Elect. Comm. in Probab. 8 (2003) 155{166 ELECTRONIC COMMUNICATIONS in PROBABILITY PATH TRANSFORMATIONS OF FIRST PASSAGE BRIDGES JEAN BERTOIN Laboratoire de Probabilit¶es et Mod`eles Al¶eatoires and Institut universitaire de France, Univer- sit¶e Pierre et Marie Curie, 175, rue du Chevaleret, F-75013 Paris, France. email: [email protected] LOijC CHAUMONT Laboratoire de Probabilit¶es et Mod`eles Al¶eatoires Universit¶e Pierre et Marie Curie, 175, rue du Chevaleret, F-75013 Paris, France. email: [email protected] JIM PITMAN Department of Statistics, University of California, 367 Evans Hall #3860, Berkeley, CA 94720-3860, USA. email: [email protected] Submitted 4 July 2003 , accepted in ¯nal form 5 November 2003 AMS 2000 Subject classi¯cation: 60J65 (60G09 60G17) Keywords: ¯rst passage bridge, path transformation, time of the minimum, uniformly dis- tributed time, exchangeable increments Abstract We de¯ne the ¯rst passage bridge from 0 to ¸ as the Brownian motion on the time interval [0,1] conditioned to ¯rst hit ¸ at time 1. We show that this process may be related to the Brownian bridge, the Bessel bridge or the Brownian excursion via some path transformations, the main one being an extension of Vervaat's transformation. We also propose an extension of these results to certain bridges with cyclically exchangeable increments. 1 Introduction Let B = (B(t); t 0) denote a standard real Brownian motion started from B(0) = 0 and ¸ Tx = inf t : B(t) > x the ¯rst passage time above level x 0. Then for ` > 0 and x = 0, the process f g ¸ 6 (B(t); 0 t `) conditioned on T = ` · · x may be called the Brownian ¯rst passage bridge of length ` from 0 to x. It is well known in the Brownian folklore (cf. e.g. [5]) that such ¯rst passage bridges may be represented in terms of the bridges of a 3-dimensional Bessel process. More precisely, for each 155 First passage bridges 156 ¸ > 0 there is the identity in distribution (B(t); 0 t ` T = `) · · j ¸ =d (¸ BES (t); 0 t ` BES (0) = ¸; BES (`) = 0) ¡ 3 · · j 3 3 where BES3 stands for a 3-dimensional Bessel process. But this representation obscures a number of fundamental properties of Brownian ¯rst passage bridges which follow directly from their interpretation in terms of a one-dimensional Brownian motion. The point of this note is to record some of these properties and to suggest that Brownian ¯rst passage bridges should be regarded as basic processes from which other more complex Brownian processes can be derived by simple path transformations. We begin with a section devoted to the discrete time analysis of the random walk case, which is based on elementary combinatorial principles. This leads in Section 3 to an extension of Vervaat's transformation for Brownian ¯rst passage bridges and other related identities in distribution. In a fourth part, we shall indicate extensions of these properties to a large class of bridges with exchangeable increments. 2 The main result in discrete time Fix two integers ¸ and n such that 1 ¸ n. Let S = (S ) be a random chain such that · · i 0·i·n S0 = 0, Sn = ¸. Suppose moreover that the increments, ¢Si = Si Si¡1, i = 1; : : : ; n take their values in the set 1; +1 and are cyclically exchangeable, that¡is, for any k = 1; : : : ; n, f¡ g d (¢S1; : : : ; ¢Sn) = (¢Sk+1; : : : ; ¢Sn; ¢S1; : : : ; ¢Sk) : Note that the latter property is equivalent to the fact that for any k = 0; 1; : : : ; n 1, the shifted chain: ¡ Si+k Sk; if i n k θk(S)i = ¡ · ¡ ; i = 0; 1; : : : ; n : ½ Sk+i¡n + Sn Sk; if n k i n ¡ ¡ · · has the same law as S. A fundamental example of such chain is provided by the simple random walk conditioned to be equal to ¸ at time n. For k = 0; 1; : : : ; ¸ 1, de¯ne the ¯rst time at which S reaches its maximum minus k as follows: ¡ mk(S) = inf i : Si = max Sj k : f 0·j·n ¡ g When no confusion is possible, we denote mk(S) = mk. Theorem 1 Let º be a random variable which is independent of S and uniformly distributed on 0; 1; : : : ; ¸ 1 . The chain θmº (S) has the law of S conditioned by the event m0 = n . Morfeover, the index¡ g m is uniformly distributed on 0; 1; : : : ; n 1 and independent fof θ (Sg). º f ¡ g mº The proof of Theorem 1 relies on a simple combinatorial argument. In this direction, denote by ¤ the support of the law of S. In particular, ¤ is a subset of n+1 (s0; : : : ; sn) R : s0 = 0; sn = ¸; ¢si 1 + 1 ; i = 1; : : : ; n : f 2 2 f¡ g g First passage bridges 157 Lemma 2 For every s ¤, de¯ne the set 2 ¤(s) = s; θ (s); : : : ; θ (s) : f 1 n¡1 g Then for any s ¤, the set ¤(s) of the paths in ¤(s) which ¯rst hit their maximum at time n, i.e. ¤(s) = x2 ¤(s) : m (x) = n , contains exactly ¸ elements and may be represented as f 2 0 g ¤(s) = θ (s); θ (s); : : : ; θ (s) : (1) f m0 m1 m¸¡1 g Proof: We can easily see from a picture that for any k = 0; 1 : : : ; ¸ 1, the path θ (s) is ¡ mk contained in ¤(s). To obtain the other inclusion, it is enough to observe that if i = mk, for k = 0; 1; : : : ; ¸ 1, then the maximum of θ (s) is reached before time n. 6 ¡ i Remark. Lemma 2 is closely related to a combinatorial lemma in Feller [10] XII.6, p.412, also known as the ballot Theorem, see [9]. Here we complete Feller's result by associating a path transformation to the combinatorial result. Note also that it may be extended to any chain with exchangeable increments, see [2]. Proof of Theorem 1. For every bounded function f de¯ned on 0; 1; : : : ; n 1 and every bounded function F de¯ned on Zn+1, we have f ¡ g 1 ¸¡1 E (F (θ (S))f(m )) = P (S = s) F (θ (s))f(m ) : mº º ¸ mj j sX2¤ Xj=0 But Lemma 2 allows us to write for any s ¤, 2 ¸¡1 n¡1 F (θmj (s))f(mj ) = F (θk(s))f(k)1Ifm0(θk(s))=ng ; Xj=0 kX=0 so that 1 n¡1 E (F (θ (S))f(m )) = P (S = s) F (θ (s))f(k)1I mº º ¸ k fm0(θk(s))=ng sX2¤ kX=0 n = E F (θ (S))f(U)1I ; ¸ U fm0(θU (S))=ng ¡ ¢ where U is uniform on 0; 1; : : : ; n 1 and independent of S. Finally, it follows from the f ¡ g cyclic exchangeability that the chain θU (S) has the same law as S and is independent of U, hence we have: E (F (θ (S))f(m )) = E(F (S) m (S) = n)E(f(U)) ; mº º j 0 which proves our result. The following transformation could be viewed as the converse of that in Theorem 1; however, it is actually a sightly weaker result. Corollary 3 Let U be uniformly distributed on 0; 1; : : : ; n 1 and independent of S. Con- ditionally on the event m (S) = n , the chain θf (S) has the¡ sameg law as S. f 0 g U First passage bridges 158 3 Main results in the Brownian setting By obvious Brownian scaling, it is enough to discuss bridges of unit length. So let F br(t); 0 t 1 =d (B(t); 0 t 1 T = 1) : x · · · · j x ¡ ¢ We refer to P. L¶evy [16], Theorem 42.5 for a proper de¯nition of this conditioning. The following fact is fundamental, and obvious by either random walk approximation or Brownian excursion theory. Write for an arbitrary real-valued process (X(t); t 0) and y 0 ¸ ¸ Ty(X) = inf t : X(t) > y and Xt = sup Xs : f g s·t Lemma 4 For each ¯xed ¸ > 0, the ¯rst passage process T (F br); 0 a ¸ a ¸ · · ¡ ¢ is a process with exchangeable increments, distributed as (T (B); 0 a ¸ T (B) = 1) ; a · · j ¸ where (T (B); a 0) is a stable(1=2) subordinator; more precisely a ¸ E (exp( ®T (B))) = exp( ap2®) ; ® 0 : ¡ a ¡ ¸ br Moreover, conditionally given the process (Ta(F¸ ); 0 a ¸), or, what is the same, given br · · br the past supremum process F (t); 0 t 1 of F br, the excursions of F F br away from ¸ · · ¸ ¸ ¡ ¸ 0 are independent Brownian³ excursions whose´ lengths are the lengths of the flat stretches of br F and correspond to the jumps of the ¯rst passage process T (F br); 0 a ¸ . ¸ a ¸ · · ¡ ¢ The next statement follows immediately from Lemma 4. Proposition 5 Let br Rbr(t) := F (t) F br(t) ; 0 t 1 : ¸ ¸ ¡ ¸ · · Then Rbr(t); 0 t 1 =d Bbr(t) ; 0 t 1 L0(Bbr) = ¸ ¸ · · j j · · j 1 br ¡ ¢ ¡ 0 br ¢ where B is a standard Brownian bridge and (Lt (B ); 0 t 1) is the usual process of local times of Bbr at level 0. Indeed, the above holds jointly with· · br F (t); 0 t 1 =d L0(Bbr); 0 t 1 L0(Bbr) = ¸ : ¸ · · t · · j 1 ³ ´ ¡ ¢ We point out that Proposition 5 can also be deduced from di®erent well known path transfor- mations. For instance, the transformation between Bbr and the Brownian meander Bme due j j me br 0 br to Biane and Yor [5]. Speci¯cally, we know that if we de¯ne B = ( B (s) + Ls(B ); 0 me me 0 j br j · s 1), then B is a Brownian meander with mint·s·1 B (s) = Lt (B ).