian motion with negative drift. In this case, the set of processes which is obtained is nothing else but the set of processes X considered in Baudoin (21a) and for which the process X =X is a diusion in its own ltration. (3) We give a last example directly deduced from the previous one by a Lamperti representation, which is the conditioning of the Brownian motion by the rst hitting time of a level. This last example is not directly covered by the general theory developed in Section 2, since here the conditioning is made at a random time. Nevertheless this can be seen as an introducing example to a more general theory.

3 F. Baudoin / Stochastic Processes and their Applications 1 (22) In Section 4, we give an application of our results to nance. More precisely, in a recent series of papers (Amendinger et al., 2; Amendinger et al., 1998; Pikovsky and Karatzas, 1996) securities markets models have been considered for which one insider possesses from the beginning extra information about the outcome of some variable Y of the prices (S t ) 6t6T. In these modelizations, one considers that this insider has the enlarged ltration F (Y ) at his disposal, where F is the public information ow. We consider here an insider who is only weakly informed on Y, meaning that the insider has only the ltration F at his disposal but knows the law of Y. More precisely, with Y we associate a probability measure on R n (assumed to be the law of Y under the eective probability of the market) which admits an almost surely bounded density with respect to P Y. Our aim is then to give the quantitative value of this weak information, and here are our results. Let U a utility function and denote by I the inverse of U. Let E be the set of probability measures Q such that (1) Q is equivalent to P (which is the martingale measure). (2) Q(Y dy)=(dy). Then, for each initial investment x; ( ( T inf sup Q U x + )) u ds u = (U I) Q E A(S) R n where A(S) is the set of admissible strategies and is dened by ( I (x) dp ) Y d (y) P(Y dy)=x: R n ( (x) dp ) Y d (y) (dy); Moreover, we obtain the remarkable fact that the probability which realizes the inmum is independent of the utility function used by the trader, and under this minimizing probability the price process is a CSDE. 2. Conditioned stochastic dierential equations 2.1. General framework We work on the Wiener space of continuous paths 1 W =(C ; (F t ) t ; (B t ) t ; P): (1) C is the space of continuous functions R + R (for T ; C T will denote the space of continuous functions [;T] R). 1 We could take any ltered probability space (; (H t) t ; ( t) 6t6T ; Q) (H being the natural ltration of ), and except for the results of Section 2.4, our discussion remains valid.

4 112 F. Baudoin / Stochastic Processes and their Applications 1 (22) (2) (B t ) t is the coordinate process dened by B t (f)=f(t). (3) (F t ) t is the natural ltration of (B t ) t. (4) P is the Wiener measure. Denition 1. A conditioning on the Wiener space is a triplet (T; Y; ) with: (1) T (; + ] a constant time horizon. It corresponds to the period of time [;T] on which a conditioning is made (2) Y a F T -measurable random variable valued in a Polish space S endowed with its Borel -algebra B(S). It corresponds to the functionals of the trajectories being conditioned (3) a probability measure on B(S). It corresponds to the conditioning. We will write simply P for P =FT, as the following study only involves the time interval [;T] and we will denote by P Y the law of Y under P. We shall always make the following assumptions (A1) and (A2) on the conditioning (T; Y; ). (A1) There exists a jointly measurable process y t ; 6 t T; y S satisfying for any random variable X bounded and F t -measurable, t T, and P Y -a.s. y S E(X Y = y)=e( y t X ): Example 2. Consider a Markov diusion Z started at. Assume its semigroup P t (x; dy) to be of the form p t (x; y)dy with p t (x; y) then y t = p T t(z t ;y) p T (;y) ; t T: Indeed; P(Z T dy F t )=p T t (Z t ;y)dy (see Fitzsimmons et al.; 1993 for further details; or Yor; 1997; p. 35). (A2) and Supp Supp P Y L 1 (S; P Y ) L 1 (S; ): 2.2. Minimal probability associated with a conditioning We rst note the following immediate consequence of the existence of a regular conditional probability given Y

5 F. Baudoin / Stochastic Processes and their Applications 1 (22) Proposition 3. On F T there exists a unique probability measure P such that (1) If X :(C T ; F T ) (S; B(S)) is a bounded random variable then E (X Y )=E(X Y ): (2) The law of Y under P is. P is given by the following disintegration formula: For A F T ; P (A)= P(A Y = y)(dy): The following remarks are worth recording. S Remark4. (1) P = P = P Y (2) If P Y then dp = d (Y )dp: dp Y (3) If A F T is P-independent of Y then it is also P -independent of Y and Supp P Supp P L 1 (C T ; P) L 1 (C T ; P ): Denition 5. We call P (T; Y; ). the minimal probability associated with the conditioning In order to justify the word minimal assigned to P we consider a convex function : R + R and denote by E ; the set of probability measures on F T which are absolutely continuous with respect to P such that (1) E ( ( dq dp ) ) + : (2) The law of Y under Q is. Proposition 6. Assume that and Then P Y S ( ) d dp Y + : dp Y ( ( )) ( dq inf E = E Q E dp ( dp dp )) :

6 114 F. Baudoin / Stochastic Processes and their Applications 1 (22) Proof. Let dq = D dp a probability measure which belongs to E. Since the law of Y under Q is ; we have E(D Y )= d (Y ): dp Y Now; from Jensen s inequality ( ) d (Y ) 6 E( (D) Y ) dp Y which implies ( ( ) ) d E (Y ) 6 E( (D)) : dp Y Example 7. With (x)=x 2 ; we see that P is the minimal variance probability; i.e. ( (dq ) ) ( 2 (dp ) ) 2 inf E = E : Q E dp dp The following easy proposition shows the transmission of the negligible sets by the map P which is trivially extended to general Borel measures. Proposition 8. Let with = a + s a P Y and s P Y the Lebesgue decomposition of with respect to P Y. Then dp = d a dp Y (Y )dp +dp s is the Lebesgue decomposition of P with respect to P. In particular P P P Y : Our aim, now, is to develop stochastic calculus under the minimal probability P.To do this, the rst step is to compute the martingale density process of P with respect to P. Proposition 9. For t T; P =F t is absolutely continuous with respect to P =Ft and dp =F t = y t (dy)dp =Ft : S

7 F. Baudoin / Stochastic Processes and their Applications 1 (22) Proof. Let t T. Let X a bounded F t -measurable random variable E (X )= E(X Y = y)(dy): From (A1); for P Y -a.s. y S S E(X Y = y)=e( y t X ): Now because of (A2); we can apply Fubini s theorem to get the expected result ( ) E (X )=E y t (dy)x which is the expected result. S Example 1. Let us consider the conditioning (T; Z T ;) where Z is a Markov diusion started at. In this case the martingale density process of the minimal probability is given by + p T t (Z t ; dy) p T (; dy) (dy); t T; where P t (z; dy)=p t (z; y)dy is the semigroup of Z. We would like to draw the attention of the reader on the fact that, as it is deeply explained in Follmer and Imkeller (1993) for the case Y = X T and = y. Proposition 11. The martingale density process D t = y t (dy); t T S S is not in general uniformly integrable. Nevertheless; it has the following P-a.s. limit lim y t (dy)= d a (Y ): t T dp Y In order to explicit the semimartingale decomposition of B under P by Girsanov s theorem, we need the following well-known lemma in the theory of initial enlargement of the Brownian ltration. Lemma 12 (see Amendinger et al., 1998; Follmer and Imkeller, 1993; Jacod, 1985). There exists a jointly measurable process [;T[ C T S R; (t;!; y) y t (!);

8 116 F. Baudoin / Stochastic Processes and their Applications 1 (22) such that (1) For P Y -a.s. y S; the process ( y t ) 6t T is F-predictable. (2) For P Y -a.s. y S and for 6 t T; ( t ) P (s y ) 2 ds + =1: (3) For P Y -a.s. y S and for 6 t T; y ;B t = t y s y s ds: Remark13. We can choose such that for P Y -a.s. y S and for 6 t T; [ t y t = exp s y db s 1 t ] (s y ) 2 ds on { y t } 2 see Follmer and Imkeller (1993). With this lemma, we can now state: Theorem 14. (B t ) 6t T is a (F t ; P ) semimartingale whose decomposition is given by S db t = y t y t (dy) S y t (dy) dt +dw t; t T; (2.1) where (W t ) 6t T is a standard (F t ; P ) Brownian motion. Proof. The process D t = y t (dy); S 6 t T is the density process of P with respect to P. By Lemma 12 and Fubini s theorem; we have ( ) d D; B t = y t y t (dy) dt: S The result is then a consequence of Girsanov s theorem. As the process ( S y t y t (dy)= S y t (dy)) 6t T predictable function F such that for all t T S y t y t (dy) = F(t; (B s ) s6t ): S y t (dy) is F-adapted, there exists a Denition 15. Let (; (H t ) 6t T ; Q) a ltered probability space on which a standard H-adapted Brownian motion ( t ) 6t T is dened. The stochastic dierential equation X t = t F(s; (X u ) u6s )ds + t ; t T (2.2)

9 F. Baudoin / Stochastic Processes and their Applications 1 (22) will be called the conditioned stochastic dierential equation (in abbreviate CSDE) associated with the conditioning (T; Y; ). Remark16. (1) By construction; the stochastic dierential equation (2.2) has always the weak solution (B; W ) dened on the ltered probability space (C T ; (F t ) 6t T ; P ) (2) If the conditioned functional Y can be simply expressed by a semimartingale (Z t ) 6t6T which is F-adapted; then it will be more convenient to work on the semimartingale decomposition of Z in the ltered probability space (C T (F t ) 6t T ; P ). Thanks to Yamada Watanabe s theorem (see Revuz and Yor, 1999, p. 368), we can now state the basic result of our work. Theorem 17. Assume that the stochastic dierential equation (2.2) enjoys the pathwise uniqueness property. Then (2.2) has a unique strong solution associated with the initial condition X = and the law of (X t ) 6t T is the minimal probability associated with the conditioning (T; Y; ) Initial enlargement of the natural ltration As a consequence of the previous paragraph, we can prove a little extension of the celebrated Jacod s theorem (see for example Follmer and Imkeller, 1993; Jacod, 1985; Jeulin, 198; Jeulin and Yor, 1985; Yor, 1997) about the initial enlargement of the Brownian ltration. Theorem 18. The process (M t ) 6t T dened by dm t = t Y dt +db t ; 6 t T (2.3) is a standard Brownian motion; not only for P but also for P in the enlarged ltration G t = F t (Y ); 6 t 6 T: Proof. We rst show that (M t ) 6t T isa(g t ; P) standard Brownian motion. Since d M t =dt it is enough to show that (M t ) 6t T isa(g t ; P) martingale; according to Paul Levy s characterization of Brownian motion. For this, we note as a consequence of Theorem 14 that (M t ) 6t T is, for P Y -a.s. y S a(f t ; P y ) martingale, where y is the Dirac measure at y. So, for s t T; A F s and B(S) we have E((M t M s )1 A (Y ) )= E y ((M t M s )1 A )P Y (dy)=: This shows that (M t ) 6t T isa(g t ; P) standard Brownian motion.

10 118 F. Baudoin / Stochastic Processes and their Applications 1 (22) Let us now show that (M t ) 6t T is also a (G t ; P ) standard Brownian motion. Since (M t ) 6t T isa(g t ; P) standard Brownian motion, it is P independent of Y. But the probabilities P and P coincide on the events which are independent from Y. Hence the law of (M t ) 6t T under P is the same as under P, i.e. the Wiener measure. Remark19. (1) Formally; we recover the decomposition (2.3) from (2.1) with = Y. This shows the analogy between Jacod s and Girsanov s theorem. This parenthood is very well explained in Yoeurp (1985); where the author understood the enlargement formula as a Girsanov formula applied on a convenient product probability space. (2) Assume that is equivalent to P Y. Let Q on F T be such that Q P and such that the law of Y under Q is. Then P is the unique probability on (M s ;s T) (Y ) such that and P = P on (M s ;s T) P = Q on (Y ): With the terminology of Amendinger et al. (1998); P is hence the martingale preserving measure associated with Q. (3) It would be interesting to give a sucient condition in order to have the decomposition (M s ;s T) (Y )=F T : In the examples which will be treated later; this will always be the case. (4) Roughly speaking; all the CSDEs associated with the same functional have the same bridges on this functional. The relationship between decompositions (2.1) and (2.3) is given by the following interesting ltering formula which is a simple consequence of Bayes formula Proposition 2. For 6 t T E (t Y S F t )= y t y t (dy) S y t (dy) : We conclude this paragraph by characterizing the set of the minimal probabilities in terms of the Brownian motion of the enlarged ltration. Theorem 21. Let Q a probability measure on C T which is locally absolutely continuous with respect to the Wiener measure (i.e. for t T; Q =Ft P =Ft ) and such that L 1 (C T ; P) L 1 (C T ; Q).

11 F. Baudoin / Stochastic Processes and their Applications 1 (22) If the process (M t ) 6t T dened by (2.3) is a standard Brownian motion under Q, then there exists a probability measure on S such that Q = P : Proof. Let y R and denote Q y the conditional probability Q( Y = y). From our assumption, the process M is, under Q y, a standard Brownian motion. Hence, by Girsanov s theorem dq y =F t = y t dp =Ft ; t T: Since we also have dp y =F t = y t dp =Ft ; t T; where P y is the conditional probability P( Y = y), we immediately deduce Q y = P y and hence Q = P ; where is the law of Y under Q Computations with Malliavin calculus and Fourier analysis In this paragraph, we give a tool to obtain in some special cases explicit computations. We assume that S = R N with N N and that Y (D 1;2 ) N. We recall (see Nualart, 1995, p. 26) that the Hilbert space D 1;2 is the closure of the class of smooth cylindric random variables S with respect to the norm F 1;2 =(E(F 2 )+E( DF 2 L 2))1=2 ; where D is the Malliavin s dierential. Proposition 22. Assume that (1) is equivalent to P Y. (2) := d=dp Y admits a continuously dierentiable version with bounded partial derivates, then; for t T y R N t y (( ) ) t (dy) y = D t ln E((Y ) F t )=E (Y ) D t Y F t : R N t (dy) Proof. Under these assumptions; we have dp = (Y )dp

12 12 F. Baudoin / Stochastic Processes and their Applications 1 (22) and so for t 6 T dp =F t = E((Y ) F t )dp =Ft : Now from the Clark Ocone formula (see Clark; 197; Malliavin; 1997, p. 183; Nualart; 1995, Proposition 1.3.5) E((Y ) F t =1+ t E(D s (Y ) F s )db s t =1+ E( (Y ) D s Y F s )db s : Hence y RN t y t (dy) y = E( (Y ) D ty F t ) R N t (dy) E((Y ) F t ) and the Bayes formula gives the expected result. Proposition 23. Assume that for almost all t T E(e iy F t ) d + (2.4) R N then; for P Y -a.s.; y R N and for all t T y t = e iy E(e iy F t )d: (2.5) R N Moreover; if for P Y -a.s.; y R N and for all t T D t E(e iy F t ) d + (2.6) R N then for P Y -a.s.; y R N and for all t T; y t Dom(D) and D t y t = y t y t : Proof. Let m a signed measure on R N such that m (dy) + : R N Let now m the Fourier transform of m dened on R N m(y)= e iy m(d): R N We have for t T But R N y t m(y)p Y (dy)=e( m(y ) F t ): E( m(y ) F t )= R N E(e iy F t )m(d) by

13 F. Baudoin / Stochastic Processes and their Applications 1 (22) hence y t m(y)p Y (dy)= E(e iy F t )m(d): R N R N Since the previous equality takes place for all m; this implies that for P Y -a.s.; y R N and for all t T y t = e iy E(e iy F t )d: R N Assume that for P Y -a.s., y R N and for all t T D t E(e iy F t ) d + : R N Because for P Y -a.s., y R N and for all t T; R, E(e iy F t ) D 1;2 it is easily seen that y t Dom(D). Moreover, let us consider a probability measure on R N such that: (1) is equivalent to P Y. (2) := d=dp Y admits a bounded continuously dierentiable version. From Clark Ocone formula, we have [ ] d y t (dy) = E(D t (Y ) F t )db t : It implies R N R N y t y t (dy)=d t E((Y ) F t )=D t R N y t (dy) and the conclusion follows easily because was arbitrary. Remark24. Of course; formula (2.5) remains true even if Y (D 1;2 ) N. 3. Examples 3.1. CSDE associated with the conditioning of a marginal law of the Brownian motion Let (; (H t ) t ; Q) a ltered probability space on which a standard (H t ; Q) Brownian motion ( t ) 6t6T is dened. Let us write the CSDE associated with the conditioning (T; B T ;). The conditioned functional is then the value of the process at the given date T +. We assume that the probability is such that + y 2 (dy) + : (3.1)

14 122 F. Baudoin / Stochastic Processes and their Applications 1 (22) Theorem 25. The stochastic dierential equation ) y 2 e 2T (y Xt)2 2(T t) (dy) dx t = + ( y Xt T t 2 y + e 2T (y Xt)2 2(T t) (dy) dt +d t ; t T (3.2) has a unique strong solution (X t ) 6t T associated with the initial condition X = and the law of (X t ) 6t T is the minimal probability associated with the conditioning (T; B T ;). Before we give the proof of this theorem, we present a number of interesting properties of the solution of (3.2) which will be proved later. Theorem 17 gives the convergence in law of (X t ) 6t T when t T, the rst question is then to study the Q-a.s. convergence. Proposition 26. For the solution (X t ) 6t T of (3.2) we have (1) For all g L 2 ([;T]; R); the process ( t g(s)dx s) 6t T converges Q-a.s. and in L 2 when t T to a random variable T g(s)dx s such that ( T ) + E g(s)dx s = y(dy) T g(s)ds; t T T and ( ( T )2 ) T + ( E g(s)dx s = g(u) 2 du + y2 (dy) T T )2 T 2 g(u)du : (2) The law of X T under Q is. Furthermore, we have the following decomposition in the enlarged ltration and a related non-canonical representation well known for a standard Brownian motion (see Meyer, 1994; Yor, 1992). Proposition 27. For the solution (X t ) 6t6T of (3.2); the process t X T X s t := X t ds; t T (3.3) T s is a standard Brownian motion in the enlarged ltration X (X T )(X is the natural ltration of X ) and the following decomposition takes place: X T = ( s ;s T) (X T ): (3.4) Moreover, the process t X s t := X t s ds; t T is well dened and is a standard Brownian motion in its own ltration which is strictly included in X. This Brownian motion is independent of X T.

15 F. Baudoin / Stochastic Processes and their Applications 1 (22) Remark28. (1) If e x2 =2T dx with a density which admits a dierentiable version ; then (3.2) can be written after an integration by parts + dx t = (y)e [(y Xt)2 =2(T t)] dy + (y)e[ (y Xt)2 =2(T t)] dy dt +d t; t T: This is the classical Doob s h-transform of a Brownian motion associated with the space time harmonic h(t; B t )= 1 + T t (y)e (y Bt)2 =2(T t) dy: This last result could have been directly derived from Malliavin s calculus. (2) For further details on decomposition (3.4) when = y for some y R we refer to Jeulin and Yor (199). (3) We note here that X is Gaussian if and only if is Gaussian (see below: Gaussian bridge). Proof of Theorem 25. Here we have [ y T y 2 t = T t exp and 2T (y B t) 2 2(T t) ] ; t T; y R y t = y B t ; t T; y R: T t Hence, Eq. (3.2) corresponds to Eq. (2.2). Assumption (3.1) implies that the function (t; x)= + ( y x T t 2 ) y 2 e 2T (y x)2 2(T t) (dy) y + e 2T (y x)2 2(T t) (dy) ; t T; x R is of class C 1 in the space variable and hence locally Lipschitz. We can hence apply Theorem 17. Proof of Proposition 26. Let g L 2 ([;T]; R) and y R. Under P y process (B t ) 6t T is a solution of the linear equation db t = y B t T t where (W y process ( t t dt +dw y t ; t T; the coordinate t ) 6t T is a standard P y Brownian motion. This implies that under P y the g(u)db u) 6t T is Gaussian; indeed for t T g(u)db u = y t t u dw y t s g(u)du g(u) T T s du + g(u)dwu y ; t T (for further details about the integration with respect to a Brownian bridge; we refer to Jeulin and Yor, 199).

16 124 F. Baudoin / Stochastic Processes and their Applications 1 (22) Its mean is given by m(t)= y t g(u)du; t T (3.5) T and its variance by t 2 (t)= g(u) 2 du 1 ( t 2 g(u)du) ; t T (3.6) T the P y -a.s. convergence of the process ( t g(u)db u) 6t T when t T is hence easily checked. Because P = + P y (dy) we deduce the convergence a.s. (and also in L 2 (P )) of ( t g(u)db u) 6t T under P. The computation of the mean and of the moment of order 2 of the limit is a direct consequence of (3.5) and (3.6). Proof of Proposition 27. The process dened by t X T X s t := X t ds; t T T s is a Brownian motion by virtue of Jacod s theorem. As this decomposition implies X t = t t T X d s T +(T t) ; t T (3.7) T s we have for t T X t ( s ;s t) (X T ): But; on the other hand; we have trivially ( s ;s t) X t (X T ): This implies immediately X T = ( s ;s T) (X T ): Now let (P t ) 6t T the standard Brownian bridge from to dened by t d s P t := (T t) T s ; t T: Let t T. Decomposition (3.7) implies that for t t X s X t s ds = X t T T + P P s t s ds: Now; as the process P P s=s ds is Gaussian; it is easy to conclude. The drift of (3.2) is not as innocent (or complicated!) as it might seem: from an analytical point of view, the set of all the CSDEs associated with the conditioning (T; B T ) is parametered by a non-linear partial dierential equation well known in potential theory and uid mechanics: the Burger s equation.

17 F. Baudoin / Stochastic Processes and their Applications 1 (22) Proposition 29. Let :[;T[ R R the function dened by ) (t; x)= + ( y x T t y 2 e 2T (y x)2 2(T t) (dy) 2 y + e 2T (y x)2 2(T t) (dy) Then is a weak (strong if + y 3 (dy) + ) solution of the Burger s + = (3.8) and we have the limit condition e x (t;s)ds x2 =2T dx x converges weakly to (dx) when t T: (3.9) e (t;s)ds x2 =2T dx Moreover; if (X t ) 6t T is the solution of (3.2) then N t = (t; X t ); t T is a Q local martingale. Remark3. The solutions of Burger s equation are related to the positive solutions of the heat equation by the Hopf Cole transformation ln This transformation explains the limit condition (3.9) because + y 2 e 2T (y x)2 2(T t) (dy) converges weakly to e x2 =2T (dx) when t T: As a rst generalization, let us show how to recover quickly in particular the well-known enlargement formula for the enlargement of the Brownian ltration by a Wiener integral (see Alili and Wu, 21; Alili, 22; Chaleyat-Maurel and Jeulin, 1985). Let (X t ) 6t T the solution of (3.2), by the time-change Y t = X t f(s)2 ds ; t R+ with f L 2 (R + ; dx) such that + f(s) 2 ds = T we immediately deduce from (3.2) Proposition 31. For the conditioning (+ ; + f(s)db s ;); for t R + and y R + [ y t = f(s) 2 ds + f(s) t 2 ds exp y f(s) 2 ds (y t f(s)db ] s) f(s) t 2 ds :

18 126 F. Baudoin / Stochastic Processes and their Applications 1 (22) and Y t = + t f(s)db s + f(s) t 2 ds : Remark32. For the CSDE associated with the functional ( + ) Y = f i (s)db s ; 16i6N where f i L 2 (R + ; dx); the computations are easily made by the techniques developed in Section 2.4. (Computations with Malliavin calculus and Fourier analysis) because D t Y =(f i (t)) 16i6N but as the expressions are complicated; we refer the interested reader to Alili (22) and Chaleyat-Maurel and Jeulin (1985). In particular; Alili (22) uses dierent techniques involving linear Volterra transforms of the Brownian motion to obtain the CSDE. We focus now our attention on some particular examples. Example 33. (1) (Gaussian bridge): Let us take (dx)= e (x m)2 =2s 2 2s s 2 6 T ; then Eq. (3.2) becomes dx t = (s2 T )X t + mt (s 2 T )t + T 2 dt +d t : The solution associated with the initial condition X = is X t = m t T t +[(s2 T )t + T 2 d u ] (s 2 T )u + T 2 dx with m R and and so; (X t ) 6t T is a Gaussian process such that (a) X t converges Q-a.s. when t T to a random variable X T such that X T has the law N(m; s 2 ). (b) E(X t )=(m=t)t. (c) E((X u (m=t)u)(x v (m=t)v)) = u v +(s 2 T)=T 2 uv. (2) Let a : For = 1 2 a a, Eq. (3.2) becomes dx t = a tanh(ax t=(t t)) X t dt +d t ; t T: T t The solution (X t ) 6t T associated with the initial condition X = converges Q-a.s. to a variable X T such that Q(X T = a)=q(x T = a)= 1 2 : Furthermore, for this process, we can show the following property: Q(X T = a X t = x)= 1 1+e ax=(t t) :

19 F. Baudoin / Stochastic Processes and their Applications 1 (22) Let : With (dx) = (cosh[x]= 2T )e x2 =2T 2 T=2 dx, Eq. (3.2) becomes dx t = tanh[x t ]dt +d t : (3.1) Here, the drift does not depend on T. This equation has only one solution associated with the initial condition X =: For this solution, we have Q(X t dx)= cosh[x] e x2 =2t (1=2) 2t dx; t : 2t This class of diusions has been studied in Benjamini and Lee (1997) Bridges related to a diusion: functional aspect On the Wiener space W =(C T ; (F t ) 6t6T ; (B t ) 6t6T ; P) we consider the process (Y t ) 6t6T which is the solution of the following stochastic dierential equation: Y t = t b(s; Y s )ds + B t ; where b :[;T] R R is a continuous function which is C 1 in the space variable (this regularity implies that pathwise uniqueness holds). In this paragraph, we write the CSDE associated with the conditioning (T; Y T ;)by means of Malliavin calculus. We make the assumption that the probability has a strictly positive density with respect to the law of Y T which admits a continuously dierentiable version such that is bounded. Lemma 34. (see Nualart (1995; p. 17) or Malliavin (1997; p. 24)). Y T Malliavin s dierential which is given by [ T D t Y T = (s; Y s)ds ; t6 T: t admits a From this, we deduce immediately with the help of Section 2.4. Proposition 35. Under the minimal probability P ; the process (Y t ) 6t6T law the following equation: Y t = t (b(s; Y s )+ (s; Y s )) ds + t ; t6 T; solves in

20 128 F. Baudoin / Stochastic Processes and their Applications 1 (22) where (1) ( t ) 6t6T is a P -standard Brownian motion. (2) ( (t; Y t )=E [ (Y T ) T ] ) (Y T ) (s; Y s)ds Y t : (3.11) Proposition 36. Under P and P ; (Y t ) 6t T is a semimartingale in the ltration F (Y T ) whose decomposition is given by Y t = t (b(s; Y s )+ (s; Y s ;Y T )) ds + M t ; where (1) (M t ) 6t6T is a P (and P ) Brownian motion adapted to F (Y T ). (2) For t T and y Supp P YT. + (t; Y t ;y)=i u e iuy E(exp[ Y t s )ds +iuy T ] Y t )du + : (3.12) e iuy E(exp[iuY T ] Y t )du In order to give an example of the previous formulas, let us look what they become in the case of the Ornstein Uhlenbeck process which is the solution of the following stochastic dierential equation: Y t = t Y s ds + B t ; R : This is a Gaussian process which can be written as t Y t =e t e s db s : Hence, after some computations [ E(e iuyt F t ) = exp iue (T t) Y t e2(t t) ] 1 u 2 : 4 Now, according to (3.11) and (3.12), the CSDE associated with the conditioning (T; Y T ;) and the decomposition in the enlarged ltration are, respectively, ( ) + dy t = Y (y e(t t) Y t ) exp e 2T y 2 1 e (e T y e t Y t) 2 t + 2T e 2t e (dy) 2T sinh((t t)) ( ) dt + exp e 2T y 2 1 e (e T y e t Y t) 2 2T e 2t e (dy) 2T +dw t and [ dy t = Y t + Y T e (T t) ] Y t dt +dm t : sinh((t t))

21 F. Baudoin / Stochastic Processes and their Applications 1 (22) Bridges related to a diusion: PDE s aspect Let T a nite time horizon. Our aim, in this paragraph, is to show that the set of the minimal probabilities associated with the conditioning (T; Y T ;) where Y is a diusion can be parametered by a non-linear partial dierential equation whose solutions are related to the space time harmonics of Y by a Hopf Cole transformation (see Cole, 1951; Hopf, 195). Let us consider on the Wiener space the diusion dy t = b(y t )dt + (Y t )db t ; t6 T; (3.13) where b and are C functions. We assume that the function (t; x)= + P T t (x; dy) P T (; dy) (dy); t T; x R is well-dened and C, where P t is the semigroup of (3.13). Proposition 37. Let (Y t ) 6t6T the unique strong solution of (3.13) associated with the initial condition Y =. Under the minimal probability P associated with the conditioning (T; Y T ;); the process (Y t ) 6t6T solves in law the following equation: dx t =[b(x t )+ 2 (X t )(t; X t )] dt + (X t )d t ; t T; (3.14) where (1) ( t ) 6t6T is a P -standard Brownian motion. (2) :[;T[ R R is solution of the partial dierential (2 2 )+ 1 ) =: Proof. Here; the density process of P with respect to P is given by + p T t (Z t ; dy) (t; Z t )= (dy): p T (; dy) As ( (t; Z t )) 6t6T is a martingale; from Itô s formula; we @x = ln is a solution of (3.15) and we conclude by means of Girsanov s theorem. Remark38. The solutions of (3.15) are related to the positive space time harmonics of (3.13) by the Hopf Cole transformation ln

22 13 F. Baudoin / Stochastic Processes and their Applications 1 (22) We also have: Proposition 39. Under P and P ; (Y t ) 6t T is a semimartingale in the ltration F (Y T ) whose decomposition is given by dy t =(b(t; Y t )+ 2 (Z t ) (t; Y t ;Y T )) dt + (Z t )dm t ; where (1) (M t ) 6t T is a P (and P ) Brownian motion adapted to F (Y T ). (2) ( ; ;y):[;t[ R R is solution for all y Supp P YT of (3.15). As a corollary, we deduce the following result of Fitzsimmons (1998), which is a generalization of Benjamini and Lee (1997). Corollary 4. Let h be a strictly positive C function such that and + Lh = h h(x)p T (; dx) + ; T + 2 (x)h (x) 2 P t (; dx)dt + for some R (L = b@=@x =@x 2 is the generator of Y ). Let us now consider the diusion ) dzt h = (b(z ht )+ 2 (Z ht ) h h (Zh t ) dt + (Zt h )db t for which; we assume there exists a unique solution (Zt h ) 6t6T associated with the initial condition Z h =. Then Z and Zh have the same bridges; namely for t T; P(Zt h dx ZT h = y)=p(z t dx Z T = y). Conversely, if the drift in (3.14) is homogeneous, then there exists a strictly positive C function h such that Lh = h with R and (t; x)= h h (x): Proof. Let h be a function which satises the assumptions of the corollary. Consider the process D t = h(z t )e t : From Itô s formula; Dis a martingale. From Girsanov s theorem; Dis then the density process of Z h with respect to Z where is a normalization constant. This proves the rst part of our assumption.

23 F. Baudoin / Stochastic Processes and their Applications 1 (22) Conversely, assume that given by (3.14) is homogeneous. 2 ln hence, there exist f and h strictly positive C functions such that (t; x)=f(t)h(x): It implies on f and h and hence f (t)h(x)+b(x)f(t)h (x)+ 1 2 (x)2 f(t)h (x)= f f (t)=lh h (x)= with R. Example 41. For the Brownian motion; we get the class of diusions with generator L = 1 d 2 d + tanh(x + ) 2 dx2 dx : We recover then the result of Benjamini and Lee (1997). By comparing the results of this section with the results of the previous one (bridges over a diusion: functional aspect), we deduce the following interesting result. Corollary 42. Assume that =1. For y Supp(P YT ); the fundamental solution ( ; ;y):[;t[ R R of the following partial dierential equation: @x (2 )+ 2 2 =: e x (t;s;y)ds p T (;x)dx e x (t;s;y)ds p T (;x)dx converges weakly to y when t T admits the following probabilistic representation: + (t; Y t ;y)=i ue iuy E(exp[ Y t s )ds +iuy T ] Y t )du + ; t T; e iuy E(exp[iuY T ] Y t )du where (Y t ) 6t6T is the unique strong solution of (3.13) associated with the initial condition Y = CSDE associated with the quadratic variation of a geometric Brownian motion Let (; (H t ) t ; Q) a ltered probability space on which a standard (H t ; Q) Brownian motion ( t ) t is dened. Let and x.

24 132 F. Baudoin / Stochastic Processes and their Applications 1 (22) It is well-known (see Dufresne, 199) that under the Wiener measure the functional + e 2Bt 2t dt is distributed, up to a multiplicative constant, as the inverse of a gamma law. The computations made in Baudoin (21a, b) allow us to obtain the CSDE associated with the conditioning (+ ; + e 2Bt 2t dt; ). We assume that admits with respect to the law 1=2 a C 2 density : R + R + which is almost surely bounded. Lemma 43 (see Baudoin; 21a; Donati-Martin et al.; 21). We denote by the probability measure dened on B(R +) by (dt)= x2 e x2 =2t 2 () t 1+ dt; t and denote by P ;x the law of the geometric Brownian motion (x e Bt t ) t. On the space C(R + ; R + ) R + endowed with the ltration F B(R +) the law of ((B u ) 6u6t ; + B 2 s ds) under P ;x is absolutely continuous with respect to the law P ;x =F t and a jointly measurable version of the density is given by ( ) ( ) 2 1+ y Bt y t = x y t e x2 =2y B2 t =2(y t B2 s ds) 1 t B2 s ds B2 s ds y; t ; y : Theorem 44. The stochastic dierential equation [( dx t = X t e u u ( t X ) ] s 2 ds + Xt 2 =2u)du + e u u 1 ( t X dt +d t ; t s 2 ds + Xt 2 =2u)du (3.16) has a unique strong solution (X t ) t associated with the initial condition X = x (1) and X is such that Q( t ; X t )=1: (2) The law of the process ( t ln X ) t x t is the minimal probability associated with the conditioning (+ ; + e 2Bt 2t dt; ). As in the previous subsection, we have to check a Q-a.s. convergence, which can here be shown by the dominated convergence theorem.

25 F. Baudoin / Stochastic Processes and their Applications 1 (22) Proposition 45. For the solution (X t ) t of (3.16) the process ( X t ) t converges Q-a.s. when t + to a random variable X and Q( X dx)=c (x)e x2 =2x x 1+ ; x ; where C is a normalization constant. As in the case Y = B T, we have an explicit formula for the decomposition in the enlarged ltration and a related non-canonical representation (see Baudoin, 22) for which we give a new proof. Proposition 46. For the solution (X t ) t of (3.16); in X ( X )(X is the natural ltration of X ); we have the following decomposition: [( dx t = X t Xt 2 ) ] dt +d t ; t ; (3.17) X X t where ( t ) t is a Q-standard Brownian motion in the enlarged ltration. And the following decomposition takes place: ( ) X = ( u ;u6t) ( X ): t Moreover the process ( X t =X t ) t is a diusion in its own ltration which is strictly included in X and this diusion is independent of X. Proof. The decomposition (3.17) is well-known (see Baudoin; 21a; Matsumoto and Yor; 21); it easily implies rst the decomposition ( ) X = ( u ;u6t) ( X ) t and then ( X t =e t+t 1 X ) t ; t : (3.18) X Hence Xt 2 (1 X t = X ) 2 =e2(t+t) : This implies by integrating along the trajectories: e () t = X X t (3.19) X X t with e () t := e t+t : If we eliminate X between the two relations (3.19) and (3.18) we nd X t X t = e() t e () : t

26 134 F. Baudoin / Stochastic Processes and their Applications 1 (22) We get then the expected result from the classical Matsumoto Yor s result (see Baudoin; 21a; Matsumoto and Yor; 1999). The diusion X =X is independent of X because is a Brownian motion for all the disintegrated probabilities Q( X = x); x R +. Here again, the set of the CSDEs associated with the conditioning (+ ; + e 2Bt 2t dt) is parametered by a non-linear partial dierential equation. Proposition 47. Let : R + R + R the function dened by (t; x)= 2 x 2 + e u u (t + x 2 =2u)du x + e u u 1 (t + x 2 =2u)du : Then is a solution of the Burger s type ) ( = x and we have the limit condition lim x e (t;x)dx = C(t); where C is a normalization constant. Remark48. (1) As in the case Y = B T there is also a Hopf Cole ln with + ) (t; x)= e u u 1 (t + x2 du 2u positive solution =: (2) If has a continuously dierentiable version; then we have + e u u 2 (t + x 2 =2u)du (t; x)=x + e u u 1 (t + x 2 =2u)du : Proposition 49. Let and x. By taking (x)=ce (2 =2)x we get: The stochastic dierential equation dx t = X t [( + 1 ) ] 2 X K 1+ (X t ) t dt +d t ; t K (X t )

27 F. Baudoin / Stochastic Processes and their Applications 1 (22) (1) admits one and only one non-explosive solution (X t ) t associated with the initial condition X = x and we have Q( t ; X t )=1: (2) The process t X s 2 ds converges P-a.s. when t + to a random variable + s ds which satises X 2 ( + Q X 2 s ds dx ) = x e (2 =2)x x 2 =2x 2 K (x ) x 1+ dx; x : For further details on this class of diusions we refer to Baudoin (21a, 22) CSDE associated with the rst hitting time of a level by the Brownian motion Let (; (H t ) t ; Q) a ltered probability space satisfying the usual conditions on which a standard Brownian motion ( t ) 6t6T is dened. In this paragraph, we give the CSDE associated with the functional T a = inf {t ; B t = a}; a : Of course, we are not directly in the assumptions of Section 2, so that this example treated by hands can be seen as introduction to a little more general theory concerning random times. Here, we can use the results of the previous subsection because of the following Lamperti representation. Lemma 5. Let (X t ) t the solution of (3.16) with = 1 2 and associated with the initial condition X = a; then the law P Z of the process (Z t ) 6t6Ta dened by Z t X s 2 ds = a X t; t satises dp Z =F Ta = (T a )dp =FTa : Proof. On the Wiener space; the following absolute continuity relation takes place: ( + ) dp X =F = Bs 2 ds dp 1=2;a =F ; where P X is the law of X and P 1=2;a the law of (ae Bt (1=2)t ;t ). Now, the process Z dened on the Wiener space by Z t e2bs s ds = a aebt (1=2)t ; t is a standard Brownian motion under the Wiener measure considered up to its rst hitting time of a. In order to be homogeneous in our computations, we write the CSDE associated with the conditioning (T a ;) where is the Borel measure dened on R + by ( + ) (dt)= e t2 =2+a m(d) (dt)

28 136 F. Baudoin / Stochastic Processes and their Applications 1 (22) with m a probability measure on R + such that + 2 m(d) + and (dt)= a =2t 2t 3 e a2 dt: With the change of variable of the previous lemma, we deduce: Theorem 51. The stochastic dierential equation + e t2 =2+X t m(d) dx t = + dt +d t ; t (3.2) e t2 =2+X tm(d) has a unique strong solution (X t ) t associated with the initial condition X = and the law of (X t ) t is the minimal probability associated with the conditioning (T a ;). Namely; the law P X of X satises dp X =F Ta = + e Ta2 =2+a m(d)dp =FTa : Corollary 52. For the solution (X t ) t of (3.2) the stopping time a = inf {t ; X t = a} is Q-a.s. nite and satises Q( a dt)=(dt); t : Corollary 53. For the solution (X t ) t of (3.2); in the ltration ((X t {t a }) ( a )) t (X is the natural ltration of X ) we have the following decomposition: ( dx t = 1 + a X ) t dt +d t ; 6 t a ; a X t a t where is a standard Brownian motion in the enlarged ltration. Remark54. More generally; we can give the CSDE associated with the rst hitting time of for a Bessel process with index where is a strictly positive constant: it suces to apply the same Lamperti representation to the process dened by (3.16) for a general. As in our rst example (CSDE associated with the conditioning of a marginal law of the Brownian motion), the drift of the CSDE is a solution of a Burger s equation. Proposition 55. Let :]; + [ R R the function dened by + e t2 =2+x m(d) (t; x)= + e t2 =2+x m(d) :

29 F. Baudoin / Stochastic Processes and their Applications 1 (22) Then is a solution of the Burger s + = and we have the limit condition lim x a e (t;x)dx = C (dt) (dt) ; where C is a normalization constant. Example 56. Let. With m = ; Eq. (3.2) becomes dx t = tanh[x t ]dt +d t : This equation has only one solution associated with the initial condition X = and for this solution; we have hence Q( a dt)= a cosh(a) e a2 =2t t 2 =2 dt; t : 2t 3 4. Application to nance: the value of a weakinformation in a complete market 4.1. Framework Let us consider an investor who trades in a complete nancial market so as to maximize the expected utility of his wealth at a prespecied time. We assume that he is in the following position: his portfolio decisions are based on a public information ow but he possesses extra information about the law of some functional of the future prices of a stock. Our basic question is then: what is the value of this information? Let T a constant nite time horizon. We work on a complete and free of arbitrage nancial market, i.e. we assume that there exists a ltered probability space (; (F t ) 6t6T ; P) satisfying the usual conditions and such that: (1) The price on the time interval [, T] of a given contingent claim realizes a trajectory of a continuous F-adapted positive local martingale (S t ) 6t6T. (2) For each F-adapted martingale (M t ) 6t6T there exists A(S) such that M t = M + t u ds u ; t6 T; where A(S) is the space of R-valued F-predictable processes integrable with respect to the price process S, such that ( t ) u ds u 6t6T isa(p; F) martingale.

30 138 F. Baudoin / Stochastic Processes and their Applications 1 (22) Remark57. t represents the number of shares of the risky asset S held by an investor at time t and the wealth process associated with the strategy A(S) with initial capital x is given by V t = x + t u ds u : In particular; strategies are assumed to be self-nancing. A utility function is a strictly increasing, strictly concave and continuously dierentiable function U :]a; + [ R with a [ ; ] and which satises lim U (x)=; x + lim U (x)=+ : x a + We use the convention that U(x) = for x 6 a. We shall denote I the inverse of U. Let Y : R n a F T -measurable random variable (it will be a functional of the trajectories of the price process). We denote by P Y the law of Y and assume, in order to be in the same kind of assumptions as in Section 2, that Y admits a regular disintegration with respect to the ltration F, i.e. that there exists a jointly measurable process y t ; 6 t T; y R n satisfying dt P Y almost surely 6 t T and y R n P(Y dy F t )= y t P(Y dy): 4.2. The value of a weak information on a functional of the price With Y we associate a probability measure on R n (assumed to be the law of Y under the eective probability of the market) which admits an almost surely strictly positive bounded density with respect to P Y. will be called a weak information on the functional Y. Let U be a utility function. Let E the set of probability measures Q on such that (1) Q is equivalent to P. (2) Q(Y dy)=(dy). In this setting, we dene the value of the weak information on the functional Y for an investor with initial investment x a and who uses the utility function U by ( T )) u(x; ) = inf sup E (U Q x + u ds u : Q E A(S)

31 F. Baudoin / Stochastic Processes and their Applications 1 (22) Theorem 58. Assumed that integrals below are convergent; then for each initial investment x a ( u(x; )= (U I) (x) dp ) Y R d (y) (dy); n where is dened by ( I (x) dp ) Y d (y) P(Y dy)=x: R n Before giving the proof of this theorem, let us look what it becomes in the most commonly used utility functions. Example 59. (1) Let (; 1) and U(x)=x = then (x)= x 1 [ R n (dp Y =d(y)) 1=( 1) ] 1 and u(x; )= x [ R n ( ) ] 1=(1 ) 1 d (y) P(Y dy) : dp Y (2) Let U(x)=lnx then (x)= 1 x and u(x; )=lnx + R n d dp Y (y)ln d dp Y (y)p(y dy): (3) Let U(x)= e x with then ( (x)=exp x + ln d ) (y)p(y dy) dp Y R n and ( u(x; )= exp x + ln d ) (y)p(y dy) : R dp n Y Proof of Theorem 58. Let Q = DP E. From classical result on complete market by martingale approach (see e.g. Karatzas et al.; 1987 or Karatzas and Shreve; 1999;

32 14 F. Baudoin / Stochastic Processes and their Applications 1 (22) Section 5.8); we know that the optimal solution to ( T )) sup E (U Q x + u ds u A(S) is given by A(S) such that T ( ) y x + u ds u = I ; D where y satises ( ( )) y E I = x: D Hence sup E (U Q A(S) ( ( ( T x+ u ds u ))= inf (xy+e P d (Y )Ũ y a dp Y where Ũ is the strictly convex function dened by Ũ(y) = sup(u(x) xy): y a We can then apply Proposition 6 to obtain the expected results The minimizing probability A(S) y d dp Y (Y ) ))) : Denition 6. We say that a probability measure P E is a minimizing probability associated with (Y; ; U) if for each initial investment x R + ( T )) u(x; ) = sup P (U x + u ds u : From the proof of Theorem 58 and from Section 2.2 (minimal probability associated with a conditioning), we immediately deduce the following interesting theorem. Theorem 61. There exists a unique minimizing probability measure P E associated with (Y; ; U); and the following equivalence relation takes place: dp = d (Y )dp dp Y thus P is independent of the utility function used. under the mini- In order to explicit the semimartingale decomposition of (S t ) 6t6T mizing probability, we need the following extension of Lemma 12. Lemma 62. There exists a jointly measurable process [;T[ R n R; (t;!; y) y t (!);

33 F. Baudoin / Stochastic Processes and their Applications 1 (22) such that (1) For P Y -a.s. y R n ; the process ( y t ) 6t T is F-predictable. (2) For P Y -a.s. y R n and for 6 t T; ( t ) P (u y ) 2 d S u + =1: (3) For P Y -a.s. y R n and for 6 t T; y ;S t = t y u y u d S u : Remark63. As in Lemma 12 we can choose such that for P Y -a.s. y S and for 6 t T; [ t y t = exp u y ds s 1 t ] (u y ) 2 d S u on { y t }: 2 We can now conclude by the same proof as the one of Theorem 14 that: Theorem 64. Under the probability P the price process (S t ) 6t6T is a semi-martingale adapted to the ltration F whose decomposition is given by y R ds t = n t y t (dy) R y n t (dy) d S t +dm t ; t6 T; where (M t ) 6t6T is a local martingale The minimizing probability in a Markov setting We consider now the case Y = S T and we assume that there exists a C function :[;T] R + R + anda(f; P) standard Brownian motion (X t ) 6t6T such that ds t = S t (t; S t )dx t ; 6 t 6 T (4.1) and ( T E ) St 2 (t; S t ) 2 dt + : We can note here that, as shown by Dupire (1997), the volatility is completely given by the prices of the European calls. More precisely, the price C(t; K) of the call with strike K and time horizon t [;T] is given by C(t; K)=E((S t K) + ) and hence satises (under suitable conditions) the partial = 1 2 K 2 2 (t; 2 : We conclude that the knowledge of C gives and hence characterizes P.