Markovian Bridges: Construction, Palm Interpretation, and Splicing

Fitzsimmons, P. J.; Pitman, Jim; Yor, Marc

doi:10.1007/978-1-4612-0339-1_5

Cited by 152 publications

(187 citation statements)

References 30 publications

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“…Indeed, let (X,P x ) := (−X, P x ) be the dual Lévy process, then the following identity in law between the bridge and its time reversed version is well known, see [9] for instance:…”

Section: Small Tail Of First Passage Times Of Stable Lévy Processesmentioning

confidence: 99%

Conditioned stable Lévy processes and the Lamperti representation

Caballero

Chaumont

2006

Journal of Applied Probability

136

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By killing a stable Lévy process when it leaves the positive half-line, or by conditioning it to stay positive, or by conditioning it to hit 0 continuously, we obtain three different positive self-similar Markov processes which illustrate the three classes described by Lamperti [10]. For each of these processes, we compute explicitly the infinitesimal generator from which we deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0 which we describe here. As an application, we give the law of the minimum before an independent exponential time of a certain class of Lévy processes. It provides the explicit form of the spacial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

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“…Indeed, let (X,P x ) := (−X, P x ) be the dual Lévy process, then the following identity in law between the bridge and its time reversed version is well known, see [9] for instance:…”

Section: Small Tail Of First Passage Times Of Stable Lévy Processesmentioning

confidence: 99%

Conditioned stable Lévy processes and the Lamperti representation

Caballero

Chaumont

2006

Journal of Applied Probability

136

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“…Conversely, suppose that (13) and (14) hold. Then (17) and (18) are valid when f is the potential of some measure ν. Now the conclusion follows because any excessive function is the increasing limit of a sequence of potentials of the form Gh n , where h n are nonnegative functions.…”

Section: Proof (I) For Any Z ∈ E the Function Y → G(y Z)mentioning

confidence: 99%

General gauge and conditional gauge theorems

Chen¹,

Song²

2002

Ann. Probab.

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General gauge and conditional gauge theorems are established for a large class of (not necessarily symmetric) strong Markov processes, including Brownian motions with singular drifts and symmetric stable processes. Furthermore, new classes of functions are introduced under which the general gauge and conditional gauge theorems hold. These classes are larger than the classical Kato class when the process is Brownian motion in a bounded C 1,1 domain.1. Introduction. Given a strong Markov process X and a potential q, the conditional expectation u(x, y) of the Feynman-Kac transform of X by q is called the conditional gauge function. (The precise definition will be given later.) The function u is important in studying the potential theory of the Schrödinger-type operator L + q, as it is the ratio of the Green's function of L + q and that of L, where L is the infinitesimal generator of X. The conditional gauge theorem says that under suitable conditions on X and q, either u is identically infinite or u is bounded between two positive numbers. The conditional gauge theorem was first proved for Brownian motions (see [12] for a history). Very recently it was established in [8] for symmetric stable processes. The proofs of the conditional gauge theorem for symmetric stable processes in [8] and [10] are quite different from that for Brownian motion, due to the complication that the sample paths of symmetric stable processes are discontinuous. See also [7].A few years ago, Professor Kai Lai Chung suggested to one of the authors that the conditional gauge theorem for Brownian motion might be proved via the gauge theorem for the conditional processes. In this paper, we show that it is indeed possible to prove the conditional gauge theorem via the gauge theorem. This new approach not only simplifies the proof but also yields a quite general conditional gauge theorem that is applicable to a large class of strong Markov processes having strong duals, including Brownian motions with singular drifts and symmetric stable processes. Furthermore, we introduce new classes of functions K 1 (X) and S 1 (X) so that the gauge and conditional gauge theorems hold for q in K 1 (X) and in S 1 (X), respectively. The classes K 1 (X) and S 1 (X) are larger than the (classical) Kato class when X is Brownian motion in a bounded C 1,1 domain. Now let us lay out the setting of this paper carefully.

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“…The identity ||B (24) is also quite easily derived from (28). In fact, the bridge and excursion sampling identities (14) and (15) are so closely related to Vervaat's identity (28) that any one of these three identities is easily derived from the other two.…”

Section: Summary Of Resultsmentioning

confidence: 95%

“…In fact, the bridge and excursion sampling identities (14) and (15) are so closely related to Vervaat's identity (28) that any one of these three identities is easily derived from the other two. The uniform variable U which appears in passing from bridge to excursion via (28) explains why the excursion sampling identity is most simply compared to its bridge counterpart with a sample size of n for the bridge and n + 1 for the excursion, as presented in Theorem 1.…”

Section: Summary Of Resultsmentioning

confidence: 96%

“…Let B me := (B me (u)) 0≤u≤1 be a standard Brownian meander, and let B me,r be the standard meander conditioned to end at level r ≥ 0. Informally, these processes are defined as These definitions of conditioned Brownian motions have been made rigorous in many ways: for instance by the method of Doob h-transforms [69,28], by weak limits of suitably scaled and conditioned lattice walks [52,45,75], and by weak limits as ε ↓ 0 of the distribution of B given A ε for suitable events A ε with probabilities tending to 0 as ε ↓ 0, as in [17], for instance…”

Section: Introductionmentioning

confidence: 99%

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Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times

Pitman

1999

Electron. J. Probab.

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For a random process X consider the random vector defined by the values of X at times 0 < U n,1 < ... < U n,n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n,i are the order statistics of n independent uniform (0, 1) variables, independent of X. The joint law of this random vector is explicitly described when X is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at n independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae are Brownian representions of various special functions, including Bessel polynomials, some hypergeometric polynomials, and the Hermite function. Various combinatorial identities involving random partitions and generalized Stirling numbers are also obtained.

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Markovian Bridges: Construction, Palm Interpretation, and Splicing

Cited by 152 publications

References 30 publications

Conditioned stable Lévy processes and the Lamperti representation

Conditioned stable Lévy processes and the Lamperti representation

General gauge and conditional gauge theorems

Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times

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